Compute a Wald test given an L vector or matrix of numeric contrasts. Requires that the
model information matrix be computed (by passing SE = TRUE when estimating the model). Use
wald(model) to observe how the information matrix columns are named, especially if
the estimated model contains constrained parameters (e.g., 1PL).
Arguments
- object
estimated object from
mirt,bfactor,multipleGroup,mixedmirt, ormdirt- L
a coefficient matrix with dimensions
nconstrasts x npars.estimated, or a character vector giving the hypothesis in symbolic form (syntax format borrowed from thecarpackage; seeDetailsbelow). Omitting this value will return the column names of the information matrix used to identify the (potentially constrained) parameters- C
a constant vector of population parameters to be compared along side L, where
length(C) == row(L). By default a vector of 0's is constructed. Note that when using the syntax input forLthis argument is ignoredThe following description is borrowed from
carpackage documentation pertaining to the character vector input to the argumentL: "The hypothesis matrix can be supplied as a numeric matrix (or vector), the rows of which specify linear combinations of the model coefficients, which are tested equal to the corresponding entries in the right-hand-side vector, which defaults to a vector of zeroes.Alternatively, the hypothesis can be specified symbolically as a character vector with one or more elements, each of which gives either a linear combination of coefficients, or a linear equation in the coefficients (i.e., with both a left and right side separated by an equals sign). Components of a linear expression or linear equation can consist of numeric constants, or numeric constants multiplying coefficient names (in which case the number precedes the coefficient, and may be separated from it by spaces or an asterisk); constants of 1 or -1 may be omitted. Spaces are always optional. Components are separated by plus or minus signs. Newlines or tabs in hypotheses will be treated as spaces. See the examples below."
References
Chalmers, R., P. (2012). mirt: A Multidimensional Item Response Theory Package for the R Environment. Journal of Statistical Software, 48(6), 1-29. doi:10.18637/jss.v048.i06
Author
Phil Chalmers rphilip.chalmers@gmail.com
Examples
# \donttest{
# View parnumber index
data(LSAT7)
data <- expand.table(LSAT7)
mod <- mirt(data, 1, SE = TRUE)
#>
Iteration: 1, Log-Lik: -2668.786, Max-Change: 0.18243
Iteration: 2, Log-Lik: -2663.691, Max-Change: 0.13637
Iteration: 3, Log-Lik: -2661.454, Max-Change: 0.10231
Iteration: 4, Log-Lik: -2659.430, Max-Change: 0.04181
Iteration: 5, Log-Lik: -2659.241, Max-Change: 0.03417
Iteration: 6, Log-Lik: -2659.113, Max-Change: 0.02911
Iteration: 7, Log-Lik: -2658.812, Max-Change: 0.00456
Iteration: 8, Log-Lik: -2658.809, Max-Change: 0.00363
Iteration: 9, Log-Lik: -2658.808, Max-Change: 0.00273
Iteration: 10, Log-Lik: -2658.806, Max-Change: 0.00144
Iteration: 11, Log-Lik: -2658.806, Max-Change: 0.00118
Iteration: 12, Log-Lik: -2658.806, Max-Change: 0.00101
Iteration: 13, Log-Lik: -2658.805, Max-Change: 0.00042
Iteration: 14, Log-Lik: -2658.805, Max-Change: 0.00025
Iteration: 15, Log-Lik: -2658.805, Max-Change: 0.00026
Iteration: 16, Log-Lik: -2658.805, Max-Change: 0.00023
Iteration: 17, Log-Lik: -2658.805, Max-Change: 0.00023
Iteration: 18, Log-Lik: -2658.805, Max-Change: 0.00021
Iteration: 19, Log-Lik: -2658.805, Max-Change: 0.00019
Iteration: 20, Log-Lik: -2658.805, Max-Change: 0.00017
Iteration: 21, Log-Lik: -2658.805, Max-Change: 0.00017
Iteration: 22, Log-Lik: -2658.805, Max-Change: 0.00015
Iteration: 23, Log-Lik: -2658.805, Max-Change: 0.00015
Iteration: 24, Log-Lik: -2658.805, Max-Change: 0.00013
Iteration: 25, Log-Lik: -2658.805, Max-Change: 0.00013
Iteration: 26, Log-Lik: -2658.805, Max-Change: 0.00011
Iteration: 27, Log-Lik: -2658.805, Max-Change: 0.00011
Iteration: 28, Log-Lik: -2658.805, Max-Change: 0.00010
#>
#> Calculating information matrix...
