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Kidney Failure and the Federal Government (1991)

Chapter: Appendix D: Survival Analysis Methods for the End-Stage Renal Disease (ESRD) Program of Medicare

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Suggested Citation:"Appendix D: Survival Analysis Methods for the End-Stage Renal Disease (ESRD) Program of Medicare." Institute of Medicine. 1991. Kidney Failure and the Federal Government. Washington, DC: The National Academies Press. doi: 10.17226/1818.
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D
Survival Analysis Methods for the End-Stage Renal Disease (ESRD) Program of Medicare

ROBERT A. WOLFE, Ph.D.*

This document addresses two specific areas related to survival analysis of Medicare ESRD data. The first focus is a critical review of methods of survival analysis for Medicare ESRD data, including methods used in the past by HCFA and the USRDS, methods that have been proposed by other members of the renal community, and methods that are potentially useful for future analyses. The second focus is a review of the results of international comparisons of mortality rates with the objective of determining what conclusions can be drawn from such comparisons.

While this paper gives a critical review of methods of analysis of ESRD mortality data, it does not report the results of any new analyses of empirical data. Instead, this paper is intended to help in the interpretation of the results of previous data analyses and to give directions for future analyses and data collection that address some of the limitations of the research carried out to date. Although a review of statistical methods must unavoidably involve some degree of abstraction, I have tried to tie abstract concepts to specific issues whenever possible.

The analysis of mortality among ESRD patients is complicated by the nature of the data available for analysis and by the heterogeneity of the patients receiving therapy. These issues are addressed in some detail in this paper in order to show how to avoid potential limitations of analyses and errors in their interpretation. Some of the specific issues are listed below.

The data for ESRD patients are complicated because they are collected over time and involve a sequence of events for each subject being studied. The sequence of relevant medical information for some of the subjects in

*  

 Department of Biostatistics, University of Michigan, Ann Arbor, Michigan

Suggested Citation:"Appendix D: Survival Analysis Methods for the End-Stage Renal Disease (ESRD) Program of Medicare." Institute of Medicine. 1991. Kidney Failure and the Federal Government. Washington, DC: The National Academies Press. doi: 10.17226/1818.
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the data set may be incompletely documented. For example, patient followup of younger ESRD patients is not typically tracked in the Medicare data system until 90 days after first therapy for ESRD. Analyses of such incomplete data are susceptible to subtle forms of bias.

Patients with treated ESRD exhibit a wide variety of characteristics that influence mortality patterns. An evaluation of the importance of any one of these characteristics must also account for the potential impact on the results of the other characteristics.

The Medicare data collection system was designed primarily for reimbursement purposes and, consequently, has some limitations for research purposes. Some patient characteristics related to mortality, such as previous medical history, are not regularly recorded in it. Other characteristics, such as patient treatment history, are derived from billing records rather than from dedicated data collection instruments and, consequently, are subject to error.

Interpretation of the results of mortality analyses is complicated by the variety of analytical methods and types of numerical summaries that can be reported. Analytical methods include adjusted and unadjusted results, cross-tabulations and multiple regression models, parametric and nonparametric methods, Cox models and logistic regression models, and other methods discussed below. The results of statistical analyses can be summarized as death rates, death proportions, mortality ratios, and expected lifetimes.

An overview of several crucial issues central to the analysis of mortality data is presented in the first section as a series of questions. Each question is followed by some of the issues that should be addressed when answering the question. These issues recur in more specific forms in subsequent sections of the document.

In the second section, general strategies for adjusting statistical analyses for patient characteristics are discussed. Patient characteristics that are currently measured or that would be useful to measure are examined, and two approaches toward adjusting statistical analyses for patient characteristics are provided.

In the third section, several methods of survival analysis that are relevant to ESRD data are reviewed with the intent of showing how to compare, interpret, and synthesize the results of survival analyses. Most of the methods reviewed in this section have been used, or proposed for use, by other members of the renal research community. Each method has qualities that make it appropriate for specific purposes. Some proposals are also made in this section for analysis methods that have not yet been widely used in renal research. In addition, several different numerical parameters that are used to summarize the results of survival analyses are discussed.

The fourth section reviews several problems associated with the interpre-

Suggested Citation:"Appendix D: Survival Analysis Methods for the End-Stage Renal Disease (ESRD) Program of Medicare." Institute of Medicine. 1991. Kidney Failure and the Federal Government. Washington, DC: The National Academies Press. doi: 10.17226/1818.
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tation of international comparisons of mortality rates. Although this section focuses on the analysis recently reported by Held et al. (1990), the issues are largely relevant to any international comparisons.

GENERAL ISSUES IN SURVIVAL ANALYSIS

Overview

Survival analysis of ESRD patient data can help to identify factors, such as etiology, that are related to differences in patient mortality. For example, a comparison of the one-year survival proportions for diabetic and nondiabetic patients shows that mortality rates differ by etiology. With the identification of such factors, survival analysis also yields estimates of the magnitude of the differences in survival associated with those factors. Although the comparison of mortality figures for two groups of patients gives a direct evaluation of the importance of the factor distinguishing the two groups, it seldom leads to a complete understanding of the mechanisms causing the difference. Since any two groups of patients may differ with respect to several factors simultaneously, it is of some interest to determine how much effect any one factor would have on mortality, if all other factors were held constant. For example, the average age among diabetic and nondiabetic patients is different, so we want to know by how much the mortality rates would differ for diabetic and nondiabetic patients, if the ages were similar in the two groups. Survival analysis can help to answer hypothetical questions such as ''By how much would the mortality patterns in two groups of patients differ, if they were to differ from each other with respect to only one characteristic at a time?" Although much this appendix describes examples related to the study of mortality differences for several treatment groups, the concepts and statistical methods that are presented apply equally well to the comparison of outcomes for groups of patients defined by other characteristics, as well.

Survival analysis can only yield results concerning factors that are measured and recorded in the available data. Unfortunately, some of the most important determinants of survival may not be recorded in the Medicare data base. Survival analysis cannot account for the potential effect of unmeasured or unmeasurable factors on patient mortality. Many of the important questions related to policy and scientific research involve factors that have not or cannot be measured. In such cases, expert opinion blended with indirect or imperfect evidence must be relied upon. Although decisions that are based on inconclusive evidence can be wrong, the available data still should be evaluated and weighed carefully when making policy decisions.

Sir Ronald Fisher, one of the great statisticians and scientists of this century, argued throughout much of his life that there was no definitive

Suggested Citation:"Appendix D: Survival Analysis Methods for the End-Stage Renal Disease (ESRD) Program of Medicare." Institute of Medicine. 1991. Kidney Failure and the Federal Government. Washington, DC: The National Academies Press. doi: 10.17226/1818.
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evidence showing that tobacco smoking causes lung cancer. He argued that there could be an underlying factor that caused certain individuals both to smoke and to be more susceptible to lung cancer. Although his reasoning was correct (the evidence is not conclusive unless a randomized controlled clinical trial is performed), the resulting policy decision based on his reasoning would undoubtedly have been a poor one.

Statistical analysis is just one tool used in the weighing of empirical evidence. The process involves the formulation of a question, the assembling of relevant data to address the question, and careful interpretation of the results of analysis of the data. An evaluation of the strengths and weaknesses of the conclusions derived from the analysis sometimes leads to a reformulation of the question, collection of new data, or reanalysis of the data. The choice of appropriate method of statistical analysis cannot be discussed in isolation. Just as the specific question that we want to answer determines the way in which we try to answer it, the choice of appropriate statistical methodology depends strongly upon the specific purposes that motivate the analysis.

Examples

Several major types of research objectives and questions can be addressed through the collection and analysis of national ESRD data. The examples listed below were selected to highlight specific issues in the appropriate interpretation of survival analysis results. Some of the examples are based on hypothetical or simplistic situations but illustrate fundamental issues that arise in more realistic situations, as well.

The examples show that statistical comparisons are central to the interpretation of statistical analyses. Reduction and avoidance of bias, through the selection of appropriate comparison groups and the control of confounding factors, are fundamentally important to most analytic research. Minimizing and evaluating the impact of random variability on the results of research is also important. The selection of methods that yield interpretable numerical summaries is an essential aspect of the dissemination of results.

Identification of the Study Population

What is the death rate among ESRD patients? The death rate among untreated ESRD patients is very high; with no kidney function, they will surely die within days. Data are available in the Medicare system for treated ESRD patients after they become eligible for Medicare payments. However, victims with undiagnosed kidney disease are not counted in the Medicare data base. Further, because of eligibility requirements, the first 90 days of ESRD therapy for many younger patients in the United States are

Suggested Citation:"Appendix D: Survival Analysis Methods for the End-Stage Renal Disease (ESRD) Program of Medicare." Institute of Medicine. 1991. Kidney Failure and the Federal Government. Washington, DC: The National Academies Press. doi: 10.17226/1818.
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not captured by the Medicare data system. Among the elderly, the fraction of deaths attributable to withdrawal from therapy can be substantial (10 percent or more), but these deaths are included in most reports of ESRD mortality. Even among treated ESRD patients, the length of survival of patients with some residual kidney function is likely to depend upon the amount of impairment and the rate of progression of the disease.

If the deaths of never-treated ESRD victims were included in reported mortality rates, then the death rates likely would be elevated above those currently reported. In contrast, if the deaths among those withdrawing from treatment were excluded, then the death rates would likely be substantially lower than those currently reported. Different definitions distinguishing between reduced kidney function and ESRD could also have a substantial effect on reported death rates among the ESRD population.

Thus, any evaluation of mortality rates among ESRD patients must identify which patient population is being considered. Without such identification, it is difficult to compare different mortality results.

Evaluation of the death rate in a group can depend strongly upon who is included in the group and which period of patient follow-up is included in the evaluation.

The Importance of a Comparison Group

Are mortality rates among treated ESRD patients very high? Mortality rates among dialyzed ESRD patients in the United States are typically 24 percent per year (USRDS, 1990, p. E.31). Although of some value in isolation, this fact is of most interest when compared to other death rates.

For example, the 24 percent annual death rate among dialyzed ESRD patients is exceptionally high in comparison to that of a non-ESRD population with the same age distribution (approximately 2 percent per year). However, the discrepancy would be somewhat smaller if the comparison were made to a non-ESRD population with the same history of diabetes and hypertension as is found among incident ESRD patients. This comparison would be especially useful for the evaluation of programs designed to prevent the progression of hypertension and diabetes to ESRD.

Although death rates among treated ESRD patients are high relative to those in a healthy population, they are low compared to those among untreated ESRD victims.

Furthermore, the comparison of death rates among ESRD patients receiving different forms or amounts of therapy would be useful for comparing the relative efficacy of the various therapies.

When evaluating ESRD death rates, it is useful to make comparisons to other death rates.

Suggested Citation:"Appendix D: Survival Analysis Methods for the End-Stage Renal Disease (ESRD) Program of Medicare." Institute of Medicine. 1991. Kidney Failure and the Federal Government. Washington, DC: The National Academies Press. doi: 10.17226/1818.
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Biased Comparisons

Spurious Differences Is the death rate for dialyzed ESRD patients higher with continuous ambulatory peritoneal dialysis (CAPD) or with center hemodialysis (CH)? CAPD was preferentially given to insulin-dependent diabetic ESRD patients in some regions of the United States in the early 1980s because it offers a convenient method for administration of insulin to the patient. Because of this, diabetes is more prevalent among CAPD patients than among CH patients in the United States. Thus, even if CAPD and CH were inherently equally efficacious therapies for ESRD (analyses to date have been inconclusive), then the high death rate among diabetic ESRD patients and the higher prevalence of diabetes among CAPD patients would cause unadjusted death rates to be higher among CAPD patients than among CH patients. Comparison of crude (unadjusted for patient characteristics) death rates for CAPD and CH patients would not account for this fact and would erroneously lead to the conclusion that death rates were higher among CAPD patients than among CH patients.

Observed differences in outcome may be due entirely to differences in patient characteristics.

Biased Comparison and Age-Specific Comparisons and Inappropriate Comparison Group How much lower are death rates among transplant patients than among dialysis patients? The annual death rate is typically 6 percent per year among ESRD patients with transplants (USRDS, 1990, p. E.40) and is typically 24 percent per year among dialysis patients (p. E.32). The death rate among transplant recipients is much lower than among dialyzed ESRD patients. However, transplant recipients are also younger, on average, than dialyzed patients. Comparison of the age-specific 5-year survival probabilities for dialyzed patients and transplant recipients indicates a substantial variation in death rates with the age of the patient (p. E.32, E.40). Thus, a large part of the difference in overall (crude) death rates between transplant recipients and dialyzed patients is due to the difference in ages between the two groups.

Part of an observed difference in outcomes may be due to differences in patient characteristics. Comparison of specific mortality rates for homogeneous subgroups of patients yields a less biased evaluation.

In addition to the effect of age, which is known and recorded for each patient, other differences between dialyzed patients and transplant recipients may account for part of the difference in death rates for these two treatment modalities. Dialyzed ESRD patients who are on the transplant waiting list are distinguished, in a variety of ways, from dialyzed ESRD patients who are not on the waiting list. Some of these distinctions may be

Suggested Citation:"Appendix D: Survival Analysis Methods for the End-Stage Renal Disease (ESRD) Program of Medicare." Institute of Medicine. 1991. Kidney Failure and the Federal Government. Washington, DC: The National Academies Press. doi: 10.17226/1818.
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difficult to identify or measure. Therefore, in order to compare the death rates of dialyzed patients and transplant recipients, it would be more appropriate to use the survival experience of dialyzed patients who are on the waiting list rather than that of all dialyzed patients.

