fmacs returns the fMACS effect size statistics given a set of loadings
and intercepts.
Usage
fmacs(
intercepts,
loadings = NULL,
pooled_item_sd,
num_obs = NULL,
weights = 0 * intercepts + 1,
group_factor = NULL,
contrast = contr.sum(nrow(intercepts)),
latent_mean = 0,
latent_sd = 1
)
fmacs_ordered(
thresholds,
loadings,
thetas = 1,
num_obs = NULL,
weights = 0 * loadings + 1,
group_factor = NULL,
contrast = contr.sum(nrow(thresholds)),
link = c("probit", "logit"),
pooled_item_sd = NULL,
latent_mean = 0,
latent_sd = 1
)Arguments
- intercepts
A \(G \times p\) matrix of measurement intercepts.
- loadings
A \(G \times p\) matrix of factor loadings, where p is the number of items and G is the number of groups.
- pooled_item_sd
A numeric vector of length p of the pooled standard deviation (SD) of the items across groups.
- num_obs
A vector of length \(G\) of sample sizes. If not
NULL, the weights will be proportional to sample sizes, assuming the same weights across items.- weights
A \(G \times p\) matrix of weights. Default assumes equal weights across groups.
- group_factor
A vector of length \(G\) indicating grouping for contrast. For example,
c(1, 1, 2)means contrasting Group 1 & 2 vs. Group 3. The default is to not combine any groups, meaning the omnibus effect is computed.- contrast
A \(p \times k\) contrast matrix where
colSums(contrast)= 0. Default iscontr.sum(p)ifgroup_factoris not specified.- latent_mean
latent factor mean for the reference group. Default to 0.
- latent_sd
latent factor SD for the reference group. Default to 1.
- thresholds
A matrix with G rows for measurement thresholds. The matrix must have column names indicating to which item index each column corresponds.
- thetas
Not currently used.
- link
Link function for the model (probit or logit).
Details
The \(f_\text{MACS}\) effect size is defined as $$f_{\text{MACS}, i} = \frac{1}{\mathit{SD}_{iP}} \sqrt{\int [(\nu_{ij} - \bar{\nu}_j) + (\lambda_{ij} - \bar{\lambda}_j) \eta]^2 f(\eta) d \eta}$$ where \(\lambda\) is the loading and \(\nu\) is the intercept, and j indexes group. The effect size reflects the square root of the ratio between the variance in observed item score due to measurement noninvariance and the variance of the observed item scores. \(f_\text{MACS}\) is analogous to the Cohen's f effect size. When there are two groups with equal sample sizes, \(f_\text{MACS}\) = \(f_\text{MACS}\) / 2
Examples
lambda <- rbind(c(.7, .8, .7, .9),
c(.7, .8, .7, .8),
c(.8, .7, .7, .5))
nu <- rbind(c(0, .5, 0, 1),
c(0, .2, 0, 1.1),
c(0, .3, 0, 1.2))
fmacs(lambda,
loadings = nu,
pooled_item_sd = c(1, 1, 1, 1),
latent_mean = 0,
latent_sd = 1)
#> [,1] [,2] [,3] [,4]
#> fmacs 0.04714045 0.1333333 0 0.1885618
# With contrast (Group 1 & 2 vs. Group 3)
fmacs(lambda,
loadings = nu,
pooled_item_sd = c(1, 1, 1, 1),
group_factor = c(1, 1, 2),
latent_mean = 0,
latent_sd = 1)
#> [,1] [,2] [,3] [,4]
#> fmacs 0.04714045 0.05270463 0 0.1795055
# Thresholds
lambda <- rbind(c(.8, .5, .7, .5),
c(.8, .5, .4, .6),
c(.8, .7, .7, .5))
tau <- rbind(c(-0.5, 0, 1, -0.3, 0.1, 0.5, -0.5, 1.5),
c(-0.5, 0, 1, -0.5, 0.3, 0.5, -1, 1.5),
c(-0.5, 0, 1, -0.5, 0.3, 0.5, -1, 0.5))
# three thresholds for items 1 and 2; one threshold for items 3 and 4
colnames(tau) <- c(1, 1, 1, 2, 2, 2, 3, 4)
fmacs_ordered(tau,
loadings = lambda,
pooled_item_sd = c(1, 1, 1, 1),
latent_mean = 0,
latent_sd = 1)
#> [,1] [,2] [,3] [,4]
#> fmacs 0 0.06906871 0.08713051 0.1169068
# With contrast (Group 1 & 2 vs. Group 3)
fmacs_ordered(tau,
loadings = lambda,
pooled_item_sd = c(1, 1, 1, 1),
group_factor = c(1, 2, 1),
latent_mean = 0,
latent_sd = 1)
#> [,1] [,2] [,3] [,4]
#> fmacs 0 0.04029493 0.06394493 0.05383212