coef(mod)
#> $Item.1
#> a1 d g u
#> par 0.988 1.856 0 1
#> CI_2.5 0.641 1.598 NA NA
#> CI_97.5 1.335 2.114 NA NA
#>
#> $Item.2
#> a1 d g u
#> par 1.081 0.808 0 1
#> CI_2.5 0.750 0.629 NA NA
#> CI_97.5 1.412 0.987 NA NA
#>
#> $Item.3
#> a1 d g u
#> par 1.706 1.804 0 1
#> CI_2.5 1.078 1.404 NA NA
#> CI_97.5 2.334 2.205 NA NA
#>
#> $Item.4
#> a1 d g u
#> par 0.765 0.486 0 1
#> CI_2.5 0.502 0.339 NA NA
#> CI_97.5 1.028 0.633 NA NA
#>
#> $Item.5
#> a1 d g u
#> par 0.736 1.855 0 1
#> CI_2.5 0.440 1.630 NA NA
#> CI_97.5 1.032 2.079 NA NA
#>
#> $GroupPars
#> MEAN_1 COV_11
#> par 0 1
#> CI_2.5 NA NA
#> CI_97.5 NA NA
#>
# see how the information matrix relates to estimated parameters, and how it lines up
# with the parameter index
(infonames <- wald(mod))
#> a1.1 d.2 a1.5 d.6 a1.9 d.10 a1.13 d.14 a1.17 d.18
#> 0.988 1.856 1.081 0.808 1.706 1.804 0.765 0.486 0.736 1.855
index <- mod2values(mod)
index[index$est, ]
#> group item class name parnum value lbound ubound est const nconst
#> 1 all Item.1 dich a1 1 0.988 -Inf Inf TRUE none none
#> 2 all Item.1 dich d 2 1.856 -Inf Inf TRUE none none
#> 5 all Item.2 dich a1 5 1.081 -Inf Inf TRUE none none
#> 6 all Item.2 dich d 6 0.808 -Inf Inf TRUE none none
#> 9 all Item.3 dich a1 9 1.706 -Inf Inf TRUE none none
#> 10 all Item.3 dich d 10 1.804 -Inf Inf TRUE none none
#> 13 all Item.4 dich a1 13 0.765 -Inf Inf TRUE none none
#> 14 all Item.4 dich d 14 0.486 -Inf Inf TRUE none none
#> 17 all Item.5 dich a1 17 0.736 -Inf Inf TRUE none none
#> 18 all Item.5 dich d 18 1.855 -Inf Inf TRUE none none
#> prior.type prior_1 prior_2
#> 1 none NaN NaN
#> 2 none NaN NaN
#> 5 none NaN NaN
#> 6 none NaN NaN
#> 9 none NaN NaN
#> 10 none NaN NaN
#> 13 none NaN NaN
#> 14 none NaN NaN
#> 17 none NaN NaN
#> 18 none NaN NaN
# second item slope equal to 0?
L <- matrix(0, 1, 10)
L[1,3] <- 1
wald(mod, L)
#> W df p
#> 1 41.0005 1 1.521906e-10
# same as above using character syntax input
infonames
#> a1.1 d.2 a1.5 d.6 a1.9 d.10 a1.13 d.14 a1.17 d.18
#> 0.988 1.856 1.081 0.808 1.706 1.804 0.765 0.486 0.736 1.855
wald(mod, "a1.5 = 0")
#> W df p
#> 1 41.0005 1 1.521906e-10
# simultaneously test equal factor slopes for item 1 and 2, and 4 and 5
L <- matrix(0, 2, 10)
L[1,1] <- L[2, 7] <- 1
L[1,3] <- L[2, 9] <- -1
L
#> [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10]
#> [1,] 1 0 -1 0 0 0 0 0 0 0
#> [2,] 0 0 0 0 0 0 1 0 -1 0
wald(mod, L)
#> W df p
#> 1 0.1528867 2 0.9264054
# Again, using more efficient syntax
infonames
#> a1.1 d.2 a1.5 d.6 a1.9 d.10 a1.13 d.14 a1.17 d.18
#> 0.988 1.856 1.081 0.808 1.706 1.804 0.765 0.486 0.736 1.855
wald(mod, c("a1.1 = a1.5", "a1.13 = a1.17"))
#> W df p
#> 1 0.1528867 2 0.9264054
# log-Liklihood tests (requires estimating a new model)
cmodel <- 'theta = 1-5
CONSTRAIN = (1,2, a1), (4,5, a1)'
mod2 <- mirt(data, cmodel)
#>
Iteration: 1, Log-Lik: -2668.786, Max-Change: 0.18243
Iteration: 2, Log-Lik: -2663.808, Max-Change: 0.13888
Iteration: 3, Log-Lik: -2661.502, Max-Change: 0.10551
Iteration: 4, Log-Lik: -2659.196, Max-Change: 0.03290
Iteration: 5, Log-Lik: -2659.083, Max-Change: 0.02744
