RMSD effect size statistic to quantify category-level DIF

This function computes a set of RMSD "badness-of-fit" statistics when investing DIF across a set of grouping variables. In a first step, a (potentially highly constrained) multiple group model is fitted, while in a second step the item (and person) parameters are estimated based on all examines across all groups. Category level DIF is assessed based on how well the pseudo-table of counts match the (constrained) probability functions implied by the original multiple group model (while also weighing across the implied density function of the latent traits). If the RSMD fit is poor, indicating non-ignorable DIF, then the multiple-group model should be adjusted to better account for the large response bias due to using a pooled model. See Lee and von Davier (2020) and Buchholz and Hartig (2019) for details.

RMSD_DIF(pooled_mod, flag = 0, probfun = TRUE, dentype = "norm")

Arguments

pooled_mod: a multiple-group model (used to compute the model-implied probability in the goodness-of-fit test)
flag: a numeric value used as a cut-off to help flag larger RMSD values (e.g., flag = .03 will highlight only categories with RMSD values greater than .03)
probfun: logical; use probability functions to compute RMSD? If FALSE, the expected score functions will be integrated instead, which may be useful for collapsing across the categories in polytomous items
dentype: density to use for the latent trait. Can be 'norm' to use a normal Gaussian density where the mean/variance are extracted from the model object(default), 'snorm' for a standard normal distribution, or 'empirical' to use the density estimate obtained via the E-table

References

Buchholz, J., and Hartig, J. (2019). Comparing Attitudes Across Groups: An IRT-Based Item-Fit Statistic for the Analysis of Measurement Invariance. Applied Psychological Measurement, 43(3), 241-250. doi:10.1177/0146621617748323

Lee, S. S., and von Davier, M. (2020). Improving measurement properties of the PISA home possessions scale through partial invariance modeling. Psychological test and assessment modeling, 62(1):55-83.

Author

Phil Chalmers rphilip.chalmers@gmail.com

Examples


# \donttest{

#----- generate some data
set.seed(12345)
a <- a2 <- matrix(abs(rnorm(15,1,.3)), ncol=1)
d <- d2 <- matrix(rnorm(15,0,.7),ncol=1)

# item 1 has DIF
d2[1] <- d[1] - .5
a2[1] <- a[1] + 1

itemtype <- rep('2PL', nrow(a))
N <- 1000
dataset1 <- simdata(a, d, N, itemtype)
dataset2 <- simdata(a2, d2, N, itemtype)
dat <- rbind(dataset1, dataset2)
group <- c(rep('D1', N), rep('D2', N))

#-----

# fully pooled model
pooled_mod <- multipleGroup(dat, 1, group=group,
   invariance = c(colnames(dat), 'free_mean', 'free_var'))
coef(pooled_mod, simplify=TRUE)
#> $D1
#> $items
#>            a1      d g u
#> Item_1  1.441  0.358 0 1
#> Item_2  1.136 -0.590 0 1
#> Item_3  0.996 -0.237 0 1
#> Item_4  0.902  0.921 0 1
#> Item_5  1.060  0.153 0 1
#> Item_6  0.440  0.637 0 1
#> Item_7  1.170  1.023 0 1
#> Item_8  0.904 -0.353 0 1
#> Item_9  0.875 -0.978 0 1
#> Item_10 0.665 -1.120 0 1
#> Item_11 0.939  1.243 0 1
#> Item_12 1.336 -0.194 0 1
#> Item_13 1.250  0.481 0 1
#> Item_14 1.050  0.444 0 1
#> Item_15 0.778 -0.064 0 1
#> 
#> $means
#> F1 
#>  0 
#> 
#> $cov
#>    F1
#> F1  1
#> 
#> 
#> $D2
#> $items
#>            a1      d g u
#> Item_1  1.441  0.358 0 1
#> Item_2  1.136 -0.590 0 1
#> Item_3  0.996 -0.237 0 1
#> Item_4  0.902  0.921 0 1
#> Item_5  1.060  0.153 0 1
#> Item_6  0.440  0.637 0 1
#> Item_7  1.170  1.023 0 1
#> Item_8  0.904 -0.353 0 1
#> Item_9  0.875 -0.978 0 1
#> Item_10 0.665 -1.120 0 1
#> Item_11 0.939  1.243 0 1
#> Item_12 1.336 -0.194 0 1
#> Item_13 1.250  0.481 0 1
#> Item_14 1.050  0.444 0 1
#> Item_15 0.778 -0.064 0 1
#> 
#> $means
#>    F1 
#> -0.08 
#> 
#> $cov
#>       F1
#> F1 1.167
#> 
#> 

