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Physician Staffing for the VA: Volume I 4 THE EMPIRICALLY BASED PHYSICIAN STAFFING MODELS The underlying premise of this chapter is that empirical observations on the current practice of medicine in the VA can be useful in helping to determine how many physicians the VA should have in order to meet its patient care and physician training commitments. The basic idea is that statistical models can be developed describing the relationships between patient care workload, physician Full-Time-Equivalent Employees (FTEE) (by specialty and including residents), and other productivity-influencing factors. With data drawn from the current system, these models can be empirically estimated, i.e., their unknown parameters are assigned specific values. From these estimated models, predictions can be derived about the amount of physician FTEE required to meet projected future workload levels. Such analyses can be performed on a specialty-specific basis and at different levels of aggregation—from the hospital-ward level all the way to derivation of national estimates. These statistical models are grounded in the current practice of medicine in the VA and provide a base against which expert judgment models can be evaluated. Two alternative, yet complementary, variants of what the committee has termed the Empirically Based Physician Staffing Models (EBPSM) will be presented and analyzed in some detail in this chapter. A quick overview follows. In the production function (PF) variant of the EBPSM, the rate of production of patient workload (e.g., bed-days of care) for a given patient care area (PCA) (e.g., the medicine bed service) at a VA medical center (VAMC) is hypothesized to be related to such factors as physician FTEE allocated expressly to patient care in that PCA; the number of residents, by postgraduate year, assigned to that PCA; nurse FTEE per physician FTEE there; support-staff FTEE per physician FTEE there; and other variables possibly associated with physician productivity in that PCA (e.g., the VAMC's affiliation status). Each VAMC is divided into 14 or fewer (depending on the scope of services offered) PCAs: inpatient care—medicine, surgery, psychiatry, neurology, rehabilitation medicine, and spinal cord injury; ambulatory care—medicine, surgery, psychiatry, neurology, rehabilitation medicine, and other physician
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Physician Staffing for the VA: Volume I services (including emergency care and admitting & screening); and long-term care—nursing home and intermediate care. A PF is estimated statistically for each PCA. To derive the total physician FTEE in a given specialty (e.g., neurology) or program area (e.g., ambulatory care) required for patient care at a given VAMC, one must solve for the FTEE required to meet patient workload on each relevant PCA, then sum across PCAs. In the inverse production function (IPF) variant of the EBPSM, specialty-specific rather than PCA-specific models are estimated. For a given specialty (e.g., neurology), the quantity of physician FTEE devoted to patient care and resident education across all PCAs at the VAMC is hypothesized to be a function of such factors as total inpatient workload associated with that specialty (e.g., total bed-days of care for patients assigned a neurology-associated diagnosis-related group); total ambulatory care workload associated with the specialty; total long-term care workload associated with the specialty; the number of residents in that specialty at the VAMC, by postgraduate year; and other variables possibly associated with physician time devoted to patient care and resident education. There are separate facility-level IPFs for each of the following 11 specialty groups: medicine, surgery, psychiatry, neurology, rehabilitation medicine, anesthesiology, laboratory medicine, diagnostic radiology, nuclear medicine, radiation oncology, and spinal cord injury. (Included in this latter group are physicians in any specialty who are assigned to the spinal cord injury "cost center" in the VA personnel data system.) For each specialty, to derive the total number of physicians required for patient care and resident education on the PCAs, one must substitute the appropriate values of workload, resident FTEE, and other control variables into that specialty's IPF, then solve directly for the corresponding physician FTEE level. The statistical confidence limits on the prediction also can be computed directly (which is not possible for the PF-based FTEE estimate, as will be seen). Both the PF and the IPF deal with only a portion of total physician FTEE at the VAMC, albeit a very important and quantitatively significant portion in each case. The fraction of physician FTEE allocated to patient care only—the focus of the PF variant—will vary by specialty and facility, of course, but it rarely falls below 65 percent and generally lies in the 70-95 percent range (see Table 9.1 in chapter 9). The sum of FTEE devoted to patient care and resident education—the focus of the IPF variant—generally lies in the 80-95 percent range. (The rationale for including both patient care and resident education in the IPF and only patient care in the PF is discussed in the section on Formal Presentation of the EBPSM.) It follows that, under either the PF or IPF variant, total FTEE required at the facility is the sum of the model-derived estimate plus separate estimates for FTEE components not incorporated in the model. Included in the latter would be FTEE for research, continuing education, and other miscellaneous assignments. The process of deriving total physician FTEE for a given specialty
