Handling missing data

Anyone who does data analysis will eventually run into the problem of missing data, either the dependent variable is missing or one or more of the independent variables is missing. The problem of handling missing data is far too complex to cover it in its entirety within this book and many excellent books are available on the subject for readers who wish greater detail. These include books by Allison (2002), Little and Rubin (2002), and Schafer (1997). 

Methods for Handling Missing Data Missing Dependent Variables 

2 The intent to treat principle essentially states that all patients are analyzed according to the treatment they were randomized to, irrespective of the treatment they actually received. Hence, a patient is included in the analysis even if that patient never received the treatment. 

If the covariates show a monotone pattern of missingness (Fig. 2.13) then the imputation procedure can be done sequentially. For instance, suppose that xj, X2, X3, and X4 are the co variates that exhibit monotone missingness and that x and X2 have no missing data. In the first step, X3 would be imputed based on the regression of x3 against Y, xi, and X2. Then given imputed values for X3, X4 would be imputed using the 

Multi-way Analysis With Applications in the Chemical Sciences 

In most modeling situations it is not a priori known how many components to choose. The main difficulty in using NIPALS is therefore to be able to detect whether the found model is real, or, as in this case, just an artifact reflecting the position of the missing data rather than the variation in present data. 

A natural way to handle missing data is to fit the model only to the nonmissing data. Thus, only optimizing the loss function over nonmissing elements. The loss function for 

Louwerse et al. [1999] also compared the imputation approach to another method for handling missing data specifically designed for cross-validation. 

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One very simplistic way of handling missing data is to remove those patients with missing data from the analysis in a complete cases analysis or completers analysis. By definition this will be a per-protocol analysis which will omit all patients who do not provide a measure on the primary endpoint and will of course be subject to bias. Such an analysis may well be acceptable in an exploratory setting where we may be looking to get some idea of the treatment effect if every subject were to follow the protocol perfectly, but it would not be acceptable in a confirmatory setting as a primary analysis. 

In all cases and particularly where the extent of missing data is substantial, several analyses will usually be undertaken to assess the sensitivity of the conclusions to the method used to handle missing data. If the conclusions are fairly consistent across these different analyses then we are in a good position. If, however, our conclusions are seen to change, or to depend heavily on the method used for dealing with missing data, then the validity of those conclusions will be drawn into question. 

How should we handle missing data what alternatives exist to LOCF ... 

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. 

Adequate response rate and methodology for handling missing data... 

Figure 6.7. Left scores and loadings from a five-component least squares PCA model. Right similar result using the NIPALS method for handling missing data.

In order to obtain independent model estimates in component models a different cross-validation technique has to be used. Instead of leaving out a complete sample/row, it is possible to leave out only one (or a few) elements. Using an algorithm that handles missing data, it is possible to estimate the relevant component model without the left-out element. The estimate of this element is obtained from the model of the whole data array (tirPjr in case of leaving out element Xij in PCA) where there is no dependence between the left-out element and the model. This is the basis for the cross-validation routines in several papers [Louwerse et al. 1999],... 

A special feature of the analysis of de Ligny et al. is that they have used an expectation maximization algorithm for handling missing data (see Chapter 6). Using the calculated model parameters, it is possible to provide predictions of the missing values. Moreover, error variances of these predictions are provided based on local linearizations of the model around the parameters. 

A parametric method for handling missing data is maximum likelihood. Recall that in linear regression maximum likelihood maximizes the likelihood function L(.)... 

In the last chapter, the theory behind nonlinear mixed effects models was introduced. In this chapter, practical issues related to nonlinear mixed effects modeling will be introduced. Due to space considerations not all topics will be given the coverage they deserve, e.g., handling missing data. What is intended is a broad coverage of problems and issues routinely encountered in actual population pharmacokinetics (PopPK) analyses. The reader is referred to the original source material and references for further details. 

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