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Eye Movements, Strabismus, Amblyopia, and Neuro-Ophthalmology: Ryan D. Niederkohr and Leonard A. Levin A Bayesian Analysis of the True Sensitivity of a Temporal Artery Biopsy Invest. Ophthalmol. Vis. Sci. 2007; 48: 675-680 [Abstract] [Full text] [PDF]

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Bayesian Estimation of Sensitivity of Temporal Artery Biopsies

James Stamey   (7 August 2009)

Bayesian Estimation of Sensitivity of Temporal Artery Biopsies 7 August 2009
James Stamey Send letter to journal: Re: Bayesian Estimation of Sensitivity of Temporal Artery Biopsies James_Stamey{at}baylor.edu James Stamey	It was with great interest that we read the paper by Niederkohr and Levin1 in which they estimate the sensitivity of the temporal artery biopsy (TAB). Accounting for errors in diagnostic tests when estimating prevalences is important in many areas of medical statistics, including ophthalmology. However, we feel the title, "A Bayesian analysis of the true sensitivity of a temporal artery biopsy," is misleading and some corrections would lead to improved understanding and be of interest to the ophthalmological community. Bayesian methods have been used in Investigative Ophthalmology & Visual Science, so we feel this correction is important and timely. Recently, considerable work has been done on estimating prevalences and diagnostic test properties from both the Bayesian and frequentist approaches.2,3,4 Niederkohr and Levin1 consider what they term a "Bayesian" approach to estimating the sensitivity of the TAB. The methods they use would be more accurately termed moment based estimators for the point estimation with a nonparametric bootstrap for the confidence intervals. Below, we describe a fully Bayesian approach to the problem. Test #2 Result   Positive Negative   Positive X1 S1 x S2 x X2i S1 x (1-S2) x   Negative X2j S2 x (1-S1) x X3 (1-S1) x (1-S2) x         Ntotal Table 1. Results of two tests for TAB. In analysis, X2 = X2i + X2j. We begin with the model proposed by Niederkohr and Levin. Assuming perfect specificity, the resulting 2x2 table is provided in Table 1. We assume a simultaneous approach to the biopsies, where the surgeon samples both sides regardless of the result of the initial test, as opposed to not taking the second sample if the first test results in a positive. We consider the two scenarios of Niederkohr and Levin, specifically the case where the tests are conditionally independent and conditionally dependent. Conditional independence is applicable when, conditional on true disease status, the test results themselves are uncorrelated. When tests have a similar biological basis, the resulting outcomes are possibly related. For the simultaneous bilateral TAB, the biological basis is the exact same for each test, so the conditional dependence model is considered as well. We denote the overall prevalence of temporal arteritis (TA) as , the sensitivity as S and the specificity as C. Following the results of Dendukuri and Joseph,2 Black and Craig,3 and others, the cell probabilities for the combinations of test results are as follows assuming conditional independence. Both tests will be positive if the subject is truly diseased and both tests are correct or if the subject is truly disease free and both tests are wrong. Thus the cell probability is p++ = x S x S + (1-) x (1-C) x (1-C). Similarly the other cells can be computed as p+- = x S x (1-S) + (1-) x (1-C) x C, p-+ = x (1-S) x S + (1-) x C x (1-C), p-- = x (1-S) x (1-S) + (1-) x C x C. Of course, for the case of TA where we assume the specificity is 100%, the cells simplify to p++ = x S x S, p+- = x S x (1-S), p-+ = x (1-S) x S, p-- = x (1-S) x (1-S) + (1-), where we assume p+- and p-+ are indistinguishable, and counts in those categories combine. Thus for a sample of size n, the joint distribution is where x1 is the number of subjects declared positive by both tests, x2 is the number of subjects declared positive by exactly one of the tests, and x3 is the number of subjects declared negative by both tests. A Bayesian analysis proceeds by combining the data model and prior distributions on the parameters to obtain the posterior distribution; see for instance Gelman et al.5 for more details. The prior distributions are used to reflect all knowledge about the parameters before observing data. For the conditional independence model, we have two parameters, both of which are probabilities. A commonly used prior for probabilities is the beta distribution.2,3 Specifically, ~ beta(,β) and S~ beta (S,βS). The beta distribution is very flexible and can accomadate a wide range of prior beliefs about the prevalence and sensitivity. If no prior information exists, a beta(1, 1), which is a uniform distribution, is often used. We obtain the Bayesian estimators using WinBugs,6 which is a free statistical package used recently in ophthalmology applications by Rudnicka et al.7 and Morgan et al.8 The WinBugs code used to perform all analyses in this paper is available from any of the authors. Though Niederkohr and Levin are able to get closed form point estimates using the moment estimation technique, they rely on a computational procedure, specifically the bootstrap, in order to obtain interval estimates. Thus ease of computation is not a strong reason to prefer the Niederkohr and Levin approach over the purely Bayesian procedure we discuss here, especially since WinBugs is free software and relatively easy to use. The conditional dependence model adds an extra parameter to model the covariance between the two administrations of the TAB. For tests with specificity less than 1, a correlation parameter for the specificities would be required as well. We denote the covariance parameter . To account for this relationship, the cell probabilities become p++ = x (S x S + ), p+- = x (S x (1-S)-), p-- = x ((1-S) x (1-S) + ). This model, as discussed in Niederkohr and Levin is over-parameterized. In other words, the