Illustration of missing data handling technique generated from hepatitis C induced hepatocellular carcinoma cohort study

4.1 Conventional missing value imputation strategy

The calculated mean value is defined as straightforward to replace the missing value. This is a quick step to fix the missing data problem.

Different missing data handling techniques known as the last observation carried forward (LOCF) (Laird, 1988; Roy et al., 2018) and baseline observation carried forward (BOCF) (Woolley et al., 2009). These are ad hoc imputation methods for longitudinal data. The previously observed data values as an alternative to missing data are considered. It is a procedure that the idea is to take the already found amount as a replacement for the missing data. When multiple values are missing in succession, the method searches for the last observed value.

A complete case analysis can result in biased estimates, inefficient or unrealistic standard errors. It only analyzes subjects with available data on each variable. Even though the study has simplicity, it reduces the statistical power and doesn’t use all the information. Listwise deletion and Pairwise deletion will not provide any bias in parameter estimates if the data are MCAR (Roy et al., 2018). If the data are MAR (not MCAR), then these methods will produce biased parameter estimates. Simple imputation methods such as mean imputation provide biased estimates (Woolley et al., 2009). Instead of imputing a single value for each missing value, a multiple imputation technique substitutes each missing observation.These multiply imputed data sets are then analyzed using standard statistical procedures and pooling the results from these analyses (Liu-Seifert et al., 2010; Allison, 2001).

A Model could be either the logistic or linear model decided by the type of response variable (Glasser, 1964).Thereafter, predictions for the incomplete cases are inversely calculated from the fitted model. It serves as the replacements for the missing data.

Stochastic regression is a specific step of the regression imputation technique. It used through exploring the correlation bias. The noise of the predictions is incorporated by stochastics regression modelling (Wallace et al., 2010).

4.2 Software packages for missing value imputation

The indicator method also comes with the regression method. If the covariate is missing, then each missing value can be replaced by zero and thereafter extends by the regression model with the response indicator. Separately, each incomplete observed covariate can be imputed. The ”mice’ package is useful to work on indicator method (Buuren and Groothuis-Oudshoorn, 2010).

The data set has an arbitrary missing data pattern, and it is assumed that the missing data are missing at random (MAR), that is, the probability that an observation is missing may depend on $Y_{\mathit{obs}}$ but not on $Y_{\mathit{miss}}$ . If any liver cirrhosis subject has the missing observation, the reason for this missing observation might only depend on their observed meld score observations and not on unobserved meld score observations. The purpose of this analysis was to compare the estimates from different imputation techniques namely Fully conditional specification (FCS) regression, FCS predictive mean matching, Markov chain Monte Carlo (MCMC) and expectation- maximization (EM) algorithm. The PROC MI procedure available in Statistical analysis software (SAS) was used to implement these techniques with statements such as FCS REGRESSION, FCS REGPMM, MCMC and EM with nimpute = 50. The chain = multiple had been used with method = MCMC for each imputation, and the option INITIAL = EM was used. The means and standard deviations from the available cases were the initial estimates for the EM algorithm using proc MI. The correlations are set to zero. The resulting estimates are used to initiate the MCMC process. Joint modelling, time-dependent cox model, and Generalized estimating equations were carried out to examine the predictors of death in the presence of time-dependent covariate meld score and baseline characteristics such as age, gender, Alkaline Phosphates, SGPR, and SGOT. The results of joint modelling analysis on 50 imputed datasets were combined to derive an overall effect in each imputation method: i.e., SAS procedure proc mianalyze was used to connect the parameter estimates in each imputation method. Now we illustrated different steps with multiple imputations. Especially, measures that are compatible with different study design.

