Editing Imputation (statistics) (section)

===Mean substitution===
Another imputation technique involves replacing any missing value with the mean of that variable for all other cases, which has the benefit of not changing the sample mean for that variable. However, mean imputation attenuates any correlations involving the variable(s) that are imputed. This is because, in cases with imputation, there is guaranteed to be no relationship between the imputed variable and any other measured variables. Thus, mean imputation has some attractive properties for univariate analysis but becomes problematic for multivariate analysis.

Mean imputation can be carried out within classes (e.g. categories such as gender), and can be expressed as <math>\hat{y}_i = \bar{y}_h</math> where <math>\hat{y}_i </math> is the imputed value for record <math>i</math> and <math>\bar{y}_h</math> is the sample mean of respondent data within some class <math>h</math>. This is a special case of generalized regression imputation:

<math display="block">
\hat{y}_{mi} = b_{r0} + \sum_j b_{rj} z_{mij} + \hat{e}_{mi}
</math>

Here the values <math>b_{r0}, b_{rj}</math> are estimated from regressing <math>y</math> on <math>x</math> in non-imputed data, <math>z</math> is a [[dummy variable (statistics)|dummy variable]] for class membership, and data are split into respondent (<math>r</math>) and missing (<math>m</math>).<ref>{{cite journal | last1 = Kalton | first1 = Graham | title = The treatment of missing survey data | journal = Survey Methodology | volume = 12 | year = 1986 | pages = 1–16}}</ref><ref>{{cite journal | last1 = Kalton |first1 = Graham | first2 = Daniel | last2 = Kasprzyk | title = Imputing for missing survey responses | journal = Proceedings of the Section on Survey Research Methods | publisher = [[American Statistical Association]] | volume = 22 | year = 1982 |s2cid = 195855359 | url = https://pdfs.semanticscholar.org/58f9/8fcc52333348a63b9e6dd5fabbdcc6fefe0e.pdf | archive-url = https://web.archive.org/web/20200212025249/https://pdfs.semanticscholar.org/58f9/8fcc52333348a63b9e6dd5fabbdcc6fefe0e.pdf | url-status = dead | archive-date = 2020-02-12 }}</ref>