RMSD_DIF(pooled_mod)
#> $D1
#>           P.0   P.1
#> Item_1  0.056 0.056
#> Item_2  0.018 0.018
#> Item_3  0.014 0.014
#> Item_4  0.019 0.019
#> Item_5  0.010 0.010
#> Item_6  0.021 0.021
#> Item_7  0.016 0.016
#> Item_8  0.017 0.017
#> Item_9  0.019 0.019
#> Item_10 0.018 0.018
#> Item_11 0.013 0.013
#> Item_12 0.014 0.014
#> Item_13 0.018 0.018
#> Item_14 0.009 0.009
#> Item_15 0.011 0.011
#> 
#> $D2
#>           P.0   P.1
#> Item_1  0.053 0.053
#> Item_2  0.021 0.021
#> Item_3  0.013 0.013
#> Item_4  0.016 0.016
#> Item_5  0.008 0.008
#> Item_6  0.022 0.022
#> Item_7  0.017 0.017
#> Item_8  0.016 0.016
#> Item_9  0.017 0.017
#> Item_10 0.014 0.014
#> Item_11 0.021 0.021
#> Item_12 0.013 0.013
#> Item_13 0.011 0.011
#> Item_14 0.004 0.004
#> Item_15 0.014 0.014
#> 
RMSD_DIF(pooled_mod, dentype = 'empirical')
#> $D1
#>           P.0   P.1
#> Item_1  0.056 0.056
#> Item_2  0.018 0.018
#> Item_3  0.014 0.014
#> Item_4  0.019 0.019
#> Item_5  0.010 0.010
#> Item_6  0.021 0.021
#> Item_7  0.016 0.016
#> Item_8  0.017 0.017
#> Item_9  0.019 0.019
#> Item_10 0.018 0.018
#> Item_11 0.013 0.013
#> Item_12 0.014 0.014
#> Item_13 0.018 0.018
#> Item_14 0.009 0.009
#> Item_15 0.011 0.011
#> 
#> $D2
#>           P.0   P.1
#> Item_1  0.052 0.052
#> Item_2  0.021 0.021
#> Item_3  0.013 0.013
#> Item_4  0.016 0.016
#> Item_5  0.009 0.009
#> Item_6  0.022 0.022
#> Item_7  0.017 0.017
#> Item_8  0.016 0.016
#> Item_9  0.017 0.017
#> Item_10 0.014 0.014
#> Item_11 0.021 0.021
#> Item_12 0.013 0.013
#> Item_13 0.011 0.011
#> Item_14 0.004 0.004
#> Item_15 0.014 0.014
#> 
RMSD_DIF(pooled_mod, flag = .03)
#> $D1
#>           P.0   P.1
#> Item_1  0.056 0.056
#> Item_2     NA    NA
#> Item_3     NA    NA
#> Item_4     NA    NA
#> Item_5     NA    NA
#> Item_6     NA    NA
#> Item_7     NA    NA
#> Item_8     NA    NA
#> Item_9     NA    NA
#> Item_10    NA    NA
#> Item_11    NA    NA
#> Item_12    NA    NA
#> Item_13    NA    NA
#> Item_14    NA    NA
#> Item_15    NA    NA
#> 
#> $D2
#>           P.0   P.1
#> Item_1  0.053 0.053
#> Item_2     NA    NA
#> Item_3     NA    NA
#> Item_4     NA    NA
#> Item_5     NA    NA
#> Item_6     NA    NA
#> Item_7     NA    NA
#> Item_8     NA    NA
#> Item_9     NA    NA
#> Item_10    NA    NA
#> Item_11    NA    NA
#> Item_12    NA    NA
#> Item_13    NA    NA
#> Item_14    NA    NA
#> Item_15    NA    NA
#> 

