Multiple Imputation of Missing Data

Frequently, statistical analyses of survey data deal with missing data by dropping any observations with missing covariates, a "complete case analysis" (Marshall, Billingham, & Bryan, 2009). However, this approach implicitly assumes that the missing data is "missing completely at random" (MCAR); that missing values are not systematically dierent from complete-case values and, therefore, inference based only on complete cases would not dier from inference based on a

hypothetical complete dataset. One alternative to this approach is to instead

assume that data is missing at random (MAR); that the likelihood of an observation being missing is related to observed data. Assuming MAR, single imputation

approaches use relationships between variables to predict and ll in missing values, leaving observed values unchanged. Statistical analysis can then be carried out using the full dataset, without dropping any observations. However, this analysis will not capture the uncertainty in parameter estimates stemming from uncertainty in imputed values, i.e. the reported standard errors will be underestimates.

Alternatively, multiple imputation approaches involve creating many imputed datasets, where each imputed value is drawn from a distribution conditional on the other values in that observation. This results in multiple datasets, each with missing values lled in with dierent draws from their specic conditional distribution. Observed variable values remain the same. Therefore, the variance in an imputed value over datasets represents the uncertainty in predicting that value from other variables in the dataset. Once multiple complete datasets have been created,

statistical analysis is carried out on each, and inferential statistics calculated. These are then combined, nally producing statistical estimates that (to the extent that the MAR assumption holds) are representative of all observations in the dataset, as well as the uncertainty incorporated from the missing value imputation (Marshall et al., 2009). This is the approach used here.

Several dierent methods for multiple imputation exist. In this analysis I used the R package MICE, which implements the fully conditional specication (FCS) method, alternatively known as multivariate imputation by chained equations (MICE). This method is less theoretically elegant than the other primary multiple imputation option, Schafer's joint model, which produces a parametric joint distribution of all variables in the dataset (van Buuren & Groothuis-Oudshoorn, 2011). However, FCS allows both non-continuous variables, including ordered and

binary data, and specication of deterministic relationships between variables (van Buuren & Groothuis-Oudshoorn, 2011). The rst step in FCS is user specication of imputation models for each variable. For example, for a continuous variable one might specify a simple linear regression model. Then, using a rst imputation of "starting values" for missing values (here created by the MICE software), it cycles over variables, calculating and then drawing from a conditional distribution for each imputation. Each time all the missing values are imputed is considered one iteration of the multiple imputation algorithm. After sucient iterations, the imputed values are reported. This process is then repeated to produce multiple imputed datasets. Inferential statistics from models t to imputed datasets are combined according to Rubin's rules (D. B. Rubin, 1987), which specify that the nal estimate of a

regression coecient ˆβ∗ _{is simply the average of the coecients from each}

imputation ˆβm_{, where m is the imputation number and M is the total number of}

imputations: ˆ β∗ = Σ M m=1βˆm M

The overall variance of ˆβ∗ _{is a combination of the within imputation variance W}

and the between imputation variance B, as specied below. Um _{is the variance}

estimate of ˆβm _{from regression m.}

V = W + M + 1 M B W = Σ M m=1Um M B = Σ M m=1( ˆβm− ˆβ ∗₎2 M − 1

missingness are shown in Table B.3 in Appendix B. The most frequently missing variable is income, at about 11%, with social and scal conservative level close behind at 7-8%. These respondents are likely to be systematically dierent from others (Whitehead, 1994), a problem which multiple imputation will help overcome. The data have both numeric and categorical variables; the numeric variables are modelled using predictive mean matching, and the binary variables are modelled using logistic regression. I specify 20 iterations for each imputed dataset, as

recommended by van Buuren and Groothis-Oudshoorn (2011). I create a total of six imputed datasets, a number that is within Schafer's (1999) recommendations. I then run each of my six regressions (for each of three forest management settings, one regression with only demographics, and one with demographics and activities) on each of six imputed datasets. For each regression, I combine the results in accordance with Rubin's (1987) rules to get nal estimates for coecients and standard deviations. I also conduct these regressions on the dataset using only complete case observations to demonstrate the sensitivity of these results to multiple imputation. The results of this sensitivity analysis are included in Appendix B, Tables B.4 and B.5.

I chose not to impute information about activities respondents participate in. This is because almost every respondent who skipped any questions about activity participation skipped all questions about activities. This meant that for those respondents, I had very little information about activities, and imputations therefore seemed less likely to reect the truth. This meant that regressions

including activity information will have approximately 15% fewer observations, and, therefore, reduced power for inference. Additionally, because it would be

inappropriate to impute values of the dependent variables, the number of observations was somewhat reduced (10-13% fewer observations), even for regressions with only demographic covariates.