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The distribution of claims for professional malpractice: some statistical and public policy aspects.

The Distribution of Claims for Professional Malpractice: Some Statistical and Public Policy Aspects

Introduction

Many physicians, over periods of five or ten years, experience several claims for medical professional liability. However, since most physicians have no claims over periods of comparable length, the usual presumption is that those physicians with several claims are incompetent. The calls for action have included pleas for the development of merit rating, for systematic professional review (Hofflander and Nye, 1985), and even for criminal prosecution (Stein, 1986). The purpose of this article is to examine the implications of public policy measures in this area.

The incidence of accidental events within "groups" of individuals seldom exhibits the traditional Poisson distribution. This finding has been reported for automobile accidents (Stanford Research Institute, 1976), health insurance claims (Lundberg, 1940), and professional liability claims (Hofflander and Nye, 1985, and Rolph, 1981). Most studies have attempted to explain this phenomenon by appealing to the notion of differences in "accident proneness" of the individuals in the particular group under analysis. The typical model presumes that each individual within the group has an inherent accident rate and that the number of accidents he/she will experience in a given period has a Poisson distribution with expectation equal to this accident rate. Accident rates are presumed to differ among individuals according to some probability distribution, such as the gamma distribution. This "compound Poisson model" has been successfully fitted to the accident or claim experience of groups of individuals.

The individual Poisson accident rate provides an expected accident rate per unit time; i.e., a weighted measure that recognizes both inherent ability (claims per opportunity) and how often the task is performed (opportunities per time unit). In the case of automobile drivers then, the expected number of automobile accidents which one individual might have in a year obviously depends on the number of miles that the individual drives. Similarly, the number of claims that a physician might expect could depend on the number of patients that are treated by that physician. Thus variation in the Poisson parameter among individuals does not directly imply variation in inherent ability. It could be explained equally well by the alternative hypothesis that all individuals have identical inherent ability but differ in their level of activity.

If all observed variation in individual Poisson accident rates is due to variation in physician activity levels, then the only direct effect of a public policy restricting the practice of physicians with higher claims rates would be to substitute the services of one professional for those of another who has exactly the same probability of contributing to a mishap on any given opportunity. Under these circumstances a public policy that creates barriers for professionals with high numbers of malpractice claims would have no benefits and would have substantial costs.

The notion that all variation in observed claims rates is due either to variation in level of inherent ability or variation in activity level is, of course, simplistic. In all likelihood, professionals differ both with respect to the probability that they will suffer an untoward event during a given procedure, and also differ in the number of procedures they conduct in a month or year. Data on activity levels would permit a study of the relation between the level of activity and the number of claims and lead to the development of better models to guide public policy decisions. Since health insurers have information on the services provided by physicians it does not appear infeasible to collect reasonable information in this area.

An important assumption in public policy measures based on inferences drawn from fitting the incidence of professional liability claims with a compound Poisson model is that the individual Poisson parameters or accident rates are stable over time. Disciplinary procedures aimed at certain physicians are prospective in nature and any direct social benefit from the application of such procedures depends on parameter stability. If physicians with high accident propensities in a particular time interval cannot be shown to have similarly high propensities in later periods, then disciplinary procedures based on past accident rates will not contribute directly to the elimination of future problems in terms of reduced malpractice by disciplined physicians. Of course, disciplinary incentives to avoid malpractice may have indirect benefits, but such incentives are beyond the scope of this article. (1)

The hypothesis that claim propensities are constant over time is discussed and tested, followed by an examination of the possible influence of activity level on the expected claims rates of individuals that can be inferred from the data. Next, measures are developed that permit inference of the impact of public policies that restrict the practice of physicians who have experienced various numbers of claims in selected periods of time under the strict assumptions of Poisson parameter constancy and no influence of physician activity levels. Finally, the article provides a summary discussion of the findings, their relation to prior work, and the types of data needed to establish sound public policy in this area.

