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Physician Staffing for the VA: VOLUME II
SECTION IV
FURTHER EXAMINATION OF THE EMPIRICALLY BASED PHYSICIAN STAFFING MODELS

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FURTHER EXAMINATION OF THE EMPIRICALLY BASED PHYSICIAN STAFFING MODELS
INTRODUCTION
From the study's inception, an underlying premise has been 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 to meet its mission-related demands. How this can be accomplished in practice has been demonstrated in detail in chapter 4 of Volume I.
The primary purpose of this concluding section of the Supplementary Papers is to examine further the statistical validity of selected physician staffing models presented in chapter 4. In addition, for a subset of these models a certain physician productivity index frequently derived in microeconomic analyses of input-output relationships— the “marginal productivity” of an input—will be developed; this index will be used to investigate the output gain expected from incremental increases in physician staffing in a given patient care setting.
The basic idea undergirding the analyses in chapter 4 of Volume I 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 VA system, these models can be statistically estimated, i.e., their unknown parameters are given specific values. From these estimated models, predictions can be derived about the amount of physician FTEE required to meet both current and projected future workload levels. Such analyses can be performed on a specialty-specific basis and at different levels of aggregations—from the hospital ward level all the way to the derivation of national-level estimates.
The empirically based physician staffing models (EBPSM) are grounded in the current practice of medicine in the VA and provide a base against which expert judgment models can be evaluated. Specifically, the VA decision maker can use these EBPSM in conjunction with the expert judgment models to arrive at decisions about the appropriate level of physician staffing in any given setting.

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See chapters 6 and 7 of Volume I for a detailed exposition and justification of the committee's proposed “Reconciliation Strategy” for balancing empirically based and expert judgment physician staffing estimates.
As a prelude to the new studies reported here, the nature and scope of these empirically based models will be briefly summarized.
OVERVIEW OF THE EBPSM
In Volume I the committee defined, developed, and ultimately endorsed two alternative, complementary variants of the empirically based physician staffing models.
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 (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 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

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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. (This latter group consists of 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.
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 Volume I). 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 detail in chapter 4 of Volume I.)
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 are FTEE for research, continuing education, and other miscellaneous assignments. The process of deriving total physician FTEE for a given specialty or program area at a VAMC is illustrated in chapter 6 of Volume I.
Reported in chapter 4 were estimated PF models for all 14 PCAs and IPF models for all 11 specialties, with several equations singled out for additional analysis. For selected PFs and IPFs, the model-derived physician FTEE at a given VAMC in FY 1989 was compared with the actual FTEE reported there for that specialty. These calculations were performed for four actual VAMCs (whose identities were masked). These PFs and IPFs were then used as the centerpieces of an algorithm to derive specialty-specific physician requirements for the four selected VAMCs for two future fiscal years, 2000 and 2005. Chapter 4 concluded with the committee's recommendations on a range of future data gathering and statistical analyses to improve the EBPSM over time.

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In this concluding section of Volume II, the committee demonstrates how an important subset of these proposed statistical analyses might proceed and the types of insights that can be gained.
This section is divided into several parts. Following this introduction, the next part illustrates two alternative tactics for examining the statistical plausibility of the estimated PFs and IPFs: (1) through a simple examination of residual plots (a standard approach already utilized in Volume I) and (2) through a newer, more complex approach termed “bootstrapping,” which has been applied in a number of domains recently but not (to the committee's knowledge) to health manpower staffing models. This part concludes with a brief examination of one other modeling issue: the robustness and stability of the PF and IPF equations to alternative specifications of the variables used to measure inpatient, outpatient, and long-term care workload.
The final part focuses on the PF variant of the EBPSM and the calculation of the marginal product (MP) of the internist, the surgeon, and the psychiatrist in the inpatient medicine, surgery, and psychiatry PCAs, respectively. While such MP calculations are of particular interest to researchers working in the area, it must be acknowledged that the index itself is not instrumental in the calculation of physician requirements. Rather, it may provide some insight into whether the PF model from which it is derived is a reasonable representation of the production process.
All analyses below focus either on the PFs estimated for inpatient medicine, inpatient surgery, or inpatient psychiatry, or on the IPFs estimated for the specialty groupings of medicine, surgery, or psychiatry. This concentration allows for a demonstration of the major analytical points, while keeping the presentation tractable; moreover, these three specialty groupings accounted nationwide for roughly 75 percent of all VA staff physician FTEE in FY 1989 (see Table 2.1 in Volume I).
STATISTICAL EVALUATION
To set the stage for the analysis, the general functional specification of both the PF and the IPF will be reviewed, along with the behavioral assumptions that justified the approach to statistical estimation adopted in each case. The multi-step regression strategy employed in the selection of the “final” version of each PF and IPF, i.e., the version appearing in Volume I, will be described. Next, for ease of reference the six estimated models that will be examined here in some detail are reproduced from Volume I: the inpatient medicine, inpatient