Comparison groups should be similar in all aspects other than the characteristic being studied, especially in those other aspects that cannot be measured.

Bias Due to Unobserved Factors How can a new therapy for ESRD patients be evaluated without bias? The ultimate evaluation of a treatment protocol should be based upon how well it works and how much it costs in comparison to other protocols. Specific outcomes (mortality is an overriding outcome) can be selected as the basis for comparison of two protocols, and the outcomes can be evaluated for two series of patients treated according to the two protocols. Quantitative comparison of the outcomes for the two protocols leads to conclusions about the size of the difference in patient outcomes, which then can be evaluated in comparison to the relative costs of the protocols.

An observed difference in outcomes cannot be ascribed to the difference in treatments unless the two patient groups are equivalent in other major regards. The interpretation of the differences should account for any preexisting differences between the two groups of patients.

Ideally, evaluation of a new treatment should be based on the comparison of patient outcomes under the new and old treatment regimens, all else equal. This ideal can never be realized because it would require two identical series of patients for study. In practice, during patient enrollment, the two treatment groups can be deliberately balanced for important factors that are likely to have substantial effects on patient outcome. However, in order to ensure balance with respect to unknown factors, the only practical solution is randomization (Campbell and Stanley, 1963).

Randomization can be used to control for unforeseen differences between patient groups. The controlled randomized clinical trial offers the only study design that can be guaranteed (with high probability) to be free of bias. All other study designs are subject to potential bias because the groups being compared may differ with respect to several factors that affect patient outcome.

Interpreting Standard Errors for Population Data

When there is no sampling error because a statistic is reported for all ESRD patients, how should the standard error be interpreted? The standard error does not reflect uncertainty about the specific population being described, since the statistic precisely summarizes the experience of the whole population. However, results from apparently similar populations do vary

Suggested Citation:"Appendix D: Survival Analysis Methods for the End-Stage Renal Disease (ESRD) Program of Medicare." Institute of Medicine. 1991. Kidney Failure and the Federal Government. Washington, DC: The National Academies Press. doi: 10.17226/1818.
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from each other, and this variability is reflected in the standard error. For example, even in an apparently stable population, the number of deaths varies from day to day. There is a corresponding variability in the number of events that occur each year as well. The amount of variability in the number of certain types of events, such as ESRD incidence and death, is closely approximated by the Poisson distribution. The standard error of the statistic reflects the amount of variability that typically occurs in the value of the statistic upon repeated observations of similar populations.

The value of the standard error of a statistic measures the typical amount of variability due to random causes that occurs in the value of the statistic.

Accounting for Random Variation

Among ESRD patients over age 65, the proportion of patients surviving for 4.75 years decreased from 20.6 percent to 16.3 percent between 1982 and 1983 (USRD, 1989, p. D.21). Does this signify a trend for this age group? Even in otherwise similar populations, incidence counts and death counts vary from year to year apparently because of random variation. A calculation using the standard errors (USRDS, 1989 p. D.22) indicates that, even in two stable populations with identical death rates, a difference in survival proportions bigger than 4.3 percent would be likely to occur frequently just by chance. The expression P > .20 indicates that a difference as large as that observed could occur between successive years more than 20 percent of the time, just by chance, for two groups of patients with identical death rates. When a difference could plausibly occur by random chance, the difference is called insignificant. A difference that would not likely occur by chance is called a significant difference. The probability that a difference as big as or bigger than the one observed could occur by chance is called the P-value of the difference. (P-values less than 5 percent are labeled as significant whereas those greater than 5 percent are labeled as insignificant.) Using this criterion, the difference reported above is insignificant.

The reported difference could represent a trend, or it could represent random variability. In this case, the change was noted for a single age group between two consecutive years. On the basis of those limited data, there is no way to know whether it is a trend or a chance occurrence. However, if the trend persisted in subsequent years, or was also seen in other age groups, then the P-value could be recomputed taking into account all the evidence. If the resulting P-value was small, then it would be unlikely that the difference had occurred by chance, so it would be more plausible that the difference represented a true trend. If the evidence from other age groups and years was inconsistent with the trend noted previously, then the recomputed P-value would tend to remain large and the difference could plausibly be attributed to chance.

Suggested Citation:"Appendix D: Survival Analysis Methods for the End-Stage Renal Disease (ESRD) Program of Medicare." Institute of Medicine. 1991. Kidney Failure and the Federal Government. Washington, DC: The National Academies Press. doi: 10.17226/1818.
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When a large change is noted that can be plausibly ascribed to chance, it is prudent to look for further evidence in order to determine whether the change represents a true trend. If a trend over time has been found to be significant, then the next step would be to try to determine what was causing the trend.

Random occurrences can lead to apparently substantial differences in outcomes, especially with a small, narrowly defined study group. Statistical and probabilistic evaluation of the chances of such differences can help to distinguish unimportant random fluctuations from more persistent patterns.

Important Versus Significant

Is it important that the difference between 5-year survival probabilities of 0.388 for females and 0.411 for males (averaged for 1977–83; see USRDS, 1989, p. D.21) is not likely to have occurred by chance (approximate P < .10)?

The difference (2.1 percent) is small relative to the survival probabilities (average 40 percent) and is therefore uninteresting. Even a small difference will be statistically significant if there are enough data documenting the consistency of the difference. In this case, the difference is statistically significant because the sample size is so large (all ESRD data for patients incident between 1977 and 1983).

Although the difference reported here is not significant at the 5 percent level (P < .05), it is significant at the 10 percent level (P < .10). Such a difference is often called marginally significant.

When a difference is significant, it is appropriate to interpret the difference as a real one rather than one that was likely to have occurred by chance. Once found to be significant, the importance of the difference must be evaluated.

The decision process about the importance of a significant difference is largely subjective. A 2 percent difference is small in comparison to other differences that have been found between patient subgroups, but it still represents a substantial number of individuals and consequently might be judged to be important. For example, a treatment change that lengthens life by 3 months for 4,000 patients yields a numerical benefit of 1,000 person-years of extra life. In contrast, special therapy that extends the life of a single person for 10 years is of special significance to the individual involved, but the numerical benefit is 10 person-years of life.

A statistically significant result is not always an important one if it is small. However, a small difference can be important if it affects a large number of individuals.

Analysis of Provider Versus Patient

Suppose that the annual death rate is 15 percent at institution A and 25 percent at institution B, that the difference is significant (P <.001), and that

Suggested Citation:"Appendix D: Survival Analysis Methods for the End-Stage Renal Disease (ESRD) Program of Medicare." Institute of Medicine. 1991. Kidney Failure and the Federal Government. Washington, DC: The National Academies Press. doi: 10.17226/1818.
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the patient characteristics are similar at the two institutions. Institution A reuses dialyzers while institution B does not. Does this prove that all institutions should reuse dialyzers?

The statistical significance reported from the analysis was based on the sample size of patients at each institution. Thus, the difference in mortality rates for the two groups of patients is unlikely to have occurred by chance. That is, there is a true difference in mortality between the two institutions. The appropriate generalization is to patients at the two institutions. However, we cannot generalize reliably to other institutions because the sample size of institutions is only two. When the P-value is calculated on the basis of an analysis of the institutional data, with a sample size of 2, the P-value is 0.50, which is not significant. Differences between institutions A and B other than in dialysis reuse may be responsible for the reported difference in death rates (Donner and Donald, 1987.

The statistical significance of an analysis based on patients should not be used to make conclusions about the population of providers.

Choice of Parameter for Mortality Summaries

If one study reports a 50 percent increase in death rates for one group relative to another group, whereas another study reports that the fractions surviving at 12 months differ by only 8.9 percent, which is right?

The results might be entirely consistent with each other. Monthly death rates of 2 percent and 3 percent will lead to surviving fractions of 78.7 percent and 69.8 percent, respectively, at 12 months. A death rate of 3 percent is accurately described as being 50 percent higher than a death rate of 2 percent. The difference in surviving fractions is accurately summarized as 8.9 percent. Note that the fractions dead in this example would be 21.3 and 30.2 percent, respectively, corresponding to a nearly 50 percent increase in the fraction dead. Part of the apparent discrepancy between the original reports from the studies was due to the fact that one comparison was made using a ratio whereas the other comparison was made using subtraction. Another, less important, cause of the apparent discrepancy can be illustrated by the analogies of death rates to compound interest rates and of death proportions to simple interest rates.

The proportion of individuals, P, surviving through an interval of time is related to the death rate, R, per unit interval by the equation

P = exp(-R).

For example, if the death rate per year is 20 percent, then the fraction that survives through the year is 81.9 percent [= exp(-0.2)]. The fraction that dies during the year is 18.1 percent, slightly less than 20 percent.

The death rate, fraction dead, and fraction surviving are all different ways

Suggested Citation:"Appendix D: Survival Analysis Methods for the End-Stage Renal Disease (ESRD) Program of Medicare." Institute of Medicine. 1991. Kidney Failure and the Federal Government. Washington, DC: The National Academies Press. doi: 10.17226/1818.
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to summarize mortality experience. Each is useful for particular purposes. Proper interpretation of results requires an understanding of the meaning of each.

Type I and Type II Error Issues

A clinical trial based on 120 patients found no significant difference between two treatment therapies (P > 0.05). Does this prove that the therapies are equally effective?

The only conclusion that should be reached from the insignificant P-value is that the difference between treatment groups seen in the clinical study could have arisen by chance. There may be a substantial difference between the therapies, but there were not enough data in the trial to document the difference.

The results of a comparative study should include, in addition to P-values, confidence intervals for the values of important outcome parameters. The confidence interval helps in the evaluation of the potential importance of the difference between two groups whereas the P-value tells whether the difference could have arisen by chance.

A nonsignificant difference should not be interpreted as a definitive result by itself. Confidence intervals for the size of the difference are more useful for interpretation. If the confidence interval includes large differences, then the true difference might also be large. If the confidence interval includes only small differences, then the true difference is likely to be small.

Projections and Extrapolations

The one-year surviving fraction among cadaveric transplant recipients has been increasing every year since 1977 (USRDS, 1989, p. E. 19). Can this trend be extrapolated to give an estimate for 1990?

Extrapolations and projections of trends are very susceptible to bias because circumstances can change over time. Projections work well if the nature of the process that is being predicted does not change over time. The usual standard errors reported in a statistical analysis reflect only the uncertainty due to random fluctuations and measurement error, not the uncertainty due to bias or to change in the nature of the problem. Thus, the trend can be extrapolated but there is no way to evaluate the accuracy of the projection.

Extrapolations and projections of trends are very susceptible to error.

Accuracy of Counts

The counts of incident ESRD patients reported by the USRDS for 1987 (USRDS, 1989, pp. A. 1, A. 11) do not agree. How can any data analysis from the Medicare data system be trusted?

Suggested Citation:"Appendix D: Survival Analysis Methods for the End-Stage Renal Disease (ESRD) Program of Medicare." Institute of Medicine. 1991. Kidney Failure and the Federal Government. Washington, DC: The National Academies Press. doi: 10.17226/1818.
×

Data reporting and analysis are based on a sequence of operations and decisions that differs from system to system, from analyst to analyst, and from purpose to purpose. The discrepancy of 611 patients out of close to 33,000 represents less than a 2 percent difference. In many cases, it may not be worth the trouble to find the reason for such a small difference. In this case, the discrepancy is due to the fact that U.S. territories are not included in one of the counts (p. A. 11).

Small discrepancies in counts of subjects are to be expected as definitions and methods of data reporting change over time. Interpretations based on percentages are often more useful than those based on counts.

ADJUSTING MORTALITY ANALYSES FOR PATIENT CHARACTERISTICS

A variety of methods have been used to summarize mortality results for the U.S. ESRD population. The summaries produced by HCFA (Eggers, various years) and USRDS (Held et al., 1990) have shown the broad patterns of ESRD mortality in the United States and point out many of the major factors that need to be considered in the design of more focused analyses of specific hypotheses.

The most important aspect of survival analysis is to have the relevant data available for analysis. Several patient characteristics collected by the Medicare system are known to be related to patient survival. Using appropriate methods to compare mortality rates should account for these factors. Other potentially important patient characteristics not currently collected should be investigated in order to determine their importance.

Patient Characteristics Related to Mortality

Several patient characteristics that are, or might be, related to mortality rates among ESRD patients are discussed below. For many of these characteristics, there are difficulties in the appropriate interpretation of their effects on mortality, and these issues are briefly discussed.

Currently Available Data

The Medicare data collection system for ESRD patients includes a wealth of information that is useful for evaluating survival of the ESRD patient population. Several of the most important items of data that are currently collected are discussed below.

Diagnosis This measure is an extremely important predictor of patient mortality. Death rates among diabetic patients are elevated by close to a

Suggested Citation:"Appendix D: Survival Analysis Methods for the End-Stage Renal Disease (ESRD) Program of Medicare." Institute of Medicine. 1991. Kidney Failure and the Federal Government. Washington, DC: The National Academies Press. doi: 10.17226/1818.
×

factor of 2 compared to other diagnoses. Unfortunately, the data for diagnosis have not always been collected reliably. Through the early part of the 1980s, this measure was missing for a substantial number of patients. Since diagnosis is an important determinant of mortality, it is difficult to evaluate trends in mortality before 1982 using national data. There is still a fundamental problem with the interpretation of this measure because it relies heavily upon a subjective evaluation, by the physician, of the patient's medical history. It is often difficult to determine the underlying cause of ESRD for patients who are first diagnosed after their kidney function is already minimal.

