Multiple imputation

One drawback to this transformation is that a log-ratio cannot be taken for elements not present or below detection limits. Sophisticated multiple imputation techniques, simple replacement, and variable omission are strategies for dealing with these zeroes (24). In this preliminary study, elements not present in all samples were simply omitted from the analysis as most of the relevant transition metals were present in all samples. Due to these restrictions on missing abundances, 20 elements (As, Ba, Ca, Ce, Cs, Dy, Eu, Hf, K, Lu, Nd, Ni, Rb, Sr, Ta, Tb, U, Yb, Zn, Zr) were excluded from the analyses. 

This chapter provides an overview of imputation, gives a description of incomplete data types, and reviews the standard methods of handling missing data, with a focus on multiple imputation. 

Traditionally, incomplete (missing) data have been handled by deletion from analysis of cases that contain missing values (single imputation) and most recently by using multiple imputation techniques. In this section single imputation techniques are discussed. Multiple imputation and the paradigm for multiple imputation are discussed in separate sections because of the broad scope of these topics. 

The three disadvantages of MI when compared with other imputation methods are (a) more effort to create the multiple imputations, (b) more time to run the analyses, and (c) more computer storage space for Ml-created data sets (6). These are hardly issues with current development in computer technology. The MI approach is computationally simpler than the ML and Bayesian approaches for most practical situations. Once the imputed data is generated, the data can be analyzed with any data analysis software of choice. 

Multiple imputations are generated by assuming a particular imputation model. Therefore, the success or failure of MI depends on the propriety of the assumed imputation model. Assumptions required in MI are (a) a model for the data values, (b) a prior distribution for parameters of the data model, and (c) the nonresponse mechanism. However, with nonparametric methods of MI, minimal distributional assumptions are required (see Section 9.6.6). 

Assuming a probability model that relates the complete response (or dependent) data Y (the combination of observed values Tobs and the missing values Ynus) to a set of parameters is the first and most important step to obtaining multiple imputations. With the probability model and the prior distribution on parameters (see Section 9.6.3), a predictive distribution P(Y s Yobs) for the missing values conditional on the observed values is found, and the imputations are then generated from the predictive distribution. 

A very useful approach to augmenting informative dropout or truncated data considered here is the propensity-adjusted multiple imputation approach (25). This approach utilizes the method of reducing a multivariate stratification to a univariate stratification using the propensity score (26, 27). The propensity score is the conditional probability of assignment to a particular treatment given a vector of observed covariates. That is, at time t a subject s propensity score is defined as the probability of the subject to remain in the study through time t given the subject s... 

FIGURE 9.1 Performance of the conditional multiple imputation and fractional single multiple imputation (LLOQ/ ) methods under the fourth scenario (i.e., assuming 45% interindividual variability and 25% residual variability) for the following parameters (A) AUCo-inf, (B) %AUC extrapolated, and (C) terminal half-life (Lambda Z HL). 

D. B. Rubin, Multiple Imputation for Nonresponse Surveys. Wiley, Hoboken, NJ, 1987. 

P.D. Allison, Multiple Imputation for Missing Data A Cautionary Ta/e (1998). Retrieved June 14, 2002, from University of Pennsylvania web site http //www.ssc.upenn.edu/ -allison. 

D. B. Rubin and N. Schenker, Multiple imputation for interval estimation from simple random samples with ignorable nonresponse. J Am Stat Assoc 81 366-374 (1986). 

P. W. Lavori, R. Dawson, and D. Shera, A multiple imputation strategy for clinical trials with truncation of patient data. Stat Med 14 1913-1925 (1995). 

Important attributes may be missing if the database was not designed with PMKD in mind. A possible solution is to use multiple imputations or other imputation techniques as dictated by the type of missingness to create a complete data set for PMKD (17-19, 40, 41). 

The Bayesian bootstrap was introduced by Rubin (26) in 1981 and subsequently used by Rubin and Schenker (29) for multiple imputation in missing-data problems. The Bayesian bootstrap is not covered because its application is for multiple imputation of missing data and this is addressed in Chapter 9. 

Rubin (1987) proposed that if m-imputed data sets are analyzed that have generated m-different sets of parameter estimates then these m-sets of parameter estimates need to be combined to generate a set of parameter estimates that takes into account the added variability from the imputed values. He proposed that if 0 and SE(0 ) are the parameter estimates and standard errors of the parameter estimates, respectively, from the ith imputed data set, then the point estimate for the m-multiple imputation data sets is... 

So the multiple imputation parameter estimate is the mean across all m-imputed data sets. Let U(0 ) be the variance of 0 , i.e., the standard error squared, averaged across all m-data sets... 

The multiple imputation standard error of the parameter estimate 0j is then the square root of Eq. (2.106). Examination of Eq. (2.106) shows that the multiple imputation standard error is a weighted sum of the within-and between-data set standard errors. As m increases to infinity the variance of the parameter estimate becomes the average of the parameter estimate variances. 

Table 2.13 Parameter Estimates and standard errors from simulated multiple imputation data set.

Missing and censored data should be handled exactly as in the case of linear regression. The analyst can use complete case analysis, naive substitution, conditional mean substitution, maximum likelihood, or multiple imputation. The same advantages and disadvantages for these techniques that were present with linear regression apply to nonlinear regression. 

The results of the simulation are presented in Tables 8.6-8.10. No difference was observed between any of the imputation methods. Each of the imputation methods resulted in mean parameter estimates that could not be distinguished from the case where no data were missing. Even mean imputation, which is frequently criticized in the literature, did well. A tendency for the variability of the parameter estimates to increase as the percent missing data increased was observed across all methods, although little difference was observed in the variability of the parameter estimates across methods. If anything, multiple imputation performed the worst as the variability in some of the variance components was large compared to the other methods. 

Table 8.10 Results from missing data analysis using multiple imputation.

Allison, P.D. Multiple imputation for missing data A cautionary tale. Sociological Methods and Research 2000 28 301-309. 

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