Data

The data used to explore the theoretical aspects of this paper were collected during the course of a study by Hofflander and Nye (1985) of the medical professional liability insurance market in Pennsylvania. In Pennsylvania, health care providers buy professional liability coverage up to state specified primary coverage limits from private insurers. The Pennsylvania Medical Professional Liability Catastrophe Loss Fund (CAT Fund) provides $1,000,000 in excess coverage above the primary limits. Primary limits were $100,000 from 1976 through 1982, $150,000 in 1983, and $200,000 in 1984. Events covered under claims-made policies are subject to the primary limits in effect when the claim is made, those covered under occurrence policies are subject to the limits in effect on the date of occurrence.

The data represent all known claims against the CAT Fund, including both open and closed claims. Open claims were included because claims are tendered to the CAT Fund only when primary carriers concede liability at primary limits. Thus, open CAT Fund claims can be considered largely equivalent to closed claims in terms of their validity. One potential problem with CAT Fund claims is that primary limits have varied in a non-economically based fashion. Claims of similar real magnitude in successive years would be those exceeding a primary limit that grew with claim cost inflation. From 1976 through 1984, primary limits grew at a compound annual rate of 9.05%, while the Medical Care Index (MCI) grew at a rate of 9.41%. Thus, over the entire period, primary limits grew at what can be construed as an economically reasonable pace. However, the first increase did not occur until 1983 implying a decline in real primary limits through 1982. Fortunately, only roughly 50 claims, out of 2036, would fail to exceed primary limits growing at the rate of increase in the MCI, and all of these fell within $20,000 of exceeding such economic primary limits. Since attaining CAT Fund status is highly desirable, in that liability at primary limits is conceded by such status, it can certainly be argued that many of these claims would have been adjusted in value to exceed primary limits even if limits had been economically adjusted with the MCI. Thus, no adjustment to the data was made based on potentially varying real primary limits.

In addition to large claims, the CAT Fund also provides full coverage of claims reported more than four years from date of occurrence (i.e., "605" claims). 605 claims that exceed primary coverage limits thus differ from normal claims only to the extent that CAT Fund liability for these claims includes primary limits. Since losses are not dealt with in this article large 605 claims require no special consideration or adjustment. Smaller 605 claims (roughly 100 of 250 total 605 claims), however, were included as well, and as a result, the analyses presented in this article strictly apply to CAT Fund claims, which are predominantly large but 5% of which do not exceed primary limits.

The analysis of parameter stability would have been more satisfactory if it had been possible to restrict the data to that for physicians who practiced for the whole period under analysis. However, it was not possible to identify physicians who did not practice the full period from the data available to the authors. Also, the different way in which primary limits are handled for claims-made versus occurrence policies introduces the potential for inconsistencies in the size of claims included in the sample, e.g., a claim for $125,000 that occurred in 1976 but was reported in 1982 would be included if against an occurrence policy and excluded if against a claims-made policy. However, such cases were few because of the tendency for only those claims which markedly exceeded primary limits to be ceded by primary carriers to the CAT Fund. Thus no censoring of the data was performed on these bases.

The data were collected in 1985, and the full set of data was used in most of the analyses. The only exception is the relative claim rate stability analysis, in which the data were restricted to those claims with occurrence dates prior to 1984.

A Test of Individual Poisson Parameter Stability

The assumption of parameter stability was tested via a method first developed by Lundberg (1940) for use with data from a single population studied in successive time periods. Lundberg showed that, for a general compound Poisson process in which the Poisson parameter of each individual, [M.sub.i], is equal to an individual parameter, [lambda.sub.i], times the average rate for all individuals in any given subinterval of time, r, the probability that an individual will have m claims in the subinterval [t.sub.2] and n claims in subinterval [t.sub.1], given that he had m + n claims in the interval [t.sub.3] = [t.sub.1] + [t.sub.2], is given by: [Mathematical Expression Omitted] where [[Theta].sub.1] = [r.sub.1.t.sub.1./r.sub.3.t.sub.3], [[Theta].sub.2] = [r.sub.2.t.sub.2./r.sub.3.t.sub.3], [r.sub.3.t.sub.3] = [r.sub.1.t.sub.1] + [r.sub.2.t.sub.2], [r.sub.j] is the average claim rate in period j, and [t.sub.j] is the duration of period j. (2, 3)