Iteration: 6, Log-Lik: -2659.016, Max-Change: 0.02305
Iteration: 7, Log-Lik: -2658.892, Max-Change: 0.00707
Iteration: 8, Log-Lik: -2658.888, Max-Change: 0.00613
Iteration: 9, Log-Lik: -2658.886, Max-Change: 0.00488
Iteration: 10, Log-Lik: -2658.882, Max-Change: 0.00116
Iteration: 11, Log-Lik: -2658.882, Max-Change: 0.00166
Iteration: 12, Log-Lik: -2658.882, Max-Change: 0.00069
Iteration: 13, Log-Lik: -2658.882, Max-Change: 0.00058
Iteration: 14, Log-Lik: -2658.882, Max-Change: 0.00236
Iteration: 15, Log-Lik: -2658.882, Max-Change: 0.00040
Iteration: 16, Log-Lik: -2658.882, Max-Change: 0.00036
Iteration: 17, Log-Lik: -2658.881, Max-Change: 0.00027
Iteration: 18, Log-Lik: -2658.881, Max-Change: 0.00030
Iteration: 19, Log-Lik: -2658.881, Max-Change: 0.00024
Iteration: 20, Log-Lik: -2658.881, Max-Change: 0.00025
Iteration: 21, Log-Lik: -2658.881, Max-Change: 0.00022
Iteration: 22, Log-Lik: -2658.881, Max-Change: 0.00021
Iteration: 23, Log-Lik: -2658.881, Max-Change: 0.00018
Iteration: 24, Log-Lik: -2658.881, Max-Change: 0.00018
Iteration: 25, Log-Lik: -2658.881, Max-Change: 0.00016
Iteration: 26, Log-Lik: -2658.881, Max-Change: 0.00015
Iteration: 27, Log-Lik: -2658.881, Max-Change: 0.00014
Iteration: 28, Log-Lik: -2658.881, Max-Change: 0.00013
Iteration: 29, Log-Lik: -2658.881, Max-Change: 0.00012
Iteration: 30, Log-Lik: -2658.881, Max-Change: 0.00011
Iteration: 31, Log-Lik: -2658.881, Max-Change: 0.00010
Iteration: 32, Log-Lik: -2658.881, Max-Change: 0.00010
# or, equivalently
#mod2 <- mirt(data, 1, constrain = list(c(1,5), c(13,17)))
anova(mod2, mod)
#> AIC SABIC HQ BIC logLik X2 df p
#> mod2 5333.763 5347.616 5348.685 5373.025 -2658.881
#> mod 5337.610 5354.927 5356.263 5386.688 -2658.805 0.152 2 0.927
#####
# test equality of means in multi-group model:
# H0: (mu1 - mu2) = (mu3 - mu4)
set.seed(12345)
a <- matrix(abs(rnorm(15,1,.3)), ncol=1)
d <- matrix(rnorm(15,0,.7),ncol=1)
itemtype <- rep('2PL', nrow(a))
N <- 500
dataset1 <- simdata(a, d, N, itemtype)
dataset2 <- simdata(a, d, N, itemtype, mu = .5)
dataset3 <- simdata(a, d, N, itemtype, mu = -1)
dataset4 <- simdata(a, d, N, itemtype, mu = -.5)
dat <- rbind(dataset1, dataset2, dataset3, dataset4)
group <- factor(rep(paste0('D', 1:4), each=N))
levels(group)
#> [1] "D1" "D2" "D3" "D4"
models <- 'F1 = 1-15'
# 3 means estimated
mod_free <- multipleGroup(dat, models, group = group, SE=TRUE,
invariance=c('slopes', 'intercepts', 'free_var','free_means'))
#>
Iteration: 1, Log-Lik: -17745.920, Max-Change: 0.38568
Iteration: 2, Log-Lik: -17572.697, Max-Change: 0.14556
Iteration: 3, Log-Lik: -17560.767, Max-Change: 0.05810
Iteration: 4, Log-Lik: -17558.317, Max-Change: 0.02706
Iteration: 5, Log-Lik: -17556.791, Max-Change: 0.01811
Iteration: 6, Log-Lik: -17555.521, Max-Change: 0.01676
Iteration: 7, Log-Lik: -17553.494, Max-Change: 0.06689
Iteration: 8, Log-Lik: -17550.635, Max-Change: 0.01867
Iteration: 9, Log-Lik: -17550.091, Max-Change: 0.01255
Iteration: 10, Log-Lik: -17549.198, Max-Change: 0.02525
Iteration: 11, Log-Lik: -17548.652, Max-Change: 0.01024
Iteration: 12, Log-Lik: -17548.384, Max-Change: 0.00795
