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LECTURE 6

LECTURE 6. RELIABILITY. RELIABILITY. Reliability is a proportion of variance measure (squared variable) Defined as the proportion of observed score (x) variance due to true score (  ) variance:  2 x =  xx’ =  2  /  2 x. VENN DIAGRAM REPRESENTATION. Var(). Var(e).

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LECTURE 6

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  1. LECTURE 6 RELIABILITY

  2. RELIABILITY • Reliability is a proportion of variance measure (squared variable) • Defined as the proportion of observed score (x) variance due to true score ( ) variance: • 2x = xx’ = 2 / 2x

  3. VENN DIAGRAM REPRESENTATION Var() Var(e) Var(x) reliability

  4. PARALLEL FORMS OF TESTS • If two items x1 and x2 are parallel, they have • equal true score variance: • Var(1 ) = Var(2 ) • equal error variance: • Var(e1 ) = Var(e2 ) • Errors e1 and e2 are uncorrelated: (e1 , e2 ) = 0 • 1 = 2

  5. Reliability: 2 parallel forms • x1 =  + e1 , x2 =  + e2 • (x1 ,x2 ) = reliability = xx’ = correlation between parallel forms

  6. Reliability: parallel forms x1 x2 x x  e e xx’ = x * x

  7. Reliability: 3 or more parallel forms • For 3 or more items xi, same general form holds • reliability of any pair is the correlation between them • Reliability of the composite (sum of items) is based on the average inter-item correlation: stepped-up reliability, Spearman-Brown formula

  8. Reliability: 3 or more parallel forms Spearman-Brown formula for reliability rxx = k r(i,j) / [ 1+ (k-1) r(i,j) ] Example: 3 items, 1 correlates .5 with 2, 1 correlates .6 with 3, and 2 correlates .7 with 3; average is .6 rxx = 3(.6) / [1 + 2(.6) ] = 1.8/2.2 = .87

  9. Reliability: tau equivalent scores • If two items x1 and x2 are tau equivalent, they have • 1 = 2 • equal true score variance: • Var(1 ) = Var(2 ) • unequal error variance: • Var(e1 )  Var(e2 ) • Errors e1 and e2 are uncorrelated: (e1 , e2 ) = 0

  10. Reliability: tau equivalent scores • x1 =  + e1 , x2 =  + e2 • (x1 ,x2 ) = reliability = xx’ = correlation between tau eqivalent forms (same computation as for parallel, observed score variances are different)

  11. Reliability: Spearman-Brown Can show the reliability of the parallel forms or tau equivalent composite is kk’ = [k xx’]/[1 + (k-1) xx’ ] k = # times test is lengthened example: test score has rel=.7 doubling length produces rel = 2(.7)/[1+.7] = .824

  12. Reliability: Spearman-Brown example: test score has rel=.95 Halving (half length) produces • xx = .5(.95)/[1+(.5-1)(.95)] • = .905 Thus, a short form with a random sample of half the items will produce a test with adequate score reliability

  13. Reliability: KR-20 for parallel or tau equivalent items/scores Items are scored as 0 or 1, dichotomous scoring Kuder and Richardson (1937): special cases of Cronbach’s more general equation for parallel tests. KR-20 = [k/(k-1)] [ 1 - piqi / 2y ] , where pi = proportion of respondents obtaining a score of 1 and qi = 1 – pi . pi is the item difficulty

  14. Reliability: KR-21 for parallel forms assumption Items are scored as 0 or 1, dichotomous scoring Kuder and Richardson (1937) KR-21 = [k/(k-1)] [ 1 - kp. q. / 2c ] p. is the mean item difficulty and q. = 1 – p. KR-21 assumes that all items have the same difficulty (parallel forms) item mean gives the best estimate of the population values. KR-21  KR-20.

  15. Reliability: congeneric scores • If two items x1 and x2 are congeneric, 1. 1  2 2. unequal true score variance: Var(1 )  Var(2 ) 3. unequal error variance: Var(e1 )  Var(e2 ) 4. Errors e1 and e2 are uncorrelated: (e1 , e2 ) = 0

  16. Reliability: congeneric scores x1 = 1 + e1 , x2 = 2 + e2 jj = Cov(t1 , t2 )/ x1x2 This is the correlation between two separate measures that have a common latent variable

  17. Congeneric measurement structure x2 x1 12 x11 x22 e1 1 e2 2 xx’ = x1 112 x22

  18. Reliability: Coefficient alpha Composite=sum of k parts, each with its own true score and variance C = x1 + x2 + …xk  ≤ 1 - 2k / 2c est= k/(k-1)[1 - s2k / s2c ]

