Psychology 405: Psychometric Theory Validity
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1 Preliminaries Psychology 405: Psychometric Theory Validity William Revelle Department of Psychology Northwestern University Evanston, Illinois USA May, /17
2 Preliminaries Outline Preliminaries 2/17
3 Preliminaries X Observed Variables Y X 1 Y 1 X 2 Y 2 X 3 Y 3 X 4 Y 4 X 5 Y 5 X 6 Y 6 3/17
4 Preliminaries Latent Variables /17
5 Preliminaries Theory: A regression model of latent variables 1 m m 2 5/17
6 Preliminaries A measurement model for X Correlated factors X X 1 X 2 X 3 k X 4 X 5 X 6 k + 2 6/17
7 Preliminaries A measurement model for Y - uncorrelated factors Y s s Y 1 Y 2 Y 3 Y 4 Y 5 Y /17
8 Preliminaries A complete structural model X Y X 1 X 2 X 3 X 4 X 5 X 6 k + k m s m 2 s Y 1 Y 2 Y 3 Y 4 Y 5 Y /17
9 Preliminaries X 1 Face/Faith Types of Validity Y 1 X 2 X 3 X 4 Convergent X 5 X 6 Convergent X 7 X 8 Concurrent k + k + Predictive @@R - Construct s m 2 Y 2 Y 3 Y 4 Y 5 Y 6 Y 7 Y 8 9/17
10 Preliminaries X 1 Face/Faith Face Validity Representative Content Seeming relevance 10 / 17
11 Preliminaries Concurrent Validity X 2 Concurrent Y 2 Does a measure correlate with the criterion? Need to define the criterion. Assumes that what correlates now will have predictive value. 11 / 17
12 Preliminaries Predictive Validity X 3 - Predictive Y 3 Does a measure correlate with the criterion? Need to define the criterion. Allow time to pass 12 / 17
13 Preliminaries Prediction 1. Continuous predictor, continuous criterion Regression, multiple regression, correlation Slope of regression implies how much change for unit change in predictor 2. Continuous predictor, dichotomous criterion point bi-serial correlation 3. Dichotomous predictor, dichotomous outcome Phi 13 / 17
14 Preliminaries Classics in Prediction and selection 1. Gideon s selection of soldiers 2. OSS and Army Air Corps selection studies 3. Kelly and Fiske (1950) selection of psychology students 4. Astronaut selection 5. Peace Corps selection 14 / 17
15 Preliminaries Gideon s assessment 15 / 17
16 Preliminaries The assessment of pilots how to show a.45 correlation makes a di erence Percentage passing S1 S2 S3 S4 S5 S6 S7 S8 S9 Ability by Stanine 16 / 17
17 Preliminaries Predicting clinical psychologists Kelly and Fiske 1. Multiple predictors of graduate school performance: Kelly and Fiske (1950), Multiple predictors 2. Ability, Interests, temperament (each with r ) have multiple R of Are they able, interested and stable? 4. Kuncel et al. (2001) 17 / 17
18 Predictive and Concurrent Validity and Decision Making Hit Rate = Valid Positive + False Negative Selection Ratio = Valid Positive + False Positive FN VN VP FP HR 1-HR 1-SR SR Phi =(VP - HR*SR) /sqrt(hr*(1-hr)*(sr)*(1-sr)
19 Validity as decision making probability of group membership probability of group membership Predictor Predictor Trading off Valid positives for False Positives probability of group membership probability of group membership
20 Decision Theory and Signal Detection Probability VP Sensitivity (correlation) Probability FP
21 Signal detection theory d prime and beta d prime maps to the correlation beta maps to selection ratio type I and type II error Need to consider utility of types of error
22 Predictive Validity and Decision Theory State of world FN VP Hit rate VN FP 1-HR Decision 1-SR Selection Ratio
23 Predictive Validity, Utility and Decision Theory State of world FN *UFN VP *UVP Hit rate VN *UVN FP* UFP 1-HR Decision 1-SR Selection Ratio Utility of test = VP *UVP + VN *UVN + FN *UFN + FP* UFP - Cost of test
24 Decisions for institutions, advice for individuals Decision State of world FN *UFN VP *UVP Hit rate VN *UVN FP* UFP 1-HR 1-SR Selection Ratio Utility of test = VP *UVP + VN *UVN + FN *UFN + FP* UFP - Cost of test
25 Decision making and the benefit of extreme selection ratios Typical traits are approximated by a normal distribution. Small differences in means or variances can lead to large differences in relative odds at the tails Accuracy of decision/prediction is higher for extreme values. Do we infer trait mean differences from observing differences of extreme values? (code for these graphs at personality-project.org/r/extreme.r)
26 Odds ratios as f(mean difference, extremity) Difference =.5 sigma Normal density for two groups Difference =1.0 sigma Normal density for two groups probability of x x Odds ratio of G1 vs G2 Odds of G2>G probability of x x Odds ratio of G1 vs G2 Odds of G2>G x x
27 The effect of group differences on likelihod of extreme scores Difference =.5 sigma Difference =1.0 sigma Cumulative normal density for two groups Cumulative normal density for two groups probability of x x Odds ratio that person in Group exceeds x Odds of G2>G probability of x x Odds ratio that person in Group exceeds x Odds of G2>G x x
28 The effect of differences of variance on odds ratios at the tails variance of two groups differ by 10% Variance of two groups differs by 20% probability of x x Odds ratio of G1 vs G2 Odds of G2>G probability of x x Odds ratio of G1 vs G2 Odds of G2>G x x
29 Restriction of range Validity of SAT is partially limited by range restriction. (see Lubinski and Benbow) Consider giving SATs to year olds SAT M 390 or SAV V 370 (top 1 in 100) SAT M 500 or SAV V 430 (top 1 in 200) SAT M 700 or SAV M 430 (top 1 in 10,000) 29
30 Predictions within top student group Validity continues even among top 1% 30
31 Validity over 25 years 31
32 X1 Construct Validity: Convergent, Discriminant, Incremental X2 X3 L1 Y Y1 Y2 X4 Y3 X5 L2 X6
33 Multi-Trait, Multi-Method Matrix T1M1 T2M1 T3M1 T1M2 T2M2 T3M2 T1M3 T2M3 T3M3 T1M1 T1M1 T2M1 M1 T2M1 T3M1 M1 M1 T3M1 T1M2 T1 T1M2 T2M2 T2 M2 T2M2 T3M2 T3 M2 M2 T3M2 T1M3 T1 T1 T1M3 T2M3 T2 T2 M3 T2M3 T3M3 T3 T3 M3 M3 T3M3 Mono-Method, Mono trait = reliability Hetero Method, Mono Trait = convergent validity Hetero Method, Hetero Trait = discriminant validity
34 Traits T1 T2 T3 T1M1 T2M1 T3M1 T1M2 T2M2 T3M2 T1M3 T2M3 T3M3 Methods M1 M2 M3
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