4.3 Missing value imputation in high-dimensional data

Data generated with several thousands of covariates and the minimal sample size is known as high dimensional data. Generally, genomics data comes as high dimensional data. The data reduction is the main challenge for high dimensional data analysis. It is a task to reduce several thousand covariates to a minimal number of covariates. There is a different statistical technique to work with the data reduction mechanism. The widely used one is the principal component analysis(PCA). Especially continuous gene expression covariates can be reduced by PCA. The multiple correspondence analysis (MCA) reduced the categorical covariates data. Simultaneously the continuous and categorical variables are reduced by multiple factor analysis (MFA) (Lin, 2010). Suppose the number of variables are $X=\left[x_{\mathit{ik}}\right]$ . The parameters are contained with p variables of n individuals. The PCA helped to generate ${\widehat{X}}_S\left(\mathit{np}\right)$ , by specific approximation of X rank S through minimizing the least square. $\left|\right|X-\widehat{X}|{|}^2$ . It works through singular value decomposition of X as

Conventionally, PCA is used as a data reduction technique. The only complete case analysis in possible to work with PCA. PCA works by maximizing the variance by individual-level diversity. Simultaneously, the Euclidean distance between the individual is derived. It helps to reduce the matrix size $X_{\mathit{np}}$ with n individuals and p variables.

The PCA is performed by an orthogonal transformation to identify principal components. It worked by equal a linear combination of the gene expression levels and are linearly uncorrelated with each other (Roy et al., 2018). Sometimes, gene expression measurement for some individuals becomes missing. It is difficult to discard those individual’s observations due to missing information. The challenge is to have PCA compatible with missing data and perform PCA by imputing missing data. The ”missMDA” package available in R is suitable to solve the challenge. It helps to understand data visualization (Figure1). But singular value decomposition (SVD) plays a crucial role in all these data reduction techniques like PCA, MCA, and factor analysis. Now the arising of missing data are common in high dimensional data analysis. But discarding the missing data and perform analysis on complete case analysis(CCA) is the standard practice. Fortunately, the missMDA package available in the R system is useful to handle with missing data obtained from high dimensional data (Ilin and Raiko, 2010; Josse and Husson, 2016). It is compatible to work with PCA. This package imputes the missing data. Continuous, categorical, and mixed continuous and categorical can be attributed by the missMDA package. There are PCA can be performed into the imputed data. However, it is to be noted that the variance of the estimator obtained by PCA may be underestimated. The variability due to the imputation of missing values is not taken into account through this imputation. We illustrated the missMDA package application in handling missing data section in Table 2.

5.1 Regression method

Sometimes it gets difficult to understand about the actual probability of missing data. It requires to have missing data process for each visit by logistics regression having response variable with each data that is observed or not. Similarly, the independent variables can be linked with the missing data. The regression model works as a suitable approach in this context (Lee et al., 2018.) In this method, missing observations for each variable is imputed using the posterior predictive distribution of the parameters. Let, the continuous variable $Y_j$ , is the response observation of $j^{\mathit{th}}$ patient with missing observations, and the model is defined as

The variable is $Y_j$ and the covariates are $X_1,X_2,\dots \dots ,X_K$ .. The fitted model includes the regression parameter estimates ${\widehat{\beta }}_i=({\widehat{\beta }}_0,{\widehat{\beta }}_1,\dots \dots ,{\widehat{\beta }}_k)$ and the associated covariance matrix ${\widehat{\sigma }}_j^2{V}_j$ , where $V_j$ is the usual $X^{\prime }X$ inverse matrix derived from the intercept and covariates $X_1,X_2,\dots ,X_k$ . Where ${\widehat{\sigma }}_j^2$ is the estimated variance of $j^{\mathrm{th}}$ patient. The imputation model is defined as follows:

Where ${\beta }_0$ is the intercept and ${\beta }_1,{\beta }_2,\dots .,{\beta }_{11}$ are regression coefficients. Multiple imputations in SAS involves three procedures. The first one is proc MI, in which the user writes the imputation model to be used and the number of imputed datasets to be produced (use FCS REG statement in proc MI and nimpute = 50). The second procedure runs the analytic model of interest (for example, proc genmod) within each of the imputed datasets. The third procedure is proc mianalyze, which pools all the estimates (coefficients and standard errors) across all the imputed datasets and produce pooled parameter estimates for the model of interest.

