STRONG TRUE SCORE THEORY- IRT

LECTURE 12

EPSY 625

Strong True Score Theory

Equivalent to g-theory:subject abilityitem difficulty

Extension of true score theoryUses form of logistic regression:

eDag( - bg )

Pr(1) = 1 + eDag( - bg )

Strong True Score Theory

Equivalent to g-theory:subject abilityitem difficulty

Extension of true score theoryUses form of logistic regression:

eDag( - bg )

Pr(1) = 1 + eDag( - bg )

Pg()

ABILITY

1.0

.50

0

Difficulty bg

Probability of Correct Answer

Item Response ModelDiscrimination ag

Difficulty: the ability score needed for a 50% probability of getting the item right

Discrimination: slope of the IRT curve at the 50% probability intersection

Assumptions: .local independence of items.single ability true score.logistic model for items:

eDag( - bg )

Pr(1) = 1 + eDag( - bg )

MODELS

One parameter model- only bg varies across items

Two parameter model- both ag and bg vary across items

1-PARAMETER ESTIMATION MPLUS:TITLE: this is an example of a one –parameter logistic item

response theory (IRT) modelDATA: FILE IS ex5.5.dat;VARIABLE: NAMES ARE u1-u5;

CATEGORICAL ARE u1-u5;ANALYSIS: ESTIMATOR = MLR;MODEL: f BY u1 (1)

u2 (1) u3 (1) u4 (1) u5 (1);

OUTPUT: TECH1 TECH8;

MPLUS 5.5 OUTPUTThresholds

Estimates S.E. Est./S.E.

F BY

U1 1.000 0.000 0.000

U2 0.982 0.243 4.042

U3 0.982 0.243 4.042

U4 0.982 0.243 4.042

U5 0.982 0.243 4.042

Thresholds

U1$1 -0.355 0.109 -3.256

U2$1 -0.431 0.108 -4.005

U3$1 -0.441 0.108 -4.080

U4$1 0.294 0.107 2.752

U5$1 0.459 0.108 4.256

Fixed slopes

Item difficulties

2-PARAMETER ESTIMATION MPLUS:TITLE: this is an example of a two-parameter

logistic item response theory (IRT) modelDATA: FILE IS ex5.5.dat;VARIABLE: NAMES ARE u1-u20;

CATEGORICAL ARE u1-u20;ANALYSIS: ESTIMATOR = MLR;MODEL: f BY u1-u20;OUTPUT: TECH1 TECH8;

MPLUS 5.5 OUTPUT MODEL RESULTS

Estimates S.E. Est./S.E.

F BY U1 1.000 0.000 0.000 U2 1.035 0.204 5.085 U3 0.893 0.173 5.156 U4 1.127 0.233 4.829 U5 0.955 0.205 4.657 U6 0.506 0.142 3.572 U7 1.100 0.223 4.923 U8 1.017 0.213 4.769 U9 0.995 0.209 4.770 U10 0.945 0.194 4.870 U11 1.205 0.227 5.298 U12 0.957 0.188 5.104 U13 0.982 0.203 4.838 U14 0.741 0.168 4.396 U15 0.772 0.156 4.938 U16 0.926 0.195 4.740 U17 1.116 0.229 4.879 U18 1.097 0.212 5.180 U19 0.761 0.165 4.604 U20 1.067 0.211 5.046

Thresholds U1$1 -0.366 0.111 -3.301 U2$1 -0.440 0.113 -3.882 U3$1 -0.324 0.107 -3.031 U4$1 -0.330 0.115 -2.862 U5$1 -0.439 0.111 -3.957 U6$1 -0.430 0.097 -4.415 U7$1 -0.450 0.115 -3.902 U8$1 -0.418 0.111 -3.747 U9$1 -0.435 0.112 -3.890 U10$1 -0.447 0.110 -4.064 U11$1 0.597 0.122 4.890 U12$1 0.555 0.112 4.942 U13$1 0.468 0.111 4.195 U14$1 0.280 0.102 2.747 U15$1 0.283 0.103 2.745 U16$1 0.401 0.109 3.689 U17$1 0.602 0.119 5.071 U18$1 0.463 0.116 3.992 U19$1 0.661 0.108 6.134 U20$1 0.479 0.115 4.172

Slopes (a parameters) difficulties (b parameters)

Three parameter model

ag and bg vary across items

parameter cg for guessing is added:

• Empirical studies indicate cg is usually lower than guessing rate

• Requires 5,000 - 10,000 cases for stable estimation (ETS, ACT or NAEP samples)

Pg()

ABILITY

1.0

.50

0

Probability of Correct Answer

ag

bg

cg

Pg()

1

.5

(1,2)

Pg()

MULTIDIMENSIONAL IRT - CONCEPTS AND ISSUES

- Difficulty in getting estimates

- Inconsistent with factor model analysis