> xzwG `bjbjَ pP\f]8$W2B.pppVWXWXWXWXWXWXW$Y[PW"WUpp+2UUULpp V6VWUPU#L VpAO LU8UNITED STATES DISTRICT COURT
DISTRICT OF MASSACHUSETTS
(Western Division)
__________________________________________
)
ROSIE D., et al., )
)
Plaintiffs, )
)
v. ) Civil Action No.
) 0130199MAP
MITT ROMNEY, et al., )
)
Defendants. )
___________________________________________)
Rebuttal Report of Richard Goldstein, Ph.D.
This report sets forth my opinions in response to certain statements contained in the Expert Report of James W. Conroy (hereafter Rpt) and Dr. Conroys November 22, 2004 Deposition (hereafter Dep).
I have been a selfemployed consulting statistician since 1979. My qualifications, including my publications, are set forth on the attached CV.
In the field of statistics there are certain accepted methods for arriving at conclusions about a larger group based on a sample of the group. Based on my training and experience, I believe it is useful to summarize these in the following principles: (1) set up, or find, a list (sampling frame) that can be used to access every unit to which one wants to generalize (the population); (2) each unit should have a known, positive, probability of being selected; note that Dr. Conroy uses a slightly narrower principle, namely, that each unit should have the same probability of being selected; the principle used by Dr. Conroy has an advantage over the more general rule: the analysis of the data is much simpler; (3) the sampling design should result in a sample that allows one to make a valid estimate of sampling variability from the sample data; (4) the sample size should be chosen to give the appropriate level of precision (margin of error). Dr. Conroy failed to apply these principles in a number of respects, as I will describe. Consequently, his sample is not a basis from which conclusions can be drawn about a larger group.
Problems Related to Sampling Frame
A larger group from which a sample is drawn is known as a sampling frame. According to Dr. Conroy, the sampling frame for the sample he drew for purposes of this case was a group of 3,226 children whose names were compiled from lists of children who received certain services (Rpt p 5, Dep pp. 47, 112, 176177). From my review of his work, I believe it is more accurate to say that there are four sampling frames, one for each group (crisis, home, inpatient, and residential); see paragraph 22 below. However, even this may be too strong. The best sampling frame is complete (see next paragraph) and has no errors (see paragraph 9) or duplicates (there are duplicates; see paragraph 8). It is sometimes possible to make adjustments for errors and duplicates, but Dr. Conroy did not even try to do so. These are violations of principle 1, above.
A valid sample can be a basis for conclusions about the sampling frame from which it was drawn. A valid sample is not however a basis for conclusions about larger groups than the sampling frame. For that reason, Dr. Conroys statements that his sample could be a basis for conclusions about the class and a much larger group of children with serious mental health needs in Massachusetts (Rpt p 3) or about the overall experiences of children in the Massachusetts mental health care system (Rpt p 12) or about children in the class (Rpt p 12) were appropriately corrected by him at his deposition, where he described those statements as a mistake and indicated that his sample was a basis only for conclusions about the sampling frame of 3,226 children (Dep pp 4143, 50, 173). To the extent that he qualified his correction and suggested that his sample could be a basis for conclusions about a larger group of unknown size (Dep pp 4344, 4748), his statements do not have a sound basis in statistics, in light of his testimony that he had no way to know whether there is a relation between his sampling frame and any larger group (Dep pp 47, 70). This is related to principle 1: in his report, Dr. Conroy clearly violates this principle.
Using Dr. Conroys definition of a valid sample as one in which [e]very individual has an equal probability of being selected (Dep pp 51, 126), the sample he drew for purposes of this case does not meet that definition, and for that reason is not a valid basis for conclusions about the sampling frame. See paragraph 8, below. That is, his sample violates principle 2, above.