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Physician Staffing for the VA: Volume I or program area at a VAMC is illustrated below in the section "Using VA Data to Assign Values to Variables." This chapter is organized as follows: Simplified versions of both the PF and the IPF are presented to explain the intuition behind the workings of both models. The models are then formally stated, and the data used for defining the variables in each model are discussed. Estimated PF models for all 14 PCAs and IPF models for all 11 specialties are reported, with several equations singled out for additional analysis. Then, the estimated IPF is applied to compare the model-derived physician FTEE level at a given facility in FY 1989 with the actual FTEE found there in that specialty. A similar analysis is performed using the estimated PF equations. Then, for selected PF equations, the model-derived workload at a given facility in FY 1989 is compared with the actual workload generated there. These calculations are performed for four actual (though unidentified) VAMCs. The estimated PF and IPF models are used, alternatively, as the centerpieces of an algorithm to derive facility-specific physician requirements for two selected future years, 2000 and 2005. For illustration, the analyses focus again on the same four VAMCs. In the final section, the committee presents recommendations for future data gathering and statistical analyses by the VA, aimed at improving the models. Overseeing the development of both variants of the EBPSM was the committee's data and methodology panel, which worked closely with the study's staff and statistical consultants. HOW THE EMPIRICALLY BASED MODELS WORK The purpose of this section is to give the statistically oriented, but time-limited reader a basic understanding of the PF and the IPF variants of the EBPSM. Throughout this section, simplifications are made in two respects. First, the hypothetical statistical models constructed below are smaller and generally simpler than the PF and the IPF equations presented in the next two sections. Second, our interpretations of statistical concepts are somewhat informal and intuitive; at various points, the reader is referred elsewhere for a more rigorous statement of definition or principle. Nonetheless, most of the methodological issues arising in the larger equations, whether regarding model specification or statistical interpretation, can be well illustrated through the simpler equations. PF and IPF variants are now considered, in turn, with some concentration on the former to introduce statistical concepts; the choice between the PF and the IPF for this purpose was entirely arbitrary and not intended to suggest a prior preference for one variant over the other.
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Physician Staffing for the VA: Volume I Anatomy of the PF Variant In building and testing a statistical model of a behavioral relationship, several steps are involved. A prior hypothesis is formed about the nature of the behavioral relationship—a process frequently inspired by a formal knowledge of, or general ''feel'' for, the relevant data. The hypothesis is transformed into a model, which requires both selecting and operationally defining the model's variables, and choosing the model's functional form—that is, a mathematical statement about the way the variables are thought to interact. A model will have one or more parameters; once these are determined, the model is fully determined. With the available data, empirical values are assigned to all variables in the model. Statistical techniques are used to estimate the model's parameters. Both the statistical strength and the theoretical plausibility of the parameter estimates, and of the model as a whole, are noted and a decision is made as to whether to accept the present model as the best available or to continue searching for a better one. Such a search could involve developing new data, specifying additional variables, or trying different functional forms. For simplicity, in the PF models discussed below, no distinction is made between PCAs or specialties, and the variables are not defined with the specificity required in later sections. Suppose the prior hypothesis is that the rate of production of patient care workload is positively related to the quantity of physician FTEE, and not related systematically to any other factor. The choice of variables for the corresponding model is clear: workload (W) and physician FTEE (Phys). A functional form must be selected; in the absence of additional information, the simplest choice is a linear relationship. Thus, where b0 and b1 are the parameters to be estimated, and ERROR is a random error term that reflects the net influence of all factors not included in the model. It is a feature of all regression models. The equation says that workload is a function of one systematic influence—physician FTEE—and a large number of nonsystematic, random influences whose net effect is captured by ERROR. Necessary conditions for Equation 4.1 to be a valid model are that its systematic part be correctly
