[R-sig-ME] IRT: discrimination parameters from LMER?

[Paul Johnson]

Hello, everybody.

I've fallen into a crowd of IRT users.  I remembered seeing the IRT
with lmer paper.

## Doran, Harold, Douglas Bates, Paul Bliese,
## Maritza Dowling, 2007. “Estimating the
## Multilevel Rasch Model: With the lme4 Package.”
## Journal of Statistical Software 20(2): 1–18.

I had not understood that a Rasch model is a very specialized kind of
logistic regression.   The items are all assumed to have the same
discrimination parameter.

I started to wonder if I could get a 2 parameter logistic IRT from
lmer, and I'm a bit stuck. I'm comparing against the output from the
rasch function in the "ltm" package.  I'm pasting in the code below
for comparison, and after that I have the output so you can compare.

I found a nice BUGS survey for the various kinds of IRT, in case you
wonder what all of these are.  The argument there is the same one that
Doran et al make, in favor of the idea of using general purpose
software for IRT.

Curtis, S. McKay. 2010. “BUGS Code for Item Response Theory.” Journal
of Statistical Software, Code Snippets 36(1): 1–34.

In the most restrictive form of the Rasch model, the discrimination
parameter is assumed to be 1, but there is an estimate of the standard
deviation of the latent ability coefficient.  In other implementations
of Rasch, it is assumed the latent trait is N(0,1) and then a
discrimination coefficient is estimated.

I *believe* that is the point of comparison from  ltm's rasch to
lmer's random effect estimate. ltm rasch assumes the individual
ability parameter is N(0,1) and estimates a discrimination coefficient
common to all items

disc * N(0,1)

and the lmer estimates disc as the standard deviation of the random effect.

N(0, disc^2).

So the random effect standard deviation estimated by lmer is one
estimate of that same disc parameter.

This little simulation shows that lmer's fixed effects for the items
are very close to the same as the difficulty parameters in rasch.  The
one disc parameter estimate from ltm is usually closer to the "true"
discrimination value than the standard deviation of the random effect
in lmer.  However, they are "in the ballpark".

Partly I'm curious to know if you agree I'm comparing the numbers properly.

But, also, I want to estimate a different discrimination parameter for
each item, so I can get an IRT 2PL model. It would be nice to get a
3PL to allow a guessing parameter.

Once I realized this mismatch between lmer and the more general IRT, I
started to think either 1) we really do need specialized IRT software,
or 2) perhaps I need to learn to write family code for lmer.

PJ

## Paul Johnson Oct. 30, 2010
## Item Response Theory, compare the
## IRT-specific R package "ltm" with mixed-effct GLM
## perspective of lmer in "lme4" package.

## The argument against IRT specific software
## is that it restricts the user. May be better
## to adapt general purpose programs to
## estimate same models.  That's argued in:

library(ltm)

library(lme4)


### Create Data: 300 respondents with 3 items each.
N <- 300
Adiff <- c(-1, 2, 2) ### actually, easiness
Bdisc <- rep(2.2, 3) ### rasch requires all same!

z <- rnorm( N, 0, sd=1)

#create one long column for latent trait
z3 <- kronecker(Adiff, rep(1,N)) - kronecker(Bdisc, z)
## caution about signs of parameters. Very
## confusing, standard IRT is
## Bdisc ( theta - Adiff)
## So if Adiff = c(-1, 2, 2), IRT standard
## difficulty estimates will be sign reversed
## c(0.5, -0.84, -0.84)


pr <- 1/(1 + exp(-1*(z3)))
y <- rbinom(length(pr), prob=pr, size=1)

id <- rep(1:N, 3)
item <- c( rep(1, N), rep(2, N), rep(3, N))

### the item matrix, N by 3
maty <- matrix(y, ncol=3)


### Long version of data frame for use with lmer
dflong <- data.frame("id"=id, "item"=item, "value"=y)

### convert "item" to factor variable
dflong$item <- as.factor(dflong$item)
### create indicators for items, just experimenting
dflong$item1 <- ifelse(dflong$item == "1",1,0)
dflong$item2 <- ifelse(dflong$item == "2",1,0)
dflong$item3 <- ifelse(dflong$item == "3",1,0)

### fit with rasch in ltm,
### Transform estimates with IRT.param=F,
### gives values comparable to lmer, it gives
### parameters in form b0 + b1*theta
### rather than IRT code like b1(b0 - theta)

dfrasch <- rasch(maty, IRT.param=F)
summary(dfrasch)

### Compare to IRT standard coding, for fun
dfrasch2 <- rasch(maty)
summary(dfrasch2)
### Note discrim param is same as dfrasch, but
### difficulties are different. Can convert
### the dfrasch2 difficulties by multiplying by
### discrimination.