The operational time intervals ([r.sub.j.t.sub.j]) contain the average claim rates ([r.sub.j]), which are not known, along with the calendar time ([t.sub.j]). The ratios may be estimated by the ratio of the total number of claims in each subperiod to the total number of claims. Once the parameters are known, the conditional distributions for all relevant values of m + n can be computed and compared to the observed data via a chi-squared test. The degrees of freedom are one less than the number of cells used in the test. Lundberg recommends grouping cells so that the expected number of claims is five or more. It is important to note that stability of individual physician parameters is tested, i.e., [[lambda].sub.i], and not necessarily the stability of individual Poisson parameters. Thus, a finding of stability strictly implies that the individual physician is above or below the average in each of the sub-periods which result is identical to a test of individual Poisson parameter stability only in the case that the average Poisson parameter is identical in both sub-periods. The test is thus robust to any frequency trend that may have occurred over the eight year period of analysis.

Lundberg used this test on health insurance claim data in Sweden and found that the results did not reject the hypothesis of a compound Poisson distribution with stable parameters. Venezian (1985) has shown that data from automobile accidents in North Carolina reject the hypothesis. The same test can be applied to claims for medical professional liability. The available data relate to physicians practicing in Pennsylvania (Hofflander and Nye, 1985) and are restricted to large claims, as discussed in Appendix 1. The large minimum dollar value used to define a claim restricts the number of claims so it was deemed best to deal with all specialties together. Such grouping is permissible because the test does not assume any form for the distribution of the Poisson parameter. Grouping would affect the test only if there were substantial switching by individuals among specialties with different claim rates. Restricting the test to each specialty would, on the other hand, reduce rather sharply the power of the test. The data are shown in Table 1.

The data include 953 claims filed from 1976 through 1979, and 1083 claims filed from 1980 through 1983. The total number of claims filed was 2036. Parameter estimates are given by:

[[Theta].sub.1] = [r.sub.1.t.sub.1] / [r.sub.3.t.sub.3] = 955 / 2038 = .4686

[[Theta].sub.2] = [r.sub.2.t.sub.2] / [r.sub.3.t.sub.3] = 1083 / 2038 = .5314

The usable data consists of all physicians with one, two, or three claims from 1976 through 1983. For these groups the observed and predicted number of physicians with zero, one, two, and three claims from 1976 through 1979 are shown in Table 2.

The contributions to chi-squared are 0.52, 3.79, and 5.89, respectively. The total value of chi-squared is 10.20. Since there are eight degrees of freedom, this value does not imply the rejection of the hypothesis that the distribution of claims follows a compound Poisson distribution with stable parameters. In the case of large claims for medical professional liability, past experience may be useful as a predictor of future claims experience. (4)

Interpretation of the Excess Variance

Under the compound Poisson model, when the number of claims observed for each of many members of the group is analyzed, the variance should approximately equal the mean if all members have the same probability of experiencing a claim in a unit of time and exceed the mean if individual members differ in this respect. A positive difference between the variance and the mean, or "excess variance," results from differences in the Poisson parameters among the members of the group.

The Poisson parameter for individual i over period T is expressed by equation 2.

[M.sub.i] = [[lambda].sub.i.T.sub.i.r]

where r is the claim rate for individuals in the group, [[lambda].sub.i] is the ratio of individual i's claim rate to that for individuals in the group, and [T.sub.i] is the period over which individual i is observed.

No matter how the parameter [M.sub.i] varies across individuals, a sample of I individuals observed over a period T will provide estimates of the mean, [E.sub.I.(N)], and variance, [V.sub.I.(N)] of the number of claims per individual. The sample value of the excess variance, [X.sub.I.(N)] is defined as:

[X.sub.I.(N)] = [V.sub.I] (N) - [E.sub.I] (N)

The characteristics of [X.sub.I] (N) have been derived and provided, for the special case of the Poisson distribution (Venezian, 1980). The expected value of this statistic is obtained by recognizing that the sample mean and variance are unbiased estimators of the corresponding population values.