# more freely estimated model (item 1 has 2 parameters estimated)
MGmod <- multipleGroup(dat, 1, group=group,
                       invariance = c(colnames(dat)[-1], 'free_mean', 'free_var'))
coef(MGmod, simplify=TRUE)
#> $D1
#> $items
#>            a1      d g u
#> Item_1  1.071  0.525 0 1
#> Item_2  1.178 -0.615 0 1
#> Item_3  1.031 -0.258 0 1
#> Item_4  0.933  0.902 0 1
#> Item_5  1.099  0.130 0 1
#> Item_6  0.455  0.627 0 1
#> Item_7  1.217  1.001 0 1
#> Item_8  0.926 -0.372 0 1
#> Item_9  0.905 -0.997 0 1
#> Item_10 0.692 -1.136 0 1
#> Item_11 0.967  1.222 0 1
#> Item_12 1.373 -0.222 0 1
#> Item_13 1.302  0.456 0 1
#> Item_14 1.079  0.421 0 1
#> Item_15 0.804 -0.080 0 1
#> 
#> $means
#> F1 
#>  0 
#> 
#> $cov
#>    F1
#> F1  1
#> 
#> 
#> $D2
#> $items
#>            a1      d g u
#> Item_1  2.100  0.083 0 1
#> Item_2  1.178 -0.615 0 1
#> Item_3  1.031 -0.258 0 1
#> Item_4  0.933  0.902 0 1
#> Item_5  1.099  0.130 0 1
#> Item_6  0.455  0.627 0 1
#> Item_7  1.217  1.001 0 1
#> Item_8  0.926 -0.372 0 1
#> Item_9  0.905 -0.997 0 1
#> Item_10 0.692 -1.136 0 1
#> Item_11 0.967  1.222 0 1
#> Item_12 1.373 -0.222 0 1
#> Item_13 1.302  0.456 0 1
#> Item_14 1.079  0.421 0 1
#> Item_15 0.804 -0.080 0 1
#> 
#> $means
#>     F1 
#> -0.036 
#> 
#> $cov
#>       F1
#> F1 1.033
#> 
#> 

# RMSD in item.1 now reduced (MG model accounts for DIF)
RMSD_DIF(MGmod)
#> $D1
#>           P.0   P.1
#> Item_1  0.007 0.007
#> Item_2  0.015 0.015
#> Item_3  0.017 0.017
#> Item_4  0.015 0.015
#> Item_5  0.009 0.009
#> Item_6  0.020 0.020
#> Item_7  0.010 0.010
#> Item_8  0.020 0.020
#> Item_9  0.017 0.017
#> Item_10 0.018 0.018
#> Item_11 0.017 0.017
#> Item_12 0.010 0.010
#> Item_13 0.017 0.017
#> Item_14 0.013 0.013
#> Item_15 0.011 0.011
#> 
#> $D2
#>           P.0   P.1
#> Item_1  0.005 0.005
#> Item_2  0.020 0.020
#> Item_3  0.018 0.018
#> Item_4  0.011 0.011
#> Item_5  0.007 0.007
#> Item_6  0.022 0.022
#> Item_7  0.011 0.011
#> Item_8  0.017 0.017
#> Item_9  0.016 0.016
#> Item_10 0.016 0.016
#> Item_11 0.022 0.022
#> Item_12 0.008 0.008
#> Item_13 0.006 0.006
#> Item_14 0.009 0.009
#> Item_15 0.014 0.014
#> 
RMSD_DIF(MGmod, flag = .03)
#> $D1
#>         P.0 P.1
#> Item_1   NA  NA
#> Item_2   NA  NA
#> Item_3   NA  NA
#> Item_4   NA  NA
#> Item_5   NA  NA
#> Item_6   NA  NA
#> Item_7   NA  NA
#> Item_8   NA  NA
#> Item_9   NA  NA
#> Item_10  NA  NA
#> Item_11  NA  NA
#> Item_12  NA  NA
#> Item_13  NA  NA
#> Item_14  NA  NA
#> Item_15  NA  NA
#> 
#> $D2
#>         P.0 P.1
#> Item_1   NA  NA
#> Item_2   NA  NA
#> Item_3   NA  NA
#> Item_4   NA  NA
#> Item_5   NA  NA
#> Item_6   NA  NA
#> Item_7   NA  NA
#> Item_8   NA  NA
#> Item_9   NA  NA
#> Item_10  NA  NA
#> Item_11  NA  NA
#> Item_12  NA  NA
#> Item_13  NA  NA
#> Item_14  NA  NA
#> Item_15  NA  NA
#> 