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surgery, and inpatient psychiatry PFs, and the medicine, surgery, and psychiatry IPFs. With these matters in hand, the discussion then proceeds to residual plots and bootstrapping.
Production Function: General Form
The general form of the PF variant endorsed by the committee is presented below; for ease of cross-reference, the equation numbers assigned in Volume I are used below wherever appropriate.
Wij=f[{StaffPhysij}, {ConPhysij}, {Resij},
C&Aij, WOCij, {NPPij}, Nurseij,
Supportij, Prodfactij, ERRORij], (4.9)
where
Wij
=the annual rate of production of workload in PCA j of VAMC i;
{Staffphysij}
=a set of variables, each of which takes the form Staffphysijk=the amount of FTEE allocated to direct patient care in PCA j of VAMC i for staff physicians based in cost center k, where each k corresponds to one of the 11 specialty groups examined here in detail;
{ConPhysij}
=a set of variables for physicians under contract to VAMC i, such that ConPhysijk=the contract physician FTEE from specialty k devoted to PCA j;
{Resij}
=a set of variables to account for the net productive contribution of residents, with each variable of the form Resijy=the amount of postgraduate year y resident FTEE allocated to PCA j at VAMC i;
C&Aij
=for non-VA physicians who perform consulting and attending duties on a fee-for-visit basis, the amount of FTEE allocated to PCA j at VAMC i;

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WOCij
=for non-VA physicians who perform consulting and attending duties without (monetary) compensation, the amount of FTEE allocated to PCA j at VAMC i;
{NPPij}
=a set of variables, each of the form NPPijm=the amount of FTEE of nonphysician practitioner type m (e.g., physician assistant) assigned to PCA j at VAMC i;
Nurseij
=the amount of nursing service FTEE allocated to PCA j at VAMC i;
Supportij
=for all personnel categories excluding physicians and nurses (and, as appropriate, also excluding psychologists and social workers), the total FTEE allocated to PCA j at VAMC i;
Prodfactij
=one or more variables for factors (e.g., capital equipment) influencing the productive efficiency of physicians and other providers in PCA j at VAMC i;
ERRORij
=the random-error term for PCA j at VAMC i, assumed to be normally distributed with mean zero and constant variance.
Each of the 14 PFs was specified as a flexible quadratic functional form (see Jensen and Morrisey (1986a,b) and estimated by ordinary least squares.
This single-equation approach assumes, among other things, that each independent variable (e.g., internist FTEE) is a nonstochastic determinant of the dependent variable, workload; thus, it is assumed ipso facto that inputs and outputs are not jointly determined in a mutually interactive fashion. In production models of the profit-maximizing firm, inputs and output are jointly determined, in theory; hence, single-equation approaches are problematic, yielding biased model estimates unless special assumptions are imposed (which is often the case).
Moreover, profit maximization implies cost minimization, which implies that the “firm” being modeled is operating with maximum productive efficiency. That is, given the level of inputs chosen, the resulting output rate is the maximum attainable. Interpreted literally, this string of assumptions leads to additional difficulties in applying the standard ordinary least squares regression

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procedure to estimate the PF. Since, for given input levels, observed output can never exceed, by definition, the output rate indicated by the production function (i.e., along the production “frontier”), ERROR is nonpositive and hence cannot be normally distributed with mean zero. Unbiased predictions of workload, conditional on input levels, can be obtained only if certain additional restrictions are imposed on ERROR (Kmenta, 1986).
In response, certain “frontier” production function models and estimation techniques have been developed (see Forsund et al., 1981). But these approaches have not been applied to health manpower analysis, with one partial exception: a recent effort to use “data envelopment analysis” —a nonstatistical procedure built around linear programming models—to study technical efficiency in VA hospitals (Sexton et al., 1989).
These technical concerns notwithstanding, the committee concluded that the nature of medical services production in the VA, in both its behavioral and technical dimensions, justifies the decision to estimate PFs in the most straightforward fashion, using ordinary least squares.
A VAMC is a public-sector organization charged with a multiobjective mission, but maximizing profits is not one of them. Rather, it is assumed that each VAMC attempts to meet its patient care mission in a way that balances several concerns: that eligible veterans are treated in a timely manner; that the quality of care is acceptable; and that budget, other resource, and administrative constraints are met. Consistent with this, it is assumed for the PF analysis that a VAMC adjusts inputs and workload in a step-sequence process: Subject to resource and budget availability, the VAMC sets input levels for each fiscal year in accordance with projected workload. Then, in the course of the year, it attempts to modulate (up or down) the rate of workload so that a clinically acceptable relationship is maintained between workload and inputs. If it is assumed that this workload adjustment process is subject to random error—that is, the VAMC will typically over- or undershoot a bit in trying to match workload to available resources—then ERROR will be normally distributed with zero mean.
The net result is that certain problems of bias and inefficiency that threaten PF estimates from the for-profit sector should not be similarly expected here.
Although there is an expectation that VAMCs will produce patient care services with high technical efficiency, the incentives to do so may be weaker than in the private sector (though they have been strengthened in recent years). The recent application of data envelopment analysis indicated that about one-third of all VAMCs were not delivering patient care with maximum cost efficiency (Sexton et al., 1989).