5.2 Predictive mean matching method

Multiple imputations is a commonly used method for handling incomplete covariates as it can provide valid inference when data are missing at random. Imputation by predictive mean matching (PMM) borrows an observed value to obtain the imputed values. It works with parametric imputation with greater robustness. It depends on being able to correctly specify the parametric model used to impute missing values, which may be difficult in many realistic settings (Morris et al., 2014). The variables with missing observations are imputed using predictive mean matching method. A linear regression model is estimated and generated a new set of coefficients $\widehat{\beta }=({\widehat{\beta }}_0,{\widehat{\beta }}_1,\dots ,{\widehat{\beta }}_k)$ randomly from the posterior predictive distribution of $\widehat{\beta }=({\widehat{\beta }}_0,{\widehat{\beta }}_1,\dots ,{\widehat{\beta }}_k)$ . Predicted observations for observed data are calculated from $\widehat{\beta }=({\widehat{\beta }}_0,{\widehat{\beta }}_1,\dots ,{\widehat{\beta }}_k)$ where as the predicted observations for unobserved data are calculated from $\widehat{\beta }=({\widehat{\beta }}_0,{\widehat{\beta }}_1,\dots ,{\widehat{\beta }}_k)$ . For each subject with a missing observation, found the closest predicted observations among the subjects with observed observation. Randomly selected one of these closest predicted observations and imputed the observed observation for the closest predicted observation. This method may be more appropriate than the regression method if there is a violation of normality assumption (Lin, 2010). In SAS, FCS REGPMM statement to be used in proc MI with nimpute = 50.

5.3 EM algorithm

The EM algorithm is an iterative procedure that finds the maximum likelihood estimate of the parameter vector by replicating the following steps: (I). The E-step calculates the conditional expectation of the complete-data log-likelihood given the observed data and the parameter estimates and (II) The M-step finds the parameter estimates to maximize the complete-data log-likelihood from the E-step (Lin, 2010; McLachlan and Krishnan, 2008). In the EM process, the observed-data log-likelihood is non-decreasing at each iteration. For multivariate normal data, suppose there are groups with distinct missing data patterns. Then the observed-data log-likelihood being maximized can be expressed as

Where, $n_g$ is the number of observations in the $g^{\mathrm{th}}$ group, $y_{\mathit{ig}}$ is a vector of observed observations corresponding to observed variables, ${\mu }_g$ is the corresponding mean vector, and ${\mu }_g$ is the associated covariance matrix.

5.4 Markov Chain Monte Carlo (MCMC) Method

The MCMC simulation shifts the computer for experimental laboratory. It helps to control various conditions to obtain the outcomes (Carsey and Harden, 2013). It documented that the MCMC simulation works as a strong tool to perform statistical methods under different setting with violated assumptions. (Takahashi, 2017). The MCMC method is used to create pseudo random draws from multidimensional data and or else intractable probability distributions through Markov chains. Suppose the data follows the multivariate normal distribution, data augmentation is applied to Bayesian inference with missing data by replicating the following steps: (I) The missing observations for observation j is denoted as $Y_{\left(j\right(\mathrm{mis})}$ and the variables with observed observations by $Y_{\left(j\right(\mathrm{obs})}$ , then the I-step draws observations for $Y_{\left(j\right(\mathrm{mis})}$ from a conditional distribution $Y_{\left(j\right(\mathrm{mis})}$ given $Y_{\left(j\right(\mathrm{obs})}$ . (II) The P-step simulates the posterior population mean vector and covariance matrix of the complete sample estimates. These new estimates are used in the I-step. If lack of prior information about the parameters, a non-informative prior is used. We may use other informative priors as well. In SAS, MCMC statement to be used in proc MI with nimpute = 50.

6.1 Generalized estimating equation

The repeatedly measured observations are always correlated. The generalized estimating equation(GEE) is technique to deal with correlated observations. The GEE is well developed and established statistical methodology (Zeger et al., 1988). Suppose there is a sample of $i=1,\dots .,K$ independent multivariate measurments and defined as (Halekoh et al., 2006)

Now i shows the cluster of $n_i$ observations. The expectation $E(Y_{\mathit{it}}={\mu }_{\mathit{it}})$ are linked with p-dimensional regression vector $x_{\mathit{it}}$ by mean-link function (Halekoh et al., 2006)

Here, $\varphi$ is defined as scale parameter and $a_{\mathit{it}}=a\left({\mu }_{\mathit{it}}\right)$ is a known variance function (Halekoh et al., 2006). Now $R_i\left(\alpha \right)$ presented as a working correlation matrix completely described by the parameter vector $\alpha$ of length m. Let

The corresponding working covariance matrix of $Y_i$ . Further, the diagonal matrix is $A_i$ .

