1. STRONG TRUE SCORE THEORY- IRT LECTURE 12 EPSY 625

2. Strong True Score Theory Equivalent to g-theory: subject ability item difficulty Extension of true score theory Uses form of logistic regression: e Dag(  - bg ) Pr(1) = 1 + e Dag(  - bg )

3. Strong True Score Theory Equivalent to g-theory: subject ability item difficulty Extension of true score theory Uses form of logistic regression: e Dag(  - bg ) Pr(1) = 1 + e Dag(  - bg )

4. Pg()Pg()  ABILITY 1.0.50 0 Difficulty b g Probability of Correct Answer Item Response Model Discrimination a g 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: e Dag(  - bg ) Pr(1) = 1 + e Dag(  - bg )

5. MODELS One parameter model- only b g varies across items Two parameter model- both a g and b g vary across items

6. 1-PARAMETER ESTIMATION MPLUS: TITLE:this is an example of a one –parameter logistic item response theory (IRT) model DATA: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;

7. MPLUS 5.5 OUTPUT Thresholds 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

8. 2-PARAMETER ESTIMATION MPLUS: TITLE:this is an example of a two- parameter logistic item response theory (IRT) model DATA: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;

9. 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)

10. Three parameter model a g and b g vary across items parameter c g for guessing is added: Empirical studies indicate c g is usually lower than guessing rate Requires 5,000 - 10,000 cases for stable estimation (ETS, ACT or NAEP samples)

11. Pg()Pg()  ABILITY 1.0.50 0 Probability of Correct Answer agag bgbg cgcg

12. Pg()Pg() 11.5 (1,2)(1,2) Pg()Pg() MULTIDIMENSIONAL IRT - CONCEPTS AND ISSUES - Difficulty in getting estimates - Inconsistent with factor model analysis