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Physician Staffing for the VA: Volume I specified, with both Phys (the independent variable) and ERROR meeting certain well-defined conditions.1 Suppose there are paired observations on W and Phys from a sufficiently large number of VAMCs.2 Given Equation 4.1, the aim now is to use these data to determine the best-fit linear relationship between W and Phys. The standard statistical technique for doing this is the least-squares method.3 This can be assumed to lead to the following estimated model, with its accompanying indicators of statistical goodness of fit: where b0 and b1 have been replaced by their estimated values, 3.41 is the t-statistic indicating the statistical strength of the estimated coefficient above it, and is an overall measure of the equation's goodness of fit. The sample size (N) of VAMC PCAs used in estimating the equation is often displayed as well; for the PF equations presented later in this chapter, N varies from about 80 to 160 depending on the type of PCA. This equation, and the hypothetical data points "used" in estimating it, are pictured symbolically in Figure 4.1. 1 Basically, it is required that ERROR be a normally distributed random variable, with a mean of zero and a variance that is constant; this implies that the variance cannot vary with either W (the dependent variable), or Phys (the independent variable). (ERROR is normally distributed with these properties if, and only if, the dependent variable W is normally distributed with constant mean and variance.) It is also required that Phys be nonprobabilistic (nonstochastic), that not all Phys values in the sample are the same, and that Phys does not grow or decline in value without limit as the sample size grows (without limit). For models with more than one independent variable, i.e., multivariate models, it is also required that there be no perfectly linear relationship between any two variables (in fact, among any subset of independent variables). For a detailed discussion of these conditions, see Kmenta (1986). 2 Strictly speaking, the number of observations must only exceed the number of parameters being estimated by one. But for stable estimates, a larger sample size than this is required. For a univariate model such as Equation 4.1, analysts typically want at least 20 data points. The larger the number of independent variables, the larger the sample size usually required (Kmenta, 1986). 3 The best-fit model under the least-squares method has the following defining property: It minimizes the sum of the squared deviations between the actual values of W and the corresponding model-predicted values of this dependent variable. To explore this, refer to Figure 4.1. For the ith value of physician time (Phys,), there is a paired observation on workload (W1), and a model-predicted workload value . The model error for this ith case is defined as . This term is squared to get This is repeated for all N observations; then the N squared terms are summed. The least-squares regression line is the particular line so positioned that it forces this sum of squares to be as small as possible. The formulas for the least-squares regression method use the data to compute parameter estimates—call them and , that effectively achieve this positioning (Kmenta, 1986).
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Physician Staffing for the VA: Volume I Of the two estimated coefficients, the more important by far is Given the positive algebraic sign on this estimate, it can be interpreted as follows: for a small increment (decrement) in physician FTEE (ΔPhys), workload can be expected to increase (decrease) by (8.42 × ΔPhys). That is, (8.42 × ΔPhys) = ΔW, which implies that (Δw/ΔPhys) = 8.42 is the slope of the PF in Figure 4.1. For example, if W was defined in terms of patient days generated per day in the PCA, the addition of one full-time physician is expected to increase workload production by 8.42 patient days per day. Thus, 8.42 can be viewed as the productivity multiplier that transforms changes in physician FTEE into changes in the rate of workload production. It can be shown that as ΔPhys decreases (in absolute value) and as these physician FTEE levels more closely approach the sample mean of Phys, the statistical reliability of this multiplier increases. Roughly speaking, the larger the t-statistic in absolute value, the greater the statistical strength of the estimated coefficient; the absolute-value proviso is required since t and the estimated coefficient take on the same sign, which can be negative. A common rule of thumb is that an estimate is significant if its t-statistic is about 2.00 or greater in absolute value. However, there is no unconditional rule for determining how large t must be for the estimate to be declared statistically significant. Under common rules of thumb, t-statistics ranging from about 1.7 to 2.6 (in absolute value) may be taken to indicate that the associated coefficient estimate is statistically significant.4 The overall goodness-of-fit measure is a statistic, taking on values between 0 and 1, indicating the fraction of the total variation in the dependent variable 4 More typically, a t-statistic such as that shown in Equation 4.1' is used to test the null hypothesis that its associated coefficient (b1) is different from 0 (sometimes referred to as a two-tail test of significance). For a given value of this statistic (t*), one rejects the hypothesis that b1 = 0 with a certain degree of statistical confidence (c*) stated in percentage terms. The larger that t* is, the larger is c*, all else equal. For example, if a sample size of about 30 or greater is assumed, a value of t = 1.96 allows one to reject the hypothesis that b1 = 0 with about 95 percent confidence; if t = 2.58, c is about 99 percent. If the null hypothesis is rejected, then is declared statistically significant and used as the (least-squares) estimate of b1 (Kmenta, 1986). In some cases it may be more reasonable to test the null hypothesis that b1 = 0 against the alternative that b1 > 0 (referred to as a one-tail test of significance). In that case, a value of t = 1.65 allows one to reject the hypothesis that b1 = 0 with 95 percent confidence. In sum, whether a given t value is interpreted to indicate "statistical significance" depends on the confidence level chosen, the sample size used to estimate the model, and whether a two-tail or one-tail test is selected. (For additional commentary on this issue, see footnote 10 in this chapter.)