It can be estimated as $(\varphi ,\widehat{\alpha })$ as the solution of the equation

Finally, the covariance matrix can be formulated as $\sum ={\mathit{lim}}_{k\to \infty }K{\sum }_0^{-1}{\sum }_0^{-1}$ where

It help to solve work with correlated measurements by statistical modeling.

6.2 Time dependent cox regression

The cox model with time dependent covariates is defined as

The baseline hazard function defined as $h_0\left(t\right)$ .The measurement of covariates at time t is defined as $x\left(t\right)$ and regression coefficient is $\beta \left(t\right)$ . The covariates can be defined with different parts. The covariates may be time-independent or dependent. In our example, the prognostic biomarker is defined as time-dependent covariates. It is observed as a continuous variable. It is expected that the Prognostic biomarker will carry a time course trajectory over the follow-up period. The time-varying component helps to get an idea about covariates at baseline measurement. The regression parameter will change over time. The distributional assumption and test also play a crucial role to define covariates as prognostics marker or not. However, the distributional assumption may be avoided through the application of the conditional model by available information at time point $t=s$ .

6.3 Joint longitudinal modeling

Time to event analysis help us to identify covariates that are predictable for death. Now survival analysis can be extended by the inclusion of time-dependent covariates. It is expected that the covariates should be error-free. It is relatively easy to promote the application of joint longitudinal and time to event data modelling in clinical research due to the recent advancement of computational flexibility. This work aims to explore the relationship between the imputed longitudinal covariates and explore their impact of death. Only the application of a longitudinal model may raise some biased outcomes. Some recent work in the modelling serves to work with multiple time-dependent covariates to the multiple-endpoint outcome (Nath et al., 2016; Bhattacharjee et al., 2018; Bhattacharjee, 2019).

6.4 Predictive mean matching method

The variables with missing observations are imputed using predictive mean matching method. A linear regression model is estimated and generated a new set of coefficients $\widehat{\beta }=(\widehat{{\beta }_0},\widehat{{\beta }_1},\dots ..,\widehat{{\beta }_k})$ randomly from the posterior predictive distribution of $\widehat{\beta }=(\widehat{{\beta }_0},\widehat{{\beta }_1},\dots ..,\widehat{{\beta }_k})$ . Predicted observations for observed data are calculated from $\widehat{\beta }=(\widehat{{\beta }_0},\widehat{{\beta }_1},\dots ..,\widehat{{\beta }_k})$ whereas the predicted observations for unobserved data are calculated from $\widehat{\beta }=(\widehat{{\beta }_0},\widehat{{\beta }_1},\dots ..,\widehat{{\beta }_k})$ . For each subject with a missing observation, found the closest predicted observations among the subjects with observed observation. Randomly selected one of these closest predicted observations and imputed the observed observation for the closest predicted observation. This method may be more appropriate than the regression method if there is a violation of normality assumption (Lin, 2010).

6.5 Markov chain monte carlo (MCMC) Method

The MCMC method is used to create pseudo random draws from multidimensional data and or else intractable probability distributions through Markov chains (Ilin and Raiko, 2010). It is assumed that the multivariate normal distribution follows for the response varable, data augmentation is applied to Bayesian inference with missing data by replicating the following steps: (I) The missing observations for observation j is denoted as $Y_{\left(j\right(\mathit{mis}\left)\right)}$ and the variables with observed observations by $Y_{\left(j\right(\mathit{obs}\left)\right)}$ , then the I-step draws observations for $Y_{\left(j\right(\mathit{mis}\left)\right)}$ from a conditional distribution $Y_{\left(j\right(\mathit{mis}\left)\right)}$ given $Y_{\left(j\right(\mathit{obs}\left)\right)}$ . (II) The P-step simulates the posterior population mean vector and covariance matrix of the complete sample estimates.These new estimates are then used in the I-step. If the lack of prior information about the parameters, a non-informative prior can be used.