[E[X.sub.I.(N)]] = V (N) - E (N)

Note that even if [[lambda].sub.i] * 1, *i, [E[X.sub.I.(N)]] [is greater than] 0 if the [T.sub.i] differ across individuals. The variance of the statistic [X.sub.I.(N)] may be expressed as follows, (where C (X,Y) denotes the covariance of random variables X and Y.)

V [[X.sub.I] (N)] = V [[V.sub.I.(N)]] - 2C [[V.sub.I.(N)], [E.sub.I.(N)]] + V [[E.sub.I.(N)]]

This expression can be simplified by using standard results on the moments of sampling distributions (see, for example, Cramer, 1945):

V [[X.sub.I.(N)]] = [[mu].sub.4.(N)] - [[mu].sup.2.sub.2.(N)] - [2[mu].sub.3.(N)] - [[mu].sub.2.(N)] / N + 0 ~ 1 / [N.sup.2] ~

where [[mu].sub.j] is the jth central moment of the variable N. Under the null hypothesis that N is Poisson distributed and the population homogeneously with respect to [M.sub.i], we find that

[E[X.sub.I.(N)]] = 0

and

V [[X.sub.I.(N)]] = 2 / I [[E.sub.I.(Tk.sub.i.p.sub.i)].sup.2] = 2 / I [[E.sub.I.(N)].sup.2]

Under the assumption of a stable compound Poisson process, the key issues are whether the data indicate the presence of significant heterogeneity in a given specialty group and, if so, the interpretation of this heterogeneity. The appropriate test requires the computation of the observed excess variance and the "variance of excess variance" via equation 3 and 5. The sample estimate of the excess variance is asymptotically normally distributed, which implies that the ratio of the sample value to its standard deviation is distributed as a standard normal variate.

Table 3 presents the number of physicians with a given number of claims during the period of the Hofflander and Nye study (1985) in each of the specialties identified in that study. The total number of physicians practicing in any given specialty varies slightly from year to year and there is no exact measure of the number of physicians who were in practice throughout the period. The total number of physicians in practice is thus an estimate based on the available data (Hofflander and Nye, 1985). The number of physicians with zero claims was obtained as the difference between estimated total physicians and those with one or more claims. In the case of ophthalmic surgeons, two sets of data are considered; one includes all physicians, the other excludes one physician who had the unusually large number of eight claims during the observation period.

The estimated excess variance and relevant standard variation for each of the eight medical specialties analyzed are shown in Table 4. The results indicate that excess variance is present in each of the specialties considered, with the exception of internal medicine and anesthesiology. In these two specialties the evidence for departure from a Poisson distribution is not strong, and any inference based on the existence of variability in claims proneness among physicians in those specialties must be interpreted with care. The fact that the claims which constitute the data base are relatively large claims should, however, be kept in mind; it is possible that heterogeneity would become apparent if data about smaller claims were available. Note that ophthalmic surgeons exhibit excess variance even if the single surgeon with eight claims during the observation period is eliminated from the group.

The statistical significance of the excess variance is of importance in evaluating private policy issues such as the merit rating of physicians. From that perspective it is important to know whether the data suggest that the Poisson parameter differs among individual members of the group. As long as the stability of the parameter for an individual is established, the existence of variability among individuals suggests that differential pricing based on past experience may be useful in achieving an equitable allocation of future costs. From the point of view of public policy issues such as restricting the ability of individuals to practice, however, this information is not sufficient because the variability could be due to differences in other individual considerations rather to differences in inherent ability, e.g., differences in activity level [5], inadequate specialty classification, differences in so-called "bedside manner," or differences in typical case difficulty. Of these, activity level would seem to be the most influential, and is assumed to be the sole other consideration in the following discussions.

It is not possible to draw firm conclusions about the relative importance of level of activity and accident propensity based on the available data. However, substantial insight into the relative importance of these two considerations can be obtained. The line of inference begins by noting that the excess variance measures the variance of the Poisson parameter. The square root of the excess variance therefore provides a measure of the standard deviation of the Poisson parameter among the members of a group.

