Item Parameter Estimation: Does WinBUGS Do Better Than BILOG-MG?

Bayesian Statistics, Fall 2009Chunyan Liu & James Gambrell

Introduction3 Parameter IRT ModelAssigns each item a logistic function with

a variable lower asymptote.

Purpose

Compare BILOG-MG and WinBUGS estimation of item parameters under the 3 parameter logistic (3PL) IRT model

Investigate the effect of sample size on the estimation of item parameters

BILOG – MG (Mislevy & Bock 1985)Propriety softwareUses unknown estimation shortcutsSometimes gives poor results“Black Box” programVery fast estimationProvides only point estimates and standard errors for

model parametersEstimation method

◦Marginal Maximum Likelihood◦Expectation-Maximization algorithm (Bock and Aitkin,

1981)

WinBUGSMore open-source (related to OpenBugs)More widely studiedMight give more robust resultsMuch more flexibleProvides full posterior densities for model

parametersMore output to evaluate convergenceVery slow estimation!

Literature ReviewMost researchers have used custom-built

MCMC samplers using Metropolis-Hastings- within-Gibbs algorithm ◦ as recommended by Cowles, 1996!

Patz and Junker (1999a & b)◦Wrote MCMC sampler in S plus◦Found that their sampler produced estimates

identical to BILOG for the 2PL model, but had some trouble with 3PL models.

◦Found MCMC was superior at handling missing data.

Literature ReviewJones and Nediak (2000)

◦Developed “commercial grade” sampler in C++◦ Improved the Patz and Junker algoritm◦Compared MCMC results to BILOG using both real and

simulated data◦Found that item parameters varied substantially, but the

ICCs described were close according to the Hellinger deviance criterion

◦MCMC and BILOG were similar for real data◦MCMC was superior for simulated data◦Note that MCMC provides much more diagnostic out to

assess convergence problems

Literature ReviewProctor, Teo, Hou, and Hsieh (2005

project for this class!)◦Compared BILOG to WinBUGS◦Fit a 2PL model◦Only simulated a single replication◦Did not use deviance or RMSE to assess error

DataTest: 36-item multiple choice

Item parameters (a, b and c) come from Chapter 6 of Equating, Scaling and Linking (Kolen and Brennan)◦ Treated as true item parameters (See Appendix)

Item responses simulated using 3PL model

a – slope b – difficulty c – guessing – examinee ability

1( )1 exp( 1.7 ( ))

cp ca b

Methods1. N (N=200, 500, 1000, 2000) θ values were generated

from N(0,1) distribution.

2. N item responses were simulated based on the θ’s generated in step 1 and the true item parameters using the 3PL model.

3. Item parameters (a, b, c for the 36 items) were estimated using BILOG-MG based on the N item responses.

4. Item parameters (a, b, c for the 36 items) were estimated using WinBUGS based on the N item responses using the same prior as specified by BILOG-MG.

5. Repeat steps two and four 100 times. For each item, we have 100 estimated parameter sets from both programs

Priors

a[i] ~ dlnorm(0, 4) b[i] ~ dnorm(0, 0.25) c[i] ~ dbeta(5,17)

Same priors used in BILOG and WinBUGS

Criterion-Root Mean Square Error (RMSE)

For each item, we computed the RMSE for a, b, and c using the same formula

where and

Here could be , , or and x could be the parameter of a, b or c

2 2ˆ ˆ ˆ( ) ( ) ( )RMSE x Bias x sd x

ˆ ˆ( ) ( )Bias x E x x

100

1

ˆˆ( )

100

ii

xE x

a b cx

Results1. Deciding the number of Burn-in Iterations- History Plots

a[28] chains 1:3

iteration

1 2000 4000 6000

0.0

2.0

4.0

6.0

b[28] chains 1:3

iteration

1 2000 4000 6000

-4.0

-2.0

0.0

2.0

4.0

c[28] chains 1:3

iteration

1 2000 4000 6000

0.0

0.1

0.2

0.3

a[28] chains 1:3

lag

0 20 40

-1.0 -0.5 0.0 0.5 1.0

b[28] chains 1:3

lag

0 20 40

-1.0 -0.5 0.0 0.5 1.0

c[28] chains 1:3

lag

0 20 40

-1.0 -0.5 0.0 0.5 1.0

a[28] chains 1:3

start-iteration

1051 2000 3000

0.0 0.5

1.0 1.5

b[28] chains 1:3

start-iteration

1051 2000 3000

0.0 0.5

1.0 1.5

c[28] chains 1:3

start-iteration

1051 2000 3000

0.0 0.5

1.0 1.5

Results-cont.1. Deciding the number of Burn-in Iterations- Autocorrelation and BGR plots

