I have two question regarding the intercept setting and Embretson's multidimensional latent trait model for measuring learning and change. 1 The first question is: Does setting the mean at dim1 or dim2 to zero change the mean differences between the two dims(time points)? As far as I know, the absolute value of the mean difference between two time points should be the same or quite similar. However, when I run TAM and SIRT on the same data, the result is quite different? What is the reason for this?
# When using SIRT, the follwing syntax was used except for a minor modification for fixing mean at T2 or T1 to zero.
#mu.fixed <- cbind( 2,0 ) #fixing mean at T2
#or
#mu.fixed <- cbind( 1,0 ) #fixing mean at T1
# When using TAM, I use the following syntax except for a minor modification listed as follows.
#beta.fixed <- cbind( 1, 2, 0 ) #fixing mean at T2
#or
#beta.fixed <- cbind( 1, 1, 0 ) #fixing mean at T1
note: the original syntax is from: https://rdrr.io/cran/sirt/man/data.long.html
data(data.long)
dat <- data.long
dat <- dat[,-1]
I <- ncol(dat)
#*************************************************
# Model 1: 2-dimensional Rasch model
#*************************************************
# define Q-matrix
Q <- matrix(0,I,2)
Q[1:6,1] <- 1
Q[7:12,2] <- 1
rownames(Q) <- colnames(dat)
colnames(Q) <- c("T1","T2")
# vector with same items
itemnr <- as.numeric( substring( colnames(dat),2,2) )
# fix mean at T2 to zero
mu.fixed <- cbind( 2,0 )
#--- M1a: rasch.mml2 (in sirt)
mod1a <- sirt::rasch.mml2(dat, Q=Q, est.b=itemnr, mu.fixed=mu.fixed)
summary(mod1a)
#--- M1c: tam.mml (in TAM)
# assume equal item difficulty of I3T1 and I3T2, I4T1 and I4T2, ...
# create draft design matrix and modify it
A <- TAM::designMatrices(resp=dat)$A
dimnames(A)[[1]] <- colnames(dat)
A1 <- A[,, c(1:6, 11:12 ) ]
A1[7,2,3] <- -1 # difficulty(I3T1)=difficulty(I3T2)
A1[8,2,4] <- -1 # I4T1=I4T2
A1[9,2,5] <- A1[10,2,6] <- -1
dimnames(A1)[[3]] <- substring( dimnames(A1)[[3]],1,2)
# estimate model
# set intercept of second dimension (T2) to zero
beta.fixed <- cbind( 1, 2, 0 )
mod1c <- TAM::tam.mml( resp=dat, Q=Q, A=A1, beta.fixed=beta.fixed)
summary(mod1c)
Based on the above analysis, using SIRT showed the mean of [-1.574, 0]'[0,1.667]. However, the result showed the mean of [-1.574,0],[0,-0.453] when using the TAM.
Q2 The second question is: May I model the learning and change model proposed by Embretson in 1991 using the above syntax except for a minor change of using the follwing Q matrix? The initial theta is loading on every items. while, the modifiability ability is only loading on the items measured in time 2.
Q <- matrix(0,I,2)
Q[1:6,1] <- 1
Q[1:12,2] <- 1