.pool.estimates <- function(Qhat, Uhat, m, diagonal = FALSE, df.com = NULL, nms = NULL) {
# pool point estimates
Qbar <- apply(Qhat, 1, mean)
# pool variances and inferences
if (!is.null(Uhat)) {
Ubar <- apply(Uhat, 1:2, mean)
B <- tcrossprod(Qhat - Qbar) / (m-1)
T <- Ubar + (1 + m^(-1)) * B
se <- sqrt(diag(T))
t <- Qbar/se
r <- (1 + m^(-1)) * diag(B) / diag(Ubar)
# compute degrees of freedom
v <- vm <- (m-1) * (1 + r^(-1))^2
if (!is.null(df.com)) {
lam <- r / (r+1)
vobs <- (1-lam) * ((df.com+1) / (df.com+3)) * df.com
v <- (vm^(-1) + vobs^(-1))^(-1)
}
fmi <- (r + 2 / (v+3)) / (r+1)
pval <- 2 * (1 - pt(abs(t), df = v))
# create output
out <- matrix(c(Qbar, se, t, v, pval, r, fmi), ncol = 7)
colnames(out) <- c("Estimate", "Std.Error", "t.value", "df", "P(>|t|)", "RIV", "FMI")
if(!diagonal) attr(out, "T") <- T
} else {
# create output
out <- matrix(Qbar, ncol = 1)
colnames(out) <- "Estimate"
}
# parameter names
rownames(out) <- nms
attr(out, "par.labels") <- attr(nms, "par.labels")
return(out)
}
.D1 <- function(Qhat, Uhat, df.com) {
# pooling for multidimensional estimands (D1, Li et al., 1991; Reiter, 2007)
k <- dim(Qhat)[1]
m <- dim(Qhat)[2]
# D1
Qbar <- apply(Qhat, 1, mean)
Ubar <- apply(Uhat, c(1, 2), mean)
B <- cov(t(Qhat))
r <- (1+m^(-1)) * sum(diag(B %*% solve(Ubar))) / k
Ttilde <- (1 + r) * Ubar
val <- t(Qbar) %*% solve(Ttilde) %*% Qbar / k
# compute degrees of freedom (df2)
t <- k*(m-1)
if(!is.null(df.com)) {
# warn about poor behavior for t<=4
if (t <= 4) {
warning("Degrees of freedom (df.com) may not be trustworthy, because the number of imputations is too low (m \u2264 5). To obtain trustworthy results, re-run the procedure with a larger number of imputations.")
}
# small-sample degrees of freedom (Reiter, 2007; Eq. 1-2)
a <- r * t / (t-2)
vstar <- ( (df.com+1) / (df.com+3) ) * df.com
c0 <- 1 / (t-4)
c1 <- vstar - 2 * (1+a)
c2 <- vstar - 4 * (1+a)
z <- 1 / c2 +
c0 * (a^2 * c1 / ((1+a)^2 * c2)) +
c0 * (8*a^2 * c1 / ((1+a) * c2^2) + 4*a^2 / ((1+a) * c2)) +
c0 * (4*a^2 / (c2 * c1) + 16*a^2 * c1 / c2^3) +
c0 * (8*a^2 / c2^2)
v <- 4 + 1/z
} else {
if (t > 4) {
v <- 4 + (t-4) * (1 + (1 - 2*t^(-1)) * (r^(-1)))^2
} else {
v <- t * (1 + k^(-1)) * ((1 + r^(-1))^2) / 2
}
}
return(list(F = val, k = k, v = v, r = r))
}
.D2 <- function(d, k) {
# pooling for multidimensional estimands (D2, Li, Meng et al., 1991)
m <- length(d)
# D2
dbar <- mean(d)
r <- (1 + m^(-1)) * var(sqrt(d))
val <- (dbar/k - (m+1)/(m-1) * r) / (1+r)
# compute degrees of freedom (df2)
v <- k^(-3/m) * (m-1) * (1 + r^(-1))^2
return(list(F = val, k = k, v = v, r = r))
}
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