Age at First Treatment Mortality rates generally increase with patient age, except possibly at very young ages. To date, most analyses of mortality rates have accounted for age by assuming that it has the same effect on mortality for all patients. However, there are indications that mortality increases at differential rates with age for different types of patient, especially for patients with different diagnoses. That is, mortality rates increase more quickly with age among diabetic patients than they do among patients with other diagnoses. In addition, most analyses of mortality have accounted for the effect of age at first treatment, but not for the progressive effect as a patient gets older. The impact on mortality rates of 5 years of aging is substantially higher for a patient whose ESRD therapy begins at age 65 than it is for a patient whose therapy starts at age 20. The proportional hazards survival models account for some of these differential effects, but such effects should be carefully accounted for in any analyses used for the formulation of policy.

Year of First Treatment The year of first ESRD treatment is not of major interest in its own right, but is important because it is a surrogate measure for other important factors that have changed over time. Unfortunately, treatment patterns as well as patient characteristics have changed over the years, and both could affect patient outcomes. Some of the changes, such as the availability of transplantation therapy and the aging of the treated ESRD patient population, are documented in the data and can be adjusted for through statistical analysis. Other changes in patient and treatment characteristics are less well documented in the Medicare data system, so it is difficult to isolate their effects on patient mortality.

  • Treatment Methods: The use of cyclosporine for transplant patients has become pervasive, and the technology for CAPD therapy has also changed. New methods of CH, such as high-flux dialysis with shorter dialysis times and reuse of dialyzers, have not been unanimously adopted but may have affected patient survival at specific institutions. The detailed data necessary to evaluate such treatment choices are not readily available in the Medicare data base. Similarly, the effect of EPO on patient outcomes will

Suggested Citation:"Appendix D: Survival Analysis Methods for the End-Stage Renal Disease (ESRD) Program of Medicare." Institute of Medicine. 1991. Kidney Failure and the Federal Government. Washington, DC: The National Academies Press. doi: 10.17226/1818.
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be difficult to evaluate because the specific measures, such as hematocrit, that are likely to be most affected by EPO have only recently been added to the Medicare data collection forms.

Data collection instruments related to new treatments have historically been introduced in the Medicare system after the new treatment method is in widespread use. This prevents comparison of the new and old treatment methods during the crucial period of transition. If the times of treatment change were known precisely for each patient, or even for each provider, then comparisons could be made of mortality rates before and after the change, and these could be used to evaluate the change during the time of transition while the new treatment is still in its formulative stages. With the current data collection patterns, the comparison of patient outcomes for the new and old treatments is complicated by the fact that data typically are collected only after a consensus has largely been reached concerning appropriate methods of therapy.

  • Patient Characteristics: The number of older patients and of diabetic patients accepted in the Medicare ESRD program has increased dramatically in recent years (USRDS, 1990). Both of these characteristics are associated with higher mortality rates. Careful statistical analysis methods can be used to account for such changes in patient characteristics if the characteristics are recorded in the data base. However, there are likely to be other patient characteristics that have changed over the years that are also associated with mortality but which have not been recorded in the Medicare data base. If such characteristics are not recorded in the data base and accounted for in statistical analyses, then their effects would appear as an unexplained general trend in mortality rates over the years. A current special study by the USRDS may help to identify some of the other important patient characteristics that are related to mortality.

Number of Years of Treatment Many of the survival analyses performed to date for ESRD patients have used years since first ESRD treatment as the fundamental measure of time (USRDS, 1990), although some have used the calendar year instead (HCFA, Eggers, 1984, 1987). Typically, analyses based on the years since initiation of ESRD therapy have summarized the experience of a cohort of patients whose ESRD therapy started during a particular year. Analyses based on calendar time have typically summarized the experience of all patients who received any treatment during a particular year. Each method of analysis has specific applications.

Death rates among ESRD patients vary substantially depending upon the time since first treatment. There is a general short-term decrease in death rates as the number of years of therapy increases, although aging has a counterbalancing long-term effect. Such patterns of decreasing mortality rates are to be expected when the population at risk is heterogeneous (Vaupel et al., 1985). It is important to account for such short-term trends in the

Suggested Citation:"Appendix D: Survival Analysis Methods for the End-Stage Renal Disease (ESRD) Program of Medicare." Institute of Medicine. 1991. Kidney Failure and the Federal Government. Washington, DC: The National Academies Press. doi: 10.17226/1818.
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evaluation of mortality patterns. The prognosis for individual patients is most easily characterized in terms of time since first ESRD therapy.

Norms of treatment patterns are likely to change nationally at a particular calendar time in response to innovations in technology or to changes in policy. A series of analyses, based on all patients receiving treatment during successive years, would be most sensitive for detecting such an effect (National Medical Care, personal communication, 1990). Such a change in treatment patterns would not be detected easily by a series of analyses based on the year of first ESRD therapy, because the treatment change would have different effects at different times relative to first ESRD therapy for the different cohorts of patients.

Multiple Measures of Time Statistical methods cannot isolate the unique effects of each of several factors when they are related to each other by a linear equation. For example, the simultaneous effects on mortality of patient year of birth, current patient age, and current year of therapy cannot be simultaneously evaluated because the following equation holds for all patients:

Current Year of Therapy = Year of Birth + Current Age

Such limitations of statistical methodology require that choices be made about which factors are to be included in a statistical analysis. Three time measures—current patient age, the number of years since first ESRD therapy, and the current calendar year of therapy—can be evaluated simultaneously because in a given year, there are ESRD patients of various ages who are in their third year of ESRD therapy. That is, age cannot be determined from the current year and the number of years of therapy. In survival analyses, age and current year of therapy are relevant because death rates vary dramatically with both of these measures. Current year of therapy is relevant because it is likely to reflect national norms of treatment practice.

Race For dialyzed patients, mortality rates are generally lower among black ESRD patients than among white ESRD patients for a given age and diagnosis. The reasons for this difference are not well understood. Recent analyses have shown that the difference is not uniform across diagnoses but is prominent only among hypertensive and diabetic patients. Further analyses may yield detail that leads to an understanding of the pathophysiological mechanism for the difference in mortality between races.

Sex Gender does not appear to have a substantial effect on mortality, although treatment patterns are gender related. However, when comparisons are made among treatment groups, adjustment for gender should be made in order to avoid any potential bias in the comparison.

Suggested Citation:"Appendix D: Survival Analysis Methods for the End-Stage Renal Disease (ESRD) Program of Medicare." Institute of Medicine. 1991. Kidney Failure and the Federal Government. Washington, DC: The National Academies Press. doi: 10.17226/1818.
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Unavailable or Difficult-to-Evaluate Data

There are several data items that may be related to patient mortality that are not currently collected by the Medicare data system. Other items are collected, but not in an ideal format for analysis. Several of the most important of these are discussed in this section.

Treatment Modality A national registry for ESRD data offers the potential for a comparison of patient outcomes by treatment modality. Such comparisons would be of substantial utility in determining patterns of optimal care for ESRD patients. A national registry could also offer timely evaluation of outcomes by treatment modality in order to allow monitoring of the quality of care being provided to the ESRD patient population. Both goals require accurate information concerning the pattern of treatment given to each patient.

Currently, the Medicare data system collects information concerning treatment modality through a variety of methods. For transplantation information, a series of dedicated data collection forms are filled out for each transplant patient. Billing information from the providers is the primary data collection instrument for dialysis treatment. Since dialysis is the most common form of therapy, billing data are a key component of the Medicare data base. Unfortunately, the bills are not designed to track patient treatments over time; consequently, it is often difficult to determine treatment histories for each patient. The timing of treatment modality changes can only be approximated from quarterly dialysis reports. It is likely that the current handling of the billing data misses some of the treatment modality changes for some patients.

Such inaccuracies in the treatment history data make it difficult to discern differences in patient outcomes for different therapies. Many of the statistical techniques discussed in this report are designed to yield comparisons of outcomes for groups of patients receiving different treatments. The statistical techniques can account for differences in patient characteristics that might otherwise bias the comparison, but they do not account for errors in the classification of patients into treatment groups.

The weakest link in the current Medicare data system may well be the relatively imprecise data available concerning the timing of dialytic treatment changes for each patient.

Medical History The medical histories of ESRD patients may have a substantial effect on their subsequent mortality rates. The important aspects of medical histories are only partially measured by the etiology of ESRD; they also include medical events and health practices for which there are little data in the Medicare data base.

Suggested Citation:"Appendix D: Survival Analysis Methods for the End-Stage Renal Disease (ESRD) Program of Medicare." Institute of Medicine. 1991. Kidney Failure and the Federal Government. Washington, DC: The National Academies Press. doi: 10.17226/1818.
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Differences in past medical histories should be accounted for when comparing the mortality experience of two or more groups of ESRD patients. The choice of treatment modality for ESRD patients is based on an evaluation, by the physician and patient, of the convenience and efficacy of the therapy. Thus, the choice of therapy may well depend upon the past medical history of the patient. This leads to the assignment of different types of patients to different types of treatments, making the comparison of outcomes for the treatments more difficult to interpret. Data concerning the differences in medical history between groups of patients receiving different therapies could be used in a statistical comparison of outcomes to account for those differences in medical history.

It is useful to attempt to distinguish between the past medical history of a patient and the medical outcomes that are caused by ESRD, although the distinction can be difficult to make in some cases. It is important to distinguish between characteristics that were present when a treatment was begun and those that were caused by the treatment. Differences present when therapy was started should generally be adjusted for when comparing outcomes, whereas those differences that are caused by the treatment are to be evaluated. For example, the frequency of hospitalization prior to the start of a therapy is a measure of the morbidity experienced by the patient and should be adjusted for, if possible, when comparing therapies because it is likely related to subsequent mortality rates. In contrast, the frequency of hospitalization after the start of therapy may well be a result of the inadequacy of the therapy and should itself be analyzed as a patient outcome, but not as a predictor of patient outcomes.

For example, hypertension can be an essential cause of ESRD, but most ESRD patients also exhibit hypertension that has resulted from ESRD. The presence of hypertension after therapy has started should not generally be adjusted for when comparing outcomes of several treatments. However, the presence of essential hypertension should be adjusted for in a comparison of patient mortality because it is likely to be associated with cardiac disease that would occur regardless of which treatment a patient receives for ESRD.

Generally, since many useful summaries of ESRD mortality are based on survival after first treatment for ESRD, data should be recorded at or before the time of first treatment for ESRD.

Social Support Systems (Family Arrangements) There are several aspects of social and family support that are known to be associated with survival in the general population. Both marital status and whether or not a person lives alone have been found to have important relationships with mortality. Since ESRD is a disease that can have substantial burdens on both the time and the financial resources of the patient, it is likely that aspects of family and social support are even more important determinants of mortality for

Suggested Citation:"Appendix D: Survival Analysis Methods for the End-Stage Renal Disease (ESRD) Program of Medicare." Institute of Medicine. 1991. Kidney Failure and the Federal Government. Washington, DC: The National Academies Press. doi: 10.17226/1818.
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the ESRD patient than for the general population. If two treatment groups differed with respect to social and family support arrangements, then a comparison of mortality for the two groups would be biased unless the analysis could be adjusted for that difference. Such differences are likely to occur when comparing different treatment modalities. Treatment modalities appear to have differential use according to gender, race, and income, all of which are likely to be associated with different patterns of family and social support.

The current Medicare data base does not include data on social and family arrangements. In order to ensure equal access to care, it is unlikely that such factors will be considered in policymaking. However, for the reasons cited above, such measures might be extremely valuable to consider in the evaluation of medical treatment methods. Although such data are not currently available at the national level, they are routinely collected by some providers through the efforts of a social worker. If the data are available at the provider level, including them in the national data base or in a sample from it may not be difficult or expensive.

Multivariable Methods

The examples in the previous sections of this paper have shown that it is important to adjust for differences in patient characteristics when comparing patient survival for several treatment groups of ESRD patients. Without adjustment, the presence or absence of differences in survival might be due to the differing characteristics of the patients rather than to the treatments. Typically, adjustments may have to be made simultaneously for several patient characteristics. Multivariable statistical methods are often used to make such adjustments, and the patient characteristics that are adjusted for in such comparisons are called confounding factors.

Generally, a measure, Z, is potentially a confounding factor in a study of association between two other factors, X and Y, if Z is associated with both X and Y. Often, a study attempts to quantify the amount by which an outcome, Y, is changed if a factor, X, is varied and all other conditions are left unchanged. For example, a study may attempt to quantify the amount of change in survival probabilities (Y) when the treatment method (X) is varied, all else (Z) equal. Except with a randomized trial, it is difficult to design an experiment that involves differences in the study factor, X, without differences in other factors, Z, also being present. If the factors, Z, can cause a change in the outcome of interest, Y, then the differences in Y observed at the end of the experiment could be due simultaneously to differences in both X and Z. One objective of statistical analysis is to attempt to isolate the separate effects of X and Z on the value of Y.

