Required background

• Missing data
• Multiple imputation
• Biplot visualisation

Missing data

Missing data mechanisms (MDMs):

• Missing Completely At Random (MCAR): Missing values are independent of observed and missing observations.

• Missing At Random (MAR): Missing values depend on observed values, but not on missing observations.

• Missing Not At Random (MNAR): Missing values depend on both observed and missing observations.

Focus placed on MAR scenarios - assumed to be the standard occurrence in practice.

Multiple imputation (MI)

• Each missing value is replaced with multiple plausible values.
• Result in multiple completed data sets for standard complete-case analysis.
• Estimates are combined using Rubin’s rules (Rubin 1987).
• Realistic representation of variation due to distribution of multiple responses per missing value.

MI strategies

Two general approaches to impute missing values in multiple variables:

• Joint modelling (JM): The same imputation model is used for all variables.

  • Multiple imputation with multiple correspondence analysis (MIMCA)

  • Multilevel joint modelling multiple imputation (jomo)

  • Dirichlet process mixture of products of multinomial distribution model (DPMPM)

• Fully conditional specification (FCS) / sequential regression / chained equations: Imputation models are specified per variable that is conditioned on the other variables.

  • Multivariate imputation by chained equations (mice).

Biplot visualisation

• Multivariate categorical data visualisation.
• Representation of two modes of data (samples and variables) in a single display.
• Multiple Correspondence Analysis (MCA) biplots.

If \(\mathbf{X}\) is categorical with \(n\) rows (samples) and \(p\) columns (variables) with a total of \(q\) category levels:

\[\mathbf{R}^{-\frac{1}{2}}\mathbf{G}\mathbf{C}^{-\frac{1}{2}} = \mathbf{U}\mathbf{ \Sigma}\mathbf{V}^\prime\]

• \(\mathbf{G}\Rightarrow\) indicator matrix of \(\mathbf{X}\) with \(n\) samples and \(q\) columns.

• \(\mathbf{R}^{-\frac{1}{2}}\),\(\mathbf{C}^{-\frac{1}{2}}\Rightarrow\) diagonal matrix of row and column weights of \(\mathbf{G}\).

Generally, plot the first two columns of:

• \(\mathbf{U\Sigma}\) (principal coordinates) for the sample coordinates and

• the first two columns of \(\mathbf{V}\) (standard coordinates) for the category level point coordinates.