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Proportions in Two-Stage Prevalence Designs. Blase Gambino. Massachusetts Council on Compulsive Gambling. In response to Abbott and Volberg's (in press) ...
Estimating Confidence Intervals and Sampling Proportions in Two-Stage Prevalence Designs Blase Gambino Massachusetts Council on Compulsive Gambling

In response to Abbott and Volberg’s (in press) rejoinder to my epidemiologic note on verification bias and estimation of prevalence rates (Gambino, in press), I provide the formulas for computing confidence intervals for the results of second-stage verification. In addition, I provide the appropriate equation for determining confidence intervals when prevalence is near zero or one. Finally, we present formulas for determining the most efficient sample sizes needed to minimize second-stage variance estimates. These allow the investigator working under a fixed budget to determine the relative value of sampling negative screens to test for false negatives. We close with an observation on the interpretability of evidence.

In their reply to my epidemiologic note on verification bias (Gambino, in press), Abbott and Volberg (in press) touch on two important related issues. The first is the technical question of obtaining accurate point and interval estimates of the prevalence of pathological gambling. The second issue is more conceptual, but still methodological in nature, and concerns the important question of the interpretability of these estimates. They correctly point out that these two issues cannot be separated. With respect to the first issue, Abbott and Volberg properly take me to task for neglecting the question of confidence intervals. A number of investigators have studied the mathematical properties of the two-stage double-sampling scheme (Deming, 1977; Shrout & Newman, Address correspondence to Blase Gambino, Ph.D., Massachusetts Council on Compulsive Gambling, 190 High Street, Suite #6, Boston, MA 02110-3031. Journal of Gambling Studies Vol. 15(3), Fall 1999 © 1999 Human Sciences Press, Inc.

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1989; Shrout, Skodal & Dohrenwend, 1986; Tenenbein, 1970). These investigators have shown that the large-sample variance estimate of P', as adapted from Shrout and Newman, is given by Var (P') =

where N = total sample size, Pp = the sample prevalence estimate, Vp= positive predictive value, Vn = negative predictive value, Op = the proportion of first stage positive screens sampled for retesting, On = the proportion of first stage negative screens sampled for retesting and the prime sign signifies that this is also an estimate of P. A corollary issue raised by Abbott and Volberg (in press) is the question of dealing with confidence intervals when prevalences are near zero or near unity. Failure to take this into account will result in an interval that will be shifted to the left (towards zero). Letting c = the criterion level, e.g., a/2 = .025, the appropriate equations are given by Fleiss (1981, p.14) as:

Abbott and Volberg (in press) also expressed some concern about the number of false negatives who may be missed by restricting testing to weekly or more frequent gamblers. This is not simply a matter of statistical analysis. An important consideration is the possibility that non-weekly pathological gamblers differ in important clinical and/or epidemiologic ways, such as, response to treatment, etiology, and so forth. There will always be a trade off between the loss in generalizability to this group, which must be weighed against the costs of acquiring the information, and the benefits to be obtained from the knowledge gained. As an aid in deciding, Shrout and Newman (1989) and others (Deming, 1977) have provided expressions for determining the optimal proportion of first stage respondents to be sampled under the constraints of a fixed budget. Let each screening test cost Cs and each

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second stage diagnostic test cost Cd, then given the total cost of the survey is fixed, then the sample proportions needed to minimize are given by:

This brings me to the second issue. Statistics, while a necessary tool, can only summarize the data. The value of any interpretation of the results, which is inherently subjective, lies in the knowledge of the researcher of its reasonableness in terms of alternative explanations, flaws in the comparisons, flaws in the data, and such criteria as plausibility, consistency with the literature, and violations of logic, such as the fact that current estimates are embedded in lifetime estimates and therefore cannot be greater. I look forward to seeing the results of Abbott and Volberg's (this issue) continuing study of the New Zealand general population. We expect this contribution to help move us forward in our understanding of the etiology and natural history of pathological gambling.

REFERENCES Abbott, M.W. & Volberg, R.A. (in press). A reply to Gambino's 'An epidemiologic note on verification bias: Implications for estimation of rates. Journal of Gambling Studies. Deming, W.E. (1977). An essay on screening, or on two phase sampling, applied to surveys of a community. International Statistical Review, 45, 29-37. Fleiss.J.L. (1981). Statistical Methods for Rates and Proportions (2nd edition). New York: John Wiley & Sons. Gambino, B. (in press). An epidemiologic note on verification bias: Implications for estimation of rates. Journal of Gambling Studies. Shrout, P.E. & Newman, S.C. (1989). Design of two-phase prevalence surveys of rare disorders. Biometrics, 45, 549-555. Shrout, PE., Skodol, A.E. & Dohrenwend, B.P. (1986). A two-stage approach for case identification and diagnosis: First stage instruments. In J.E. Barrett (Ed.) Mental Disorders in the Community: Progress and Challenge, (pp. 287-303). New York: Guilford Press. Tenenbein, A. (1970). A double sampling scheme for estimating from binomial data with misclassifications. Journal of the American Statistical Association, 65, 1350-1361. Received September 16, 1999; accepted.