multinomial distribution with the above probabilities does not have the necessary degrees of freedom to estimate all the parameters. The approach of Niederkohr and Levin was to assume was known and perform a sensitivity study by allowing it to vary across possible values and monitoring the impact on the other estimators. Dendukuri and Joseph2 and Black and Craig3 use informative priors in order to estimate all parameters in this model. Vacek9 shows that the covariance parameter ranges from (S-1) x (1-S) to S-S2, thus a beta prior over this range can be utilized. Specifically, ~ beta[a,b](,β), a = (S-1) x (1-S) and b = S - S2. Whereas the procedure of Niederkohr and Levin required that the covariance parameter be fixed, which leads to an understatement of total model uncertainty, the Bayesian approach accounts for the uncertainty in the estimate of . However, this does yield an over-parameterized model and requires use of at least one informative prior (Dendukuri and Joseph,2 Black and Craig3). Niederkohr and Levin analyze data sets from four previously published articles. We utilize the data found in Boyev et al.10 We first provide Bayesian estimates for the conditional independence model. As can be seen in Table 2, the posterior means and 95% intervals are quite close to the moment estimators and the 95% intervals for the Boyev data. In general, if the conditional independence assumption is reasonable, the moment approach discussed in Niederkohr and Levin and the Bayesian approach discussed here will provide estimators that are very similar. The Bayesian procedure we employ allows for straightforward computation of quantities such as the probability the parameter of interest exceeds some value of interest. For instance, suppose interest lies in the quantity P( > 0.25|x), that is the probability the prevalence exceeds 0.25 given the data. We find this probability to be 0.0124 indicating there is little evidence the prevalence is this large. Similarly, it might be of interest that the sensitivity is at least 0.8, or, conversely, the concern might be that the sensitivity is less than 0.8. We find P(S < 0.8|x) to be 0.138 indicating this parameter is likely to be below 0.8. Point Estimate 95% Interval 0.172 (0.166) 0.114, 0.239 (0.109, 0.228) S 0.862 (0.884) 0.735, 0.949 (0.759, 0.973) Table 2. Bayesian point estimates and intervals (frequentist in parentheses). Niederkohr and Levin investigate the impact of conditional dependence by assuming it is known measuring the change in the estimates of the prevalence and sensitivity as the covariance parameter varies across the range of possible values. While this sort of sensitivity analysis is useful, it underestimates the uncertainty in the parameter estimates by assuming this parameter is known. The Bayesian approach is to include the covariance parameter as an unknown parameter and including at least one informative prior on the three parameters. For instance, suppose that an expert on the sensitivity of TAB is interviewed and the prior beliefs can be summarized with a beta(15.0, 2.6) which has a prior 95% interval of (0.66, 0.97). We note this is not a substantially informative prior and is essentially equivalent to a prior sample of 18 subjects. Summaries of the posteriors are provided in Table 3. The main difference in the estimates in the dependence model as opposed to the independence is that the sensitivity estimate is reduced slightly and the 95% intervals are slightly wider. Neither of these results is surprising. It is well known that ignoring the dependence leads to an overestimation of the sensitivity and by including the covariance parameter as unknown, the precision of the estimates is suitably lower than in the conditional independence model. Point Estimate 95% Interval 0.183 0.119, 0.263 S 0.834 0.674, 0.935 0.044 -0.015, 0.149 Table 3. Bayesian point estimates and intervals for dependence model. In conclusion, we applaud Niederkohr and Levin for bringing to the attention of researchers the importance of accounting for diagnostic test error when estimating population prevalences, and the method of moments procedure they discuss is a reasonable statistical method for the conditional independence model. However, if prior information on parameters is available or if correlation between administrations of the diagnostic test is expected, we believe the Bayesian procedure we have described is the preferred method. James Stamey1 Daniel Beavers2 John Seaman III2 1Baylor University, Waco, Texas 2Wake Forest School of Medicine, Winston-Salem, North Carolina References 1. Niederkohr RD, Levin LA. A Bayesian analysis of the true sensitivity of a temporal artery biopsy. Invest Ophthalmol Vis Sci. 2007;48:675-680. 2. Dendukuri N, Joseph L. Bayesian approaches to modeling the conditional dependence between multiple diagnostic tests. Biometrics. 2001;57:208-217. 3. Black MA, Craig BA. Estimating disease prevalence in the absences of a gold standard. Stat Med. 2002;21:2653-2669. 4. Boese D, Young DM, Stamey JD. Confidence intervals for a binomial parameter based on binary data subject to false-positive misclassification. Comput Statist Data Anal. 2006;50:3369-3385. 5. Gelman A, Carlin JB, Stern HS, Rubin DB. Bayesian Data Analysis. New York: Chapman and Hall; 2004. 6. Spiegelhalter DJ, Thomas A, Best NG. WinBUGS Version 1.2 User Manual. MRC Biostatistics Unit; 1999. 7. Rudnicka AR, Mt-Isa S, Owen CG, Cook DG, Ashby D. Variations in primary open-angle glaucoma prevalence by age, gender, and race: A Bayesian meta-analysis. Invest Ophthalmol Vis Sci. 2006;47:4254-4261. 8. Morgan W, Hazelton M, Azar S, et al. Retinal venous pulsation in glaucoma and glaucoma suspects. Ophthalmology. 2005;111:1489-1494. 9. Vacek PM. The effect of conditional dependence of the evaluation of diagnostic tests. Biometrics. 1985;41:959-968. 10. Boyev LR, Miller NR, Green R. Efficacy of unilateral versus bilateral temporal artery biopsy in giant cell arteritis. J Neuroophthalmol. 1999;128:211-215.