Iteration: 13, Log-Lik: -17547.951, Max-Change: 0.02867
Iteration: 14, Log-Lik: -17547.362, Max-Change: 0.00836
Iteration: 15, Log-Lik: -17547.242, Max-Change: 0.00565
Iteration: 16, Log-Lik: -17547.038, Max-Change: 0.01271
Iteration: 17, Log-Lik: -17546.901, Max-Change: 0.00473
Iteration: 18, Log-Lik: -17546.842, Max-Change: 0.00380
Iteration: 19, Log-Lik: -17546.746, Max-Change: 0.01310
Iteration: 20, Log-Lik: -17546.618, Max-Change: 0.00371
Iteration: 21, Log-Lik: -17546.591, Max-Change: 0.00251
Iteration: 22, Log-Lik: -17546.545, Max-Change: 0.00613
Iteration: 23, Log-Lik: -17546.513, Max-Change: 0.00213
Iteration: 24, Log-Lik: -17546.500, Max-Change: 0.00183
Iteration: 25, Log-Lik: -17546.478, Max-Change: 0.00613
Iteration: 26, Log-Lik: -17546.449, Max-Change: 0.00164
Iteration: 27, Log-Lik: -17546.443, Max-Change: 0.00120
Iteration: 28, Log-Lik: -17546.432, Max-Change: 0.00304
Iteration: 29, Log-Lik: -17546.424, Max-Change: 0.00095
Iteration: 30, Log-Lik: -17546.422, Max-Change: 0.00088
Iteration: 31, Log-Lik: -17546.416, Max-Change: 0.00288
Iteration: 32, Log-Lik: -17546.410, Max-Change: 0.00071
Iteration: 33, Log-Lik: -17546.409, Max-Change: 0.00058
Iteration: 34, Log-Lik: -17546.406, Max-Change: 0.00156
Iteration: 35, Log-Lik: -17546.404, Max-Change: 0.00043
Iteration: 36, Log-Lik: -17546.404, Max-Change: 0.00042
Iteration: 37, Log-Lik: -17546.402, Max-Change: 0.00135
Iteration: 38, Log-Lik: -17546.401, Max-Change: 0.00030
Iteration: 39, Log-Lik: -17546.401, Max-Change: 0.00028
Iteration: 40, Log-Lik: -17546.400, Max-Change: 0.00085
Iteration: 41, Log-Lik: -17546.400, Max-Change: 0.00021
Iteration: 42, Log-Lik: -17546.399, Max-Change: 0.00019
Iteration: 43, Log-Lik: -17546.399, Max-Change: 0.00059
Iteration: 44, Log-Lik: -17546.399, Max-Change: 0.00014
Iteration: 45, Log-Lik: -17546.399, Max-Change: 0.00013
Iteration: 46, Log-Lik: -17546.399, Max-Change: 0.00040
Iteration: 47, Log-Lik: -17546.399, Max-Change: 0.00009
#>
#> Calculating information matrix...
wald(mod_free) # obtain parameter names
#> a1.1.63.125.187 d.2.64.126.188 a1.5.67.129.191 d.6.68.130.192
#> 1.234 0.643 1.189 -0.573
#> a1.9.71.133.195 d.10.72.134.196 a1.13.75.137.199 d.14.76.138.200
#> 0.981 -0.141 1.058 0.864
#> a1.17.79.141.203 d.18.80.142.204 a1.21.83.145.207 d.22.84.146.208
#> 1.146 0.248 0.499 0.537
#> a1.25.87.149.211 d.26.88.150.212 a1.29.91.153.215 d.30.92.154.216
#> 1.274 1.204 0.997 -0.368
#> a1.33.95.157.219 d.34.96.158.220 a1.37.99.161.223 d.38.100.162.224
#> 0.899 -1.032 0.735 -1.087
#> a1.41.103.165.227 d.42.104.166.228 a1.45.107.169.231 d.46.108.170.232
#> 1.031 1.345 1.603 -0.160
#> a1.49.111.173.235 d.50.112.174.236 a1.53.115.177.239 d.54.116.178.240
#> 1.272 0.578 1.228 0.501
#> a1.57.119.181.243 d.58.120.182.244 MEAN_1.123 COV_11.124
#> 0.850 -0.080 0.382 0.865
#> MEAN_1.185 COV_11.186 MEAN_1.247 COV_11.248
#> -1.062 0.864 -0.594 0.936
# View(mod2values(mod_free))
# reference group mean = 0 by default
wald(mod_free, c("0 - MEAN_1.123 = MEAN_1.185 - MEAN_1.247"))
#> W df p
#> 1 0.7142209 1 0.3980461
# }