#################
# polytomous example
set.seed(12345)
a <- a2 <- matrix(rlnorm(20,.2,.3))

# for the graded model, ensure that there is enough space between the intercepts,
# otherwise closer categories will not be selected often (minimum distance of 0.3 here)
diffs <- t(apply(matrix(runif(20*4, .3, 1), 20), 1, cumsum))
diffs <- -(diffs - rowMeans(diffs))
d <- d2 <- diffs + rnorm(20)

# item 1 has slope + dif for first intercept parameter
d2[1] <- d[1] - .5
a2[1] <- a[1] + 1

itemtype <- rep('graded', nrow(a))
N <- 1000
dataset1 <- simdata(a, d, N, itemtype)
dataset2 <- simdata(a2, d2, N, itemtype)
dat <- rbind(dataset1, dataset2)
group <- c(rep('D1', N), rep('D2', N))

#-----

# fully pooled model
pooled_mod <- multipleGroup(dat, 1, group=group,
         invariance = c(colnames(dat), 'free_mean', 'free_var'))
coef(pooled_mod, simplify=TRUE)
#> $D1
#> $items
#>            a1     d1     d2     d3     d4
#> Item_1  1.708  1.205  0.527 -0.453 -0.948
#> Item_2  1.517 -0.518 -0.992 -1.653 -2.434
#> Item_3  1.187  1.757  0.681  0.375 -0.638
#> Item_4  1.167  2.719  1.868  1.539  0.808
#> Item_5  1.410  0.485 -0.466 -0.867 -1.581
#> Item_6  0.683 -0.984 -1.396 -1.940 -2.910
#> Item_7  1.387  2.128  1.234  0.341 -0.386
#> Item_8  0.985  2.332  1.994  1.374  0.440
#> Item_9  1.043  1.597  0.850  0.024 -0.373
#> Item_10 0.906 -0.284 -1.281 -1.573 -2.038
#> Item_11 1.128  1.269  0.470 -0.460 -1.184
#> Item_12 2.016  0.268 -0.318 -1.128 -1.716
#> Item_13 1.356 -0.202 -0.835 -1.233 -2.016
#> Item_14 1.261  3.169  2.581  1.916  1.334
#> Item_15 0.928  2.307  1.956  1.151  0.268
#> Item_16 1.479  2.080  1.345  0.682 -0.305
#> Item_17 0.815  2.163  1.208  0.410 -0.557
#> Item_18 1.113  0.333 -0.481 -1.357 -2.072
#> Item_19 1.604  1.256  0.584  0.240 -0.145
#> Item_20 1.333  1.876  1.439  0.858  0.241
#> 
#> $means
#> F1 
#>  0 
#> 
#> $cov
#>    F1
#> F1  1
#> 
#> 
#> $D2
#> $items
#>            a1     d1     d2     d3     d4
#> Item_1  1.708  1.205  0.527 -0.453 -0.948
#> Item_2  1.517 -0.518 -0.992 -1.653 -2.434
#> Item_3  1.187  1.757  0.681  0.375 -0.638
#> Item_4  1.167  2.719  1.868  1.539  0.808
#> Item_5  1.410  0.485 -0.466 -0.867 -1.581
#> Item_6  0.683 -0.984 -1.396 -1.940 -2.910
#> Item_7  1.387  2.128  1.234  0.341 -0.386
#> Item_8  0.985  2.332  1.994  1.374  0.440
#> Item_9  1.043  1.597  0.850  0.024 -0.373
#> Item_10 0.906 -0.284 -1.281 -1.573 -2.038
#> Item_11 1.128  1.269  0.470 -0.460 -1.184
#> Item_12 2.016  0.268 -0.318 -1.128 -1.716
#> Item_13 1.356 -0.202 -0.835 -1.233 -2.016
#> Item_14 1.261  3.169  2.581  1.916  1.334
#> Item_15 0.928  2.307  1.956  1.151  0.268
#> Item_16 1.479  2.080  1.345  0.682 -0.305
#> Item_17 0.815  2.163  1.208  0.410 -0.557
#> Item_18 1.113  0.333 -0.481 -1.357 -2.072
#> Item_19 1.604  1.256  0.584  0.240 -0.145
#> Item_20 1.333  1.876  1.439  0.858  0.241
#> 
#> $means
#>     F1 
#> -0.022 
#> 
#> $cov
#>       F1
#> F1 1.182
#> 
#> 