The problem of adjusting for confounding factors is a recurrent issue in

Suggested Citation:"Appendix D: Survival Analysis Methods for the End-Stage Renal Disease (ESRD) Program of Medicare." Institute of Medicine. 1991. Kidney Failure and the Federal Government. Washington, DC: The National Academies Press. doi: 10.17226/1818.
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statistical comparison of mortality patterns for two or more groups. In comparing mortality rates for two or more treatments, the observed results from the patients receiving the two types of treatment are compared. However, if the characteristics of the comparison groups of patients other than treatment differ (confounding factors), then the results should be adjusted to isolate that component of the difference that is attributable to the treatment alone. In practice, only adjustments for confounding factors that are recorded for each patient in the data base can be made.

Stratification and modeling are two major approaches used to adjust statistical analyses for confounding factors. The stratification approach is based on the principal of ''divide and conquer.'' With stratification methods, patients are classified into subgroups (strata) that are homogeneous with respect to the confounding factors. In a single stratum, any observed treatment difference in patient outcome is attributable to the treatment, since the patients are otherwise similar to each other in each homogeneous stratum. The differences in patient outcome are often not reported for each stratum of patient but are usually summarized with an overall average value.

Modeling is based on the principal of "synthesize and approximate." A statistical model approximates the relationship between patient outcomes and patient characteristics using an equation. Since outcomes vary from patient to patient, individual patient outcomes cannot be predicted precisely with an equation. A statistical model describes the average patient outcome on the basis of patient characteristics. Modeling methods use an equation (model) to summarize the evidence of treatment differences in patient outcomes across various types of patients.

Stratification methods are less prone to bias than are modeling methods, but stratification requires larger amounts of data in order to obtain precise estimates of adjusted rates because many subgroups of patients may be needed in order to form the homogeneous strata. Modeling methods may be biased if the wrong model is used in the analysis, but they usually yield more precise estimates if the correct model is used.

Stratification

Stratification involves tabulating and combining comparative summaries of rates across specific homogeneous subgroups of patients. For any particular patient subgroup, the specific value of the summary for that group can be looked up in the tabulation. Adjusted summaries can then be derived by using a weighted average of estimates in specific cells of the table. Stratification methods work best when there is a substantial amount of data in each cell of the cross-classification of the data.

The tables in the 1989 USRDS Annual Data Report (USRDS, 1989, p. A.4) give an example of a stratified analysis of incidence rates with direct

Suggested Citation:"Appendix D: Survival Analysis Methods for the End-Stage Renal Disease (ESRD) Program of Medicare." Institute of Medicine. 1991. Kidney Failure and the Federal Government. Washington, DC: The National Academies Press. doi: 10.17226/1818.
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standardization. A comparison of crude (unadjusted) incidence rates by race does not account for the fact that the black populations and white populations in the United States have different age structures. The table allows a comparison of the incidence rates for both populations in equal age intervals. The specific rates for groups of patients defined by race, age, and gender are reported in one part of the table. The incidence rates can be compared by race for each age-gender subgroup. Since there are many age-gender subgroups, it is useful to derive a single comparison that summarizes the results from the individual comparisons. A weighted average of the specific rates for different age-gender groups is computed for each race by gender group. The resulting adjusted rates allow comparisons across racial groups that are adjusted for age and gender. The age-gender adjusted rates for the two races can be interpreted as though the age-gender distributions for the two races were the same. Even though the age distributions for the two races are not the same in the original data, the age-gender adjusted rates estimate the size of the difference in rates that would have been observed had the age (and gender) distributions been the same.

In the example above, the relationship between race and incidence rates was examined with adjustment for the confounding factors of age and sex. Similar methods could be used to yield adjusted estimates of relationships between any two study factors with adjustment for confounding factors. The same type of cross-classification could theoretically be used for mortality rates, as well. For example, the table presented in the 1989 USRDS Annual Data Report (USRDS, 1989, p. D.9) is based on direct standardization of one-year death proportions with adjustment for age, race, gender, and primary diagnosis.

Stratified analyses based on indirect standardization use a different computational method to adjust for confounding factors, although the objective of adjustment is the same as it is for direct standardization. The results with indirect adjustment are often similar to those with direct standardization. Indirect standardization can yield more precise summary comparisons when there are not enough data to yield precise results with direct standardization (Breslow and Day, 1987).

Modeling

In survival analysis, a model is an equation that relates the numerical values of a set of patient characteristics to the numerical value of a summary mortality measure for the corresponding group of patients. A hypothetical model for death rates, h(t), based on age (X1) and race (X2) could be written as

Suggested Citation:"Appendix D: Survival Analysis Methods for the End-Stage Renal Disease (ESRD) Program of Medicare." Institute of Medicine. 1991. Kidney Failure and the Federal Government. Washington, DC: The National Academies Press. doi: 10.17226/1818.
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Such a simple equation often gives a poor fit to the data for death rates. Models for death rates often use the exponential function (inverse of the natural logarithm) in an equation of the form:

An important stage of analysis, called identification of the equation, consists of determining which type of equation to use (equations 1 and 2 are two different types) and which measures to include on the right-hand side of the equation (Box and Jenkins, 1976).

The numerical values appearing in the equations above are called coefficients. The values of the coefficients are determined in each analysis so that the equation fits the data as well as possible. The calculated values of the coefficients in the model are called estimates of the coefficients, in recognition of the fact that they only approximate the values that would result if more data were available.

Model building in statistical analysis often involves iterative stages of model identification and estimation. A variety of criteria have been used in order to assess how well an equation fits the observed data. The results of modeling usually include the form of the equation, the values of the coefficients in the equation, and the precision of the estimated coefficients.

The effect on mortality due to changing only one patient characteristic (such as treatment modality) can be estimated by changing the value of only that characteristic on the right-hand side of the equation. The resulting change in the computed value of the summary mortality measure estimates the effect of changing that one factor while all others are held constant.

The detailed interpretation of a model requires an understanding of the form of the equation and knowledge of the units of each measure in the equation. This allows the numerical value of the survival measure to be interpreted for any combination of the characteristics on the right-hand side of the equation. A rough interpretation of a model often involves only the directions of the effects of each factor on mortality and some evaluation of the importance of those effects.

In the hypothetical examples above, the numerical value of the death rate is computed from an equation based on the numerical value of patient age and numerical codings of the race of a patient. The equations can be used to estimate the mortality rate for any type of patient that can be characterized by the variables in the model equation. In equation 2, assume that the units for death rates are per year, that age is given in years, and that race is coded as 0 or 1 if the patient is black or white, respectively. Then the estimated annual death rates for black patients and white patients of age 60 are 0.230 [= exp (-1.95 + 0.008 · 60 + 0)] and 0.267 [= exp(-1.95 + 0.008 · 60 + 0.15 · 1)], respectively. According to the hypothetical model, death rates are about 16 percent higher for white patients than for black patients of identical ages.

Suggested Citation:"Appendix D: Survival Analysis Methods for the End-Stage Renal Disease (ESRD) Program of Medicare." Institute of Medicine. 1991. Kidney Failure and the Federal Government. Washington, DC: The National Academies Press. doi: 10.17226/1818.
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Simultaneous Effects of Variables

In order to estimate the unique effect of each of several factors on mortality rates when the others are held constant, a multivariable analysis using data for all of the factors must be performed. Results must be simultaneously adjusted for all relevant factors. For example, it is not sufficient to present a set of results that are adjusted for age and another set of results that are adjusted for diagnosis; results must be adjusted for both age and diagnosis simultaneously. As the number of relevant factors increases, the detail needed to summarize the results can increase dramatically.

Research based on data from small sources cannot usually provide the detail that is available from a national registry. Important factors and interactions often appear to be random noise when the sample size is small, and so the results are not reported. Thus, the clinical literature cannot be expected to provide a reliable reference for the level of detail that can be available from a national registry.

Determination of the appropriate level of detail in a statistical summarization requires a balancing of the value of interpretability (the results should not be obscured by the presentation), consistency (broad summaries are more useful than are summaries for each of many subgroups), precision (small standard errors are desirable), and accuracy (the results should not be biased).

Constraints on the Adjustment Process

The major limitation on the adjustment for confounding factors is that the confounding factors must be measured for the patients in the data base. The mortality rates in the USRDS data base cannot be adjusted for patient comorbidity because no such data have been collected. A retrospective collection of such data from patient medical records will allow a study to be performed for some patients, but the validity of the results depends upon the reliability and completeness of the medical records.

A second limitation is specific to the nature of survival analysis. Survival analysis is based on modeling the occurrence of future events (death) on the basis of current patient characteristics. Thus, comparison groups should not be defined in terms of events that occur in the future, but should be defined on the basis of past events. Definitions of patient characteristics cannot be based on the future events for the patients. At any time, the death rates for a group can be modeled in terms of the complete history of the group up to that time, but should not be allowed to depend upon future events.

An example makes the limitations imposed by this second constraint clearer. In order to study the effect of switching treatment modalities on

Suggested Citation:"Appendix D: Survival Analysis Methods for the End-Stage Renal Disease (ESRD) Program of Medicare." Institute of Medicine. 1991. Kidney Failure and the Federal Government. Washington, DC: The National Academies Press. doi: 10.17226/1818.
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mortality rates, data are needed on the sequence of treatments for a patient. Then, mortality rates at a particular time for patients who have not yet switched are compared to mortality rates for patients who have already switched. Note that the comparison groups are defined in terms of what has already taken place. It is inappropriate to compare mortality rates among patients who do not switch treatment modalities in the future to rates among patients who do switch sometime in the future. The difficulty with the second comparison is that patients who die quickly have less chance to switch treatments than do patients who survive a long time, so the death rate of the non switching group would appear to be higher than that of the switching group.

Many attempts have been made to try to make valid comparisons of mortality for groups that are defined in terms of future events. None has been entirely successful.

Case-control studies at first appear to be an exception to the discussion above but are in fact consistent with it. In a case-control study, the comparison groups are defined in terms of the patient outcome itself. That is, survivors are compared to dead patients to see how they differ. This type of study requires great care in its design in order to avoid biased results. Specifically, the controls (survivors) must be selected so that they have been followed for the same length of time as the cases (dead patients). The characteristics of the controls must be evaluated on the basis of their history up to the time corresponding to the death of the case. With this design, the dead patients and the surviving patients are being compared at the same time of follow-up, so their characteristics are effectively being defined on the basis of past (at the time of the dead patient's death) history.

STATISTICAL METHODS OF ANALYSIS FOR ESRD MORTALITY DATA

Statistical survival analyses are designed to summarize the distribution of lifetimes for a population of patients. The summaries characterize the overall pattern of mortality in the population rather than give details of the individual lifetimes which vary from person to person in the population.

One of the distinctive aspects of survival analysis is that although the lifetimes of individuals in a population are to be summarized, not all of the people in the population have died when the summarization is made. Special methods, derived from actuarial concepts, are used in survival analysis to yield an estimate that summarizes lifetimes even though not all of the lifetimes have been observed. The lifetimes that are not completely observed are called censored data, because they are hidden from the analysis by the nature of the observation process.

There are several ways to summarize patterns of mortality in a popula-

Suggested Citation:"Appendix D: Survival Analysis Methods for the End-Stage Renal Disease (ESRD) Program of Medicare." Institute of Medicine. 1991. Kidney Failure and the Federal Government. Washington, DC: The National Academies Press. doi: 10.17226/1818.
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tion, including death proportions, death rates, survival curves, and average lifetimes. These summaries are discussed in more detail below. Each summary is called a parameter. Survival analysis methods use data from a sample of a population in order to yield estimates of the value of the parameter for the whole population. The word parameter is often reserved for a single numerical value, but its meaning has been extended in this appendix to allow it to encompass a function of time or a curve that is plotted versus time (e.g., survival curve).

Some parameters characterize the way in which mortality patterns differ among subgroups of a population. For example, regression coefficients in a statistical model can summarize the relationship between death rates and patient characteristics.

In addition to parameters that characterize the pattern of mortality in a single population, other parameters measure the difference in mortality for two populations. These include differences, ratios, and log ratios.

Each parameter listed above is a useful summary of mortality and can be used to compare mortality for two or more groups of patients. To appropriately interpret the results of analysis, the meaning of the parameters that are being reported must be understood. Mathematical calculations can sometimes be used to relate the values of different parameters to each other in order to compare results from different studies that have used different sets of parameters.

Descriptive Parameters for One Group

A parameter that summarizes mortality for a single group is measured on an absolute scale, and its numerical value can be interpreted without reference to other values. For example, the fraction of a population that has died after one year (i.e., death proportion) is directly interpretable for that population. Other examples of parameters are death rates, surviving fractions, and expected lifetimes.

Death Proportions

The fraction, or proportion, of a group of individuals who die during a specific interval of time is a widely used summary of mortality.

Since most ESRD survival analyses include censored data (some subjects were withdrawn alive from follow-up before the end of the interval under study), this fraction should be estimated using the Kaplan-Meier method or other actuarial type of estimator. This fraction should not be estimated as a simple proportion if the data include censored observations.

Death proportions should not be compared unless they correspond to equivalent time intervals. Thus, 1-year death fractions can be usefully

Suggested Citation:"Appendix D: Survival Analysis Methods for the End-Stage Renal Disease (ESRD) Program of Medicare." Institute of Medicine. 1991. Kidney Failure and the Federal Government. Washington, DC: The National Academies Press. doi: 10.17226/1818.
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compared to each other, but 1-year death fractions cannot be compared directly to 2-year death fractions.

