Fig. 1
Data structure and notation for a single treatment group. Note n is the total sample size, n1. is the number of survey respondents, and n2. is the number of survey non-respondents. Then among survey respondents, denote n11 as the number of observed self-report abstinence and n21 as the number of imputed self-report abstinence. Similarly, n12 and n22 represent the number of observed failures and imputed failures based on the self-report data, respectively. For the urine samples, u(obs) and u(imp) represent the number of observed and estimated (based on the imputed survey data) urine samples being provided; similar notations, v(obs) and v(imp) are used for the number of unavailable urine samples. For the urine data, analogous notations are defined as for the survey data except for using f, instead of n, to denote the numbers of subjects under different conditions (with the superscript 11, 12, 21, and 22 having the same meaning). In addition, we used f11(imp) to denote the abstinence and f12(imp) to denote the failure obtained from the estimated available urine samples u(imp)).Then we combined the f11(obs) and f11(imp) to obtain the number of urine-verified abstinence f11. among the urine samples what were actually provided or could have been provided if all surveys were completed, whereas combined f12(obs) and f12(imp) to obtain the urine-verified failure f12..Denote OR1 as the assumed odds ratio between missing and smoking for self-report data and OR2 for urine data. Dashed lines indicate where missing data are reallocated based on certain assumptions or estimations. Bolded notation denotes values that are not observed