How do I calculate cronbach's alpha on multiply imputed data? - r

Question

I have run a multiple imputation (m=45, 10 iterations) using the MICE package, and want to calculate the cronbach's alpha for a number of ordinal scales in the data. Is there a function in r that could assist me in calculating the alpha coefficient across the imputed datasets in a manner that would satisfy Rubin's rules for pooling estimates?

Answer 1

We may exploit pool.scalar from the mice package, which performs pooling of univariate estimates according to Rubin's rules.
Since you have not provided a reproducible example yourself, I will provide one.
set.seed(123)
# sample survey responses
df <- data.frame(
x1 = c(1,2,2,3,2,2,3,3,2,3,
1,2,2,3,2,2,3,3,2,3,
1,2,2,3,2,2,3,3,2,3),
x2 = c(1,1,1,2,3,3,2,3,3,3,
1,1,1,2,3,3,2,3,3,3,
1,2,2,3,2,2,3,3,2,3),
x3 = c(1,1,2,1,2,3,3,3,2,3,
1,1,2,1,2,3,3,3,2,3,
1,2,2,3,2,2,3,3,2,3)
)
# function to column-wise generate missing values (MCAR)
create_missings <- function(data, prob) {
x <- replicate(ncol(data),rbinom(nrow(data), 1, prob))
for(k in 1:ncol(data)) {
data[, k] <- ifelse(x[, k] == 1, NA, data[,k])
}
data
}
df <- create_missings(df, prob = 0.2)
# multiple imputation ----------------------------------
library(mice)
imp <- mice(df, m = 10, maxit = 20)
# extract the completed data in long format
implong <- complete(imp, 'long')
We need a function to compute cronbach's alpha and obtain an estimate of the standard error of alpha, which can be used in a call to pool.scalar() later on. Since there is no available formula with which we can analytically estimate the standard error of alpha, we also need to deploy a bootstrapping procedure to estimate this standard error.
The function cronbach_fun() takes the following arguments:
list_compl_data: a character string specifying the list of completed data from a mids object.
boot: a logical indicating whether a non-parametrical bootstrap should be conducted.
B: an integer specifying the number of bootstrap samples to be taken.
ci: a logical indicating whether a confidence interval around alpha should be estimated.
cronbach_fun <- function(list_compl_data, boot = TRUE, B = 1e4, ci = FALSE) {
n <- nrow(list_compl_data); p <- ncol(list_compl_data)
total_variance <- var(rowSums(list_compl_data))
item_variance <- sum(apply(list_compl_data, 2, sd)^2)
alpha <- (p/(p - 1)) * (1 - (item_variance/total_variance))
out <- list(alpha = alpha)
boot_alpha <- numeric(B)
if (boot) {
for (i in seq_len(B)) {
boot_dat <- list_compl_data[sample(seq_len(n), replace = TRUE), ]
total_variance <- var(rowSums(boot_dat))
item_variance <- sum(apply(boot_dat, 2, sd)^2)
boot_alpha[i] <- (p/(p - 1)) * (1 - (item_variance/total_variance))
}
out$var <- var(boot_alpha)
}
if (ci){
out$ci <- quantile(boot_alpha, c(.025,.975))
}
return(out)
}
Now that we have our function to do the 'heavy lifting', we can run it on all m completed data sets, after which we can obtain Q and U (which are required for the pooling of the estimates). Consult ?pool.scalar for more information.
m <- length(unique(implong$.imp))
boot_alpha <- rep(list(NA), m)
for (i in seq_len(m)) {
set.seed(i) # fix random number generator
sub <- implong[implong$.imp == i, -c(1,2)]
boot_alpha[[i]] <- cronbach_fun(sub)
}
# obtain Q and U (see ?pool.scalar)
Q <- sapply(boot_alpha, function(x) x$alpha)
U <- sapply(boot_alpha, function(x) x$var)
# pooled estimates
pool_estimates <- function(x) {
out <- c(
alpha = x$qbar,
lwr = x$qbar - qt(0.975, x$df) * sqrt(x$t),
upr = x$qbar + qt(0.975, x$df) * sqrt(x$t)
)
return(out)
}
Output
# Pooled estimate of alpha (95% CI)
> pool_estimates(pool.scalar(Q, U))
alpha lwr upr
0.7809977 0.5776041 0.9843913