Compute dMACS effect size described in Nye & Drasgow (2011) for two groups.

dmacs returns the dMACS effect size statistics given a set of loadings and intercepts.

Usage

dmacs(
  intercepts,
  loadings = NULL,
  pooled_item_sd = NULL,
  latent_mean = 0,
  latent_sd = 1,
  uniqueness = NULL,
  ns = NULL,
  item_weights = NULL,
  pooled_test_sd = NULL
)

dmacs_ordered(
  thresholds,
  loadings,
  thetas = 1,
  link = c("probit", "logit"),
  pooled_item_sd = NULL,
  latent_mean = 0,
  latent_sd = 1,
  item_weights = NULL,
  pooled_test_sd = NULL
)

Arguments

intercepts: A $2 \times p$ matrix of measurement intercepts.
loadings: A $2 \times p$ matrix of factor loadings, where p is the number of items.
pooled_item_sd: A numeric vector of length p of the pooled standard deviation (SD) of the items across groups.
latent_mean: latent factor mean for the reference group. Default to 0.
latent_sd: latent factor SD for the reference group. Default to 1.
uniqueness: A vector of length $p$ of uniqueness.
ns: A vector of length $p$ of sample sizes.
item_weights: Default is NULL. Otherwise, one can specify a vector of length $p$ of weights; if so, test-level dMACS will be computed for the weighted sum of the items (unit weights give the plain sum).
pooled_test_sd: The pooled standard deviation of the weighted item sum. Only used when item_weights is not NULL. If not supplied, it is approximated from loadings, pooled_item_sd, and latent_sd, assuming the same latent variance in both groups (and equal group pooling weights, or the weights in ns when it has one entry per group). For dmacs_ordered() and fmacs_ordered() only the weighted sum of the item variances is used, as the cross-item covariances are not identifiable from the inputs.
thresholds: A matrix with two 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).

Value

A 1 x p matrix of dMACS effect size. If item_weights is not NULL, $p$ = 1.

Details

The $d_\text{MACS}$ effect size is defined as (Nye & Drasgow, 2011, p. 968) $$d_{\text{MACS}, i} = \frac{1}{\mathit{SD}_{iP}} \sqrt{\int [(\nu_{iR} - \nu{iF}) + (\lambda_{iR} - \lambda_{iF}) \eta]^2 f(\eta) d \eta}$$ where $\lambda$ is the loading and $\nu$ is the intercept, F and R denote the focal and the reference group. The effect size reflects the standardized mean difference on an item due to measurement noninvariance, and is analogous to the Cohen's d effect size.

References

Nye, C. & Drasgow, F. (2011). Effect size indices for analyses of measurement equivalence: Understanding the practical importance of differences between groups. Journal of Applied Psychology, 96(5), 966-980.

Examples

lambdaf <- c(.8, .5, .7, .5)
lambdar <- c(.8, .5, .4, .6)
nuf <- c(0.1, 0, 0.2, 0)
nur <- c(0.2, 0, 0, 0)
dmacs(rbind(nuf, nur),
      loadings = rbind(lambdaf, lambdar),
      pooled_item_sd = c(1, 1, 1, 1),
      latent_mean = 0,
      latent_sd = 1)
#>       [,1] [,2]      [,3] [,4]
#> dmacs  0.1    0 0.3605551  0.1
dmacs(rbind(nuf, nur),
      loadings = rbind(lambdaf, lambdar),
      pooled_item_sd = c(1, 1, 1, 1),
      latent_mean = 0,
      latent_sd = 1,
      item_weights = c(1, 1, 1, 1))
#>         item_sum
#> dmacs 0.07784989
# Thresholds
lambda <- rbind(c(.8, .5, .7, .5),
                c(.8, .5, .4, .6))
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))
# 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)
dmacs_ordered(tau,
              loadings = lambda,
              pooled_item_sd = c(1, 1, 1, 1),
              latent_mean = 0,
              latent_sd = 1)
#>       [,1]       [,2]      [,3]       [,4]
#> dmacs    0 0.01711254 0.2019787 0.02276356