Experience rating in medical professional liability.

Experience Rating in Medical Professional Liability Insurance: Comment

Abstract

Hofflander and Nye [3] (HN) attempt to make a case for experience rating of individual physicians for medical liability insurance. This is a timely issue that deserves careful analysis. The authors begin well by obtaining reliable data (something not easy to do in this area) on claim frequencies. Unfortunately, the data set is inadequate for answering the key question--is experience rating worthwhile? The purpose of this comment is to demonstrate this inadequacy and then indicate what needs to be done to resolve the issue.

Experience Rating

Any experience rating procedure depends on striking a balance between the variability from one physician to the next (the between variance) and the variability of the claims from a single physician (the within variance). If the between variance is small, the physicians are essentially all alike and there is no point in constructing an experience rating system. If the within variance is large then the experience for a specific physician will vary greatly over time and therefore the past will be inadequate for predicting the future. This would also indicate that experience rating is of little value. The second case could arise either because of a high natural variability, in which case the estimats from past data will be unreliable, or because the underlying propensity of the physician to produce claims is varying over time, in which case the future is unlikely to reflect the past.

This notion can be represented by the following two state model. Let Xi, be the number of claims for the ith physician. The first stage gives probabilities for Xi given a parameter that reflects the ith physician's propensity for producing claims. It is customary to assume that given these parameters the observations are independent. So we have Pr(Xi = x) = p(x/ i), x = 0, 1,..., i = 1,..., k. The second stage describes how the parameters vary from physician to physician. The multivariate density is f(0/ ) where is a vector of parameters of this second level distribution. It is also customary to assume that 01,...,0k are independent. The two variances are

Between: Var[E(X/0)]

Within: E[Var(X/0)]. It is well known that the sum of these two quantities is Var(X/ ) the unconditional variance of the observations. The key, then, is to split this variance into its two components.

Three Models

In this section three models for the claims process will be presented. The first is the one given by HN. It postulates the Poisson distribution with parameter for the first stage (so the parameter 0 is univariate) and the gamma distribution with parameters k and m/k. The two variances are

Between: Var[E(X/ )] = Var( ) = m2/k

Within: E[Var(X/ )] = E( ) = m. This model would be appropriate if it could be established that individual physicians actually do generate claims according to a homogenous Poisson process and claims from a randomly selected physician follow the negative binomial distribution (for this is the unconditional distribution of X in this model).

The second model produces one of the extreme cases from an experience rating viewpoint. Suppose that individual physicians generate claims according to the negative binomial distribution with parameters k and m/(m + k). The mean and variance for this distribution are m and m(m + k)/k. Further assume that all physicians are identical. That is, they all have the same values of m and k. In this case the two variances are

Between: Var[E(X/m,k)] = Var(m) = 0

Within: E[Var(X/m,k)] = E[m(m + k)/k] = m(m + k)/k. In this case experience rating is of no value since all physicians are identical. While the hypothesis that all physicians are identical might be hard to defend, the negative binomial model is reasonable if we believe that the physician's propensity to produce claims varies over time.

The third model presents the other extreme. The parameter for an individual physician, K, indicates exactly how many claims that physician will have each year. Further assume that this parameter is distributed throughout the population according to a negative binomial distribution with parameters k and m/(m + k). The two variances are

Between: Var[E(X/K)] = Var(K) = m(m + k)/k

Within: E[Var(X/K)] = E(0) = 0. In this case experience rating is a must. As soon as one year of experience has been observed the future experience can be predicted without error.

For the two stage model described above the best linear experience rating formula based on one year of observation is W/W + B X + B/W + B m where B is the between variance and W is the within variance. For the three models this reduces to m/k + m X + k/k + m m, X, and m respectively. The first one is the formula used by HN to produce their Table 3. All the above results are well-known in the credibility area [2, 5].

The important fact about these three models is that for all of them the unconditional (though still given m and k) distribution of X is negative binomial. This means that any test of fit of these three models will lead to identical conclusions; if one fits, they all fit. Furthermore, there is nothing in the data to distinguish these three models.

HN claim that experience rating is a good idea because they are able to reject the hypothesis that k = . If their model could be verified, this would indeed be the correct hypothesis to test. However, the proper hypothesis to test is that the ratio of the between variance to the within variance is zero. This hypothesis cannot be tested from the available data because it is impossible to separate the total variance into these two components.

Justifying Experience Rating

There are two possible approaches to justifying the experience rating of physicians. One is to verify a distributional model (such as the Poisson-gamma), estimate the parameters, find the two variances, and then apply the experience rating formula. This is the approach used by HN with the verification step left out. The other is to estimate the two variances directly from the data. If individual physicians do not vary over time the formulas of Buhlmann and Straub [1] are the standard, although to some extent they can be considered to be based on the normal distribution. In either case the hypothesis that the between variance is zero can be tested.

The key to both approaches is the existence of information on the within variance. This requires multiple observations on the same physician. Of course, this can only be obtained over time and so there are still some issues that cannot be resolved. In particular, it will be difficult to determine what part of the variation is due to year-to-year parameter variations versus the natural variation in the claims process. With such information a great many different models can be tried. Besides the three given above other possibilities are Poisson-inverse Gaussian, and Poisson-Pascal [4] as well as one that is between models one and two above. That is, assume in any one year a single physician generates claims according to a Poisson distribution but the Poisson parameter varies from year-to-year according to a gamma distribution with the yearly values being independent. Further assume that the scale parameter of this gamma distribution varies from physician to physician according to another gamma distribution. There are now three parameters to estimate by maximum likelihood. This model can be verified by first showing that individual physician's claims follow a negative binomial distribution and that all physicians in a group have the same first parameter. The next step is to verify that the complete set of observations is consistent with a Poisson-generalized Pareto model.

Conclusion

HN have opened up an important line of inquiry. The question of whether or not experience rating of physicians for medical liability insurance is a good idea remains wide open. Some suggestions for answering the question have been provided. The remaining steps are to obtain data suitable for answering the question and then performing the appropriate analysis.

Physicians should pay malpractice insurance premiums appropriate for their expected level of malpractice risk, and past malpractice experience provides valuable information about expected future losses. When physicians fully enjoy the benefits of good malpractice experience and are subject to the penalties of poor malpractice experience, the deterrent of the fault system functions to reduce malpractice incidents to everyone's benefit.
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