On the Analysis of Tuberculosis Studies with Intermittent Missing Sputum Data

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1 On the Analysis of Tuberculosis Studies with Intermittent Missing Sputum Data Daniel Scharfstein Johns Hopkins University July 12, 2013

2 Happy Birthday!

3 Collaborators Andrea Rotnitzky Maria Abraham Aidan McDermott Lawrence Geiter Richard Chaisson

4 Observed Data Moxifloxacin Ethambutol

5 Example of Patient Data Visit Line L R+1 S 1 Culture Results? +? -? Converter? N N??? Y Y Y 3 6 {3, 4, 6}

6 Observed Data Moxifloxacin Ethambutol

7 Example of Patient Data Visit Line L R+1 S 1 Culture Results? +? -? Converter? N N??? Y Y Y 3 6 {3, 4, 6} Need assumptions on the distribution of time of culture conversion given observed data. 3: V3 -, V5 -; 4: V3+, V5-; 6: V5+ Forward time: visit 3, visit 5 given 3-, visit 5 given 3+ Reverse time: visit 5, visit 3 given 5-

8 Example of Patient Data Visit Line L R+1 S 1 Culture Results? +? -? Converter? N N??? Y Y Y 3 6 {3, 4, 6} 3 Culture Results? +? Converter? N N N N N Y Y Y 6 6 {6} 5 Culture Results? +? Converter? N N? Y Y Y Y Y 3 4 {3, 4}

9 Example of Patient Data Visit Line L R+1 S 1 Culture Results? +? -? Converter? N N??? Y Y Y 3 6 {3, 4, 6} 3 Culture Results? +? Converter? N N N N N Y Y Y 6 6 {6} 5 Culture Results? +? Converter? N N? Y Y Y Y Y 3 4 {3, 4} 7 Culture Results? Converter? N N N Y Y Y Y Y 4 4 {4} 9 Culture Results? Converter? N N Y Y Y Y Y Y 3 3 {3}

10 Benchmark Assumptions We postulate that P[T = r + 1 O = o] = P[T = r + 1 O = o (r) ] (1) P[T = k T k, O = o] = P[T = k O = o (k 1) ] (2)

11 Sensitivity Analysis P[T = r + 1 O = o] = P[T = r + 1 O = o(r) ] exp(α) h r+1 (o (r) ; α) P[T = k T k, O = o] = P[T = k O = o(k 1) ] exp(α) h k (o (k 1) ; α) (3) (4)

12 Curse of Dimensionality Need to estimate for each realization O = o with S > 1, P[T = k O = o (k 1) ] These probabilities cannot be estimated non-parametrically. Postulate a parametric model for the law of the observed data O given baseline covariates X. This model induces parametric models for P[T = k O = o (k 1) ], that ultimately enable estimation of P[T = k O = o] by borrowing information across strata O = o (k 1).

13 Model for Observed Data O k = (M c k, C obs k, M s k, S obs k ); O = ( X, O K ). Model the law of O given X by modeling the distribution of O k given O k 1 and X for all k = 1,..., K. logit{p[m c k = 1 O k 1, X ]} = a(k, O k 1, X ; γ (a) ) logit{p[c obs k = 1 M c k = 0, O k 1, X ]} = b(k, O k 1, X ; γ (b) ) logit{p[mk s = 1 Mk c, Ck obs, O k 1, X ]} = c(k, Mk c, Ck obs, O k 1, X ; γ (c) ) logit{p[s obs k = 1 Mk s = 0, Mk c, Ck obs, O k 1, X ]} = d(k, Mk c, Ck obs, O k 1, X ; γ (d) )

14 Inference We can express for all realizations O = o with S > 1, the conditional probability P[T = k O = o (k 1) ] as a given functions of o (k 1) and γ = (γ (a), γ (b), γ (c), γ (d) ). We can express P[T = r + 1 O = o] and P[T = k T k, O = o] as given functions, P[T = r + 1 O = o; γ; α] and P[T = k T k, O = o; γ; α] of o, γ and α.

15 Inference Estimate γ by γ using maximum likelihood. Estimate P[T = k] by 1 P[T n i=1 i = k; α] where P[T i = k; α] equals 0 if k / S i, equals 1 if S = 1 and k S i, equals P[T = R i + 1 O = O i ; γ; α] if S i > 1 and k = R i + 1, and equals {1 P[T = R i + 1 O = O i ; γ; α]} (1 P[T = s T s, O = O i ; γ; α]) k<s<r i +1 k S i P[T = k T k, O = O i ; γ; α] if S i > 1 and k S i, k < R i + 1. Confidence intervals by non-parametric bootstrap.

16 Data Analysis Treatment groups were not balanced with respect to the cavitation status at baseline; 81.1% and 56.9% have cavitation in the moxifloxacin and ethambutol arms, respectively. Estimate for each treatment group, the distribution of time of culture conversion by a weighted average of cavitation-specific distribution of time of culture conversion. Weights are taken to be the marginal (i.e., not conditional on treatment arm) proportion of patients with and without cavitation at baseline, respectively.

17 Data Analysis Under benchmark assumption, the estimated probabilities of being a culture converter at or by week 8 are 92.5% and 75.5% in the moxafloxacin and ethmabutol arms, respectively. Estimated difference is 17.0% (95% CI: [4.5%,29.3%]). Benchmark analysis suggests a statistically significant difference in culture conversion at or by week 8 in favor of moxifloxacin.

18 Data Analysis Moxifloxacin Ethambutol Probability Probability α 1 α 0

19 Data Analysis α 1 (Moxifloxacin) Moxifloxacin α 0 (Ethambutol)

20 Data Analysis Moxifloxacin Ethambutol Probability Probability Visit Visit

21 Data Analysis Moxifloxacin Ethambutol Signed Distance Signed Distance α 1 α 0

22 Data Analysis Compare the treatment-specific distributions of time to culture conversion. Estimate a common treatment effect over time, using Cox(1972) logistic model for discrete survival data. This model assumes that h z (k) 1 h z (k) = τ k exp(βz) k = 1...., 8, z = 0, 1 where h z (k) = P z [T = k T k] and τ 1,..., τ 8 0. exp(β) is the ratio of the odds of first becoming a culture converter at visit k given culture conversion at or after visit k comparing moxifloxacin to ethambutol.

23 Data Analysis For each choice of α 0 and α 1, we minimize the following objective function: 1 8 z=0 k=1 { } 2 ĥz (k) 1 ĥz(k) τ k exp(βz) with respect τ 1,..., τ 8 0 and β, where ĥ z (k) = P z [T = k T k]. For each choice of α 0 and α 1, this method finds the closest fitting logistic model to the data : { ĥ z (k) : k = 1,..., 8, z = 0, 1}. Even if the model is incorrectly specified, still provides a valid test of the null hypothesis of no treatment effect.

24 Data Analysis Under benchmark assumption, the estimated hazard ratio is 3.41 (95% CI: [1.16,16.90]), indicating that patients treated with moxifloxacin have a statistically significant shorter time of culture conversion than those treated with ethambutol.

25 Data Analysis α Moxafloxacin α 0

26 Discussion Roshamon Coarsening at Random Everything is relative. Sensitivity analysis parameters are not scientifically interpretable. Look at induced distributions Requires scientific judgement.