  19. Reliability: Coefficient alpha Alpha = 1. Spearman-Brown for parallel or tau equivalent tests 2. = KR20 for dichotomous items (tau equiv.) = Hoyt, even for 2 x item 0 (congeneric)

  20. Hoyt reliability • Based on ANOVA concepts extended during the 1930s by Cyrus Hoyt at U. Minnesota • Considers items and subjects as factors that are either random or fixed (different models with respect to expected mean squares) • Presaged more general Coefficient alpha derivation

  21. Reliability: Hoyt ANOVA Source df Expected Mean Square Person (random) I-1 2 + 2 x items+ K2 Items (random) K-1 2 + k2 x item + I2items error (I-1)(K-1) 2 + 2 x item parallel forms => 2 x item = 0 Hoyt = { ℇ(MSpersons) - ℇ(MSerror) } / ℇ(MSpersons) est Hoyt = [ (MSpersons) - (MSerror) ] / (MSpersons)

  22. Reliability: Coefficient alpha Composite=sum of k parts, each with its own true score and variance C = x1 + x2 + …xk Example: sx1 = 1, sx2=2, sx3=3 sc = 5 est= 3/(3-1)[1 - (1+4+9)/25] = 1.5[1 – 14/25] = 16.5/25 = .66

  23. SPSS DATA FILE JOE 1 1 1 0 SUZY 1 0 1 1 FRANK 0 0 1 0 JUAN 0 1 1 1 SHAMIKA 1 1 1 1 ERIN 0 0 0 1 MICHAEL 0 1 1 1 BRANDY 1 1 0 0 WALID 1 0 1 1 KURT 0 0 1 0 ERIC 1 1 1 0 MAY 1 0 0 0

  24. SPSS RELIABILITY OUTPUT R E L I A B I L I T Y A N A L Y S I S - S C A L E (A L P H A) Reliability Coefficients N of Cases = 12.0 N of Items = 4 Alpha = .1579

  25. SPSS RELIABILITY OUTPUT R E L I A B I L I T Y A N A L Y S I S - S C A L E (A L P H A) Reliability Coefficients N of Cases = 12.0 N of Items = 8 Alpha = .6391 Note: same items duplicated

  26. TRUE SCORE THEORY AND STRUCTURAL EQUATION MODELING True score theory is consistent with the concepts of SEM - latent score (true score) called a factor in SEM - error of measurement - path coefficient between observed score x and latent score  is same as index of reliability

  27. COMPOSITES AND FACTOR STRUCTURE • 3 Manifest (Observed) Variables required for a unique identification of a single factor • Parallel forms implies • Equal path coefficients (termed factor loadings) for the manifest variables • Equal error variances • Independence of errors

  28. Parallel forms factor diagram e e x1 x2 x x e  x x3 xixj= xi * xj = reliability between variables i and j

  29. RELIABILITY FROM SEM • TRUE SCORE VARIANCE OF THE COMPOSITE IS OBTAINABLE FROM THE LOADINGS: k  =  2i = Variance of factor i=1 k = # items or subtests = k2x = k times pairwise average reliability of items

  30. RELIABILITY FROM SEM • RELIABILITY OF THE COMPOSITE IS OBTAINABLE FROM THE LOADINGS:  = k/(k-1)[1 - 1/  ] • example 2x = .8 , K=11 = 11/(10)[1 - 1/8.8 ] = .975

  31. TAU EQUIVALENCE • ITEM TRUE SCORES DIFFER BY A CONSTANT: i = j + k • ERROR STRUCTURE UNCHANGED AS TO EQUAL VARIANCES, INDEPENDENCE

  32. CONGENERIC MODEL • LESS RESTRICTIVE THAN PARALLEL FORMS OR TAU EQUIVALENCE: • LOADINGS MAY DIFFER • ERROR VARIANCES MAY DIFFER • MOST COMPLEX COMPOSITES ARE CONGENERIC: • WAIS, WISC-III, K-ABC, MMPI, etc.