In order to be interpretable, death proportions should correspond to a well-defined interval of time. Thus, the beginning and the end of the interval must be well defined. For example, the interval of time could be the first year of ESRD. A death proportion should not be interpreted on the basis of the length of the interval alone because the death proportion during the first year of ESRD is generally higher than it is during subsequent 1-year intervals. Thus, when comparing the death proportion during the first year of renal replacement therapy (RRT) in the United States to that in the first year of RRT in another country, it should be noted that the data for the United States are likely to start only 90 days after first RRT for each patient whereas data from other countries may start on the first day of RRT for each patient. Because of high mortality during the first 90 days of RRT, the mortality during the first 365 days is likely to differ from the mortality between days 90 and 465. Thus, data from the United States may not be directly comparable to data from other countries.

Although the length of the interval used to calculate a death proportion should be specified, there can be exceptions to the rule. One exception is the fraction of transplant recipients who are discharged alive from the hospital after the transplant operation. In this case, the interval of time is appropriately defined by two events (admission and discharge) rather than by the length of the interval. Such proportions cannot be directly compared to proportions that correspond to other time intervals.

Death proportions are defined relative to the population alive at the beginning of the time interval being considered. Thus, a 10 percent death proportion during the second year of RRT means that among those who survive for one year, 10 percent die during the second year, not that 10 percent of those who started RRT will die during their second year. For example, if the death proportion is 10 percent for both the first and the second year of ESRD, then it is not true that 20 percent will die by the end of 2 years; instead, only 19 percent will die during the first 2 years since 90 percent survive the first year and 10 percent of that 90 percent (or 9 percent more) die during the second year.

In order to compare death proportions based on different types of time intervals, it is sometimes possible to compute death rates that correspond to each of the proportions and then to compare the death rates to each other.

Death Rates

The death rate is approximately equal to the fraction dying during an interval of time divided by the length of the interval. The approximation becomes more precise when the length of the interval is made shorter. The

Suggested Citation:"Appendix D: Survival Analysis Methods for the End-Stage Renal Disease (ESRD) Program of Medicare." Institute of Medicine. 1991. Kidney Failure and the Federal Government. Washington, DC: The National Academies Press. doi: 10.17226/1818.
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numerical value of the death rate usually depends upon the time at which the interval starts. The death rate is thus a function of time and can be represented as a curve plotted against time. The time axis can be chosen from among age, years since first ESRD, or calendar date, depending upon the purposes of the analysis. For example, a plot of death rates versus time since first ESRD would show that the death rate at the beginning of RRT is higher than the death rate in subsequent years. The death rate can be defined as the limiting value of the following equation as the length of the interval declines.

h(t) = Pr(die between times t and t2 given alive at t)/(t2-t)

Death rates are measured per unit of time and are sometimes called hazard rates. The value of a death rate depends upon the unit of measure of times as shown by the following example: a monthly death rate of 0.01 is equal to an annual rate of 0.12.

The most commonly used estimator for the death rate during an interval of time is equal to the total number of deaths observed during the interval divided by the total length of follow-up during the interval. This is calculated as follows: A series of patients is identified who are alive at the beginning of the interval, and the number who die during the interval is counted. The lengths of time that each patient is observed to be alive during the interval are summed across patients to compute the total length of follow-up. The ratio of the death count to the follow-up time estimates the death rates during the interval. This estimator assumes that the death rate is constant during the length of the interval.

Death rates are not proportions, although they are closely related to proportions. They can be greater than 1.0 in value, for example, whereas proportions cannot be. There is a mathematical link between death proportions and death rates that involves a quantity called the cumulative hazard (Breslow and Day, 1987).

The cumulative hazard during a short interval of time is approximately equal to the length of the interval times the hazard at the beginning of the interval. Continuing with the numerical values in the example, the cumulative hazard for death during a 6-month interval would be approximately 0.06, which can be computed as 0.01 · 6 if the calculations are based on months or as 0.12 · 0.5 if the calculations are based on years. The cumulative hazard is used to relate death rates to death proportions, as discussed below.

For short intervals of time, the death proportion for the interval is numerically close to the cumulative hazard during the interval. That is, among those alive at the beginning of a short interval of time, the fraction who die during the interval is approximately equal to the cumulative death rate during the interval.

Death proportions and death rates are related mathematically. If the

Suggested Citation:"Appendix D: Survival Analysis Methods for the End-Stage Renal Disease (ESRD) Program of Medicare." Institute of Medicine. 1991. Kidney Failure and the Federal Government. Washington, DC: The National Academies Press. doi: 10.17226/1818.
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death rate is 0.01 per month, then the death proportion during each month is approximately 1 percent. If a death rate is small during an interval of time, then it is nearly equal to the death proportion per unit of time. The precise relationship involves the exponential function. For example, if the death rate is 0.1 per year, then the proportion surviving for one year is 0.9048. This is calculated as the inverse natural logarithm of the negative of the death rate [inverse ln(-0.1) = exp(-0.1)]. The death proportion is then calculated as 1 minus the surviving proportion, or 0.0952 in this example. Note that the fraction dead after one year—0.0952—is slightly different from 0.10, the death rate per year.

Survival Curves

Survival curves show the fraction of patients who are still alive at each time of follow-up and the full distribution of lifetimes in a population. They are an excellent graphical tool for summarizing patterns of mortality in a population. The horizontal axis in the survival curve measures time since the start of patient follow-up, whereas the vertical axis measures the fraction of patients who are alive at each time.

Survival curves can be reliably estimated with censored data. The Kaplan-Meier and actuarial methods are usually used to estimate unadjusted survival curves. The Kaplan-Meier estimator is more precise than the actuarial methods but is more difficult to compute. It is preferred for use on small samples of patients, whereas actuarial methods yield excellent approximations with larger data sets. An extension of the Kaplan-Meier estimator based on the Cox model (discussed below) can be used to estimate adjusted survival curves.

The estimated survival curves are composed of a series of horizontal lines, but are sometimes drawn as a smooth curve. Survival curve estimates should not be extrapolated beyond the extent of the data that are used to estimate them. Some statistical packages choose to artificially show the survival curve as dropping to 0 (indicating that everyone has died) after the longest follow-up time in the data set, even though a large fraction of the subjects are still alive at that time. The practice should be avoided.

Expected Lifetimes

The average lifetime is a very interpretable summary of mortality and can be used for making superficial comparisons of the mortality patterns for two or more groups of patients. However, several caveats should be kept in mind when interpreting reported average lifetimes. As discussed in the remainder of this section, the average lifetime gives only a superficial summary of the mortality patterns for a population and is difficult to estimate

Suggested Citation:"Appendix D: Survival Analysis Methods for the End-Stage Renal Disease (ESRD) Program of Medicare." Institute of Medicine. 1991. Kidney Failure and the Federal Government. Washington, DC: The National Academies Press. doi: 10.17226/1818.
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with censored data. Because of these limitations, and because summaries based on death rates and survival curves are more useful, use of expected lifetimes is not recommended for the routine summarization of survival analyses for ESRD patients.

The average lifetime does not capture the variability in lifetimes in a population. For example, an average lifetime of 5 years results if every person lives for exactly 5 years and also results if half the people die immediately and half live for 10 years. Thus, the average lifetime does not indicate the details of mortality patterns in a population.

There are two major methods for calculating average lifetimes: parametric and nonparametric. Both can be used with censored data. However, both methods can yield unreliable or uninterpretable answers if a large fraction of the population is estimated to be alive at the end of the study. Many statistical packages calculate a nonparametric truncated lifetime with censored data, the truncated point being the time of the longest censored lifetime in the data set. Truncated lifetimes are not comparable to each other unless the truncation times are equal to each other. Parametric models rely on extrapolation to estimate the average lifetime with heavily censored data, but estimates of the average lifetime can be very unreliable if they are based on heavily censored data.

Comparative Parameters

Summaries of individual groups are often compared to each other. Instead of reporting the individual mortality summaries for each of several groups, comparative statistics can be used to summarize just the sizes of the differences between the individual mortality summaries. Important examples include relative death rates (from a Cox or other proportional hazards regression model), differences in death proportions, and relative lifetimes. Comparative statistics cannot be directly interpreted for either of the two groups being compared. However, if a mortality measure is reported for one group and comparative summaries are reported for other groups, then the individual group summaries can usually be calculated for each group.

Two common comparative statistics are often reported: differences (subtraction) and ratios (division). It is crucial to the interpretation of the results to know which is being reported. For example, if the fractions dead in two groups are 10 percent and 20 percent, then the comparison for the second group relative to the first group could be reported as a 10 percent increase (difference), as a 100 percent increase (ratio), or as double the fraction (ratio). In addition, the comparison could be summarized by saying that the surviving fraction in the second group is only 89 percent (100 × 8/9) as high as in the first group.

Several other specific types of comparative summaries are discussed below.

Suggested Citation:"Appendix D: Survival Analysis Methods for the End-Stage Renal Disease (ESRD) Program of Medicare." Institute of Medicine. 1991. Kidney Failure and the Federal Government. Washington, DC: The National Academies Press. doi: 10.17226/1818.
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Regression Models

Several different regression models are appropriate for use with survival data, including Cox, Poisson, Weibull, Bailey-Makeham. These are discussed and compared in more detail below. All of these regression models allow multiple patient characteristics to be empirically related to mortality. Since the Cox model is the most widely used method of analysis for survival data and since it has many of the features of other methods, the interpretation of results from the Cox model is discussed in detail here.

The Cox model yields an estimate of the ratio of death rates for two groups of patients and the standard error of the ratio. The ratio represents a single numerical summary of the difference in mortality patterns for two groups. The standard error allows an evaluation of uncertainty of the estimated ratio and can be used to compute the statistical significance of the difference between the groups and a confidence interval for the magnitude of the difference in death rates between the two groups.

In addition, Cox model analyses yield adjusted survival curve estimates for several groups. The adjusted survival curves are a more complete summarization of the comparative mortality patterns among the groups than are death rate ratios. Moreover, the adjusted survival curve estimates are useful for descriptive analyses.

Adjusted Death Rate Ratios The primary results of most Cox model analyses are estimates of death rate ratios. A ratio different from 1.0 indicates that the death rates are different for the two groups being compared. The two groups being compared can be defined in terms of either quantitative (e.g., age) or qualitative (e.g., etiology) characteristics. The model yields adjusted estimates of the ratio of death rates for two groups that differ with respect to one characteristic if all other characteristics included in the model are the same for the two groups. The Cox model regression coefficient is an estimate of the log of the death rate ratio. Thus a coefficient value of 0 corresponds to a rate ratio of 1. The rate ratio corresponding to two groups of patients who differ by 1 unit with respect to the value of a specific characteristic is computed as the exponential function of the coefficient estimate for that characteristic in the regression model.

For example, in order to estimate the ratio of death rates for diabetic and nondiabetic patients, a 0–1 indicator for diabetes would be included in the Cox model. The estimated ratio of death rates would be estimated as the exponential function of the estimated coefficient of the diabetes variable.

If only the diabetes variable were included in the model, then the estimated ratio would be unadjusted and would include the effects of any other differences between the two groups. Part of the difference between the diabetic and nondiabetic patients might be explainable by age differences

Suggested Citation:"Appendix D: Survival Analysis Methods for the End-Stage Renal Disease (ESRD) Program of Medicare." Institute of Medicine. 1991. Kidney Failure and the Federal Government. Washington, DC: The National Academies Press. doi: 10.17226/1818.
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between the two groups, for example, but this effect would be included in the estimated unadjusted ratio of death rates for the two groups.

By simultaneously including several patient characteristics in a Cox model, the effect on mortality of any one factor can be estimated with adjustment for the other factors in the model. For example, in order to estimate the ratio of death rates for diabetic and nondiabetic patients who are of the same age, both diabetes and age would be included in the Cox model. The resulting coefficient for the diabetes variable estimates the log of the adjusted death rate ratio. This ratio would be adjusted only for age. In order to adjust for other potential confounding factors, they must also be included simultaneously in a Cox model.

In order to adjust appropriately for other factors in a Cox model, they must be included in the model in the correct form. As a first step, many researchers often assume that the effect on mortality of each characteristic in the model is independent of the other characteristics in the model. Such a model is called a main-effects model. However, there is substantial evidence that the effect on mortality of some factors is modified by other factors. For example, white patients have substantially higher death rates than do black patients if their diagnosis is hypertension or diabetes, but not otherwise. Models that account for these more complicated relationships are called interaction models.

The principal value of multiple regression models is to yield estimates of the effect of one factor, with adjustment for other confounding factors. Models that do not adjust for relevant factors, or that adjust for them incorrectly, yield less definitive interpretations than do more complete models. Although no model can adjust for characteristics that are unknown or unmeasured, it is useful to eliminate the effects of known important factors as accurately as possible.

Survival Curves A qualitative patient characteristic can be included in a Cox model in two different ways. If a characteristic is included as a covariate in the model, then its effect on mortality is summarized by a ratio of rates. If a characteristic is included as a stratifying variable, then its effect on mortality is summarized as a set of survival curves for the different groups that it represents. For quantitative comparisons, a patient characteristic is usually included as a covariate. For descriptive comparisons, a characteristic is usually included as a stratifying factor.