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Physician Staffing for the VA: Volume I that can be "explained" by variation in the independent variable(s). 5 The larger the is, the better is the equation's fit of the data. A value of 1.00 would indicate that the model accounts perfectly for variations in the dependent variable; in this case, all data points would fall on the estimated line. In Equation 4.1', the variation in Phys is found to explain 72 percent of the variation in W. Although no estimate for ERROR is shown, "observations" on this random component are also generated and play an important role in assessing whether the assumptions made about ERROR (see footnote 3) appear to hold (see "Estimated PF and IPF Equations," below). For the ith physician FTEE value, there is a corresponding Wi and a model-predicted . The difference between these two is termed the ith residual. Taken together, these residuals can be regarded as observations generated from the random variable, ERROR. If the assumptions about ERROR hold, these residuals should have a random appearance, that is, no discernible patterns or trends. Of obvious importance is that Equation 4.1' can be used to derive physician requirements for patient care at a given VAMC. If a projected workload for the VAMC of 100 units per time period is substituted into the equation, such that then Next, some PF alternatives to Equations 4.1 and 4.1' are considered. These would be motivated in each case by data points that appear differently configured than those in Figure 4.1. Suppose that there is an evident nonlinear relationship between physician FTEE and workload—in particular, that W rises with increases in Phys, but at a decreasing rate. This case of "diminishing marginal productivity" of physician 5 More precisely, represents a modification of the traditional goodness-of-fit measure (R2) in order to adjust for the number of independent variables included in the model. It can be shown that R2 always rises as the number of explanatory variables is increased, irrespective of the strength of their contributions. A new variable increases if and only if its associated t-statistic exceeds 1 in absolute value. For the formulas to compute , see Kmenta (1986). Many analysts advocate choosing the model specification that maximizes , on two grounds. The criterion is easy to use and simple to interpret. More important, it can be shown that choosing on this basis is equivalent to choosing the model that has minimum mean-squared error; the latter is defined as the expected value of the square of the difference between the estimated parameter value (here, ) and then its true value (b1) (Kmenta, 1986).