# Item_1 fits poorly in several categories (RMSD > .05)
RMSD_DIF(pooled_mod)
#> $D1
#>           P.1   P.2   P.3   P.4   P.5
#> Item_1  0.088 0.057 0.024 0.023 0.037
#> Item_2  0.014 0.008 0.010 0.011 0.012
#> Item_3  0.019 0.019 0.015 0.017 0.027
#> Item_4  0.024 0.010 0.010 0.014 0.028
#> Item_5  0.030 0.017 0.006 0.023 0.021
#> Item_6  0.010 0.015 0.015 0.019 0.016
#> Item_7  0.016 0.015 0.020 0.015 0.019
#> Item_8  0.014 0.008 0.012 0.021 0.022
#> Item_9  0.019 0.031 0.022 0.014 0.028
#> Item_10 0.025 0.015 0.010 0.011 0.032
#> Item_11 0.012 0.017 0.015 0.011 0.015
#> Item_12 0.013 0.011 0.021 0.013 0.018
#> Item_13 0.023 0.013 0.018 0.021 0.014
#> Item_14 0.009 0.011 0.005 0.022 0.029
#> Item_15 0.015 0.011 0.016 0.010 0.028
#> Item_16 0.016 0.025 0.020 0.022 0.012
#> Item_17 0.025 0.015 0.020 0.023 0.031
#> Item_18 0.016 0.015 0.017 0.016 0.023
#> Item_19 0.019 0.022 0.016 0.013 0.013
#> Item_20 0.033 0.011 0.012 0.012 0.025
#> 
#> $D2
#>           P.1   P.2   P.3   P.4   P.5
#> Item_1  0.101 0.059 0.036 0.018 0.040
#> Item_2  0.007 0.013 0.005 0.012 0.011
#> Item_3  0.035 0.028 0.017 0.030 0.016
#> Item_4  0.011 0.020 0.012 0.011 0.027
#> Item_5  0.017 0.017 0.010 0.013 0.016
#> Item_6  0.026 0.011 0.007 0.016 0.014
#> Item_7  0.017 0.013 0.017 0.019 0.020
#> Item_8  0.021 0.016 0.016 0.017 0.030
#> Item_9  0.019 0.016 0.014 0.015 0.020
#> Item_10 0.020 0.014 0.014 0.018 0.028
#> Item_11 0.018 0.021 0.017 0.013 0.019
#> Item_12 0.013 0.010 0.022 0.007 0.015
#> Item_13 0.018 0.013 0.013 0.012 0.017
#> Item_14 0.026 0.007 0.023 0.012 0.008
#> Item_15 0.019 0.010 0.014 0.017 0.023
#> Item_16 0.015 0.019 0.019 0.018 0.013
#> Item_17 0.017 0.018 0.028 0.024 0.012
#> Item_18 0.024 0.022 0.025 0.017 0.028
#> Item_19 0.027 0.017 0.014 0.009 0.024
#> Item_20 0.025 0.013 0.014 0.011 0.018
#> 
RMSD_DIF(pooled_mod, flag = .05)
#> $D1
#>           P.1   P.2 P.3 P.4 P.5
#> Item_1  0.088 0.057  NA  NA  NA
#> Item_2     NA    NA  NA  NA  NA
#> Item_3     NA    NA  NA  NA  NA
#> Item_4     NA    NA  NA  NA  NA
#> Item_5     NA    NA  NA  NA  NA
#> Item_6     NA    NA  NA  NA  NA
#> Item_7     NA    NA  NA  NA  NA
#> Item_8     NA    NA  NA  NA  NA
#> Item_9     NA    NA  NA  NA  NA
#> Item_10    NA    NA  NA  NA  NA
#> Item_11    NA    NA  NA  NA  NA
#> Item_12    NA    NA  NA  NA  NA
#> Item_13    NA    NA  NA  NA  NA
#> Item_14    NA    NA  NA  NA  NA
#> Item_15    NA    NA  NA  NA  NA
#> Item_16    NA    NA  NA  NA  NA
#> Item_17    NA    NA  NA  NA  NA
#> Item_18    NA    NA  NA  NA  NA
#> Item_19    NA    NA  NA  NA  NA
#> Item_20    NA    NA  NA  NA  NA
#> 
#> $D2
#>           P.1   P.2 P.3 P.4 P.5
#> Item_1  0.101 0.059  NA  NA  NA
#> Item_2     NA    NA  NA  NA  NA
#> Item_3     NA    NA  NA  NA  NA
#> Item_4     NA    NA  NA  NA  NA
#> Item_5     NA    NA  NA  NA  NA
#> Item_6     NA    NA  NA  NA  NA
#> Item_7     NA    NA  NA  NA  NA
#> Item_8     NA    NA  NA  NA  NA
#> Item_9     NA    NA  NA  NA  NA
#> Item_10    NA    NA  NA  NA  NA
#> Item_11    NA    NA  NA  NA  NA
#> Item_12    NA    NA  NA  NA  NA
#> Item_13    NA    NA  NA  NA  NA
#> Item_14    NA    NA  NA  NA  NA
#> Item_15    NA    NA  NA  NA  NA
#> Item_16    NA    NA  NA  NA  NA
#> Item_17    NA    NA  NA  NA  NA
#> Item_18    NA    NA  NA  NA  NA
#> Item_19    NA    NA  NA  NA  NA
#> Item_20    NA    NA  NA  NA  NA
#> 
RMSD_DIF(pooled_mod, flag = .1, probfun = FALSE) # use expected score function
#> $D1
#>         S(theta)
#> Item_1     0.184
#> Item_2        NA
#> Item_3        NA
#> Item_4     0.106
#> Item_5        NA
#> Item_6        NA
#> Item_7        NA
#> Item_8        NA
#> Item_9        NA
#> Item_10       NA
#> Item_11       NA
#> Item_12       NA
#> Item_13       NA
#> Item_14       NA
#> Item_15       NA
#> Item_16       NA
#> Item_17    0.101
#> Item_18       NA
#> Item_19       NA
#> Item_20    0.102
#> 
#> $D2
#>         S(theta)
#> Item_1     0.211
#> Item_2        NA
#> Item_3        NA
#> Item_4        NA
#> Item_5        NA
#> Item_6        NA
#> Item_7        NA
#> Item_8        NA
#> Item_9        NA
#> Item_10       NA
#> Item_11       NA
#> Item_12       NA
#> Item_13       NA
#> Item_14       NA
#> Item_15       NA
#> Item_16       NA
#> Item_17       NA
#> Item_18       NA
#> Item_19       NA
#> Item_20       NA
#> 

