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).
wald(object, L, C = NULL)estimated object from mirt, bfactor,
multipleGroup, mixedmirt, or mdirt
a coefficient matrix with dimensions nconstrasts x npars.estimated,
or a character vector giving the hypothesis in symbolic form
(syntax format borrowed from the car package; see Details below).
Omitting this value will return the column names of the
information matrix used to identify the (potentially constrained) parameters
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 for L this argument is ignored
The following description is borrowed from car package documentation pertaining to the character vector
input to the argument L: "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."
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
# \donttest{
# View parnumber index
data(LSAT7)
data <- expand.table(LSAT7)
mod <- mirt(data, 1, SE = TRUE)
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 prior.type
#> 1 all Item.1 dich a1 1 0.9879254 -Inf Inf TRUE none
#> 2 all Item.1 dich d 2 1.8560605 -Inf Inf TRUE none
#> 5 all Item.2 dich a1 5 1.0808847 -Inf Inf TRUE none
#> 6 all Item.2 dich d 6 0.8079786 -Inf Inf TRUE none
#> 9 all Item.3 dich a1 9 1.7058006 -Inf Inf TRUE none
#> 10 all Item.3 dich d 10 1.8042187 -Inf Inf TRUE none
#> 13 all Item.4 dich a1 13 0.7651853 -Inf Inf TRUE none
#> 14 all Item.4 dich d 14 0.4859966 -Inf Inf TRUE none
#> 17 all Item.5 dich a1 17 0.7357980 -Inf Inf TRUE none
#> 18 all Item.5 dich d 18 1.8545127 -Inf Inf TRUE none
#> prior_1 prior_2
#> 1 NaN NaN
#> 2 NaN NaN
#> 5 NaN NaN
#> 6 NaN NaN
#> 9 NaN NaN
#> 10 NaN NaN
#> 13 NaN NaN
#> 14 NaN NaN
#> 17 NaN NaN
#> 18 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)
# 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'))
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
#> 1.235 0.643 1.190 -0.573
#> a1.9.71.133.195 d.10.72.134.196 a1.13.75.137.199 d.14.76.138.200
#> 0.982 -0.140 1.059 0.865
#> a1.17.79.141.203 d.18.80.142.204 a1.21.83.145.207 d.22.84.146.208
#> 1.147 0.248 0.499 0.538
#> a1.25.87.149.211 d.26.88.150.212 a1.29.91.153.215 d.30.92.154.216
#> 1.275 1.204 0.998 -0.368
#> a1.33.95.157.219 d.34.96.158.220 a1.37.99.161.223 d.38.100.162.224
#> 0.900 -1.031 0.736 -1.087
#> a1.41.103.165.227 d.42.104.166.228 a1.45.107.169.231 d.46.108.170.232
#> 1.032 1.346 1.604 -0.159
#> a1.49.111.173.235 d.50.112.174.236 a1.53.115.177.239 d.54.116.178.240
#> 1.273 0.578 1.229 0.502
#> a1.57.119.181.243 d.58.120.182.244 MEAN_1.123 COV_11.124
#> 0.850 -0.079 0.382 0.863
#> MEAN_1.185 COV_11.186 MEAN_1.247 COV_11.248
#> -1.061 0.862 -0.594 0.935
# 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 0.7199774 1 0.3961513
# }