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Physician Staffing for the VA: Volume I time is shown in Figure 4.2. A possible (though again hypothetical) estimated regression equation corresponding to this result is where the nonlinear relationship is modeled as a quadratic equation in which W reaches a maximum for some Phys value, then diminishes absolutely beyond that. In geometric terms, the function pictured is an inverted parabola, with only the rising portion of the curve relevant to the data likely generated in the "real world" practice of medicine. That is, for sufficiently large values of Phys (not shown in Figure 4.2), the equation would indicate that workload declines with increases in physician FTEE. As portrayed, the coefficient estimates for both the linear and the quadratic terms are statistically significant. If, on the other hand, the estimate for Phys had been significant whereas the estimate for Phys2 had not been, the hypothesis of a linear relationship would have been sustained. The derivation of physician requirements from Equation 4.2 is illustrated by again setting W = 100 and solving the resulting quadratic relationship; the clinically relevant solution is Phys = 14.30. Next, a multivariate regression model is considered, in which the rate of workload production depends on more than physician FTEE, for example, also on whether the VAMC is affiliated with one or more non-VA health care institutions. To accommodate this analysis, a data set enlarged to include a variable labeled "Affil" is required. If a VAMC is affiliated, Affil = 1; otherwise, Affil = 0. (That there may be different degrees of affiliation is thus ignored here.) The use of such categorical (or dummy) variables is quite common in regression analysis. As can be seen in the following three sections, multivariate models can include any combination of continuous variables (such as Phys) and categorical variables. The simplest hypothesis here, portrayed in Figure 4.3, is that affiliated and unaffiliated VAMCs have PFs that differ only by a parallel shift; that is, for any value of Phys, the difference between the workload rates at the two types of VAMCs is a constant; the physician productivity multiplier (the slope) is the same in both cases. Thus, it is posited that there is something about being affiliated that raises, or lowers, a VAMC's overall productivity, but does not affect the marginal effect of physicians on workload. A hypothetical equation that, in conjunction with Figure 4.3, portrays these assumptions is
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Physician Staffing for the VA: Volume I which indicates that affiliated VAMCs are more productive, all else equal. The committee emphasizes that this is merely an illustration with no policy implications intended or possible; how the actual effect of affiliation status on productivity and physician requirements can be inferred is discussed later in the chapter. The amount of physician FTEE required to meet workload at a VAMC now depends on whether it is affiliated. If Affil = 1 in Equation 4.3, the FTEE required to produce W = 100 is 9.78; if Affil = 0, the required value of Phys is 13.59 FTEE. An interesting alternative hypothesis is that affiliation status affects both the VAMC's overall productivity level (for any value of Phys) and the physician productivity multiplier. Such a situation is shown symbolically in Figure 4.4 and reflected in the following (hypothetical) estimated equation: where the net impact of affiliation on productivity involves the resolution of two effects. Although the direct-effect variable (Affil) is still positive and significant, the interaction-term variable (Phys × Affil) is negative and significant. Regarding the influence of the latter, if a VAMC is unaffiliated, Affil = 0 and thus is also the interaction-term variable; the physician productivity multiplier remains 8.10. But for an affiliated facility, with Affil = 1, the multiplier is effectively reduced to (8.10-1.80) = 6.30. It can be shown that whether affiliation status is associated with higher productivity on net—that is, whether for a given Phys value, W is greater for an affiliated VAMC—depends here on the absolute level of Phys. This is evident from Figure 4.4. Based on Equation 4.4, the physician FTEE required to produce W = 100 for an affiliated facility is 10.66, whereas it is 11.86 for an unaffiliated VAMC. Anatomy of the IPF Variant The PF and the IPF are potentially complementary constructs. Each yields a well-defined answer to a well-defined question, though not the same question. The PF seeks to identify factors associated with the production of patient workload in each PCA of the VAMC. If a variable does not make an independent contribution to explaining overall productivity, it will not merit inclusion in the PF, at least by conventional statistical criteria.
