Imputation of quadratic terms

Imputes incomplete variable that appears as both main effect and quadratic effect in the complete-data model.

Usage

mice.impute.quadratic(y, ry, x, wy = NULL, quad.outcome = 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.
quad.outcome: The name of the outcome in the quadratic analysis as a character string. For example, if the substantive model of interest is y ~ x + xx, then "y" would be the quad.outcome
...: Other named arguments.

Value

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

Details

This function implements the "polynomial combination" method. First, the polynomial combination \(Z = Y \beta_1 + Y^2 \beta_2\) is formed. \(Z\) is imputed by predictive mean matching, followed by a decomposition of the imputed data \(Z\) into components \(Y\) and \(Y^2\). See Van Buuren (2012, pp. 139-141) and Vink et al (2012) for more details. The method ensures that 1) the imputed data for \(Y\) and \(Y^2\) are mutually consistent, and 2) that provides unbiased estimates of the regression weights in a complete-data linear regression that use both \(Y\) and \(Y^2\).

Note

There are two situations to consider. If only the linear term Y is present in the data, calculate the quadratic term YY after imputation. If both the linear term Y and the the quadratic term YY are variables in the data, then first impute Y by calling mice.impute.quadratic() on Y, and then impute YY by passive imputation as meth["YY"] <- "~I(Y^2)". See example section for details. Generally, we would like YY to be present in the data if we need to preserve quadratic relations between YY and any third variables in the multivariate incomplete data that we might wish to impute.

Author

Mingyang Cai and Gerko Vink

Examples

# Create Data
B1 <- .5
B2 <- .5
X <- rnorm(1000)
XX <- X^2
e <- rnorm(1000, 0, 1)
Y <- B1 * X + B2 * XX + e
dat <- data.frame(x = X, xx = XX, y = Y)

# Impose 25 percent MCAR Missingness
dat[0 == rbinom(1000, 1, 1 - .25), 1:2] <- NA

# Prepare data for imputation
ini <- mice(dat, maxit = 0)
meth <- c("quadratic", "~I(x^2)", "")
pred <- ini$pred
pred[, "xx"] <- 0

# Impute data
imp <- mice(dat, meth = meth, pred = pred, quad.outcome = "y")
#> 
#>  iter imp variable
#>   1   1  x  xx
#>   1   2  x  xx
#>   1   3  x  xx
#>   1   4  x  xx
#>   1   5  x  xx
#>   2   1  x  xx
#>   2   2  x  xx
#>   2   3  x  xx
#>   2   4  x  xx
#>   2   5  x  xx
#>   3   1  x  xx
#>   3   2  x  xx
#>   3   3  x  xx
#>   3   4  x  xx
#>   3   5  x  xx
#>   4   1  x  xx
#>   4   2  x  xx
#>   4   3  x  xx
#>   4   4  x  xx
#>   4   5  x  xx
#>   5   1  x  xx
#>   5   2  x  xx
#>   5   3  x  xx
#>   5   4  x  xx
#>   5   5  x  xx

# Pool results
pool(with(imp, lm(y ~ x + xx)))
#> Class: mipo    m = 5 
#>          term m   estimate         ubar            b            t dfcom
#> 1 (Intercept) 5 0.09523804 0.0014726259 0.0001460981 0.0016479437   997
#> 2           x 5 0.47686983 0.0009562814 0.0003835250 0.0014165114   997
#> 3          xx 5 0.49101019 0.0004636236 0.0001422658 0.0006343426   997
#>          df       riv    lambda       fmi
#> 1 252.89879 0.1190511 0.1063858 0.1133699
#> 2  35.86887 0.4812705 0.3249038 0.3596410
#> 3  51.32816 0.3682275 0.2691274 0.2960333

# Plot results
stripplot(imp)

plot(dat$x, dat$xx, col = mdc(1), xlab = "x", ylab = "xx")
cmp <- complete(imp)
points(cmp$x[is.na(dat$x)], cmp$xx[is.na(dat$x)], col = mdc(2))