Each child in Dr. Conroys sampling frame had an equal probability of being selected for the sample only if every name in the sampling frame appeared only once (or the same number of times if more than once). Dr. Conroy testified that he did not know if there were duplications (Dep pp 4448), and did not personally sort the children in the sampling frame into the four groups defined by plaintiffs clinicians or know how the person who performed that sorting handled duplications (Dep pp 8588, 175, 181). He also did not impose any quality checks on how the sorting was done (Dep p 91).
To determine whether in fact every child in Dr. Conroys sampling frame had an equal probability of being selected for his sample, I requested and received a CD ROM listing the names of the children in Dr. Conroys sampling frame. Every one of the four categories in Dr. Conroys sampling frame contains some duplication, i.e., in every category there are some childrens names that appear more than once. Dr. Conroys sample thus does not meet his own criterion for a valid sample (Dep pp 51, 126): every child did not have an equal probability of being selected for the sample; rather, children whose names appeared more than once had a greater probability of being selected. Dr. Conroys sample is therefore not a valid sample, and not a basis for conclusions about the sampling frame; the only valid conclusions that can be drawn from his sample are those about children whose cases were actually reviewed. This is a violation of principle 1, above.
An additional problem with Dr. Conroys sampling frame arises from the fact that one of the named plaintiffs, Sheena Marley, appears in one of the lists (for residential) and therefore in the sampling frame for that list. I agree with Dr. Conroys statement that the named class members should not have been part of the sample because doing so would introduce bias into the sample (Dep pp 115, 130). This is a violation of principle 1, above.
Dr. Conroys sampling frame was drawn from data from two months, November 2002 and March 2003 (Rpt p 5). A potential problem with his sample arises from the fact that he did not take any steps to investigate the possibility that the time of year could affect the need for services (Dep pp 84, 127). Kathy Betts of MBHP, when asked about possible seasonal effects, said, The only seasonal effect that we see in childrens services, is a bump in the number of hospitalizations and stuck kids in the spring that said, spring is a loose term. Sometimes it is in March, sometimes April/May. The surge usually is aligned with the change in temperature . . .. This points to a flaw in the sampling frame that prevents it being representative of the entire possible population, even if the population is seen as the four groups used by Dr. Conroy. In addition, if individual months were to be chosen, rather than the entire time period at issue, those months should have been chosen randomly. Regardless of whether there is a known seasonal effect, the purpose of randomization is to guard against effects caused by unknown factors. Purposefully, or even haphazardly, choosing a list greatly limits the ability to generalize because it loses the ability to guard against unknown factors. Since this part of the sample was not drawn randomly, this is a violation of principle 3, above.
Problems Related to Drawn Sample
Turning now to the methods Dr. Conroy used to draw a sample from his sampling frame, there are a number of problems here as well.
To generate random numbers for purposes of drawing a random sample, Dr. Conroy used a Microsoft Excel random number generator, specifically function RND of Microsoft Excel 2000 (Dep pp 9697). For many years, it has been well known among statisticians that there are potentially serious problems with Excels random number generators. Knusel, L., (2002), On the reliability of Microsoft Excel XP for statistical purposes, Computational Statistics and Data Analysis, 39: 109110, concludes the discussion of the uniform random number generator in Excel: the discrete lattice of these random number generators is not fine enough for scientific statistical purposes, and The findings in this section show that random numbers generated by Excel XP cannot suffice [sic] scientific requirements. Other relevant citations are McCollough, B.D. and Wilson, B. (2002) On the accuracy of statistical procedures in Microsoft Excel 2000 and Excel XP, Computational Statistics and Data Analysis, 40: 713721; McCullough, B.D. and Wilson, B. (1999) On the accuracy of statistical procedures in Microsoft Excel 97, Computational Statistics and Data Analysis, 31: 2737; Knusel, L., On the accuracy of statistical distributions in Microsoft Excel 97, Computational Statistics and Data Analysis, 26: 375377. Dr. Conroy was not familiar with these problems with the Microsoft Excel random number generators, and took no steps to minimize their impact (Dep pp 9798); his use of a random number generator with known statistical problems is reason to question whether his drawn sample is a valid random sample even if there were no other problems.