The Pennsylvania data refer to a relatively stable number of physicians but of changing composition. If it is assumed that the professional life of a physician spans 35 years, then 2.86 percent of the group can be expected to retire in any given year and to be replaced by new entrants. Thus, in an eight-year period, approximately 2.86 percent would have practiced for only one year during the period, 2.86 percent for two years, etc., with the remainder practicing all eight years included in the sample. The average number of years practiced by physicians then in the eight-year sample period is 7.2 years with standard deviation equal to 25 percent of this value. Thus, a ratio of standard deviation of excess variance to average claims experience of .25 would be consistent with the hypothesis that variations in activity level, in terms of number of years participating in the sample, completely explain variations in claims experience over physicians.

An additional source of claims experience variation attributable to level of activity considerations is the relative number of procedures performed by different physicians. If physician activity level is assumed to be normally distributed with standard deviation equal to 25 percent of average activity level, then the most active physicians would have activity levels in excess of seven times those of the least active physicians; i.e., those physicians with activity levels three standard deviations above the mean activity level relative to those three standard deviations below the mean. Activity level variations of this magnitude would seem to reasonably encompass actual variations in the Pennsylvania physician population. Combining both types of activity variation implies a maximum ratio of standard deviation of excess variance to mean claims experience consistent with variation in activity level being the principal source of variations in claims experience of .5, assuming perfect positive correlation between the two activity variable. Thus, for such ratios in the vicinity of one and greater, it is readily apparent that variation in activity level could not account for a major share of the variation in the Poisson parameter, and the inference that the variation is due to different levels of competence among physicians is reasonable.

Table 5 shows the relative variation in the Poisson parameter for the various specialty groups. The only specialty for which the ratio is so large as to virtually preclude an important role for level of activity is ophthalmic surgery. In that case the comment is valid even after the exclusion of the single surgeon with eight claims during the observation period. The two specialties for which the excess variance was not significantly greater than zero, internal medicine and anesthesiology, have the two lowest values for the ratio. The other specialties have ratios in the neighborhood of 1.00, a range in which differences in the level of activity could account for a significant portion of the variability in the Poisson parameter, but are not the principal consideration in explaining claims variability.

These results suggest that information on the level of activity of individual physicians in some specialties would be very useful in determining the relative roles of ability and level of activity in the generation of claims. Obstetrics and gynecology might be a fruitful area for research because the ratio is relatively low and because data from the health care sector indicate that the percentage of deliveries that experience complications tends to decline with the level of activity. In that context restrictions placed on the practice of individuals with high levels of activity could result in negative benefits to the patients. It is, of course, important to note that the inferences drawn by the reader from these data may differ from those of the authors. Claims experience does seem to be inversely related to physician competence but further research is essential if claims rates are to be used in public policy decisions affecting physician licensure.

Relation of Public Policy Choices to the Savings from Restricting Practice

If the idea is accepted that restriction of practice may be used to reduce the load of professional liability on society, the effect of different policy choices must still be investigated. One extreme form of practice restriction is revocation of a physician's license to practice. License revocation is, of course, the most severe of possible disciplinary actions short of criminal prosecution, but is highly tractable analytically and serves to provide an indication of the maximum benefits possible from increased disciplinary action. If practice by physicians who have had a given number of claims in a period of time is to be eliminated, the number of claims that will serve as a threshold and the period of time that will be used must be specified. Moreover, inquiry must be made intothe effects of using the same decision parameters for all specialties. (Note that this discussion assumes that all the variation in the Poisson parameter is due to differences in the relative competence of physicians, but activity level considerations can easily be incorporated when appropriate data become available.)