  33. e2 e1 x1 x2 x1 x2 e3 x3  x3 (x1 , x2 )= x1 * x2

  34. COEFFICIENT ALPHA • xx’ = 1 - 2E /2X • = 1 - [2i (1 - ii )]/2X , • since errors are uncorrelated •  = k/(k-1)[1 - s2i / s2C ] • where C = xi (composite score) • s2i = variance of subtest xi • sC = variance of composite • Does not assume knowledge of subtest ii

  35. COEFFICIENT ALPHA- NUNNALLY’S COEFFICIENT • IF WE KNOW RELIABILITIES OF EACH SUBTEST, i • N = K/(K-1)[1-s2i (1- rii )/ s2X ] • where rii = coefficient alpha of each subtest • Willson (1996) showed   N  xx’

  36. NUNNALLY’S RELIABILITY CASE e2 e1 x1 x2 x1 x2 s1 s2 e3 x3  x3 s3 XiXi = 2xi + s2i

  37. Reliability Formula for SEM with Multiple factors (congeneric with subtests) Single factor model:  = i2 / [i2 + ii +  ij ] >  If eij = 0, reduces to  = i2 / [i2 + ii ] = Sum(factor loadings on 1st factor)/ Sum of observed variances This generalizes (Bentler, 2004) to the sum of factor loadings on the 1st factor divided by the sum of variances and covariances of the factors for multifactor congeneric tests Maximal Reliability for Unit-weighted Composites Peter M. Bentler University of California, Los Angeles UCLA Statistics Preprint No. 405 October 7, 2004 http://preprints.stat.ucla.edu/405/MaximalReliabilityforUnit-weightedcomposites.pdf

  38. Multifactor models and specificity • Specificity is the correlation between two observed items independent of the true score • Can be considered another factor • Cronbach’s alpha can overestimate reliability if such factors are present • Correlated errors can also result in alpha overestimating reliability

  39. CORRELATED ERROR PROBLEMS e2 e1 s x1 x2 x1 x2 e3 x3  x3 Specificities can be misinterpreted as a correlated error model if they are correlated or a second factor s3

  40. CORRELATED ERROR PROBLEMS e1 e2 x1 x2 x1 x2 e3 x3  Specificieties can be misinterpreted as a correlated error model if specificities are correlated or are a second factor x3 s3

  41. SPSS SCALE ANALYSIS • ITEM DATA • EXAMPLE: (Likert items, 0-4 scale) • Mean Std Dev Cases • 1. CHLDIDEAL (0-8) 2.7029 1.4969 882.0 • 2. BIRTH CONTROL • PILL OK 2.2959 1.0695 882.0 • 3. SEXED IN SCHOOL 1.1451 .3524 882.0 • 4. POL. VIEWS • (CONS-LIB) 4.1349 1.3379 882.0 • 5. SPANKING OK • IN SCHOOL 2.1111 .8301 882

  42. CORRELATIONS • Correlation Matrix • CHLDIDEL PILLOK SEXEDUC POLVIEWS • CHLDIDEL 1.0000 • PILLOK .1074 1.0000 • SEXEDUC .1614 .2985 1.0000 • POLVIEWS .1016 .2449 .1630 1.0000 • SPANKING -.0154 -.0307 -.0901 -.1188

  43. SCALE CHARACTERISTICS • Statistics for Mean Variance Std Dev Variables • Scale 12.3900 7.5798 2.7531 5 • Items Mean Minimum Maximum Range Max/Min Variance • 2.4780 1.1451 4.1349 2.9898 3.6109 1.1851 • Item Variances • Mean Minimum Maximum Range Max/Min Variance • 1.1976 .1242 2.2408 2.1166 18.0415 .7132 • Inter-itemCorrelations • Mean Minimum Maximum Range Max/Min Variance • .0822 -.1188 .2985 .4173 -2.5130 .0189

  44. ITEM-TOTAL STATS • Item-total Statistics • Scale Scale Corrected • Mean Variance Item- Squared Alpha Total Multiple if item • Correlation R deleted • CHLDIDEAL 9.6871 4.4559 .1397 .0342 .2121 • PILLOK 10.0941 5.2204 .2487 .1310 .0961 • SEXEDUC 11.2449 6.9593 .2669 .1178 .2099 • POLVIEWS 8.2551 4.7918 .1704 .0837 .1652 • SPANKING 10.2789 7.3001 -.0913 .0196 .3655

  45. ANOVA RESULTS • Analysis of Variance • Source of • Variation Sum of Sq. DF Mean Square F Prob. • Between People 1335.5664 881 1.5160 • Within People 8120.8000 3528 2.3018 • Measures 4180.9492 4 1045.2373 934.9 .0000 • Residual 3939.8508 3524 1.1180 • Total 9456.3664 4409 2.1448

  46. RELIABILITY ESTIMATE • Reliability Coefficients 5 items • Alpha = .2625 Standardized item alpha = .3093 • Standardized means all items parallel

  47. RELIABILITY: APPLICATIONS

  48. STANDARD ERRORS • se = standard error of measurement • = sx [1 - xx ]1/2 • can be computed if xx is estimable • provides error band around an observed score: [ -1.96se + x, 1.96se + x ]

  49. x -1.96se +1.96se ASSUMES ERRORS ARE NORMALLY DISTRIBUTED

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