Dr. Conroy initially drew a sample of 165 names, and subsequently drew an additional 30 names in the crisis category (Rpt pp 67, Dep pp 161163). From that drawn sample, Dr. Conroy stated, he aim[ed] to complete reviews for about 45 (Rpt p 6). The number of 45 clinical reviews was determined by available resources (Rpt p 7), and specifically considerations of the reviewers available time and money (Dep pp 6263). In the end, only 35 children were reviewed (Rpt p 11, Dep pp 103, 118). Note that no justification is given regarding the precision of the sample information, which is a violation of principle 4, above.
I agree with Dr. Conroy that Consent in general is the first threat to pure randomness because Its possible that the people who dont consent are somehow uniformly different from the people who do consent, Dep p 98. Many of the parents of children in Dr. Conroys drawn sample did not respond to requests for their participation in the clinical review (Rpt p 7); as a result, as Dr. Conroy conceded, the reviewed sample is not randomly drawn from the drawn sample (Dep p 142). This is a separate respect in which the reviewed sample fails Dr. Conroys criteria for a valid sample, in that it was not random (Dep p 59). That the reviewed sample was not randomly drawn from the drawn sample is an additional reason why the reviewed sample is not a valid basis for conclusions about the sampling frame. This points to a problem that violates principle 3, above.
There are specific measures that can be taken to control nonresponse bias, that is, sampling bias created by a situation where some members of a drawn sample do not respond to requests for their participation. Indeed, the sole text on sampling cited by Dr. Conroy in his report, Kish, Survey Sampling (Rpt p 9 n 2) gives both proposed remedies for nonresponse and several methods of controlling nonresponse ( 13.5 and 13.6, pp. 548, 557). Dr. Conroy took no steps to avoid nonresponse bias (Dep p 98), and did not use Kishs remedies for nonresponse (Dep pp 119, 126).
The fact that an additional 30 names were pulled in the crisis group is another respect in which Dr. Conroy did not follow the Kish methodology. Kish in fact recommends against substitution in this way because it aggravates the problem of nonresponse (it generally is of little help and may actually make matters worse. Kish, p. 558).
Problems Related to Age and Gender Bias Testing
Dr. Conroy compared his drawn sample to his sampling frame with respect to age and gender to attempt to rule out the possibility of age and gender bias in the drawn sample (Rpt pp 89, Dep pp 140, 142). His tests for age and gender bias were flawed in that he did not exclude the sample from the sampling frame when he compared them; this made the drawn sample look more like the sampling frame than it should have.
In addition, Dr. Conroy tested for bias in the drawn sample, not the sample that was actually reviewed (Dep pp 139, 141). In other words, he did not exclude, or even examine, the possibility of age and gender bias from the sample of children that were actually reviewed. This is a test he could easily have performed and at one point (Dep p 141) he actually says he wishes he had run the other test. Note that I am unable to run this test since I was not provided with the ages and genders of any of the children. He conceded that he had no reason to suppose that either age or gender would create bias with respect to a need for home based services, and that he did not attempt to test for bias with respect to a host of other factors, even though doing so would have led to a stronger conclusion (Dep pp 144146).
Problems Related to Calculation of Margins of Error
I have a number of issues with Dr. Conroys margin of error calculations.
