Clustering with missing data: which equivalent for Rubin’s rules?

Abstract

Multiple imputation (MI) is a popular method for dealing with missing values. However, the suitable way for applying clustering after MI remains unclear: how to pool partitions? How to assess the clustering instability when data are incomplete? By answering both questions, this paper proposed a complete view of clustering with missing data using MI. The problem of partitions pooling is here addressed using consensus clustering while, based on the bootstrap theory, we explain how to assess the instability related to observed and missing data. The new rules for pooling partitions and instability assessment are theoretically argued and extensively studied by simulation. Partitions pooling improves accuracy, while measuring instability with missing data enlarges the data analysis possibilities: it allows assessment of the dependence of the clustering to the imputation model, as well as a convenient way for choosing the number of clusters when data are incomplete, as illustrated on a real data set.

Appendix

1.1 Partitions pooling

See Tables 3 and 4.

Table 3 Accuracy of the clustering procedure under a MCAR mechanism: average adjusted rand index (mean) and interquartile range (IQ) over the ${S}=200$ generated data sets varying by the number of individuals (${n}$), the correlation between variables ($\rho$) and the proportion of missing values ($\tau$) for various imputation methods (JM-DP or FCS-RF), various consensus methods (NMF or SAOM)

Table 4 Accuracy of the clustering procedure under a MAR mechanism: average adjusted rand index (mean) and interquartile range (IQ) over the ${S}=200$ generated data sets varying by the number of individuals (${n}$), the correlation between variables ($\rho$) and the proportion of missing values ($\tau$) for various imputation methods (JM-DP or FCS-RF), various consensus methods (NMF or SAOM)

1.2 Instability pooling

See Tables 5 and 6.

Table 5 Instability of the clustering procedure under a MCAR mechanism: average within-instability ($\bar{U}$) average between-instability (B) and average total instability (T) over the ${S}=200$ generated data sets for various number of individuals (${n}$), correlation between variables ($\rho$) and proportion of missing values ($\tau$)

Table 6 Instability of the clustering procedure under a MAR mechanism: average within-instability ($\bar{U}$) average between-instability (B) and average total instability (T) over the ${S}=200$ generated data sets for various number of individuals (${n}$), correlation between variables ($\rho$) and proportion of missing values ($\tau$)

1.3 Influence of ${M}$

1.3.1 Accuracy

See Figs. 7, 8 and 9.

Fig. 7

Accuracy of the clustering procedure according to ${M}$: adjusted rand index over the ${S}=200$ generated data sets varying by the number of individuals (${n}$), the correlation between variables ($\rho$) and the proportion of missing values ($\tau$) generated under a MAR mechanism. Data sets are imputed by JM-DP varying by the number of imputed data sets (${M}$). For each data set, clustering is performed using k-means clustering and consensus clustering is performed using NMF

Fig. 8

Accuracy of the clustering procedure according to ${M}$: adjusted rand index over the ${S}=200$ generated data sets varying by the number of individuals (${n}$), the correlation between variables ($\rho$) and the proportion of missing values ($\tau$) generated under a MCAR mechanism. Data sets are imputed by FCS-RF varying by the number of imputed data sets (${M}$). For each data set, clustering is performed using k-means clustering and consensus clustering is performed using NMF

Fig. 9

Accuracy of the clustering procedure according to ${M}$: adjusted rand index over the ${S}=200$ generated data sets varying by the number of individuals (${n}$), the correlation between variables ($\rho$) and the proportion of missing values ($\tau$) generated under a MAR mechanism. Data sets are imputed by FCS-RF varying by the number of imputed data sets (${M}$). For each data set, clustering is performed using k-means clustering and consensus clustering is performed using NMF

1.3.2 Instability

See Figs. 10, 11 and 12.

Fig. 10

Instability according to ${M}$: total instability (T) over the ${S}=200$ generated data sets varying by the number of individuals (${n}$), the correlation between variables ($\rho$) and the proportion of missing values ($\tau$) generated under a MAR mechanism. Data sets are imputed by JM-DP varying by the number of imputed data sets (${M}$). For each data set, clustering is performed using k-means clustering and consensus clustering is performed using NMF

Fig. 11

Instability according to ${M}$: total instability (T) over the ${S}=200$ generated data sets varying by the number of individuals (${n}$), the correlation between variables ($\rho$) and the proportion of missing values ($\tau$) generated under a MCAR mechanism. Data sets are imputed by FCS-RF varying by the number of imputed data sets (${M}$). For each data set, clustering is performed using k-means clustering and consensus clustering is performed using NMF

Fig. 12

Instability according to ${M}$: total instability (T) over the ${S}=200$ generated data sets varying by the number of individuals (${n}$), the correlation between variables ($\rho$) and the proportion of missing values ($\tau$) generated under a MAR mechanism. Data sets are imputed by FCS-RF varying by the number of imputed data sets (${M}$). For each data set, clustering is performed using k-means clustering and consensus clustering is performed using NMF

1.4 Application

See Table 7.

Table 7 Animals data set