fmacs: Compute fMACS effect size for two or more groups.

In marklhc/pinvsearch: Specification search for partial factorial invariance

Compute fMACS effect size for two or more groups.

Description

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,
  item_weights = NULL
)

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,
  item_weights = NULL
)

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 is contr.sum(p) if group_factor is 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.
item_weights	Default is NULL. Otherwise, one can specify a vector of length p of weights; if so, test-level dMACS will be computed.
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

Value

A 1 x p matrix of fMACS effect size.

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)
# 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)
# 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)
# 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)

marklhc/pinvsearch documentation built on June 11, 2025, 6:43 a.m.