### use lmer, estimates of fixed effect parameters
### "item1", "item2", "item3" are "difficulty
### parameters.

dflmer <- lmer (value ~ -1 + item1 + item2 + item3 + (1|id) ,
data=dflong, family=binomial)
summary(dflmer)

### How to estimate the "discrimination parameter"?
### I'm just guessing now. rasch assumes all
### discrimination parameters for all items are same.
### Recall x ~ N(0, s^2), same as x ~ s * N(0,1).
### So estimated standard deviation of random effect
### is same as one discrimination parameter "s" in
### above.  I think.


dflmer2 <- lmer (value ~ -1 + item + (1|id) , data=dflong, family=binomial)
summary(dflmer2)


ldfy <- ltm( dfy ~ z1 , IRT.param=F)
summary(ldfy)

##################################################


Here's the output:


> library(ltm)
Loading required package: MASS
Loading required package: msm
Loading required package: mvtnorm
Loading required package: polycor
Loading required package: sfsmisc

This is package 'ltm' version '0.9-5'

> library(lme4)
Loading required package: Matrix
Loading required package: lattice

Attaching package: 'Matrix'

The following object(s) are masked from 'package:base':

    det


Attaching package: 'lme4'

The following object(s) are masked from 'package:stats':

    AIC

> N <- 300
> Adiff <- c(-1, 2, 2) ### actually, easiness
> Bdisc <- rep(2.2, 3) ### rasch requires all same!
> z <- rnorm( N, 0, sd=1)
> z3 <- kronecker(Adiff, rep(1,N)) - kronecker(Bdisc, z)
> pr <- 1/(1 + exp(-1*(z3)))
> y <- rbinom(length(pr), prob=pr, size=1)
> id <- rep(1:N, 3)
> item <- c( rep(1, N), rep(2, N), rep(3, N))
> maty <- matrix(y, ncol=3)
> dflong <- data.frame("id"=id, "item"=item, "value"=y)
> dflong$item <- as.factor(dflong$item)
> dflong$item1 <- ifelse(dflong$item == "1",1,0)
> dflong$item2 <- ifelse(dflong$item == "2",1,0)
> dflong$item3 <- ifelse(dflong$item == "3",1,0)
> dfrasch <- rasch(maty, IRT.param=F)
summary(dfrasch)
>
Call:
rasch(data = maty, IRT.param = F)

Model Summary:
   log.Lik      AIC      BIC
 -483.7193 975.4385 990.2537

Coefficients:
        value std.err  z.vals
Item1 -1.1239  0.2029 -5.5392
Item2  1.8955  0.2371  7.9957
Item3  1.7567  0.2305  7.6227
z      1.8959  0.2336  8.1157

Integration:
method: Gauss-Hermite
quadrature points: 21

Optimization:
Convergence: 0
max(|grad|): 0.003
quasi-Newton: BFGS


> dfrasch2 <- rasch(maty)
> summary(dfrasch2)

Call:
rasch(data = maty)

Model Summary:
   log.Lik      AIC      BIC
 -483.7193 975.4385 990.2537

Coefficients:
                value std.err  z.vals
Dffclt.Item 1  0.5928  0.1082  5.4799
Dffclt.Item 2 -0.9998  0.1249 -8.0022
Dffclt.Item 3 -0.9266  0.1211 -7.6502
Dscrmn         1.8959  0.2336  8.1157

Integration:
method: Gauss-Hermite
quadrature points: 21

Optimization:
Convergence: 0
max(|grad|): 0.003
quasi-Newton: BFGS


> dflmer <- lmer (value ~ -1 + item1 + item2 + item3 + (1|id) , data=dflong, family=binomial)
> summary(dflmer)
Generalized linear mixed model fit by the Laplace approximation
Formula: value ~ -1 + item1 + item2 + item3 + (1 | id)
   Data: dflong
 AIC  BIC logLik deviance
 984 1003   -488      976
Random effects:
 Groups Name        Variance Std.Dev.
 id     (Intercept) 2.9100   1.7059
Number of obs: 900, groups: id, 300

Fixed effects:
      Estimate Std. Error z value Pr(>|z|)
item1  -1.1155     0.1754  -6.358 2.04e-10 ***
item2   1.8382     0.1938   9.486  < 2e-16 ***
item3   1.7049     0.1898   8.984  < 2e-16 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Correlation of Fixed Effects:
      item1 item2
item2 0.247
item3 0.255 0.310
>




-- 
Paul E. Johnson
Professor, Political Science
1541 Lilac Lane, Room 504
University of Kansas