Multiple Imputation Models in the 2002 Environmental Sustainability Index

Missing data arise in many situations and pose more than a technical problem to the analyst. Software applications often require complete datasets, but more importantly, incomplete or missing observations may impact on the validity of the statistical analysis. Ad-hoc solutions, such as case or listwise deletion, mean imputation, and simple regression methods can lead to severe bias if the population of nonrespondents/missing items differs systematically from the respondents/observed items for the variable under investigation. In other word, ad-hoc imputation models yield valid results if observations are missing completely at random (MCAR). The concept of multiply imputation missing data (Rubin 1987, Little and Rubin 1987) has stimulated the development of methods for dealing with missing observations in longitudinal studies (Laird 1988), intermittent non-response and item non-response (Robins 1997, Little and Wang 1996, Molenberghs et al. 1998, Glynn et al. 1993). Schafer (1997) shows that under the assumption that the data are missing at random (MAR), the imputation bias of the abovementioned ad-hoc methods can be eliminated. The most notable advantage of multiple imputations is that the uncertainty inherent in missing values is accounted for, allowing the calculation of standard errors and confidence intervals. In this paper we apply multiple imputations to a subset of the data used for the Environmental Sustainability Index (ESI), a composite index measuring national progress towards environmental sustainability. The ESI is an initiative of the World Economic Forum and further developed by the Yale Center for Environmental Policy and Law (YCELP) and the Center for International Earth Science Information Network (CIESIN). Systematic missing data generating processes cannot be ruled out in the ESI data. For example, missingness correlates with national wealth. According to the World Bank and IMF classification of countries by income for 1998 GDP/capita figures1, low income countries (10 in the ESI dataset) have 29% and lower middle-income countries (50) 26% missing data but upper middle-income countries (49) and high-income countries (33) have only 15% and 9% incomplete observations, respectively. This raises doubts that the data are missing completely at random and hence restricts the use of ad-hoc imputation methods. The paper investigates the application of two multiple imputation models with less restrictive assumptions on the non-response mechanisms. The first performs Bayesian Markov Chain Monte Carlo simulations (MCMC) under a MAR process and the second uses a stochastic non-ignorable censoring process for imputations using a linear regression model as described by Greenlees et al. in an application to income data (1982). The second model is extended to allow for multiple imputations. The paper proceeds with an introduction of the conceptual framework of the ESI. It then describes the criteria used for the selection of the variable subset and the missing data structure of the dataset under investigation. This is followed by the development of the MAR and non-ignorable censoring models and a concluding summary of the results.