Descriptive Statistics
Suppose \(Y = (y_1, y_2, ..., y_n)^'\) is the \((n \times p)\) matrix of complete data
- which might not be fully observed
\(n_0\) is the number of observations fully observed and \(n_j\) is the number of observations with observed values for variable \(Y_j\)
With complete cases, the sample mean vector is \(\bar{y} = \frac{1}{n_0} \sum{y_i}\)
and the CSSCP matrix is \(\sum(y_i-\bar{y})(y_i - \bar{y})^'\) Where each summation is over the fully observed observations… The sample covariance matrix is \(S=\frac{1}{n_0 -1}\sum(y_i - \bar{y})(y_i - \bar{y})^{'}\) And is an unbiased estimate of the covariance matrix
The correlation matrix \(\mathbb{R}\)
- which contains the Pearson product-moment correlations of the variables
is derived by scaling corresponding covariance matrix \(\mathbb{R}=D^{-1}S D^{-1}\) Where \(D\) is a diagonal matrix whose diagonal elements are the square roots of the diagonal elements of \(S\) With available cases, the corrected sum of squares for variable \(Y_j\) is \(\sum(y_{ij}-\bar{y}_j)^2\) Where \(\bar{y}_j = \frac{1}{n_j}\sum y_{ji}\) is the sample mean
- and each summation is over observations with observed values for variable \(Y_j\)
The variance is \(s^2_{jj} = \frac{1}{n_j - 1} \sum(y_{ij} - \bar{y}_j)^2\) The correlations for available cases contain pairwise correlations for each pair of variables
- Each correlation is computed from all observations that have nonmissing values for the corresponding pair of variables
Practical Guide to Modern Imputation
Imputation Methods
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To impute missing values for a continous variable data sets with monotone missing patterns: you should use either…
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A parametric method that assumes multivariate normality
- includes regression method and predictive mean matching method
- A nonparameteric method that uses propensity scores
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A parametric method that assumes multivariate normality
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To impute missing values for a classification variable in data sets with monoton missing patterns you should use either…
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A logistic regression method
- when the classification variable has a binary, nominal, or ordinal response
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A discriminant function method
- when the variable has binary or nominal response
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A logistic regression method
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To impute missing values for data sets with arbitrary missing patterns you can use either…
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Markov chain Monte Carlo (MCMC) method
- that assumes multivariate normality
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fully conditional specification (FCS) method
- that assumes the existence of a joint distribution for all variables
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Markov chain Monte Carlo (MCMC) method
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For continuous variables in data sets with arbitrary missing patterns you can use either…
- MCMC to impute all missing values or just enough missing values to enable monotone missing paterns
- Regression method
- Specify a different set of covariates for each imputed variable
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To impute missing values for continuous and classification variables in data sets with arbitrary missing patterns you can use either…
- FCS method
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Regression and predictive mean matching method(s)
- for continouous variables(s)
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Logistic Regression method
- for classification variable(s)
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Discriminant Function method
- for classification variable(s)
Missing data
MCAR, MAR, MNAR, Structured missingness, Planned missingness
Mulitple Imputation
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One replaces missing values with multiple predictions from an imputation model
- Those \(M > 1\) complete datasets can be used to assess the uncertainty of imputed values