M2.StructuredFactor | R Documentation |
The limited information M_2 statistic (Maydeu-Olivares and Joe, 2006) of bi-factor and second-order copula models for item response data.
M2Bifactor(y,cpar, copnames1, copnames2, gl, ngrp, grpsize) M2Second_order(y,cpar, copnames1, copnames2, gl, ngrp, grpsize)
y |
n \times d matrix with the ordinal reponse data, where n and d is the number of observations and variables, respectively. |
cpar |
A list of estimated copula parameters. |
copnames1 |
For the bi-factor copula: d-vector with the names of bivariate copulas that link each of the oberved variabels with the common factor. For the second-order factor copula: G-vector with the names of bivariate copulas that link the each of the group-specific factors with the common factor, where G is the number of groups of items. Choices are “bvn” for BVN, “bvtν” with ν = \{2, …, 9\} degrees of freedom for t-copula, “frk” for Frank, “gum” for Gumbel, “rgum” for reflected Gumbel, “1rgum” for 1-reflected Gumbel, “2rgum” for 2-reflected Gumbel. |
copnames2 |
For the bi-factor copula: d-vector with the names of bivariate copulas that link the each of the oberved variabels with the group-specific factor. For the second-order factor copula: d-vector with the names of bivariate copulas that link the each of the oberved variabels with the group-specific factor. Choices are “bvn” for BVN, “bvtν” with ν = \{2, …, 9\} degrees of freedom for t-copula, “frk” for Frank, “gum” for Gumbel, “rgum” for reflected Gumbel, “1rgum” for 1-reflected Gumbel, “2rgum” for 2-reflected Gumbel. |
gl |
Gauss legendre quardrature nodes and weights. |
ngrp |
number of non-overlapping groups. |
grpsize |
vector indicating the size for each group, e.g., c(4,4,4) indicating four items in all three groups. |
The M_2 statistic has been developed for goodness-of-fit testing in multidimensional contingency tables by Maydeu-Olivares and Joe (2006). We use the M_2 to assess the overall fit for the bi-factor and second-order copula models for item resposne data (Kadhem & Nikoloulopoulos, 2021).
A list containing the following components:
M2 |
The M_2 statistic which has a null asymptotic distribution that is χ^2 with s-q degrees of freedom, where s is the number of univariate and bivariate margins that do not include the category 0 and q is the number of model parameters. |
df |
s-q. |
p-value |
The resultant p-value. |
Sayed H. Kadhem s.kadhem@uea.ac.uk
Aristidis K. Nikoloulopoulos a.nikoloulopoulos@uea.ac.uk
Kadhem, S.H. and Nikoloulopoulos, A.K. (2023) Bi-factor and second-order copula models for item response data. Psychometrika, 88, 132–157. doi: 10.1007/s11336-022-09894-2.
Maydeu-Olivares, A. and Joe, H. (2006). Limited information goodness-of-fit testing in multidimensional contingency tables. Psychometrika, 71, 713–732. doi: 10.1007/s11336-005-1295-9.
#------------------------------------------------ # Setting quadreture points nq <- 15 gl <- gauss.quad.prob(nq) #------------------------------------------------ # TAS Data #------------------ ----------------- data(TAS) #using a subset of the data #group1: 1,3,6,7,9,13,14 grp1=c(1,3,6) #group2: 2,4,11,12,17 grp2=c(2,4,11) #group3: 5,8,10,15,16,18,19,20 grp3=c(5,8,10) #Rearrange items within testlets set.seed(123) i=sample(1:nrow(TAS),500) ydat=TAS[i,c(grp1,grp2,grp3)] d=ncol(ydat);d n=nrow(ydat);n #size of each group g1=length(grp1) g2=length(grp2) g3=length(grp3) grpsize=c(g1,g2,g3)#group size #number of groups ngrp=length(grpsize) #------------------------------------------------ # M2 #------------------------------------------------ #BI-FACTOR tauX0 = c(0.49,0.16,0.29,#0.09,0.47,0.49,0.30, 0.46,0.41,0.33,#0.29,0.24, 0.10,0.16,0.14)#,0.12,0.03,0.03,0.10,0.10) tauXg = c(0.09,0.37,0.23,#0.53,0.24,0.32,0.27, 0.53,0.58,0.20,#0.23,0.25,0.34,0.33, 0.30,0.19,0.24)#,0.29,0.43,0.26) copX0 = rep("bvt2", d) copXg = c(rep("rgum", g1), rep("bvt3", g2+g3)) #converting taus to cpars cparX0=mapply(function(x,y) tau2par(x,y),x=copX0,y=tauX0) cparXg=mapply(function(x,y) tau2par(x,y),x=copXg,y=tauXg) cpar=c(cparX0,cparXg) m2_Bifactor = M2Bifactor(y=ydat, cpar, copX0, copXg, gl, ngrp, grpsize)
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