The gamma distribution may be written in the form:

G(M) = [(k/m).sup.k] / T(k) [M.sup.k - 1.e.sup.-(k/m)M]

The second panel of Table 5 shows the maximum likelihood estimates of the parameters and the standard deviation of these estimates for each of the specialties considered (Nye and Hofflander, 1988). The estimated values of the parameters may be used to estimate the effect of a given policy concerning the restriction of practice. The probability that a physician in a specialty characterized by values k and m will experience n claims in a period t is given by the negative binomial distribution:

[Mathematical Expression Omitted]

The observed distribution of the number of physicians in each specialty with a given number of claims over the observation period can be compared with the theoretically expected number using a chi-squared test. The results of the test provide a measure of the goodness-of-fit. The usual rule of thumb in conducting such a test is that groups should have an expected value of at least five observations to ensure that the test is not unduly likely to reject the hypothesis of good fit. For purposes of the test, the number of physicians in each specialty with four or more claims were considered to be a single group, which resulted in a chi-squared statistic with four degrees of freedom. While there were still a number of specialties with groups too small to meet the usual rule of thumb, the chi-squared values shown in Table 6 did not indicate a lack-of-fit. The table also shows the results of performing a similar test to assess the goodness-of-fit of the simple Poisson distribution. In this case the grouping method did not eliminate the problem of low cell values for most specialties. A much more conservative value of chi-squared was therefore calculated by ensuring that no group had fewer than five expected cases. Even in this case, the Poisson distribution can be rejected for most specialties. The notable exceptions are internal medicine and anesthesiology, the specialties that were found to have no significant excess variance.

If all physicians who have experienced more than n * claims in a period of duration t are eliminated from practice, the estimated fraction of physicians eliminated, [F.sub.d.(n*,t)], is given by Equation 11.

[Mathematical Expression Omitted]

where P(n/t) is obtained from Equation 10.

A simple and well known calculation shows that physicians who have experienced exactly n claims in period t have an expected number of claims equal to:

[Mathematical Expression Omitted]

Hence, the expected fraction of claims eliminated by the policy, [F.sub.a.(n*,t)], is found to be:

[Mathematical Expression Omitted]

Given estimates of k and m, equation 11 and 14 may be used to estimate the fraction of physicians and claims that would be eliminated by various policies. Table 7 shows the results for an observation period of nine years, i.e., the entire sample interval. It is readily apparent that it is possible to devise a policy for ophthalmic surgeons that affects relatively few physicians but has a dramatic affect on number of claims. For the other specialties the effect on claims ranges from approximately twice to five times the effect on physicians.

Discussion

The first study of merit rating in medical professional liability in the United States was that by Rolph (1981). The data available at that time were inadequate for the estimation of parameters without strong assumptions. Rolph assumed that claim frequencies have a Gamma distribution for any given specialty and had to assume some relation between the parameters of the gamma distributions in order to develop estimates from data covering a single period. His assumption is equivalent to assuming that the parameter k is the same for all groups. That assumption is not borne out by this study, but it must be kept in mind that Rolph's data consisted of enumerations of doctors with specified numbers of claims of any size whereas the data for this study are restricted to very large claims. Information about the distribution of the size of claims by specialty would be required in order to determine whether the assumption made by Rolph is consistent with these findings.

This study examines the potential impact of a public policy that restricts the ability of physicians with high frequency of claims to practice their profession. The data do not reject the hypothesis that the past experience of individual physicians in various specialties may be taken as a valid measure of the individual's exposure to claims in the future. Thus, policies based on the assumption that past history is a good predictor of future experience have a reasonable foundation.

The data suggest that the relation between level of activity and relative competence of physicians need to be studied more deeply if public policy is to have a solid foundation. The available data cannot provide conclusive evidence as to whether the elimination of practicing physicians with large numbers of claims will merely impact the more active physicians or will actually eliminate incompetent physicians. In a few specialties (e.g., anesthesiology and internal medicine), variation in activity levels could account for much of the observed variation in large claims rates. In a few others (e.g., ophthalmologic surgery), it appears that variation in activity levels contributes negligibly to the overall variability in large claims rates. In the remaining specialties analyzed, variation in activity levels may play a role in the observed variation in large claim experience over physicians. If level of activity is a minor determinant of total variability, it may be useful to determine whether the nature of the practice is similar throughout the range or whether classification refinement might be in order. In the absence of classification problems, the number of claims experienced during a base period could serve as a basis of re-examining the qualifications of the professional.

