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\begin{center}
{\Large Revisiting the Political Life-Cycle Model:
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Later Maturation and Turnout Decline Among Young Adults}
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Kaat Smets\\Royal Holloway, University of London\\\\
e-mail: \texttt{kaat.smets@royalholloway.ac.uk}
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{\it Published in European Political Science Review}
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\section*{Online Appendix C}

\singlespacing
Handling missing data raises both conceptual difficulties and computational challenges. The default way in which most statistical packages approach the missing values problem -- through listwise deletion or complete case analyses -- can yield bias, is inefficient, and is therefore considered unreliable. In general, two approaches to handling missing data are recommended in the literature: maximum likelihood (ML) and multiple imputation (MI) \citep{Schafer:2002, Raghunathan:2004}.

There are different types of missing data. Unit non-response occurs when the entire data collection procedure fails because respondents e.g. refuse to participate. Item non-response occurs when data are partially missing because a respondent e.g. did not answer all questions in the survey. Although part of the missing data problem in this paper is due to item non-response, the more pressing problem is that  certain variables were not included in certain waves of the British Election Studies. Using listwise deletion would cause the sample size to vary considerably in the analyses presented. This renders comparison of results between the models difficult. 

Table~\ref{tab:missingdata} below lists all the variables used in the analyses of this paper, the ratio of missing to valid answers, the percentage of missing values\footnote{Ratio's and percentages are calculated based on the whole sample rather than the subset of young adults aged 35 years or less. The reason for this is that the average turnout of older citizens is part of the model and missing values thus need to be imputed for all respondents.},and the main reason for the lack of data. If no reason is given, data are missing due to item non-response. The variable with the largest percentage of missing values is residential stability, which was missing in 27.5\% of the cases. 


\begin{table}[H]
\centering
\scriptsize
\singlespacing
\begin{tabular}{lccl}\hline
{\bf variable} & {\bf missing/valid } &{\bf percentage of} & {\bf reason for } \\ 
{\bf name} & {\bf value ratio} &{\bf missing values} & {\bf missingness} \\ \hline%
turnout & 0/33737  & 0\% &  \\ 
age & 0/33737  & 0\% &  \\ 
left education & 203/33737 & 0.6\% &  \\ 
married & 29/33737 & 0\% &  \\ 
has child(ren) & 6915/33737 & 20.5\% & Not included in 1974 (feb), 1992 \\ 
home ownership & 2000/33737 & 5.9\% & Not included in 1966 \\ 
residential stability & 9264/33737 & 27.5\% & Not included in 1983, 1992; 1/2 sample in 2001 \\ 
works & 57/33737 & 0\% &  \\ 
gender & 0/33737 & 0\% &  \\ 
educational level & 897/33737 & 2.7\% &  \\ 
union membership & 3014/33737 & 8.9\% & Not included in 1966 \\ 
pid strength & 1656/33737 & 4.9\% &  \\ 
voted in previous elections & 4690/33737 & 13.9\% & Only posed to 1/6 of the sample in 2001 \\
average turnout older voters & 0/33737 & 0\% & \\  \hline%
\end{tabular}
\caption{Missing values of modelled variables}
\label{tab:missingdata}
\end{table}

The percentage of missing values in the sample used is quite high for certain variables. This calls for an imputation method with a high level of efficiency. Suppose $x$ is a real value and $\hat{x}$ an estimated value. While treating missing data in a sample we want to make sure that the bias between estimated and the true values is small. Moreover, we want the variance and standard deviation of the estimated values to be small. Bias and variance are often combined into one measure called mean square error, which is the squared distance between the estimated and the real values over repeated samples: ($\hat{x}$-$x)^{2}$. The mean square error is equal to the squared bias plus the variance. Bias, variance, and the mean error describe the behaviour of an estimate. However, we also want to be confident about the measures of uncertainty that we report and estimate the true $x$ with a probability of a certain predefined rate \citep[p.~149]{Schafer:2002}.

Multiple imputation (MI) is a method for handling missing data that solves the problem of uncertainty that many single imputation methods face. MI replaces each missing value by a list of $m\textgreater1$ simulated values and as such produces $m$ plausible alternative versions of the complete data set. Each of the $m$ data sets is estimated in the same fashion by a complete data method. Estimates of parameters of interest are subsequently averaged to give a single estimate. Standard errors are computed according to the `Rubin rules' (see below), allowing for between- and within-imputation components of variation in the parameter estimates.

MI does not need many rounds of estimation to reach a high level of efficiency. \citet{Rubin:1987} developed with the following way to calculate the efficiency of an estimate based on an $m$ number of imputation (see equation~\ref{eq:efficiencymi}):

\begin{equation} \label{eq:efficiencymi}
eff = (1 + \lambda/m)^{-1}
\end{equation}

\noindent where the efficiency is a function of the rate of the missing information ($\lambda$) and the number of imputations ($m$). For example, with 27.5\% of missing information (as is the case with residential stability), $m$= 5 imputations will yield results that are 100/(1 + .055) = 94.8\% efficient. A rule of thumb for selection of the number of imputation rounds is that the confidence coefficient for the worst-case parameter (in this case residential stability) should  be at least 95\% \citep[p.~239]{Royston:2004}. This means that in this particular case more than five rounds of imputation are desirable. Six rounds of imputation yield an efficiency of 95.6\% for the residential stability variable. Therefore, $m$ is set to six for the imputation procedure used to handle missing data for the analyses in this paper. The \texttt{ice} command in Stata is used to execute  the multiple imputation process \citep[see][]{Royston:2004, Royston:2005a, Royston:2005b}. 

As mentioned above, multiple imputation creates a small number of data sets (in this case six), each of which has the missing values suitably imputed. The next step is to analyze each complete data set independently and summarize the results of these independent estimations. Coefficients are simply averaged. Summarizing the standard errors requires a bit more work (see equation~\ref{eq:sesmi} taken from \citet{Rubin:1987}):

%\begin{equation} \label{eq:meanerrormi}
%\bar{e} = \frac{1}{M}\sum_{m=1}^{m}e_{m}
%\end{equation}

\begin{equation} \label{eq:sesmi}
s = \sqrt{\bar{u}_{m} + \frac{m + 1} {m} b_{m}}
\end{equation}

\noindent where $\bar{u}_{m}$ is the mean of the standard error's, and $b_{m}$ is the variance of the estimates across the imputations. The \texttt{micombine} command in Stata combines the estimates from the $m$ analyses using Rubin's rules \citep{Royston:2004, Royston:2005a, Royston:2005b}.

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