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Physician Staffing for the VA: Volume I For each specialty, the IPF seeks to identify factors associated with the total amount of physician FTEE devoted to patient care and resident education across all PCAs at the VAMC. The volume of patient workload at the facility, especially on PCAs where the specialty is active, is a likely explanatory factor. But it need not be the only such factor; and if it happened not to be statistically significant, the IPF might still prescribe a positive amount of physician FTEE for the VAMC. Two related features of the IPF become evident in later sections. First, compared with the PF variant, deriving physician requirements through the IPF is computationally more straightforward. Second, statements of statistical confidence, often summarized in terms of "prediction intervals,' can be computed around the IPF's best estimate of physician requirements; this is not possible with the PF, which permits instead the computation of prediction intervals around the level of workload that a given physician FTEE level (in conjunction with other factors) is expected to produce. The following simplified and hypothetical IPF specifications are structurally so similar to the PF equations above that the presentation can be relatively compact. The simple hypothesis that physician FTEE is linearly related to workload is depicted in Figure 4.5 and by the estimated equation which can be compared with Equation 4.1' to make an important point: Regression theory does not permit one to derive one estimated equation from the other by simple algebraic manipulation. That is, if one solves Equation 4.1' for Phys in terms of W, the result is not Equation 4.5 (Kmenta, 1986). Equation 4.5 serves to reemphasize another point: Drawing inferences from a regression can be precarious for independent-variable values lying far outside the sample range. A negative quantity of physician FTEE is predicted for values of W less than 9.3, but is of no practical relevance if workload observations in the sample—all in the range, say, of 60 through 110—are representative of VAMC workload levels generally. From Equation 4.5, the quantity of physician FTEE required for patient care and resident education at a VAMC for which W = 100 is equal to -0.84 + 0.09(100) = 8.16. An alternative hypothesis—that as workload increases, physician FTEE requirements increase at an increasing rate—is illustrated in Figure 4.6 and in the following hypothetical estimated equation:
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Physician Staffing for the VA: Volume I On the other hand, the hypothesis of a linear relationship would have been sustained had the estimated coefficient of W2 not been statistically significant. If projected workload at a VAMC is again 100 units, Equation 4.6 implies that the physician FTEE required for patient care and resident education is 10.34. The (illustrative) hypothesis that less physician time is required in response to any given workload level in an affiliated VAMC, compared with an unaffiliated facility, is depicted in Figure 4.7 and in the following equation: where the marginal (incremental) relationship between workload and physician FTEE, as captured in the estimated coefficient on W, is assumed to be the same for both types of facilities. To produce workload at a rate of W = 100, an affiliated VAMC would require 6.68 physician FTEE, according to Equation 4.7, whereas an unaffiliated VAMC would require 11.48 FTEE. An IPF specification that depicts, hypothetically, the results from testing this assumption directly is shown in Figure 4.8 and in the following equation: This equation implies that in an affiliated VAMC the marginal effect of small changes in workload on physician requirements (for patient care and resident education) is transmitted through a multiplier of 0.045. But if a facility is affiliated, so that Affil = 1, the multiplier becomes (0.045 + 0.025) = 0.07, which implies lower efficiency on the margin. As with Equation 4.4, whether an affiliated VAMC is more, or less, productive overall than an unaffiliated VAMC will depend on the net effect of the direct-effect and interaction terms, in concert, and hence will depend on the value of W at which the assessment is made. Using W = 100, it can be found from Equation 4.8 that an affiliated facility requires 8.35 physician FTEE, whereas the requirement in an unaffiliated VAMC is 7.29. Implicitly assumed in all of these examples is that the quality of care, however defined, does not vary significantly across the sample—that is, units of W are of comparable quality across VAMCs and for all rates of production. In addition, if these estimated models are to be used prescriptively to derive physician requirements consistent with high-quality care, it is necessary that paired sample observations on W and Phys reflect the delivery of high-quality
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Physician Staffing for the VA: Volume I FIGURE 4.5 IPF with Physician FTEE Linearly Related to Workload
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Physician Staffing for the VA: Volume I FIGURE 4.6 IPF with Nonlinear Relationship between Physician FTEE and Workload
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Physician Staffing for the VA: Volume I FIGURE 4.7 IPF with Affiliation Status and Workload Having Distinct (Independent) Effects on Physician FTEE
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Physician Staffing for the VA: Volume I FIGURE 4.8 IPF with Affiliation Status and Workload Having an Interactive Effect on Physician FTEE
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Physician Staffing for the VA: Volume I FIGURE 4.9 Inpatient Medicine PF Residuals Scatterplot
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Physician Staffing for the VA: Volume I FIGURE 4.10 Inpatient Surgery PF Residuals Scatterplot
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Physician Staffing for the VA: Volume I FIGURE 4.11 Inpatient Rehabilitation Medicine PF Residuals Scatterplot
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Physician Staffing for the VA: Volume I FIGURE 4.12 Ambulatory Medicine PF Residuals Scatterplot
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Physician Staffing for the VA: Volume I FIGURE 4.13 Medicine IPF Residuals Scatterplot
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Physician Staffing for the VA: Volume I FIGURE 4.14 Surgery IPF Residuals Scatterplot
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Physician Staffing for the VA: Volume I FIGURE 4.15 Psychiatry IPF Residuals Scatterplot
Representative terms from entire chapter: