Insurance Mortality Rates , Performance Indicators , and Possibly Monotonic Population Proportions

Two applications are described of a probability model that can express uncertainty regarding a pre-specified monotonicity hypothesis for binomial proportions. The model also yields a random effects overdispersion formulation where the population proportions definitely satisfy a monotonicity specification. One application concerns an insurance data set recording mortalities of clients from ages 35 to 64. Two new actuarial graduation procedures are developed. The other application derives from a Veterans’ administration hospital quality monitor and concerns the failure to return rates for psychiatric patients attending substance abuse clinics. While smoothed performance indicators are proposed, measures of their extra-binomial variation highlight problems experienced by evidence-based approaches when the data are uncontrolled.


Introduction
When investigating m population proportions θ 1 , θ 2 , . . ., θ m it is sometimes appropriate to consider monotonicity constraints based upon prior reasoning.The θ i might for example represent the true success rates for a treatment, at consecutive time periods, where the success rates are thought to be non-decreasing in time.They may alternatively denote the success rates for m multi-centered trials, where the centers have been ordered according to preliminary performance indicators.More generally, we may have an isotonic regression situation where the θ i are thought to possess the same ordering as an increasing covariate, for example, dose level.
A major theme of this paper lies in the argument that, while there may be some prior justification for the monotonicity constraints (1), the previous information may be insufficient to assume that (1) definitely holds.In section 3, hierarchical assumptions are introduced which, under binomial sampling assumptions, relax the investigator's prior belief in (1), thus permitting the observed data to refute (1).The probability model described may alternatively be interpreted as a random effects model for beta-binomial observations.The embedded extra-binomial variation then yields potentially quite different conclusions regarding the proposed monotonicity of the population proportions.

The Veterans' Administration Hospital Quality Monitor Data
We analyze part of a data set modeled by West and Aguilar (1997), Aguilar and West (1998), West et al. (1998), and Burgess et al. (2000), using Bayesian multiple time series.The subsample considered here provides information from the years 1992 and 1993 for m = 159 hospitals in the Veterans' Administration (VA) system.The 1993 data provide our dependent variables, and the 1992 data are used to calculate a set of explanatory variables.
Let y i denote the number of individuals who failed to return for an outpatient visit within 30 days of discharge during 1993 out of the total number of annual discharges at the ith hospital, for i = 1, 2, . . ., m.Then p i = y i /n i can be regarded as a performance indicator or measure of (lack of) quality for the ith hospital.The sample sizes range from 5 to 1142 with an average of n = 324.7.Let x i denote the corresponding proportion for the year 1992.For our first analysis, we attach our indices after reordering the hospitals according to increasing values x 1 < x 2 < • • • < x m .The rank ordering of the performance indicators for 1992 is thus taken into account when considering the rank ordering for 1993.Assumptions of The association between the raw performance indicators is described by the entries to the scatterplot in Figure 1, which plot the p i against the x i .There is some overall increasing trend, but with considerable random scatter.The hospitals' raw performance indicators for 1992 do not provide good predictions of the performances for 1993.The solid plot describes a piecewise linear isotonic regression, as defined in sections 4.3 and 6, and justified under beta-binomial sampling assumptions.The abscissa of this plot provide smoothed performance indicators for 1993 which are consistent with the rank ordering for 1992.The dotted plots add or subtract estimated standard errors of the 1993 sample proportions, which account for substantial extra-binomial variation.The magnitudes of the estimated standard errors provide guidance regarding the usefulness of the fitted performance indicators, for predictive rather than descriptive purposes.Further discussion is provided in section 6.The preceding explanatory variables may be replaced by the VA's diagnostic related group (DRG) predictions that, for each hospital in each year, are supposed to provide predictions of the corresponding p i .The DRG predictions for 1993 do not depend upon the sample proportions for years prior to 1993.In Figure 2, the p i are plotted against the DRG predictions.
The performances of these predictions and the previous raw performance indicators are comparable.As the labeling of the x-axis of Figure 2 is quite compressed, when compared with Figure 1, the fitted isotonic graph, while similar in shape, represents a much steeper regression.The estimated standard errors of the corresponding sample proportions are however comparable.
In other analyses, not reported here, the explanatory variables were replaced by equally weighted or unequally weighted combinations of appropriately normalized proportions for 1992 and DRG predictions for 1993.Quite surprisingly, none of these combinations yielded substantive modifications to the shape of the isotonic regression graph, and the estimated standard errors were at best only marginally reduced.The inclusion of multiplicative interaction terms failed to improve the predictive performance.

Actuarial Graduation
The data in the second, third and fourth columns of Table 1 were collected and analyzed by Broffitt (1988) and reconsidered by Carlin (1992), Liu (2000), and Yang and Schwarz (2005).For i = 1, 2, . . ., 30, the count y i denotes the number of deaths out of n i male clients of age a i , for premium paying policies issued by an insurance company, with face amounts between $10,000 and $24,900.The previous authors assume in some cases the truth of the monotonicity hypothesis for the corresponding underlying mortality rates, where m = 30.We furthermore take the y i to be realizations of random variables Y i satisfying the first stage assumptions specified in section 3, that is, the Y i are independent given the θ i , and possess binomial distributions with respective cell probabilities θ i and sample sizes n i .In section 3, a prior distribution will be described for the θ i , when they are not constrained, that expresses uncertainty in the monotonicity hypothesis (1) and permits the data to assist in the measurement of the posterior uncertainty in H 0 .The corresponding posterior means and standard deviations of the θ i are described in the sixth and seventh columns of Table 1.While substantially smoothing the raw mortality rates in the fifth column, some of the posterior means deviate quite noticeably from the monotonicity hypothesis.The magnitudes of the posterior standard deviations of the θ i nevertheless suggest that the data are reasonably consistent with H 0 under the binomial sampling model.
As the data are uncontrolled there is no particular reason, apart from simplicity, to assume the preceding product binomial sampling model.The probabilistic assumptions of section 3 can alternatively be taken to represent a sampling model incorporating overdispersion, where the mortality rates or population proportions possess a random effects distribution and definitely satisfy a monotonicity specification paralleling (1).The posterior means and standard deviations of the mortality rates under this overdispersion model are given in the eighth and ninth columns of Table 1.The posterior means are more disperse than those under the product binomial sampling model and the posterior standard deviations are noticeably larger.As mortality rates are generally thought to increase with age, the random effects overdispersion interpretation is perhaps more appealing.

A Hierarchical Model
A four stage probability model with the following first two stages is employed: Stage 2: The θ i are independent and beta distributed, given an unknown parameter γ and respective conditional means ξ i where, with the standard parameterization, With the further assumption that the unknown ξ i satisfy the monotonicity specification the preceding two stages can be interpreted in either of the following two ways: (A) Let Stage 1 represent the sampling distribution of the Y i and Stage 2 describe the first stage of a hierarchical prior distribution for the population proportions θ i (further stages for γ and the conditional means ξ i will be added below).In this case Stage 2 represents uncertainty in the belief that the monotonicity hypothesis (1) holds for the θ i , thus extending an idea introduced by O'Hagan and Leonard (1976) in a single parameter normal situation.For given ξ i and γ, the parameter θ i can be said to possess a beta distribution with mean ξ i and sample size γ, where this (prior) sample size measures the degree of belief in (1).As γ → ∞ the monotonicity constraints are completely specified for the θ i .A small value of γ represents substantial uncertainty in this hypothesis.Our formulation does not however require the specification of a definite value for γ, since the current data will typically provide considerable information regarding γ.
(B) The two stages may alternatively be combined.Unconditionally on θ i , Y i possesses a beta-binomial distribution, labeled by its parameters ξ i and γ, and sample size n i .The probability mass function of Y i , given ξ i and γ, is for where B(a 0 , a 1 ) = Γ(a 0 + a 1 )/Γ(a 0 )Γ(a 1 ) is the complete beta function with arguments a 0 and a 1 .With the ξ i now denoting our population proportions, we have a conditionally independent beta-binomial sampling model, within which the monotonicity specification in (2) is definitely satisfied as a modeling assumption.The plausibility of this specification may of course be further investigated.
In either case, the conditional distributions of the θ i , given γ, the ξ i and the observed values y i of the Y i are, for i = 1, 2, . . ., m, independently beta with respective (posterior) sample sizes n i + γ and means where p i = y i /n i and In case (A), equation ( 4) describes the conditional posterior mean of θ i .The θ * i compromise between the ξ i satisfying the monotonicity specification (2), and the p i , which can be taken to represent a general alternative hypothesis.Any data-based estimate of the average shrinkage proportion can be interpreted as an overall measure, on a unit scale, of the evidence against the monotonicity hypothesis (1), and in favor of a general alternative hypothesis.The weighted modifications ρ = ∑ n i are more sensitive to values of the larger n i .
Under the beta-binomial interpretation (B), the P i = Y i /n i are unbiased estimators of the ξ i with respective variances is the ith over-dispersion factor.These estimators do not however take account of (2).Moreover, not all of the m + 1 parameters γ and ξ 1 , ξ 2 , . . ., ξ m are identifiable from the data, as there are just m observations.We consequently extend our conditionally independent beta-binomial model, by introducing the following random effects assumption: Stage 3: Given b 0 = λη and b 1 = λ(1 − η), the ξ i possess the probability structure of the increasing order statistics based upon a random sample of size m from a Beta(b 0 , b 1 ) distribution, that is, a beta distribution with mean η and sample size λ.
Our random effects beta-binomial sampling model for case (B) possesses just three parameters γ, λ, and η.When m is moderate to large, it is therefore possible to draw sensible proper Bayes inferences regarding these three identifiable parameters, and also for the ξ i .Posterior estimates for γ and the ξ i can thereby be imputed for the parameters of the preceding conditionally independent beta-binomial model.For computational convenience, we initially take the distribution of the parameters γ and λ in the prior assessment to be discrete.The prior distribution for the three parameters of our random effects model is selected as follows: Stage 4: γ, λ, and η are independent, and η ∼ Beta(d 0 , d 1 ).The distribution of γ assigns probabilities π 1 , π 2 , . . ., π k to the points g 1 , g 2 , . . ., g k , and the distribution of λ assigns probabilities δ 1 , δ 2 , . . ., δ l to the points h 1 , h 2 , . . ., h l .
The assumption of prior independence of γ and λ can be relaxed by taking these parameters to possess a general discrete joint distribution on a k × l dimensional grid and practical choices of the prior parameters will be discussed in section 5.In a special case it will just be necessary to choose prior estimates n 0 and λ 0 for γ and λ, and, with d 0 = d 1 = 1, to then consider the sensitivity of the posterior inferences to the choices of n 0 , λ 0 , k, and l.Baseline values for n 0 and λ 0 will be recommended.Large values for k and l will yield close approximations to inferences under an interesting thick-tailed continuous prior distribution, which is effectively assumed.
In case (A), Stages 2, 3, and 4 provide a hierarchical prior distribution for the θ i .Stage 3 permits input from the data regarding the values of the Stage 2 parameters ξ i .Stage 4 facilitates input from the data regarding the value of γ, and the Stage 3 parameters b 0 = λη and b 1 = λ(1 − η).Related hierarchical models for binomial probabilities, without the constraints in (2), provide alternatives to the binomial logit/normal prior or normal random effects developments by Leonard (1972Leonard ( , 1976)), Warn et al. (2002), and many others.

Posterior Inferences
In case (A) of section 3 the marginal posterior distribution of θ i averages a beta distribution with sample size n i + γ and mean θ * i satisfying (4), with respect to the unconditional posterior distribution of γ and the ξ i .All posterior quantities of interest for both cases (A) and (B) may be calculated, subject to a minor approximation, via standard Metropolis algorithm/MCMC procedures.Please see Appendix 2 for details.Unconditional posterior densities can be computed along with the means and standard deviations reported in the current paper.
For illustrative purposes only, note that the posterior distribution of the ξ i , given γ, λ, and η, may be roughly approximated by taking the ξ i to possess independent beta distributions, with respective sample sizes D −1 i n i + λ and means where D i = (n i + γ)/(1 + γ), but then constraining these distributions to the region defined by (2).The expressions in (7) constrain the p i towards a common unknown value η.The posterior means of the ξ i are furthermore substantially influenced by the constraints in (2).As well as taking (2) into account, the unconditional posterior inferences create a partial pooling process which roughly speaking has the effect of flattening the ξ i towards a pooled estimate for η.
When d 0 = d 1 = 1, η is estimated by a slightly adjusted center of location of the p i .For example, the first posterior analysis of section 6, leading to the isotonic regression graph in Figure 1, yielded a posterior mean of 0.439 for η.This compares with the overall sample proportion p * = 0.425, and the average sample proportion p = 0.444, and accounts, via the shrinkages of the ξ i , for a flattening of the isotonic regression graph.Pooled information from across the hospitals is thus incorporated.When judging the plausibility of a monotonic relationship, via the residual analysis of sections 6, it is important to realize that our regression graph meaningfully flattens steeper monotonic graphs which may better fit the data.

Two Useful Approximations and a Parameter of Interest
In Appendix 1, an approximation to the conditional distribution of the ξ i , given the θ i , γ, η, and λ, under Stages 2 and 3 of our probability model is justified unless γ, b 0 = λη, or b 1 = λ(1−η) is small.The approximation constrains m independent beta distributions to the region (2).These distributions may, for i = 1, 2, . . ., m, be described as follows: where ξi and λ = γ + λ + 1 , This development highlights ζ in (10) as an interesting bounded function of γ and λ.As ζ approaches zero, the ξi in ( 9) approach the common unknown value η.While the shrinkage proportions ρ i in (5) relate to shrinkages of the θ i towards the ordered ξ i , the proportion ζ controls the shrinkages of the ξi towards a common value η.Our preceding approximate conditional distribution for the ξ i provides a key ingredient of the posterior computational procedures described in Appendix 2, and will be made more exact by acceptance sampling.The exact joint distribution of the ξ i , given the θ i , γ, η, and λ, initially takes the ξ i to be independent, with respective densities for 0 < ξ i < 1 and i = 1, 2, . . ., m, but then constrains the joint distribution of the ξ i to the region (2).The acceptance sampling methodology refers to (11) without simulating from the corresponding exact distribution.In Appendix 2, the approximation η|ξ, λ, y ∼ Beta{m(λ to the conditional posterior (or prior) distribution of η, given the ξ i and λ, is also motivated, with ξ denoting the average ξ i .The beta distribution in (12) possesses sample size m(λ + 1) + d 0 + d 1 , and mean which is close to ξ whenever m(λ + 1) is large compared with d 0 + d 1 .The approximation in ( 12) may be contrasted with the exact conditional density π(η|ξ, λ, y) ∝ π(η) l(η, λ|ξ) , for 0 < η < 1, where π(η) is a beta density with parameters d 0 and d 1 , and When justifying ( 12) and ( 15), it is important to note that the information provided about η and λ by fixed ordered values of the ξ i is the same as when regarding the ξ i as an unordered random sample from a beta distribution with mean η and sample size λ.This information is unaffected by knowledge of the data.

Regression Situations
The methodology underlying the isotonic regression examples of section 2.1 is now discussed.Consider case (B) of section 3, where each Y i is taken to possess a beta-binomial distribution, conditional on parameters ξ i and γ.Suppose that each Y i and corresponding population proportion ξ i is associated with a pre-specified value x i of a covariate, where Assume that the ordering in (2) of the ξ i is consistent with the ordering ( 16) of the x i .A monotonic increasing regression of the ξ i upon the x i is therefore assumed.In situations where two or more of the x i are equal, the ordering of the corresponding ξ i should be based upon prior specification.Modifications to our procedure, which set two or more of the ξ i equal, would alternatively be available.The posterior means of the ξ i under our general analysis may be plotted against the x i and connected by straight lines.If two or more of the x i are equal, then the corresponding posterior means may be weighted according to the corresponding sample sizes.The recommended graph provides our estimated isotonic regression of the ξ i upon the x i .This semi-parametric approach provides an alternative to parametric procedures, see for example, Leonard and Novick (1986) and Lee and Nelder (1996), which replace stages 3 and 4 of our probability model, and the monotonicity assumption (2) by the specification of a functional form for the regression of the ξ i upon the x i .The precise modeling of this specification might sometimes present practical difficulties.
Our semi-parametric approach is also relevant to case (A) of section 3.If the posterior deviations of the θ i from the ξ i , are small, then the preceding estimated isotonic regression of the ξ i upon the x i can be used to meaningfully describe a fitted regression of the θ i upon the x i .Otherwise it is more important to report posterior inferences for the unconstrained θ i .This contrasts with previous isotonic regression procedures for binomial data, for example, Barlow et al. (1972).
While our approach takes into account the ordering of the x i , the specific values of the x i are largely ignored in the posterior analysis, though they are re-introduced when plotting the regression of the ξ i upon the x i .Many isotonic regression procedures (e.g.,  similarly trade information regarding the x i for simplicity in the modeling procedure.Numerous possible adjustments to our method could however be considered.For example, when the regression of the ξ i upon the x i is thought to follow a segment of a concave function, (2) can be replaced by a decreasing slope specification.Information regarding the x i can also be incorporated by generalizing Stage 2 of our probability model, by an assumption that θ i |γ, ξ i ∼ Beta{a i γξ i , a i γ(1 − ξ i )}.The a i adjust the sample size γ and may be specified subjectively as functions of several adjacent x i .

Practical Prior Choices
The broad prior assumptions at Stage 4 of our probability model permit a wide spectrum of representation of prior beliefs, depending upon the information or views possessed by the statistician analyzing the data.However, in some practical situations, information external to the current data set may be sparse.In these circumstances, pragmatic choices should be made.For example, the values d 0 = d 1 = 1 lead to a uniform distribution for η on the unit interval.We will also assume that, for some specified n 0 , the parameter is a priori uniformly distributed over the equally spaced grid of points i/(k + 1) for i = 1, 2, . . ., k.Then the Stage 4 distribution for γ assigns equal prior probabilities π i = 1/k to the unequally spaced points Since E(ρ 0 ) = 1/2, n 0 provides a prior estimate for γ, which is more sensible than the prior mean of γ.As k gets large, the distribution of ρ 0 approaches a continuous uniform distribution on the unit interval.In this limiting case γ possesses a Cauchy-tail prior density π(γ) = n 0 /(n 0 + γ) 2 , for 0 < γ < ∞.No prior mean for γ exists in the limiting case owing to the extremely thick right tail of the prior distribution.The Cauchy-tail density contrasts with the log-Cauchy prior density assumed by Crook and Good (1982) for a multinomial smoothing parameter.In the current situation, the limiting conditional posterior density of γ given the ξ i is for 0 < γ < ∞, where the contributions l * i to the product on the right hand side are defined in (3).Each ξ * i converges to unity as γ → ∞, for any fixed ξ i and y i .Therefore the upper right tail of (17) invariably behaves like the upper right tail of π(γ), for large values of γ.
Quite interestingly, if an improperly unfinitely uniform distribution with density π(γ) ∝ 1, for 0 < γ < ∞, is instead assumed for γ, then the density in (17) will never represent a proper distribution, thus invalidating the entire analysis.The Cauchy-tail prior density more appropriately controls the right tail of (17).This specification nevertheless represents quite sparse prior information regarding γ.
The parameter ρ 0 plays a somewhat similar role to the ρ i satisfying ( 5) and ( 6), and can be interpreted as a shrinkage proportion relating to a hypothetical binomial experiment with sample size n 0 .Under a beta prior distribution for θ i with sample size γ and mean ξ i , the posterior mean of θ i , given only the hypothetical sample proportion p 0 , is the weighted average compromise θ o i = ρ 0 p 0 + (1 − ρ 0 )ξ i .A uniform distribution for γ rather than ρ 0 , on an equally spaced grid, is much less appealing.This will become infinitely uniform as the width of the entire grid becomes large.
The choice of k should be based partly on considerations of computational simplicity.In practice, our prior assumptions for γ will however typically be justifiable only if the posterior inferences are insensitive to the choices of k and the prior estimate n 0 .Reference will be made to a baseline value n * for n 0 , equal to the value of γ for which the average shrinkage proportion ρ in ( 6) is equal to 1/2.In pragmatic terms, n * can be regarded as the value of γ for which, given the observed sample sizes, we judge the monotonicity hypothesis and a general alternative hypothesis to possess equal weight.When all the n i are equal, n * is equal to their common value.More generally n * describes a robust center of location for the n i .
With η, γ, and λ a priori independent, it is similarly assumed that, for some specified λ 0 , the parameter is uniformly distributed over the equally spaced grid of points i/(l + 1), for i = 1, 2, . . ., l.The corresponding distribution for λ assigns equal prior probabilities δ i = i/(l + 1) to the unequally spaced points yielding the Cauchy-tail prior density π(λ) = λ 0 /(λ 0 + λ) 2 , for 0 < λ < ∞, in the limiting case, or l gets large.A sensitivity analysis with respective to the choices of l and the prior estimate λ 0 of λ should also be performed.As an alternative specification, the shrinkage proportion ζ in (10) could be taken to be uniformly distributed over the same grid.In this case γ and λ would not be independent.
When γ and λ are independent it may be reasonable to replace γ in (10) by its prior estimate n 0 before taking ζ to be uniformly distributed.This is the same as taking ζ 0 in (5.4) to be uniformly distributed, with the choice λ 0 = n 0 + 1 for the prior estimate of λ.Our prior estimate for the shrinkage proportion ζ, which controls the weighted average compromise (9), is then equal to the neutral value of 1/2.The specification λ 0 = n 0 +1 should not therefore unduly bias our investigation of the monotonicity hypothesis, and is consequently recommended as a baseline choice.The initial baseline selections n 0 = n * and λ 0 = n * + 1, when followed by a careful sensitivity analysis, promise a reasonably fair evaluation of the information regarding possible monotonicity contained in the current data.
Let ρ * and ζ * denote the posterior means of the bounded parameters ρ = n * /(n * + γ) and ζ = (n * + 1)/(n * + λ + 1) under the preceding prior assumptions, where the prior parameters n 0 and λ 0 may differ from the values n * and n * + 1.The posterior means of the unbounded parameters γ and λ invariably become arbitrarily large as k and l get large.We therefore recommend estimating γ and λ in the posterior assessment by the inverse transformations Unconditional posterior inferences for the θ i and ξ i promise to be reasonably insensitive to the choices of k and l, since their posterior distributions, given γ and λ, depend only upon bounded functions of γ and λ.

Performance Indicators for Quality Monitoring
The conclusions described in section 2.1 for the data introduced there are now discussed further.The solid plot in Figure 1 describes the piecewise linear isotonic regression, defined in section 4.3, of the ξ i , upon the 1992 raw proportions x i .Smoothed performance indicators ξ * 1 , ξ * 2 , . . ., ξ * 159 for 1993, under conditionally independent beta-binomial assumptions, are thereby available.This ordering is consistent with the rank ordering of raw proportions for 1992.The posterior standard deviations of the ξ i decrease from std(ξ 1 ) = 0.032 (with n 1 = 350 and s 1 = 0.022) to std(ξ 69 ) = 0.10 (with n 69 = 786 and s 69 = 0.017).They then increase from std( ξ * 109 ) = 0.010 (with n 109 = 301 and s 109 = 0.027) to std( ξ * 159 ) = 0.031 (with n 159 = 481 and s 159 = 0.022).They are however generally much smaller than the corresponding s i .