Results-cont.1. Deciding the number of Burn-in Iterations- Statistics

node mean sd MC error 2.5% median 97.5% start samplea[1] 0.899 0.1011 0.004938 0.7117 0.8949 1.107 2501 3500a[2] 1.339 0.1159 0.004132 1.125 1.333 1.58 2501 3500a[3] 0.7308 0.111 0.005769 0.551 0.717 0.9893 2501 3500a[4] 2.012 0.2712 0.009897 1.531 1.996 2.59 2501 3500a[5] 1.766 0.2202 0.009585 1.394 1.745 2.243 2501 3500

b[1] -1.706 0.2944 0.01793 -2.253 -1.717 -1.1 2501 3500b[2] -0.4277 0.1167 0.005916 -0.6571 -0.428 -0.1857 2501 3500b[3] -0.7499 0.3967 0.01586 -1.409 -0.7994 0.1348 2501 3500b[4] 0.4324 0.09295 0.004443 0.2363 0.4384 0.6008 2501 3500b[5] -0.05619 0.122 0.006737 -0.3127 -0.05246 0.1657 2501 3500

c[1] 0.2458 0.088 0.004718 0.09253 0.2415 0.4362 2501 3500c[2] 0.1403 0.04745 0.002158 0.05368 0.139 0.2361 2501 3500c[3] 0.2538 0.09285 0.005864 0.09991 0.243 0.4557 2501 3500c[4] 0.2669 0.035 0.001491 0.1911 0.2693 0.3282 2501 3500c[5] 0.2588 0.05029 0.002589 0.1526 0.261 0.35 2501 3500

1. Running conditions for WinBUGS

Adaptive phase: 1000 iterations

Burn-in: 1500 iterations

For computing the Statistics: 3500 iterations

Using 1 chain

Using bugs( ) function to run WinBUGS through R◦Need BRugs and R2WinBUGS packages

Results-cont.

2. Effect of Sample SizeResults-cont.

0 5 10 15 20 25 30 350

0.1

0.2

0.3

0.4

0.5

0.6BILOG-MG aN=200

N=500N=1000

Item

RM

SE

0 5 10 15 20 25 30 350

0.10.20.30.40.50.60.70.80.9

BILOG-MG bN=200N=500N=1000

Item

RM

SE

0 5 10 15 20 25 30 350

0.04

0.08

0.12

0.16

0.2BILOG-MG c

N=200N=500N=1000

Item

RM

SE

0 5 10 15 20 25 30 350

0.1

0.2

0.3

0.4

0.5

0.6WinBUGS aN=200

N=500N=1000

Item

RM

SE

0 5 10 15 20 25 30 350

0.10.20.30.40.50.60.70.80.9

WinBUGS bN=200N=500N=1000

Item

RM

SE

0 5 10 15 20 25 30 350

0.04

0.08

0.12

0.16

0.2WinBUGS c

N=200N=500N=1000

Item

RM

SE

BILOG-MG vs. WinBUGS – a parameter

Results-cont.

0 5 10 15 20 25 30 350

0.1

0.2

0.3

0.4

0.5

0.6N=200BILOG-MG

WinBUGS

Item

RM

SE

0 5 10 15 20 25 30 35-0.0999999999999994

5.82867087928207E-16

0.100000000000001

0.200000000000001

0.300000000000001

0.400000000000001

0.500000000000001

0.600000000000001N=500

BILOG-MGWinBUGS

Item

RM

SE

0 5 10 15 20 25 30 350

0.1

0.2

0.3

0.4

0.5

0.6N=1000

BILOG-MGWinBUGS

Item

RM

SE

0 5 10 15 20 25 30 350

0.1

0.2

0.3

0.4

0.5

0.6N=2000BILOG-MG

WinBUGS

Item

RM

SE

BILOG-MG vs. WinBUGS - b parameter

Results-cont.