A third method of presentation of comparative results is sometimes used in a Cox model, but it can be deceptive. In that method, a patient characteristic is included as a covariate in a Cox model, and different adjusted survival curves corresponding to different values of the covariate are estimated. Such survival curves are based on the overall experience in the whole data set, which is adjusted up or down, depending upon the value of

Suggested Citation:"Appendix D: Survival Analysis Methods for the End-Stage Renal Disease (ESRD) Program of Medicare." Institute of Medicine. 1991. Kidney Failure and the Federal Government. Washington, DC: The National Academies Press. doi: 10.17226/1818.
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the covariate. However, it is usually much more informative to estimate the survival curve for each group, based on its own data, through the use of stratification, as described previously.

Specific Models and Methods

A variety of statistical methods have been used or proposed for the analysis of mortality rates among ESRD patients. Several of these methods, all of which are regression models, are discussed below, and include Poisson, Cox, logistic, exponential, Weibull, and Bailey-Makeham. All can yield interpretable summaries of the way in which mortality is simultaneously related to several patient characteristics. All have associated methods of statistical inference that allow confidence intervals and statistical hypothesis tests to be computed.

All statistical methods are based on certain assumptions that must be checked empirically in order to ensure that the methodology is appropriate for the data being analyzed. The methods discussed below differ with regard to the types of assumptions that they make. Methods based upon either Poisson regression or Cox regression models are applicable for a variety of important issues related to the survival of ESRD patients. Methods based on logistic regression models are limited in the types of data for which they are appropriate. Fully parametric methods may prove useful for specific objectives, but they are based on more assumptions than are the Poisson and Cox regression models and therefore require more careful checks of the appropriateness of the model.

The Poisson and Cox regression models are both extremely flexible in the types of models that can be estimated. Both yield estimates of relative death rates. The major difference between them is that Poisson regression models yield direct estimates of death rates whereas Cox models yield direct estimates of survival curves. Both methods can require substantial computing resources if the specification of the model involves patient characteristics that change over time.

The methods discussed below vary according to several dimensions, including the way in which time is accounted for in the analysis, appropriateness, interpretability, ease of implementation, and level of parametrization of the method.

Poisson Regression for Death Rates

A death rate approximates the probability of dying per unit time among those still alive. Death rates among dialysis patients are typically between 3 percent and 60 percent per year for young and old patients, respectively (USRDS, 1990, Table D.29), or approximately 0.25 to 5 percent per month.

Suggested Citation:"Appendix D: Survival Analysis Methods for the End-Stage Renal Disease (ESRD) Program of Medicare." Institute of Medicine. 1991. Kidney Failure and the Federal Government. Washington, DC: The National Academies Press. doi: 10.17226/1818.
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It is clear that death rates vary among dialysis patients, depending upon the characteristics of the patients being described. Death rates also vary among ESRD transplant recipients, although they tend to be lower for transplant recipients than they are for dialysis patients. Statistical models as well as cross-tabulations can be used to summarize how the death rates vary according to the characteristics of the patients and their treatment modalities.

A statistical model for death rates uses an equation to summarize the way in which death rates vary according to the characteristics of the patients. Statistical models only approximate the broad patterns of variability in death rates and do not attempt to account for each individual patient death. As an example, Equation 3, below, was estimated using data for black patients (USRDS, 1990, Tables D.28 and D.31):

Rate = exp(-3.74 + 0.03518 × Age)

The death rates calculated by this equation agree closely with the observed death rates shown in Table 1, although the agreement is far from perfect. The equation implies that the death rates among black ESRD patients increase by a factor of 1.0358 [=exp(0.03518)] for each 1 year increase in age. The advantage of the equation is that it allows the change in death rates with age to be easily summarized as a 3.6 percent increase per year of age, instead of requiring the full tabulation of observed rates in Table D-1. The lack of agreement between the rates calculated using the equation and the observed rates might be due to random variability in the observed rates or to the fact that the equation is not a perfect representation of the true relationship between age and death rates. However, the discrepancy between the rates calculated using the equation and the observed rates is relatively unimportant compared to the substantial change in death rates with age.

The discrepancies between the rates calculated from the equation and the observed rates may be useful for identifying the exceptions to the general rule given by the equation. For example, the observed rates tend to be higher than the calculated rates at very young ages, indicating that death rates among young patients may follow a slightly different pattern than is apparent among older patients. This process of fitting a model, summarizing the broad features of the model, and then looking for discrepancies between the observed data and the model is very useful in finding a balance between useful generalizations and levels of detail.

Death rates can also be tabulated simultaneously according to several patient characteristics. USRDS Table D.31 reports death rates by age, race, and disease group. The resulting Table D.31 includes a lot of detail that is useful for looking up specific death rates but not for making general conclusions. A statistical model (not presented here) could quantitatively summarize the main features of such a table with just a few broad generalizations.

Suggested Citation:"Appendix D: Survival Analysis Methods for the End-Stage Renal Disease (ESRD) Program of Medicare." Institute of Medicine. 1991. Kidney Failure and the Federal Government. Washington, DC: The National Academies Press. doi: 10.17226/1818.
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TABLE D-1 Death Rates and Age Among All Black ESRD Patients in 1988

Age (lower limit)

Death Rate Observeda

Model

0

30.6

23.8

5

28.3

28.3

10

51.3

33.7

15

40.9

40.2

20

55.1

47.8

25

64.7

57.0

30

76.2

67.9

35

85.5

80.9

40

91.8

96.3

45

108.7

114.7

50

129.6

136.7

55

154.0

162.8

60

194.1

194.0

65

242.1

231.1

70

299.9

275.3

75

334.3

327.9

80

379.5

390.6

85

461.0

465.3

a USRDS, 1990, Table D.31.

Poisson regression models are appropriate for estimating statistical models for death rates using data from a registry such as the USRDS. In its simplest form, this methodology is descriptive; moveover, it can yield estimates of the death rate for any specific combination of patient characteristics. That is, death rates can be estimated for patient subgroups that are defined by a simultaneous specification of several patient characteristics. Poisson regression models can be implemented with GLIM or S-plus statistical packages that are available for many computers.

If the number of characteristics used to classify patients is large, then there may be few patients in any specific cross-classification. Consequently, the data used to estimate death rates for specific groups may be sparse and the resulting estimates are likely to be unreliable. For example, the death rate reported by USRDS (1990, Table D.31) for white diabetic patients age 0 to 4 years in 1988 is 1,500 per 1,000 patient years at risk. This rate is estimated on the basis of two deaths (USRDS, Table D.28) and is likely due to chance events rather than to an important phenomenon. With sparse data, multiple regression models can be used to effectively pool information across various subgroups in order to yield more precise estimates of the true

Suggested Citation:"Appendix D: Survival Analysis Methods for the End-Stage Renal Disease (ESRD) Program of Medicare." Institute of Medicine. 1991. Kidney Failure and the Federal Government. Washington, DC: The National Academies Press. doi: 10.17226/1818.
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death rate for any particular patient subgroup. In addition, statistical models can be used to summarize the adjusted effect of a single factor on mortality rates.

Death rates of ESRD patients vary with the number of years since first ESRD service, with a general decrease in death rates as the number of years since first service increases. It is speculated that this decrease is due to attrition of the less healthy ESRD patients as time goes on. The death rates discussed in this section ignored this factor and collapsed together patients with differing amounts of time since first service. The time since first service could be accounted for by including it as a patient characteristic that is entered into the Poisson regression model. Alternatively, the Cox model, discussed below, accounts for time since first service explicitly without the need to classify patient follow-up according to this measure of time.

A comprehensive Poisson regression model would include multiple patient characteristics in the estimated equation for death rates. Some of these patient characteristics, such as gender, race, year of first ESRD therapy, and primary diagnosis, do not change with time. Some important patient and treatment characteristics that can change with time include age, years since first ESRD therapy, treatment modality, and treatment facility. Poisson regression models can account for all these characteristics simultaneously, although the data management and analysis costs required to do so would be substantial.

Cox Models for Relative Rates and Survival Functions

The Poisson regression models discussed in the previous section are most appropriate for data based on short intervals of patient follow-up. If long intervals of time are used, then patient characteristics such as age and time since first ESRD service vary during the time interval, and it is less appropriate to ascribe a single death rate to the interval. The Cox models allow the periods of follow-up to be different for each patient and to be arbitrarily long or short. However, instead of yielding estimates of death rates, as with the Poisson regression models, the Cox models yield estimates of relative death rates and of the survival function, i.e., the fraction surviving at various times since entry into study.

The Cox model was designed primarily to estimate relative death rates. Thus, the ratio of death rates for any pair of patient subgroups is estimated directly by the Cox model, whereas survival functions rather than specific death rates are estimated for each patient subgroup.

The survival function gives the fraction of patients surviving at each of several times since entry into study. For ESRD patients incident in 1979, survival functions are reported in Table D-2 for several time points after first ESRD therapy. Mortality among ESRD patients is often described rela-

Suggested Citation:"Appendix D: Survival Analysis Methods for the End-Stage Renal Disease (ESRD) Program of Medicare." Institute of Medicine. 1991. Kidney Failure and the Federal Government. Washington, DC: The National Academies Press. doi: 10.17226/1818.
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TABLE D-2 Survival Probabilities for ESRD Patients Incident in 1979

Years Since 90 Days After First ESRD Service

Fraction Surviving

0

1.0000

1

0.8138

2

0.6786

5

0.4419

10

0.2576

 

SOURCE: USRDS, 1990.

tive to the number of years since first therapy for ESRD because patient data are entered into the USRDS data base near the time of first ESRD therapy. Since the first 90 days of ESRD therapy is undocumented for many patients in the USRDS, mortality is described subsequent to 90 days after the first service for ESRD in the USRDS Annual Data Report. Table D-2 summarizes the fraction of patients still alive for various intervals of time since 90 days after first ESRD service. The data for this table are abstracted from the USRDS Annual Data Report Tables E.10, E.12, E.14, and E.16.

The Cox model uses data from patients who have different periods of follow-up. The Cox model uses data from all patients to estimate 1-year survival probabilities. The Cox model then uses only the data from patients with 2 years of follow-up to estimate the probability of survival during the second year, given survival through the first year. The product of these two probabilities yields the overall probability of surviving for 2 years. Although this description of the Cox model is a simplification of the calculations that are actually performed, it captures the essential nature of the concept involved in the survival curve estimates derived from the Cox model. The model also uses data from different periods of patient follow-up to estimate relative death rates.

The ability to utilize data from patients with different periods of followup is one of the features that makes the Cox model so useful. Such data are often said to be right censored, and the Cox model is the most widely used method for multiple regression analysis with right censored data. It is adaptable to a wide variety of types of applications and data structures. Moreover, it can be used to evaluate the effect of changing treatment modalities and other time-dependent factors, such as patient age, calendar year, and time since first ESRD therapy. The results of analysis include estimates of relative death rates and survival curves.

The Cox model and the Poisson regression models have similar applica-

Suggested Citation:"Appendix D: Survival Analysis Methods for the End-Stage Renal Disease (ESRD) Program of Medicare." Institute of Medicine. 1991. Kidney Failure and the Federal Government. Washington, DC: The National Academies Press. doi: 10.17226/1818.
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tions. Both yield estimates of relative death rates. Both yield summaries of overall mortality; the Poisson regression model yields estimates of death rates whereas the Cox model yields estimates of the survival function. The Poisson regression model requires that the data be grouped into relatively short periods of follow-up time, whereas the Cox model allows the followup interval for each patient to be arbitrarily long. The calculations for the Cox model tend to be more intensive than for the Poisson regression model, especially if time-dependent covariates are included in the model. The data management costs tend to be higher for the Poisson regression model than for the Cox model.

Estimation for Cox regression models can be implemented with SAS or BMDP computer programs. If time-varying covariates are included, BMDP is preferred, although SAS will soon implement facilities for analyzing time-dependent covariates.

Logistic Regression for the Probability of Death

Logistic regression models can be used to estimate the probability of death during a specified interval of time. Such models are appropriate for analysis of prevalent cohorts and for detecting trends in mortality due to changes in treatment patterns (E. Lowrie, National Medical Care, personal communication, 1990). However, the proposed implementation has serious deficiencies.

The logistic model yields estimates of the probability of death for each set of patient characteristics. Such models must be based on data from a series of patients who are all potentially followed for the same period of time. The follow-up requirement is difficult to ensure with data from a registry such as the USRDS, and it is especially hard to ensure if treatment modality is being studied.

Since mortality patterns change with the length of ESRD treatment, a comparison of mortality proportions for a specific time interval tells only part of the story. The patient characteristics that are most important during the first year of ESRD therapy may be different from the patient characteristics that are most important during the second year of ESRD therapy. A series of probability models, one for each interval of time, would then provide a more complete tool for comparing mortality patterns. This can be more easily accomplished with a single survival analysis model (Cox model), which provides estimates for the probabilities of death at each time interval of follow-up.