First, the report uses 90% margins of error (Rpt p 11, Dep p 180). In my experience, most courts use 95% margins of error. The fact that statistical textbooks include tables with both 90% and 95% margins of error (Dep pp 154155) does not in my opinion make a 90% margin of error acceptable for use in the context of this case. In addition, in the only text that Dr. Conroy cites in his report (p 9 n 2), Kish, L. (1965), Survey Sampling , New York: John Wiley & Sons, Professor Kish specifically addresses this issue: Nevertheless, some arguments favor publishing tables of sampling errors equal to two standard errors. First, since the 95 percent level is widely understood and expected, departure from it may lead to confusion. (p. 579). Using 95% margins of error would result in margins wider than those set forth in Dr. Conroys table, Rpt p 11, a point he conceded (Dep p 180). I am unable to match (reproduce) the margins of error given in Dr. Conroys table on p. 11 of his report; however, I give a brief example here using what I believe to be a similar approximation (and also using an exact calculation) to show how much wider the margins can go when switching from 90% to 95%: for a sample size of 8, and assuming a rate of 50%, an approximate calculation has margins of error of 40.8% for 95% intervals and 32.9% for 90% intervals; using an exact calculation (using Stata, version 8.2), the margins of error are 34.3% for 95% intervals and 30.7% for 90% intervals.
A second problem with Dr. Conroys calculations of margin of error relates to his failure to take nonresponse into account. Nonresponse should be taken into account when computing a margin of error; survey sampling texts, including the Cochran (Cochran, W.G. [1977]), Sampling Techniques , New York: John Wiley & Sons, pp. 361363) text which Dr. Conroy cites (Dep p 166), point this out. This is often handled by weighting; weighting for nonresponse increases the variance and thus the margin of error. Dr. Conroy did not take nonresponse into account in computing his margins of error and stated that he didnt know you could (Dep p 159). Because he did not take nonresponse into account in calculating margins of error, his margins of error are underestimates. Note that Cochrans text says, The rapid increase in the width of the confidence interval [what Dr. Conroy calls margin of error] with increasing [nonresponse rate] is evident. (p. 362). Cochrans examples only go as high as a 20% nonresponse rate; yet, as seen from a corrected version of Dr. Conroys table on p. 11 of his report, we have nonresponse rates of 88% for Crisis, 82% for each of Home and Inpatient, and 73% for Residential. Since I am unable to match the margins of error given in his table, I do not present corrected versions here.
A third problem with Dr. Conroys margins of error is specific to his calculation of a total margin of error for all four groups, which he calculates as 15.5% (Rpt p 11). The problem arises from the fact that the four groups are not mutually exclusive. Dr. Conroy testified that, while he did not perform the sorting himself, where a child fell within more than one group, he would have advised that the child be included in both groups (Dep pp 4445, 83, 8788). This procedure was followed, and, based on my review of the list of names from which Dr. Conroy drew his sample, there are hundreds of cases in which the same name occurs in more than one group. This means, however, that Dr. Conroy should not have aggregated the groups for purposes of arriving at a total margin of error for the whole sample, since doing so violates his own criterion of each child having the same chance of exclusion. On this ground, the total sample should not be used for any purpose and Dr. Conroys inclusion of a total margin of error of 15.5% on p. 11 of his report is invalid.
Significantly, Dr. Conroys text discussion of why his margins of error are reasonable, Rpt p. 11, relies on the 15.5% figure (which is much lower than the margins of error for the four groups). Since that figure is the result of an invalid generalization, it means that his work rests on the much higher margins of error that he calculated for the four groups, which even he would have difficulty justifying (compare Dep p 120, something near 50 would be a fatal margin of error, with 40.6% margin of error for crisis group). Note that this 40.6% margin of error is narrower than it should be for two reasons discussed above: use of 90% intervals (instead of the more usual 95%) and no adjustment for nonresponse.
A fourth problem with Dr. Conroys calculations of margin of error arises from his use of approximations. Dr. Conroy used approximations in calculating his margins of error (Dep pp 155156: I used tables, Kish formula for sampling margin of error, which I have on an Excel spreadsheet, which you use to calculate margins of error under certain specified conditions. They are approximations, but Im using the Kish tables and definition.) When Professor Kish was writing his text, in 1965, it was standard practice to use approximations in calculating margins of error because computers were much less common and much less flexible. Today, however, it is easy to obtain exact confidence levels for any given rate given the sample size, and there is no reason to use approximations. In fact, it is better to use exact confidence levels; as I will explain below, Dr. Conroys use of approximations led him into error.