# more freely estimated model (item 1 has more parameters estimated)
MGmod <- multipleGroup(dat, 1, group=group,
                       invariance = c(colnames(dat)[-1], 'free_mean', 'free_var'))
coef(MGmod, simplify=TRUE)
#> $D1
#> $items
#>            a1     d1     d2     d3     d4
#> Item_1  1.293  1.471  0.527 -0.408 -0.925
#> Item_2  1.553 -0.527 -1.000 -1.659 -2.439
#> Item_3  1.226  1.753  0.675  0.368 -0.648
#> Item_4  1.201  2.716  1.864  1.534  0.802
#> Item_5  1.447  0.476 -0.475 -0.876 -1.590
#> Item_6  0.702 -0.989 -1.400 -1.944 -2.915
#> Item_7  1.426  2.122  1.227  0.333 -0.395
#> Item_8  1.009  2.325  1.987  1.368  0.433
#> Item_9  1.076  1.593  0.846  0.017 -0.380
#> Item_10 0.936 -0.290 -1.289 -1.581 -2.047
#> Item_11 1.157  1.261  0.463 -0.468 -1.191
#> Item_12 2.070  0.255 -0.331 -1.141 -1.728
#> Item_13 1.392 -0.211 -0.844 -1.242 -2.025
#> Item_14 1.290  3.158  2.571  1.905  1.324
#> Item_15 0.952  2.301  1.951  1.145  0.262
#> Item_16 1.516  2.070  1.334  0.671 -0.315
#> Item_17 0.835  2.158  1.202  0.405 -0.562
#> Item_18 1.141  0.326 -0.487 -1.363 -2.078
#> Item_19 1.643  1.245  0.573  0.230 -0.155
#> Item_20 1.371  1.869  1.432  0.851  0.234
#> 
#> $means
#> F1 
#>  0 
#> 
#> $cov
#>    F1
#> F1  1
#> 
#> 
#> $D2
#> $items
#>            a1     d1     d2     d3     d4
#> Item_1  2.523  0.906  0.519 -0.587 -1.079
#> Item_2  1.553 -0.527 -1.000 -1.659 -2.439
#> Item_3  1.226  1.753  0.675  0.368 -0.648
#> Item_4  1.201  2.716  1.864  1.534  0.802
#> Item_5  1.447  0.476 -0.475 -0.876 -1.590
#> Item_6  0.702 -0.989 -1.400 -1.944 -2.915
#> Item_7  1.426  2.122  1.227  0.333 -0.395
#> Item_8  1.009  2.325  1.987  1.368  0.433
#> Item_9  1.076  1.593  0.846  0.017 -0.380
#> Item_10 0.936 -0.290 -1.289 -1.581 -2.047
#> Item_11 1.157  1.261  0.463 -0.468 -1.191
#> Item_12 2.070  0.255 -0.331 -1.141 -1.728
#> Item_13 1.392 -0.211 -0.844 -1.242 -2.025
#> Item_14 1.290  3.158  2.571  1.905  1.324
#> Item_15 0.952  2.301  1.951  1.145  0.262
#> Item_16 1.516  2.070  1.334  0.671 -0.315
#> Item_17 0.835  2.158  1.202  0.405 -0.562
#> Item_18 1.141  0.326 -0.487 -1.363 -2.078
#> Item_19 1.643  1.245  0.573  0.230 -0.155
#> Item_20 1.371  1.869  1.432  0.851  0.234
#> 
#> $means
#>     F1 
#> -0.009 
#> 
#> $cov
#>       F1
#> F1 1.071
#> 
#> 

