• Ongoing investigation: simulation protocol (Nienkemper-Swanepoel (2025))
  • Uniform distribution
  • 1000 samples and 5 variables (categorised with 2 to 5 levels)
• Limited response profiles

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.

Only MAR considered for this presentation - assumed to be the standard occurrence in practice.

• MAR used to delete 20% observations

Quick overview of 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.

Two general approaches to impute missing values in multiple variables:

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

  • missMDA - Multiple imputation with multiple correspondence analysis (Josse and Husson (2016))

  • jomo - Multilevel joint modelling multiple imputation (Quartagno and Carpenter (2023))

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

  • mice - Multivariate imputation by chained equations, more flexible (van Buuren and Groothuis-Oudshoorn (2011)).
  • mi - Multivariate imputation by chained equations, strict Bayesian framework (Gelman and Hill (2011)).
    
    [R packages]

• 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.

• Construct MCA biplots for each multiple imputation.
• Iteratively align the coordinates of the biplots to an updated target (i.e. average coordinates) until convergence.
  • Translation, reflection, rotation and scaling.
• Average over the aligned coordinates to obtain a unified coordinate set for the GPAbin biplot.
• Application: GPAbin (Nienkemper-Swanepoel (2026)).

• Random noise added to the numerical response values representing the category levels.
• The main patterns and groupings will still be visible.
• Overplotting and emphasised artificial clusters are avoided.

Consists of 3 components (Cook and Laa (2026); Wickham et al. (2011)):

• Data - multiple imputed data sets

• Tour path - linear projections presented in a smooth sequence using geodesic interpolation

  • Grand tour: random linear projections are obtained

  • Geodesic interpolation: shortest path between planes

• Display method - scatterplot

tourr (Wickham et al. (2011)) & detourr (Hart and Wang (2025))