After an initial sensitivity analysis, it was assumed that k = 99 and l = 24.The baseline values n * = 242.77and n * + 1 = 243.77are employed for γ 0 and η 0 , and the posterior conclusions can again be shown to be reasonably insensitive to these assumptions.The posterior estimates for γ and λ are γ * = 25.99 and λ * = 106.46.As ρ has posterior mean 0.862 and standard deviation 0.013, with τ virtually equal to zero, there is negligible evidence to substantiate (1) under binomial sampling assumptions.As the shrinkage proportion ζ has posterior mean 0.214 and standard deviation 0.071, the ξ * i are substantially smoothed towards a common value.The location parameter η possesses posterior mean 0.439 and standard deviation 0.010.
An intuitive overall evaluation of our monotonicity specification may be made by reference to the average squared normalized residual In the current example, W = 1.006.A full residual analysis, though not reported here, can be roughly inferred from Figure 2.This indicates that the data are largely consistent with (2).In other words, the performance indicators for 1993 are largely consistent with the rank ordering for 1992 when sensible extra-binomial variation is permitted.The two dotted plots in Figure 1 graph the p i − s * i and the p i + s * i where for i = 1, 2, . . ., m, with D * i = (n i + γ * )/(1 + γ * ), is the estimated standard error of p i under independent beta-binomial sampling assumptions.These estimated standard errors are quite large, ranging in magnitude from 0.092 to 0.237, though mainly in the region of 0.10.For a typical sample size of 250 our extra-binomial assumptions inflate the estimated standard errors by a factor of 3.20.The predictions of sample proportions for future years, with comparable sample sizes, are likely to be subject to greater random variability.
The solid graph in Figure 2 indicates that the performances for 1993 are also largely consistent with the rank ordering of the DRG predictions.The analysis assumed the same prior parameters as for Figure 1 and yielded W = 1.005, γ * = 24.38,and λ * = 101.17.The posterior means of ρ, ζ, and η were 0.868, 0.209, and 0.443, with respective posterior standard deviations 0.013, 0.059, and 0.010.There is a remarkable similarity with the corresponding posterior quantities underlying the analysis for Figure 1.This further emphasizes the close comparability of the predictive performances of the quite different rank orderings, based upon the 1992 raw indicators, and the DRG predictions for 1993.
The accuracy of prediction from this noisy data set is open to some improvement by reference to the binomial logit/normal random effects time series formulation employed by West et al. (1998).See also Aguilar et al. (1999).This general paradigm offers considerable scope for incorporating information from years previous to 1992, and combining information across the hospitals.For, say 1993, West et al. assume a simple linear regression for the binomial logits, upon the logits of the DRG predictions.Separate fixed effects regression parameters are estimated for each year.Random error terms, expressing assumed autoregressive time dependence and the representing the substantial residual variation in the data, are added to the regression functions.Any estimated standard errors of the sample proportions should refer to appropriate marginal distributions under random effects assumptions, since these can express the extra-binomial variability inherent in the data.West et al. demonstrate that the total lower level random effects variability is very large, thus again highlighting possible difficulties with prediction.They obtain very useful descriptive conclusions regarding the regression coefficients.More generally, the usefulness of performance indicators and quality monitoring, for predictive rather than descriptive purposes, is open to further discussion when the data are not objectively generated by random sampling schemes.

Figure 1 :
Figure 1: Raw Performance Indicators for 1992 and 1993

Figure 2 :
Figure 2: DRG Predictions and Observed Proportions for 1993