# RMSDs in Item_1 now reduced (MG model better accounts for DIF)
RMSD_DIF(MGmod)
#> $D1
#>           P.1   P.2   P.3   P.4   P.5
#> Item_1  0.007 0.019 0.017 0.016 0.010
#> Item_2  0.016 0.009 0.009 0.011 0.015
#> Item_3  0.016 0.018 0.014 0.018 0.025
#> Item_4  0.021 0.010 0.010 0.014 0.022
#> Item_5  0.028 0.017 0.007 0.022 0.022
#> Item_6  0.009 0.016 0.015 0.018 0.015
#> Item_7  0.016 0.013 0.021 0.015 0.018
#> Item_8  0.012 0.008 0.012 0.022 0.021
#> Item_9  0.016 0.029 0.023 0.014 0.025
#> Item_10 0.023 0.015 0.010 0.011 0.031
#> Item_11 0.016 0.018 0.015 0.011 0.015
#> Item_12 0.010 0.010 0.022 0.012 0.020
#> Item_13 0.019 0.011 0.017 0.021 0.016
#> Item_14 0.012 0.010 0.005 0.023 0.032
#> Item_15 0.015 0.011 0.016 0.011 0.028
#> Item_16 0.013 0.024 0.020 0.022 0.015
#> Item_17 0.024 0.015 0.019 0.024 0.030
#> Item_18 0.019 0.016 0.017 0.016 0.020
#> Item_19 0.021 0.022 0.016 0.013 0.013
#> Item_20 0.029 0.011 0.012 0.012 0.022
#> 
#> $D2
#>           P.1   P.2   P.3   P.4   P.5
#> Item_1  0.026 0.003 0.019 0.007 0.010
#> Item_2  0.008 0.013 0.004 0.012 0.014
#> Item_3  0.032 0.027 0.016 0.031 0.014
#> Item_4  0.009 0.019 0.011 0.011 0.023
#> Item_5  0.021 0.017 0.010 0.014 0.017
#> Item_6  0.026 0.012 0.007 0.016 0.014
#> Item_7  0.020 0.013 0.019 0.018 0.018
#> Item_8  0.021 0.015 0.017 0.017 0.026
#> Item_9  0.017 0.015 0.015 0.015 0.019
#> Item_10 0.018 0.014 0.014 0.018 0.027
#> Item_11 0.020 0.021 0.017 0.013 0.017
#> Item_12 0.010 0.010 0.022 0.007 0.017
#> Item_13 0.015 0.012 0.013 0.013 0.014
#> Item_14 0.028 0.007 0.022 0.013 0.009
#> Item_15 0.018 0.010 0.015 0.017 0.024
#> Item_16 0.014 0.019 0.018 0.017 0.016
#> Item_17 0.017 0.017 0.028 0.025 0.013
#> Item_18 0.029 0.022 0.025 0.019 0.025
#> Item_19 0.027 0.016 0.015 0.010 0.024
#> Item_20 0.019 0.013 0.014 0.012 0.014
#> 
RMSD_DIF(MGmod, flag = .05)
#> $D1
#>         P.1 P.2 P.3 P.4 P.5
#> Item_1   NA  NA  NA  NA  NA
#> Item_2   NA  NA  NA  NA  NA
#> Item_3   NA  NA  NA  NA  NA
#> Item_4   NA  NA  NA  NA  NA
#> Item_5   NA  NA  NA  NA  NA
#> Item_6   NA  NA  NA  NA  NA
#> Item_7   NA  NA  NA  NA  NA