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Although a substantial part of this variability may be attributable to differences across VAMCs in the commitment to teaching and research (which affects the relative amount of staff physician time available for patient care), the end result is the following: in a given sample of VAMCs, variation can be expected in the efficiency with which inputs are transformed into output. Even if this study's PFs were to include variables to control for differences in teaching and research, considerable variability still could be expected across VAMCs in the rate of workload, given any fixed set of input levels. These results lend support to the assumption that ERROR is not only random, but normally distributed with zero mean in the PF models specified here.
Inverse Production Function: General Form
In the PF variant, the basic question is: What factors account for the production of patient care workload? In the IPF, the basic question is: What factors account for the observed level of staff physician FTEE?
The IPF's underlying assumption is that the amount of physician FTEE from a given specialty required for patient care and resident education is a function of the volume of patient care workload to be produced, the number of residents to be taught on the PCAs, and possibly other factors influencing the relationship among workload, resident education, and staff physician requirements.
From a cause-and-effect standpoint, the basic behavioral assumption in the IPF variant (in contrast with the PF) is that the VAMC adjusts physician FTEE levels in response to a given projected workload level, controlling for other factors. Thus, the volume of workload per period cannot be significantly modulated (i.e., it is “exogenously” determined by demand-influencing factors beyond the VAMC's control).
That the PF and IPF have different underlying assumptions does not in any way constitute an empirical contradiction. Every model has, by definition, its defining assumptions; and the PF and IPF are models offering alternative perspectives on the same underlying production process.
In concept, an IPF can be estimated for each specialty-PCA combination (e.g., for neurology requirements on the inpatient medicine PCA), or for each specialty on a facility-total basis by aggregating across PCAs (e.g., for neurology requirements for all 14 PCAs combined). However, efforts to estimate PCA-specific IPFs for each specialty produced equations frequently exhibiting poor goodness of fit and coefficient estimates whose algebraic signs

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were counterintuitive. Hence, the focus in this study is exclusively on facility-level IPFs, whose aggregated FTEE observations yield more reliable estimates.
Both the PF and the IPF are pursued because each variant has its strengths and drawbacks, conceptually and statistically, and because together they can play complementary roles in helping the VA decision maker determine appropriate staffing (see the extended discussion in chapters 3, 4, 6, and 11 of Volume I).
The general form of the IPF variant of the EBPSM is:
=g[{Wik}, {Resik}, {NPPik}, Prodfactik, ERRORik] (4.10)
where
=across all PCAs at VAMC i, the total amount of specialty k staff physician and contract physician FTEE devoted to patient care and resident education;
{Wik}
=a set of workload variables, each of the form Wijk=the level of workload on PCA j of VAMC i associated with specialty k;
{Resik}
=a set of variables, each of the form Resiky=the amount of postgraduate year y resident FTEE at VAMC i in specialty k;
{NPPik}
=a set of variables, each of the form NPPikm=the amount of FTEE of nonphysician practitioner type m associated with the PCA-related activities of physicians in specialty k at VAMC i;
Prodfactik
=one or more variables for factors influencing the productive efficiency of specialty k physicians at VAMC i;
ERRORik
=the random-error term for specialty k at VAMC i.
Originally, the general specification of the IPF also included variables for nurses and support staff. But after a number of statistical analyses, it became clear that the physician-substitutive role these providers are hypothesized to play could not be modeled adequately at the facility level. Rather, the PCA is the more appropriate level of aggregation for studying these relationships in the production of workload.

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FIGURE 15 Medicine IPF Residuals Plot

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FIGURE 16 Surgery IPF Residuals Plot

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FIGURE 17 Psychiatry IPF Residuals Plot

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FIGURE 18 Neurology IPF Residuals Plot

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FIGURE 19 Rehabilitation Medicine IPF Residuals Plot

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FIGURE 20 Spinal Cord Injury IPF Residuals Plot

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FIGURE 21 Anesthesiology IPF Residuals Plot

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FIGURE 22 Laboratory Medicine IPF Residuals Plot

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FIGURE 23 Diagnostic Radiology IPF Residuals Plot

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FIGURE 24 Nuclear Medicine IPF Residuals Plot

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FIGURE 25 Radiation Oncology IPF Residuals Plot