Equal intervals of follow-up cannot be ensured when members of the group of interest can leave the study, for example, because of treatment modality changes. As an example of this difficulty, consider the problem of estimating the 1-year survival probability among dialysis patients. In order

Suggested Citation:"Appendix D: Survival Analysis Methods for the End-Stage Renal Disease (ESRD) Program of Medicare." Institute of Medicine. 1991. Kidney Failure and the Federal Government. Washington, DC: The National Academies Press. doi: 10.17226/1818.
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to estimate this with a logistic regression model, the analysis must be limited to patients who are to be treated for at least 1 year with dialysis. Thus, it is clear that patients who receive transplants during the first year should be excluded from the analysis. The difficulty arises because among the dialysis patients who die in the first year, it is not known which among them would have received transplants had they survived. Thus, there is no way to limit the analysis to patients who would have received dialysis for a full year. Exclusion of the survivors who receive transplants without exclusion of the patients who died before receiving transplants will bias the estimated probability of death and cause it to be higher than it should be.

Models for the probability of death are especially difficult to interpret when the population of interest is defined in terms of the complete treatment history, rather than in terms of the treatment modality at the beginning of the time interval. For example, the 1-year survival probability for all ESRD patients who start on hemodialysis can be estimated as a simple fraction. This probability is based on all patients starting on hemodialysis, regardless of subsequent treatment changes. The 1-year survival probability among those dialysis patients who either remained on hemodialysis for a complete year or who died within one year with no change to another treatment modality is less interpretable. (See the section on Constraints on the Adjustment Process, above.)

The effect of year of current therapy on dialysis mortality rates can be estimated with either Poisson or Cox regression models. A unified analysis would account for the simultaneous effect on death rates of year of first therapy, year of current therapy, patient age, and other patient characteristics. Logistic regression models are not appropriate for this analysis because of the difficulties discussed above.

Conditional Logistic Regression and Sampling from the Risk Set

One of the difficulties with using either Cox or Poisson regression models to evaluate the effect of treatment modality on death rates is that treatment modality changes with time for many patients. The data management efforts required to document all of the treatment changes for all of the patients in the USRDS data base are substantial. For such analyses, the Cox model can be used with a reduced data set to yield results that are nearly as precise as those resulting from analysis of the full data set. The reduced data set is derived by sampling in a specific way from the full data set. The Cox model, then, is equivalent to a conditional logistic regression model. Although the level of detail required to describe this methodology is inappropriate for this document, details of this methodology have been described by Breslow and Day (1987) and are not too difficult to implement.

Suggested Citation:"Appendix D: Survival Analysis Methods for the End-Stage Renal Disease (ESRD) Program of Medicare." Institute of Medicine. 1991. Kidney Failure and the Federal Government. Washington, DC: The National Academies Press. doi: 10.17226/1818.
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Fully Parametric Models

Certain specific parametric models have proved useful for answering specific research questions about survival patterns. The major limitation of such models is that they start with the assumption that a particular type of equation is correct for the population being studied. If the assumption is (approximately) correct, then the conclusions based on the model are (approximately) accurate. However, if the assumed equation is incorrectly specified, then the conclusions based on the equation can be inaccurate. Parametric models can sometimes lead to useful qualitative conclusions, even when the quantitative results are inaccurate because the model is incorrectly specified.

Parametric statistical models are based on the choice of a particular type of formula or equation that might plausibly approximate the survival pattern in a population. Such equations are often specified by the numerical values of a few parameters, or coefficients. Once the values of the parameters are known, the equation can be used to compute the value of any other characteristic that can be defined in terms of the equation.

Fully parametric models are based on certain assumptions about mortality patterns which may or may not be true for the ESRD population; thus, the resulting analyses may or may not be appropriate. Nonparametric or semiparametric models are based on fewer assumptions than are fully parametric models, and the results of nonparametric analyses are correspondingly less likely to be biased or incorrect. However, if an appropriate model is used, the results of parametric analyses tend to be more precise than the results of nonparametric analyses.

Exponential Model The exponential model is based on the assumption that the death rate for a group of ESRD patients does not change with time. If this assumption is correct, then the survival curve for the patients is an exponential function of time and the curve is specified by one parameter: the death rate per unit of time. On the basis of the value of this death rate, other values can be computed, including the median lifetime, the expected lifetime, the probability of surviving for 5 years, and so on.

However, the exponential model is known to be a poor approximator of the long-term survival pattern for people because death rates rise with age. Further, the death rate among ESRD patients tends to decrease with the number of years since first ESRD therapy. Thus, average lifetimes that are calculated on the basis of an assumed exponential distribution are likely to be inaccurate if the calculated lifetime spans a large age or time range.

Weibull Model This model yields a better approximation of mortality patterns among ESRD patients than does the exponential model. The Weibull model has two parameters that must be estimated from the data, in addition to regression coefficients that relate mortality rates to patient characteris-

Suggested Citation:"Appendix D: Survival Analysis Methods for the End-Stage Renal Disease (ESRD) Program of Medicare." Institute of Medicine. 1991. Kidney Failure and the Federal Government. Washington, DC: The National Academies Press. doi: 10.17226/1818.
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tics. These two parameters allow the Weibull model to fit a variety of patterns of mortality. The model yields useful and interpretable summaries of death rates, survival functions, relative death rates, and so on. Estimates of the model can be derived easily using the SAS statistical package. It is not as easy to use the Weibull model with time-varying patient characteristics as it is to use the Cox or Poisson regression models, because the standard statistical packages have not been extended to allow time-dependent covariates with the Weibull model.

One danger in the use of this model, or any other parametric model, is that the model can be estimated on the basis of a short period of patient follow-up and then extrapolated to yield estimates of long-term survival. There is no way to check the assumptions that the model makes for long-term survival on the basis of short-term data, and consequently there is no way to be assured that the long-term extrapolations are correct. The more nonparametric Poisson and Cox regression models naturally limit their predictions to the intervals of time for which data are available and thus are less subject to the abuse of extrapolation.

Bailey-Makeham Model The Bailey-Makeham parametric model has proved to be very useful for qualitatively distinguishing between predictors of long-and short-term survival. If the model can be shown to yield a good approximation to the survival distribution for ESRD patients, then it may prove to be a particularly important analytical tool. The Bailey-Makeham model is less widely implemented on computers than are the Weibull, Poisson, and Cox models. Although the Bailey-Makeham model can be used to answer the same variety of analytic questions as can other models, it is especially attractive for its ability to quantify the differential effect of patient characteristics on long-and short-term patient survival.

Prevalent Versus Incident Cohort Analyses

In addition to the selection of an appropriate statistical methodology, it is crucial to select the appropriate group of patients to be included in the analysis. The selection depends strongly upon the objectives of the analysis of particular relevance is the distinction between prevalent and incident cohorts of patients. A prevalent cohort includes all patients treated during a specific year, including those whose therapy started prior to that year. An incident cohort includes only patients whose therapy started during a specific year. Many of the analyses performed to date have been limited to one or the other type of study group. Analysis of successive prevalent cohorts is most relevant to detecting trends in mortality over calendar time. Analysis of incident cohorts is more appropriate if the objective is to characterize how mortality changes with the time since first ESRD therapy.

Suggested Citation:"Appendix D: Survival Analysis Methods for the End-Stage Renal Disease (ESRD) Program of Medicare." Institute of Medicine. 1991. Kidney Failure and the Federal Government. Washington, DC: The National Academies Press. doi: 10.17226/1818.
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For the purposes of summarizing patient survival after ESRD therapy starts, it is appropriate to classify patients by the year in which their ESRD therapy started. Such classification accounts for the changes in acceptance patterns that might occur over time. By definition, the baseline characteristics of a patient accepted into the Medicare program for ESRD do not change subsequent to acceptance. In such a classification, each patient is in just one cohort. The USRDS Annual Data Report (1990) has reported such summaries.

If the objective of analysis is to evaluate changes in therapy that occur with calendar year, possibly in association with changing technology or with program administration, then it is more useful to classify patient follow-up according to the year in which the therapy occurs. A patient contributes information on death rates in the prevalent cohort during each of the years that the patient is treated. Further, a patient is potentially in each of several prevalent cohorts in such a classification. Analysis can be performed either with Poisson regression models or with a Cox model using time-dependent covariates or strata. The analysis is conceptually similar to a series of annual analyses of 1-year survival for all prevalent (new and continuing) ESRD patients in each year. Eggers (various years) has reported such series in some of the HCFA reports.

In order to evaluate the simultaneous effects of both year of incidence and year of treatment, the statistical model used must incorporate both time measures. Tabulation of death rates according to patient characteristic and according to year of incidence and year of treatment would not be useful because the number of cells to be examined would be too large and the data would be too sparse.

Frailty

All of the statistical methods discussed above can account for measured patient characteristics and can summarize the relationship between patient characteristics and mortality. However, none of the models can directly account for patient characteristics that are not measured. There are several unmeasured patient characteristics that are related to mortality, and patients who are at higher risk for these unmeasured characteristics will tend to die sooner than patients who are at lower risk. The ensemble of unmeasured patient characteristics has been given the name frailty in the statistical literature (Vaupel et al., 1985), and frailty is known to affect the estimation of death rates. The selection process tends to lead to apparently lower death rates as time goes on because the less frail individuals are those that survive. In counterbalance to the unmeasurable effect of decreasing frailty are the measurable effects of increasing age as time goes on. Several statistical methods are currently being developed to account for frailty.

Suggested Citation:"Appendix D: Survival Analysis Methods for the End-Stage Renal Disease (ESRD) Program of Medicare." Institute of Medicine. 1991. Kidney Failure and the Federal Government. Washington, DC: The National Academies Press. doi: 10.17226/1818.
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Treatment Modality

One important objective of survival analysis in the ESRD program is the evaluation of treatment modalities for ESRD. Such analysis is complicated because clinical trials are not commonly used for the evaluation of treatment therapies. Instead, treatment therapies are selected through a highly subjective set of decisions that involve both the patient and the provider of care. Because of this process, it is likely that different therapies have different profiles of patients assigned to them. Thus, any differences noted in patient outcomes could be due either to the different therapies or to differences in patient characteristics.

In order to reach more definitive conclusions, information is needed about the condition of each patient at the time of each therapy change. Using patient condition data collected at the start of each therapy, statistical methods could be used to yield adjusted measures of patient outcomes for a specific therapy. Further, the patient condition at the start of a therapy change could act as a measure of patient outcome for the previous therapy.

It is impractical and unnecessary to collect such detailed data for a census of the ESRD patient population. Instead, statistically valid samples could be drawn from the population of ESRD patients. Such samples could be drawn either prospectively, from newly incident cases, or retrospectively, from patients who have already received ESRD therapy. Prospective samples would be necessary if data collection were to include measures that are not readily available in the medical records. Retrospective samples could be drawn if data collection were to be limited to information that was readily available in existing records.

Publication of Standard Death Rates

The USRDS has started to publish mortality rates that can be used for small data base research. Death rates among prevalent ESRD patients in 1988 have been calculated for each major age-race-disease group classification (USRDS, 1989, Table D.31). These national rates can be used to compute the expected number of deaths for any study group. The ratio of the observed to the expected number of deaths can then be used to evaluate the mortality rates for the study group. Methods for such calculations are reviewed in detail by Breslow and Day (1982, 1987).

The rates published in the USRDS Annual Data Report are currently limited to the prevalent cohort of ESRD patients at the start of 1988. If continued for successive years, these rates will be useful for comparing death rates in small study groups to the expected rates based on national data. In addition to the patient characteristics of age, race, and disease group given in the published rates, it would be of value to extend the list of patient character-

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istics so that more precise comparisons could be made. Other patient characteristics could include gender, comorbidity, treatment, year of current prescription, and year of first prescription.

Institutional Characteristics

Analyses of patient-specific characteristics should be distinguished from analyses of facility-specific characteristics. The effective sample size for patient-specific analyses is related to the number of patients whereas the sample size for facility-specific analyses is the number of facilities studied. These two different types of analyses typically are addressed with different methods. Analysis of facility-specific outcomes should be based on a single observation per facility, as discussed by Cornfield (1978).

The type of institution or treatment protocol at an institution may affect mortality rates. In order to study such relationships, the institution rather than the patient is the unit of analysis (assuming that all patients at an institution receive the same treatment). Factors that could be or could have been studied on the basis of institutional analyses are profit-nonprofit status, dialyzer reuse, length of dialysis, and transplant technique.

Internal and External Standardization

The statistical analyses described here are based on the concept of comparison. For example, the death rates for two groups of patients can be compared. The mortality rate for one group of patients can be compared to that from another group in the same study (internal comparison) or to published mortality rates for another population (external comparison). Death rates published by the USRDS could serve as an external standard of comparison for a series of patients from a small study. In a larger study, there may be sufficient numbers of patients in several patient subgroups that their mortality rates can be usefully compared to each other.

Generally, internal comparisons are more valid than external comparisons, because bias is less likely to be a problem. However, external comparisons can provide indications of trends that may be useful for qualitative comparisons.

Analyses that involve an external comparison or standard may prove useful in understanding the effect of ESRD on mortality rates. ESRD patients with an etiology of hypertension could be compared to the general population with hypertension. Similar comparisons could be made for diabetes. For example, ESRD patients with an etiology of AIDS may have much higher mortality rates than do patients with other etiologies. However, therapy for ESRD may prove to be just as useful in extending the lifetime of ESRD AIDS patients relative to expected lifetimes among non-ESRD AIDS patients, as it is for diabetic ESRD patients relative to non-ESRD diabetic patients.