#> Item_8   NA  NA  NA  NA  NA
#> Item_9   NA  NA  NA  NA  NA
#> Item_10  NA  NA  NA  NA  NA
#> Item_11  NA  NA  NA  NA  NA
#> Item_12  NA  NA  NA  NA  NA
#> Item_13  NA  NA  NA  NA  NA
#> Item_14  NA  NA  NA  NA  NA
#> Item_15  NA  NA  NA  NA  NA
#> Item_16  NA  NA  NA  NA  NA
#> Item_17  NA  NA  NA  NA  NA
#> Item_18  NA  NA  NA  NA  NA
#> Item_19  NA  NA  NA  NA  NA
#> Item_20  NA  NA  NA  NA  NA
#> 
#> $D2
#>         P.1 P.2 P.3 P.4 P.5
#> Item_1   NA  NA  NA  NA  NA
#> Item_2   NA  NA  NA  NA  NA
#> Item_3   NA  NA  NA  NA  NA
#> Item_4   NA  NA  NA  NA  NA
#> Item_5   NA  NA  NA  NA  NA
#> Item_6   NA  NA  NA  NA  NA
#> Item_7   NA  NA  NA  NA  NA
#> Item_8   NA  NA  NA  NA  NA
#> Item_9   NA  NA  NA  NA  NA
#> Item_10  NA  NA  NA  NA  NA
#> Item_11  NA  NA  NA  NA  NA
#> Item_12  NA  NA  NA  NA  NA
#> Item_13  NA  NA  NA  NA  NA
#> Item_14  NA  NA  NA  NA  NA
#> Item_15  NA  NA  NA  NA  NA
#> Item_16  NA  NA  NA  NA  NA
#> Item_17  NA  NA  NA  NA  NA
#> Item_18  NA  NA  NA  NA  NA
#> Item_19  NA  NA  NA  NA  NA
#> Item_20  NA  NA  NA  NA  NA
#> 
RMSD_DIF(MGmod, probfun = FALSE, flag = .1) # use expected score function
#> $D1
#>         S(theta)
#> Item_1        NA
#> Item_2        NA
#> Item_3        NA
#> Item_4        NA
#> Item_5        NA
#> Item_6        NA
#> Item_7        NA
#> Item_8        NA
#> Item_9        NA
#> Item_10       NA
#> Item_11       NA
#> Item_12       NA
#> Item_13       NA
#> Item_14       NA
#> Item_15       NA
#> Item_16       NA
#> Item_17    0.104
#> Item_18       NA
#> Item_19       NA
#> Item_20       NA
#> 
#> $D2
#>         S(theta)
#> Item_1        NA
#> Item_2        NA
#> Item_3        NA
#> Item_4        NA
#> Item_5        NA
#> Item_6        NA
#> Item_7        NA
#> Item_8        NA
#> Item_9        NA
#> Item_10       NA
#> Item_11       NA
#> Item_12       NA
#> Item_13       NA
#> Item_14       NA
#> Item_15       NA
#> Item_16       NA
#> Item_17       NA
#> Item_18       NA
#> Item_19       NA
#> Item_20       NA
#> 

# }

RMSD effect size statistic to quantify category-level DIF

Arguments

References

See also

Author

Examples