An alternative procedure for imputing missing data based on principal components analysis

In general, it is always advisable to examine the socio-demographic profiles of cases with missing data very carefully. Furthermore, an anomalous incidence of missing information may indicate the emergence of problems during the data collection (difficulty in understanding the questions encountered by the interviewees, inadequate administration of the interviews by the interviewers, etc.).

In Italy this situation is typical of all surveys investigating voting intentions: nearly always the non-responses reach, and at time even surpass, 50 % of the sample.

The literature dealing with missing data substitution is ample and rapidly evolving. For further details see, among others, Enders (2010), Holenberghs and Kenward (2007), Chantala and Suchindran (2003), Akritas et al. (2002), Little and Rubin (2002), Allison (2001), Huisman et al. (1998), Little (1997) and Little and Schenker (1994).

When, due to limites economic and/or time resources, it is not possible to apply substitution, the so-called ‘mortality’ of the sample is compensated for by recourse to opportune weighting techniques used to restore the original sample size. As these operation lead to serious problems, the researcher must apply them with the utmost caution (for a critique of weighting techniques see Di Franco 2010).

Accordingly, the entire sample is divided into a number of sub-samples on the basis of socio-demographic variables (usually age and educational level) used as stratifying criteria, and each missing value is replaced by the central value of the class it belongs to. Through this procedure, one seeks to take into account the potential differences between the cases showing missing data across the different classes. For instance, it is possible that the central tendency measures of variables constructed on the basis of some opinion questions vary according to the age or educational level of the respondants. Throught this expedient, it should be possible, to improve the quality of each of the substituted data and, at the same time, limit the size of the shrinkage of variance. To avoid the reduction of variance, a random procedure has also been suggested. This may be used with any kind of variables and involves substituting each missing value with a different random value. These random values should be taken from the same socio-demographic class of the case, and should be chosen in such a way that not all the values have the same probability, but probabilities which are proportional to the frequencies of the cases which present data for that variable. This procedure leaves unaltered the original central value, the variance, and the distribution of cases both for the complete sample and the sub-types identified. Clearly, it is not claimed that the substitution of missing data for the single case comes close to the actual state on that variable.

In the case of categorical missing data imputation through regression may be carried out through logistic regression analysis (Di Franco 2011a). For nominal variables, substitution through hot-deck or cold-deck imputation is more appropriate.

The N versions of the complete data matrix are analysed using standard statistical techniques and the results combined using simple rules in order to reach single joint estimates, which formally incorporate the intrinsic uncertainty of the missing data. Therefore, the results of the estimates are averages computed on the N matrices of complete data.

The data used in the research project was drawn from European and international data banks (Eurostat, The World Bank, ITU, etc.). Some referred to structural characteristics of the countries; others were drawn from surveys regarding the use of computers as well as access to and use of Internet (Di Franco 2011b).

For further details concerning the method used to construct the index see Di Franco (2011b).

For an in-depth discussion of PCA (see Di Franco and Marradi 2003).