The data make it clear that any policy for reviewing the qualifications or restricting the practice of physicians must be designed for the specialty in question. Policies not based on data for particular specialties are likely to restrict practice of very large fractions of some specialties while leaving professionals in other specialties with no review. Finally, the results presented here provide some guidance on the additional information required and on the relative degree of refinement needed in order to resolve important remaining issues.

(1) The stability is assumed to exist under whatever disciplinary procedures may be in force. The threat of disciplinary action may continue to stability even if the underlying accident propensities are themselves unstable.

(2) Note that the relation does not require that the average claims rates, [r.sub.j] be the same in the subintervals, nor is it necessary to know the values of these rates. The statistical analysis depends only on the ratio of the products, [r.sub.j.t.sub.j], which can be estimated from the data.

(3) The relevance of Equation 1 is not limited to compound Poisson processes in which the Poisson parameter is distributed according to a gamma distribution despite its derivation under this restrictive assumption. Lundgren proves that it is indeed a general relation for compound Poisson distributions as long as the individual claim rates are stable relative to the average claim rate over the period analyzed.

(4) The fact that the data refer to a group of physicians of variable composition should be kept in mind. The test is strictly applicable to data from a fixed group. Changing composition of the group would result in more observations than predicted with the largest number of claims in either the first or the second and fewer observations than predicted with claims in both periods.

(5) The study of Hofflander and Nye (1985) found a highly significant relationship between claims and levels of activity for hospitals in Pennsylvania. No data were available on the levels of activity of private physicians.

References

[1] Cramer, Harold, 1945, Mathematical Methods of Statistics (Uppsala: Almqvist and Wicksells).

[2] Hofflander, Alfred E., and Blaine F. Nye, 1985, Medical Malpractice Insurance in Pennsylvania (Menlo Park, CA: Management Analysis Center, Inc.).

[3] Lundberg, O., 1940, On Random Processes and Their Application to Sickness and Accident Statistics (Uppsala: Almqvist and Wicksells).

[4] Nye, Blaine F., and Alfred E. Hofflander, Experience Rating in Medical Professional Liability Insurance, Journal of Risk and Insurance, 55: 150-57.

[5] Rolph, J.E., 1981, Some Statistical Evidence on Merit Rating in Medical Malpractice Insurance, Journal of Risk and Insurance 48:247-60.

[6] Seal, H.L., 1969, Stochastic Theory of a Risk Business (New York, NY: John Wiley and Sons).

[7] Stanford Research Institute, 1976, The Role of Risk Classifications in Property and Casualty Insurance: A Study of the Risk Assessment Process (Menlo Park, CA).

[8] Stein, A., 1986, Doctors Who Get Away With Killing And Maiming Must Be Stopped, The New York Times February 2, 1986:E21.

[9] Venezian, Emilio C., 1980, Good Drivers and Bad Drivers -- A Markov Model of accident Proneness, Proceedings of the Casualty Actuarial Society 68:65-85.

[10] Venezian, Emilio C., 1985, Are Automobile Accidents Poisson Distributed? (Essex Fells, NJ: Venezian Associates).

[11] Venezian, Emilio C., Blaine F. Nye, and Alfred E. Hofflander, 1986, The Distribution of Claims for Professional Malpractice: Some Statistical and Public Policy Aspects (Columbia, South Carolina: Risk Theory Seminar).

[12] Woll, Richard G., A Study of Risk Assessment, Proceedings of the Casualty Actuarial Society 67:84-138.

Emilio C. Venezian is Chairman and Associate Professor of Business Administration at Rutgers University and President of Venezian Associates. Blaine F. Nye is Professor of Finance at the University of San Francisco and President of Stanford Consulting Group, Menlo Park, California. Alfred E. Hofflander is Professor of Finance and Insurance at the University of California, Los Angeles.
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Author:Venezian, Emilio C.; Nye, Blaine F.; Hofflander, Alfred E.
Publication:Journal of Risk and Insurance
Date:Dec 1, 1989
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