0 5 10 15 20 25 30 350

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8N=200

BILOG-MGWinBUGS

Item

RM

SE

0 5 10 15 20 25 30 350

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8N=500BILOG-MG

WinBUGS

Item

RM

SE

0 5 10 15 20 25 30 350

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8N=1000

BILOG-MGWinBUGS

Item

RM

SE

0 5 10 15 20 25 30 350

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8N=2000

BILOG-MGWinBUGS

Item

RM

SE

BILOG-MG vs. WinBUGS - c parameter

Results-cont.

0 5 10 15 20 25 30 35-4.16333634234434E-17

0.03

0.06

0.09

0.12N=200 BILOG-MG

WinBUGS

Item

RM

SE

0 5 10 15 20 25 30 35-4.16333634234434E-17

0.03

0.06

0.09

0.12N=500 BILOG-MG

WinBUGS

Item

RM

SE

0 5 10 15 20 25 30 35-4.16333634234434E-17

0.03

0.06

0.09

0.12N=1000 BILOG-MG

WinBUGS

Item

RM

SE

0 5 10 15 20 25 30 35-4.16333634234434E-17

0.03

0.06

0.09

0.12N=2000 BILOG-MG

WinBUGS

Item

RM

SE

Discussion & Conclusions

Larger sample size decreased RMSE for all parameters under both programs.

For N=200, there was a significant convergence problem for BILOG-MG. No problem with WinBUGS.

Discussion & Conclusions-cont.Slope parameter “a”

◦ WinBUGS was superior to BILOG when N = 500 or less◦ More accurately estimated for items without extreme a or b

parameters by both programs.Difficulty parameter “b”

◦ BILOG was superior to WinBUGs when N = 500 or less◦ Both programs had larger error for items either too difficult or

too easyGuessing parameter “c”

◦ WinBUGs was superior to BILOG at all sample sizes, but especially at N = 1,000 or less

◦ More accurately estimated for difficult items by both programs.◦ Both programs had larger error for items with shallow slopes.

LimitationsOnly one chain is used in the simulation study.

Some of the MC errors are not less than 1/20 of the standard deviation, could use more iterations in MCMC sampler

Simulated data◦ Conforms to the 3PL model much more closely than real data

would◦ No missing responses◦ No omit problems◦ Fewer low scores

WinBUGS code for running 3PLmodel 3PL;{ for (i in 1:N) { for (j in 1:n) { e[i,j]<-exp(a[j]*(theta[i]-b[j])) p[i,j] <- c[j]+(1-c[j])*(e[i,j]/(1+e[i,j])) resp[i,j] ~ dbern(p[i,j])

} theta[i] ~ dnorm(0,1) } for (i in 1:n) { a[i] ~ dlnorm(0, 4) b[i] ~ dnorm(0, 0.25) c[i] ~ dbeta(5,17) }}

True Item Parmaetersitem a b c item a b c

1 0.5496 -1.796 0.1751 19 0.6562 0.3853 0.12012 0.7891 -0.4796 0.1165 20 1.0556 0.9481 0.20363 0.4551 -0.7101 0.2087 21 0.3479 2.2768 0.14894 1.4443 0.4833 0.2826 22 0.8432 1.0601 0.23325 0.974 -0.168 0.2625 23 1.1142 0.5826 0.06446 0.5839 -0.8567 0.2038 24 1.4579 1.0241 0.24537 0.8604 0.4546 0.3224 25 0.5137 1.379 0.14278 1.1445 -0.1301 0.2209 26 0.9194 1.0782 0.08799 0.7544 0.0212 0.16 27 1.8811 1.4062 0.1992

10 0.917 1.0139 0.3648 28 1.5045 1.5093 0.164211 0.9592 0.7218 0.2399 29 0.9664 1.5443 0.143112 0.6633 0.0506 0.124 30 0.702 2.2401 0.085313 1.2324 0.4167 0.2535 31 1.2651 1.8759 0.244314 1.0492 0.7882 0.1569 32 0.8567 1.714 0.086515 1.069 0.961 0.2986 33 1.408 1.5556 0.078916 0.9193 0.6099 0.2521 34 0.5808 3.4728 0.139917 0.8935 0.5128 0.2273 35 0.9257 3.1202 0.10918 0.9672 0.195 0.0535 36 1.2993 2.1589 0.1075

Acknowledgement

Professor Katie Cowles

Questions?