Calculates imputations for univariate missing data by Bayesian linear regression, also known as the normal model.

## Usage

mice.impute.norm(y, ry, x, wy = NULL, ...)

## Arguments

y

Vector to be imputed

ry

Logical vector of length length(y) indicating the the subset y[ry] of elements in y to which the imputation model is fitted. The ry generally distinguishes the observed (TRUE) and missing values (FALSE) in y.

x

Numeric design matrix with length(y) rows with predictors for y. Matrix x may have no missing values.

wy

Logical vector of length length(y). A TRUE value indicates locations in y for which imputations are created.

...

Other named arguments.

## Value

Vector with imputed data, same type as y, and of length sum(wy)

## Details

Imputation of y by the normal model by the method defined by Rubin (1987, p. 167). The procedure is as follows:

1. Calculate the cross-product matrix $$S=X_{obs}'X_{obs}$$.

2. Calculate $$V = (S+{diag}(S)\kappa)^{-1}$$, with some small ridge parameter $$\kappa$$.

3. Calculate regression weights $$\hat\beta = VX_{obs}'y_{obs}.$$

4. Draw a random variable $$\dot g \sim \chi^2_\nu$$ with $$\nu=n_1 - q$$.

5. Calculate $$\dot\sigma^2 = (y_{obs} - X_{obs}\hat\beta)'(y_{obs} - X_{obs}\hat\beta)/\dot g.$$

6. Draw $$q$$ independent $$N(0,1)$$ variates in vector $$\dot z_1$$.

7. Calculate $$V^{1/2}$$ by Cholesky decomposition.

8. Calculate $$\dot\beta = \hat\beta + \dot\sigma\dot z_1 V^{1/2}$$.

9. Draw $$n_0$$ independent $$N(0,1)$$ variates in vector $$\dot z_2$$.

10. Calculate the $$n_0$$ values $$y_{imp} = X_{mis}\dot\beta + \dot z_2\dot\sigma$$.

Using mice.impute.norm for all columns emulates Schafer's NORM method (Schafer, 1997).

Rubin, D.B (1987). Multiple Imputation for Nonresponse in Surveys. New York: John Wiley & Sons.

Schafer, J.L. (1997). Analysis of incomplete multivariate data. London: Chapman & Hall.