# Multiple imputation inference Combines estimates and standard errors from m complete-data analyses performed on m imputed datasets to produce a single inference. Uses the technique described by Rubin (1987) for multiple imputation inference for a scalar estimand. ```r mi.inference(est, std.err, confidence=0.95) ``` ## Arguments - `est`: a list of $m$ (at least 2) vectors representing estimates (e.g., vectors of estimated regression coefficients) from complete-data analyses performed on $m$ imputed datasets. - `std.err`: a list of $m$ vectors containing standard errors from the complete-data analyses corresponding to the estimates in `est`. - `confidence`: desired coverage of interval estimates. ## Returns a list with the following components, each of which is a vector of the same length as the components of `est` and `std.err`: - **est**: the average of the complete-data estimates. - **std.err**: standard errors incorporating both the between and the within-imputation uncertainty (the square root of the "total variance"). - **df**: degrees of freedom associated with the t reference distribution used for interval estimates. - **signif**: P-values for the two-tailed hypothesis tests that the estimated quantities are equal to zero. - **lower**: lower limits of the (100*confidence)% interval estimates. - **upper**: upper limits of the (100*confidence)% interval estimates. - **r**: estimated relative increases in variance due to nonresponse. - **fminf**: estimated fractions of missing information. ## METHOD Uses the method described on pp. 76-77 of Rubin (1987) for combining the complete-data estimates from $m$ imputed datasets for a scalar estimand. Significance levels and interval estimates are approximately valid for each one-dimensional estimand, not for all of them jointly. ## See Also `dabipf`, `imp.cat` ## Examples ```r # # Example 1 Based on Schafer's p. 329 and ss. This is a toy version, # using a much shorter length of chain than required. To # generate results comparable with those in the book, edit # the \dontrun{ } line below and comment the previous one. # data(belt) attach(belt.frame) oddsr <- function(x) { # Odds ratio of 2 x 2 table. o <- (x[1,1]*x[2,2])/ (x[1,2]*x[2,1]) o.sd <- sqrt(1/x[1,1] + # large sample S.D. (Fienberg, 1/x[1,2] + # p. 18) 1/x[2,1] + 1/x[2,2]) return(list(o=o,sd=o.sd)) } colns <- colnames(belt.frame) a <- xtabs(Freq ~ D + S + B2 + I2 + B1 + I1, data=belt.frame) m <- list(c(1,2,5,6),c(1,2,3,4),c(3,4,5,6), c(1,3,5),c(1,4,6),c(2,4,6)) b <- loglin(a,margin=m) # fits (DSB1I1)(DSB2I2)(B2I2B1I1)(DB1B2) # (DI1I2)(SI1I2) in Schafer's p. 329 s <- prelim.cat(x=belt[,-7],counts=belt[,7]) m <- c(1,2,5,6,0,1,2,3,4,0,3,4,5,6,0,1,3,5,0,1,4,6,0,2,4,6) theta <- ecm.cat(s,margins=m, # excruciantingly slow; needs 2558 maxits=5000) # iterations. rngseed(1234) # # Now ten multiple imputations of the missing variables B2, I2 are # generated, by running a chain and taking every 2500th observation. # Prior hyperparameter is set at 0.5 as in Schafer's p. 329 # est <- std.error <- vector("list",10) for (i in 1:10) { cat("Doing imputation ",i,"\n") theta <- dabipf(s,m,theta,prior=0.5, # toy chain; for comparison with steps=25) # results in Schafer's book the next # statement should be run, # rather than this one. ## Not run: theta <- dabipf(s,m,theta,prior=0.5,steps=2500) imp<- imp.cat(s,theta) imp.frame <- cbind(imp$x,imp$counts) colnames(imp.frame) <- colns a <- xtabs(Freq ~ B2 + I2, # 2 x 2 table relating belt use data=imp.frame) # and injury print(a) odds <- oddsr(a) # odds ratio and std.dev. est[[i]] <- odds$o - 1 # check deviations from 1 of std.error[[i]] <- odds$sd # odds ratio } odds <- mi.inference(est,std.error) print(odds) detach(belt.frame) ``` ## References Fienberg, S.E. (1981) **The Analysis of Cross-Classified Categorical Data**, MIT Press, Cambridge. Rubin (1987) **Multiple Imputation for Nonresponse in Surveys,** Wiley, New York, Schafer (1996) **Analysis of Incomplete Multivariate Data.** Chapman & Hall, Chapter 8.

mi.inference function