In general, margins of error express a range within which one can be confident of a result. To determine the range, you add the margin of error to the result to get the upper limit, and subtract the margin of error from the result to get the lower limit. (In this way, the margin of error shown by Dr. Conroy is symmetrical). To give a familiar example, if a public opinion poll says that 49% of voters favor candidate A with a margin of error of 3% at a 95% confidence level, then you can feel 95% confident (speaking loosely) that the percent of voters who favor candidate A is between 46% and 52% (the interval).
Dr. Conroys work in this case relates not to the percent of voters favoring a particular candidate but to the percent of children with a need for home based services. However, the margins of error set forth in Dr. Conroys table on p. 11 of his report should be able to be used as in my example with respect to voters, i.e., you should be able to add the margin of error to the result to get the upper limit, and subtract the margin of error from the result to get the lower limit.
Dr. Conroy testified that he completed his work and wrote his report before the reviewers performed their clinical review of the children in the sample (Dep pp 138, 153). For that reason, in calculating his margins of error, he did not use the reviewers results with respect to the number of children needing services. Instead, he approximated that result, apparently using an approximate result of 50% (i.e., he apparently assumed that 50% of the children would have a need for home based services).
The problem arises because the actual result differed from that assumption: according to Dr. Conroy, nine out of the 34 were people who didnt have any current need for the service (Dep p 158). In other words, according to Dr. Conroy, the result was that 25 out of 34, or approximately 74%, of the children needed home based services. (I dont know why the denominator was 34 rather than 35, the number of clinical reviews, but I am merely using Dr. Conroys numbers).
When I add and subtract Dr. Conroys margins of error from the actual result rate of 74% rather than his assumed rate of 50%, the problem is apparent. Adding the margins of error for every one of his groups results in totals of over 100%, which is impossible.
This problem could have been eliminated by the use of exact margins of error based on the reviewers results, rather than margin of error calculations based on assumptions, but Dr. Conroy did not calculate exact confidence levels (Dep pp 151152).
Overall, based on the problems with Dr. Conroys calculations of margins of error as described, in my opinion his margins of error are not reliable, and the true margins of error for his work are significantly higher than he has indicated.
Conclusion
For all the reasons stated in this report, it is my opinion that Dr. Conroys reviewed sample is not a basis for generalizations either about his sampling frame or about any other, larger group of children. The reviewed sample is only a basis for conclusions about the 35 children in the reviewed sample, and even then, the conclusions should be about the four groups, with sample sizes of 7, 8, 8, and 12, rather than about the entire 35.
In arriving at the opinions set forth in this report, I examined Dr. Conroys report and deposition. I also examined a CD ROM of Dr. Conroys sampling frame and lists of his drawn and reviewed samples. In addition, I relied on my professional training and familiarity with the research in this area.
The cases in which I have testified as an expert witness at trial or deposition during the past 4 years are as follows:
Linda A. Corsini and Alan Cantera, on behalf of themselves and all persons similarly situated, Plaintiffs, v. United HealthCareServices, Inc., et al., Defendants (File No. CA960608T, US District Court for the District of Rhode Island), 2000 (deposition), 2001 (testimony at trial);
Goldstein et al. v. Savings Bank Life Insurance Company of Massachusetts; deposed on 3/11/03 (class representative);
Dziadiewicz and Durden, et al. v. Blue Cross & Blue Shield of Rhode Island (C.A. No. 962758, U.S. District Court); Hannoian, Triangolo and Kane, et al. v. Blue Cross & Blue Shield of Rhode Island (C.A. No. PB962579, State of Rhode Island and Providence Plantations Superior Court); deposition on 7/12/04 and 7/13/04.
My compensation for my work on this case will be based on my hourly rate of SEQ CHAPTER \h \r 1$250 for normal research work, meetings, drafting reports, etc., and
$375 for testimony, deposition, preparation for either, time spent in Court, etc.
December ____, 2004
Richard Goldstein
 PAGE 16
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