Suggested Citation:"Appendix D: Survival Analysis Methods for the End-Stage Renal Disease (ESRD) Program of Medicare." Institute of Medicine. 1991. Kidney Failure and the Federal Government. Washington, DC: The National Academies Press. doi: 10.17226/1818.
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Some of the specific models described previously allow internal as well as external comparisons of mortality rates to be made simultaneously.

International comparisons of mortality among ESRD patients are a form of external comparison in which data from very disparate sources are evaluated. Some of the uses and limitations of such comparisons are discussed in the next section.

INTERNATIONAL COMPARISONS

International comparisons of mortality rates have two major objectives. The first is to document the existence of differences in mortality rates, if they exist. The second is to identify the reasons for the differences, if they exist. Using the currently available data, it is difficult to arrive at a definitive answer to the first objective and it is impossible to arrive at an answer to the second. The current data can give indications, but not proof, of differences in mortality rates for otherwise similar patients from different nations. Expert opinion can then be sought regarding hypotheses about causes of any differences that are thought to exist.

With the current system of separate registries, international comparisons can serve, at best, to point out somewhat crude differences in mortality patterns. Since only rough adjustments are possible across registries, there is no feasible way to isolate the reasons for any observed differences among nations.

The most recent and comprehensive international comparisons of mortality rates have been reported recently by Held et al. (1990). Many of the comments below are specifically motivated by the Held report but are also relevant to the interpretations of any international comparisons. Most of the limitations of international comparisons described below were recognized and acknowledged by Held report but are reviewed here in more detail. The results in the Held report are intriguing and give some indication that ESRD mortality rates are substantially lower in other nations than they are in the United States. Specific hypotheses generated by the Held comparison should be evaluated in more detail in order to determine whether a cause-and-effect explanation can be found for the differences that were found in that study. In addition, mortality rates in the United States should be closely monitored for trends over time.

Limitations

Many of the issues relevant to the use of survival analysis techniques for the analysis of U.S. data are also relevant to the international comparison of mortality rates. However, the problems are compounded in international comparisons because the data bases often have not been analyzed in a consistent way, the data often have not been collected in a consistent way, and

Suggested Citation:"Appendix D: Survival Analysis Methods for the End-Stage Renal Disease (ESRD) Program of Medicare." Institute of Medicine. 1991. Kidney Failure and the Federal Government. Washington, DC: The National Academies Press. doi: 10.17226/1818.
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there are almost certainly differences in the characteristics of patients from different nations that are not measured in the data bases. Adjustment for confounding factors is more problematical with international comparisons than it is with national analyses because the patient-specific data are not available in a unified structure.

In order to use statistical analysis of international data to determine the reasons for differences in mortality rates, it will be necessary to measure the potential causes of any differences at the national level and to correlate those measures with adjusted mortality rates across the nations.

The level of mortality observed in a national registry is strongly influenced by the criteria for acceptance into the registry. Different acceptance criteria, whether part of stated policy or influenced by the individuals who implement the policies, can have dramatic effects on mortality rates. If only healthy patients are accepted into a treatment program, then mortality rates will tend to be lower than if patients with high levels of comorbidity are accepted into the program. Different rates of diagnosis and treatment of ESRD among various nations gives some indication that acceptance criteria differ among nations, although the direction of the bias, if any, that such differences would cause is unknown. After patient age, the most important patient characteristic for predicting patient mortality may well be comorbidity, which is not recorded at the national level in the United States except for primary diagnosis. Since comorbidity is not currently recorded in the registries, it cannot be currently determined whether differences in patient morbidity are a likely cause of differences in patient mortality.

There are known differences between the types of patient accepted into the U.S. and other ESRD treatment programs. The importance of such differences is documented by the experience in the U.S. alone. The acceptance rate into the ESRD program in the U.S. has increased dramatically over recent years, with the result that the treated ESRD population is substantially older and has many more diabetic patients than it did previously. This has led to an increase in the crude mortality rate in the United States since 1977 (USRDS, 1990, Tables E.10, E.12, and E.14). However, death rates adjusted for age, race, sex, and primary diagnosis have been relatively stable during the same period (USRDS, 1990, Tables E.53, E.55, and E.57). Post hoc adjustments to international comparisons of mortality rates can also be made for known patient characteristics, such as age and etiology, but such adjustments are likely to be less accurate than would be a unified analysis of the combined data from several nations.

Etiology

The adjustments made by Held et al. (1990) to international comparisons have partially accounted for national differences in age and frequency of

Suggested Citation:"Appendix D: Survival Analysis Methods for the End-Stage Renal Disease (ESRD) Program of Medicare." Institute of Medicine. 1991. Kidney Failure and the Federal Government. Washington, DC: The National Academies Press. doi: 10.17226/1818.
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diabetes in the ESRD population. A more complete adjustment for diabetes would involve information about both the type of diabetes and the respective mortality rates among the non-ESRD diabetic patients in the nations being compared. The frequency of type of diabetes is not accounted for in the current adjustments and may differ across national boundaries. Differences in the management of diabetes may lead to different mortality rates among diabetics from different nations, even if they do not have ESRD.

Other aspects of etiology may also be important. For example, in the United States, death rates are elevated relative to glomerulonephritis if the etiology is hypertension. Since cardiovascular disease is less prevalent in Japan than in the United States, it may also be important to adjust for hypertension.

It is instructive to consider the difference between mortality rates of black patients and white patients as an example of the amount of variability that has been seen in the United States. The 5-year survival probabilities, adjusted for age, gender, and primary diagnosis, for black ESRD dialysis patients and white ESRD dialysis patients incident in 1984 are 36.5 and 30.5, respectively (USRDS, 1990, p. E.73). This unexplained difference of over 6 percentage points in survival probabilities is smaller than some of those reported by Held et al. (1990) for international comparisons, but it indicates that substantial differences can exist between identifiable groups, even within the same data collection system and nation. Other recent analyses (Wolfe et al., 1990) have shown that the difference between the mortality rates of blacks and whites is most substantial for diabetic patients and hypertensive patients, indicating that the impact of these two etiologies can vary substantially across different groups of patients. The existence of substantial differences in mortality between two groups of patients in the United States makes it clear that large differences in mortality rates among nations can be expected.

Age

The adjustment made by Held et al. (1990) for age is in 10-year (Europe) or 15-year (Japan) age groups. These are wide age intervals because the 5-year survival probability decreases dramatically with age after age 20 (USRDS, 1990, p. E. 14). Differences of just a few years in the average age of patients in corresponding age groups would cause a substantial difference in the mortality rates for the groups. If patients in one nation are older overall, then they will tend to be older in each age category as well. For example, even if the age-specific death rates were identical in two nations, but the average ages in corresponding age groups were 5 years higher in one nation than in the other, then the two nations would have age-adjusted survival proportions that differed by approximately 5 percent. (The 5-year death fraction decreases by approximately 1 percent per year of age; see USRDS,

Suggested Citation:"Appendix D: Survival Analysis Methods for the End-Stage Renal Disease (ESRD) Program of Medicare." Institute of Medicine. 1991. Kidney Failure and the Federal Government. Washington, DC: The National Academies Press. doi: 10.17226/1818.
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1990, p. E. 14.) Age differences between patients in the different nations are thus plausibly responsible for at least a portion of the difference in mortality noted by Held et al. (1990).

In addition to factors that are measured across national registries, there are likely to be substantial differences between patients with respect to other characteristics, including past medical history, comorbidity, distance from a treatment center, and level of kidney function at first treatment. These factors cannot be adjusted for with the current data; they would require special studies, but it would be difficult to evaluate the potential impact of such unmeasured factors. However, evaluation of geographic differences in mortality in the United States would give an indication of the amount of variability present nationally that could be compared to the differences seen internationally.

Withdrawal Rates

The rate of withdrawal from therapy among dialysis patients in the United States is not negligible. Port and colleagues (1989) have reported that up to 10 percent of deaths among elderly patients in Michigan follow soon after withdrawal from therapy. Furthermore, at least 8.6 percent of all ESRD deaths in the United States in 1987 can be attributed to withdrawal from therapy (P. Eggers, HCFA, personal communication, 1990). There are large differences in withdrawal rates between groups in the United States, and it is plausible that large cross-national differences in withdrawal rates might also exist. Withdrawal may be a particularly relevant issue in international comparisons because the largest international differences in mortality were reported by Held et al. (1990) in the nonpediatric age groups, the same age range in which withdrawal is common in the United States.

Patient Follow-up

Ascertainment of mortality status by the ESRD data system is largely complete because of the computer links to the Social Security System. Although patients with long-lived transplants may be temporarily lost to the Medicare data collection system, their deaths are recorded when they occur so that overall mortality rates can be accurately estimated. It would be useful to have information from other nations concerning the fraction of the ESRD population that are followed to eventual mortality.

Directions for Further Research

Although the international comparisons in death rates that are reported by Held et al. (1990) indicate that mortality may be higher in the United

Suggested Citation:"Appendix D: Survival Analysis Methods for the End-Stage Renal Disease (ESRD) Program of Medicare." Institute of Medicine. 1991. Kidney Failure and the Federal Government. Washington, DC: The National Academies Press. doi: 10.17226/1818.
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States than in some other peer nations, such differences are plausibly attributable to different data collection methods, differences in patient comorbidity and health practices, and differences in patient compliance. However, although international comparisons are not definitive, they still indicate that differences exist, for some unknown reasons. There are several areas of research that could be profitably explored to better our understanding of international comparisons:

  • The impact of differential death rates in the general population has been partially addressed by Held et al. (1990). Further study of differential mortality rates in the populations of diabetics and hypertensives from various nations may also be useful.

  • The fact that the differences in death rates among nations are largest in the nonpediatric age groups helps give some focus to the search for the reasons for such differences. Further identification of subgroups with differential death rates may help clarify the reasons for differences. Comparison of multivariable models from different nations would be an efficient method for such studies, and cause of death could be a useful measure.

  • More international communication on the methods of managing data registries could prove useful for all nations that attempt to maintain ESRD data registries. For example, methods for validation of data registries could be standardized.

  • Careful evaluation of different patterns of treatment methods among nations would be useful. Currently, much of the data on treatment patterns are derived from expert opinion rather than through data collection.

  • Differences in patient compliance should be studied in order to determine the effect of withdrawal rates.

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Breslow NE, Day NE. 1982 and 1987. Statistical Methods in Cancer Research, Vol I and II, Oxford University Press, Oxford.


Campbell DT, Stanley JC. 1963. Experimental and Quasi-Experimental Designs for Research, Rand McNally College Publishing Co., Chicago.

Cornfield J. 1978. Randomization by group: A formal analysis. Am J Epidemiology 108:100–102.

Cox DR. 1972. Regression models and life tables, JRSSB 34:187–220.

Cox DR, Oakes D. 1984. Analysis of Survival Data, Chapman and Hall, London.


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Donner A, Donald A. 1987. Analysis of data arising from a stratified design with the cluster as unit of randomization, Statist Med 6:43–52.


Eggers P. (HCFA). 1990. Personal communication concerning withdrawal rates.


Health Care Financing Research Report (P. Eggers) End Stage Renal Disease HCFA, various years.

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Held PJ, et al. 1990. Five-year survival for end stage renal disease patients in the U.S., Europe, and Japan, Am J Kid Dis 15:451–457.

Kalbfleisch JD, Prentice RL. 1980. The Statistical Analysis of Failure Time Data, New York, Wiley.


Lawless JF. 1982. Statistical Models and Methods for Lifetime Data, New York, Wiley.


Payne CD. 1987. The GLIM System Release 3.77, Royal Statistical Society, NAG, Downers Grove, IL.

Port FK, Wolfe RA, Hawthorne VM, Ferguson CW. 1989. Discontinuation of dialysis therapy as a cause of death, Am J Nephrol 9:145–149.


SAS Institute. 1988. SAS/STAT User's Guide, Release 6.03, SAS Institute Inc, Cary, NC.


USRDS (U.S. Renal Data System). 1989. Annual Data Report. National Institute of Diabetes and Digestive and Kidney Diseases, Bethesda, MD.

USRDS. 1990. Annual Data Report. National Institute of Diabetes and Digestive and Kidney Diseases, Bethesda, MD.


Vaupel JW, et al. 1985. Heterogeneity's ruses: Some surprising effects of selection on population dynamics, Am Statist 39:176–185.


Wolfe RA, Port FK, Hawthorne WM, Guire, KE. 1990. A comparison of survival among dialytic therapies of choice: In-center hemodialysis versus continuous ambulatory peritoneal dialysis at home. Am J Kidney Dis 15:433–440.

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Next: Appendix E: Institute of Medicine ESRD Study Committee Public Hearing, May 5, 1989, Chicago, Illinois »
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Since 1972, many victims of endstage renal disease (ESRD) have received treatment under a unique Medicare entitlement. This book presents a comprehensive analysis of the federal ESRD program: who uses it, how well it functions, and what improvements are needed.

The book includes recommendations on patient eligibility, reimbursement, quality assessment, medical ethics, and research needs.

Kidney Failure and the Federal Government offers a wealth of information on these and other topics:

  • The ESRD patient population.
  • Dialysis and transplantation providers.
  • Issues of patient access and availability of treatment.
  • Ethical issues related to treatment initiation and termination.
  • Payment policies and their relationship to quality of care.

This book will have a major impact on the future of the ESRD program and will be of interest to health policymakers, nephrologists and other individual providers, treatment site administrators, and researchers.

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