Todd D. Little University of Kansas Director, Quantitative Training Program

1. Why the Items versus Parcels Controversy Needn’t Be One Todd D. Little University of Kansas Director, Quantitative Training Program Director, Center for Research Methods and Data Analysis Director, Undergraduate Social and Behavioral Sciences Methodology Minor Member, Developmental Psychology Training Program crmda.KU.edu Presidential Address presented 08-04-2011 @ American Psychological Association Meeting in Washington, DC crmda.KU.edu

2. Why the Items versus Parcels Controversy Needn’t Be One I Think Todd D. Little University of Kansas Director, Quantitative Training Program Director, Center for Research Methods and Data Analysis Director, Undergraduate Social and Behavioral Sciences Methodology Minor Member, Developmental Psychology Training Program crmda.KU.edu Presidential Address presented 08-04-2011 @ American Psychological Association Meeting in Washington, DC crmda.KU.edu

3. Overview • Learn what parcels are and how to make them • Learn the reasons for and the conditions under which parcels are beneficial • Learn the conditions under which parcels can be problematic • Disclaimer: This talk reflects my view that parcels per se aren’t controversial if done thoughtfully. crmda.KU.edu 3

4. Parceling: Averaging (or summing) two or more items to create more reliable indicators of a construct ≈ Packaging items, tying them together Data pre-processing strategy What is Parceling? crmda.KU.edu 4

5. 1* 1* A CFA of Items -.29 Positive 1 Negative 2 .83 .81 .84 .77 .84 .71 .82 .82 .81 .69 .80 .76 Great Cheer ful Happy Good Glad Super Sad Down Un happy Blue Bad Terr ible .43 .31 .34 .30 .41 .30 .50 .32 .34 .35 .52 .35 Model Fit: χ2(53, n=759) = 181.2; RMSEA = .056(.048-.066); NNFI/TLI = .97; CFI = .98 crmda.KU.edu 5

6. y21 Positive 1 Negative 2 y11 1* 1* y22 11 21 31 42 52 62 Great &Glad Cheerful & Good Happy & Super Terrible &Sad Down & Blue Unhappy & Bad q66 q11 q22 q55 q33 q44 CFA: Using Parcels (6.2.Parcels) Average 2 items to create 3 parcels per construct crmda.KU.edu 6

7. -.27 Positive 1 Negative 2 1* 1* .89 .89 .91 .87 .87 .91 Great &Glad Cheerful & Good Happy & Super Terrible &Sad Down & Blue Unhappy & Bad .21 .21 .17 .18 .25 .25 CFA: Using Parcels • Similar solution • Similar factor correlation • Higher loadings, • more reliable info • Good model fit, • improved χ2 Model Fit: χ2(8, n=759) = 26.8; RMSEA = .056(.033-.079); NNFI/TLI = .99; CFI = .99 crmda.KU.edu 7

8. Philosophical Issues To parcel, or not to parcel…? crmda.KU.edu 8

9. Pragmatic View • “Given that measurement is a strict, rule-bound system that is defined, followed, and reported by the investigator, the level of aggregation used to represent the measurement process is a matter of choice and justification on the part of the investigator” • Preferred terms: remove unwanted, clean, reduce, minimize, strengthen, etc. From Little et al., 2002 crmda.KU.edu 9

10. Empiricist / Conservative View • “Parceling is akin to cheating because modeled data should be as close to the response of the individual as possible in order to avoid the potential imposition, or arbitrary manufacturing of a false structure” • Preferred terms: mask, conceal, camouflage, hide, disguise, cover-up, etc. From Little et al., 2002 crmda.KU.edu 10

11. Hobbesian View • Parcels should be avoided because researchers are ignorant (perhaps stupid) and prone to mistakes. And, because the unthoughtful or unaware application of parcels by unwitting researchers can lead to bias, they should be avoided. • Preferred terms: most (all) researchers are un___ as in … unaware, unable, unwitting, uninformed, unscrupulous, etc. crmda.KU.edu 11

12. Other Issues I Classical school vs. Modeling School Objectivity versus Transparency Items vs. Indicators Factors vs. Constructs Self-correcting nature of science Suboptimal simulations Don’t include population misfit Emphasize the ‘straw conditions’ and proofing the obvious; sometimes over generalize crmda.KU.edu 12

13. Other Issues II Focus of inquiry Question about the items/scale development? Avoid parcels Question about the constructs? Parcels are warranted but must be done thoughtfully! Question about factorial invariance? Parcels are OK if done thoughtfully. crmda.KU.edu 13

14. Measurement Measurement starts with Operationalization Defining a concept with specific observable characteristics [Hitting and kicking ~ operational definition of Overt Aggression] Process of linking constructs to their manifest indicants(object/event that can be seen, touched, or otherwise recorded; cf. items vs. indicators) Rule-bound assignment of numbers to the indicants of that which exists [e,g., Never=1, Seldom=2, Often=3, Always=4] … although convention often ‘rules’, the rules should be chosen and defined by the investigator “Whatever exists at all exists in some amount. To know it thoroughly involves knowing its quantity as well as its quality” - E. L. Thorndike (1918) crmda.KU.edu 14

15. “Indicators are our worldly window into the latent space” - John R. Nesselroade crmda.KU.edu 15

16. Classical Variants • a) Xi= Ti + Si + ei • b) X = T + S + e • c) X1= T1+ S1+ e1 • Xi : a person’s observed score on an item • Ti: 'true' score (i.e., what we hope to measure) • Si : item-specific, yet reliable, component • ei : random error or noise. • Assume: • Si and eiare normally distributed (with mean of zero) and uncorrelated with each other • Across all items in a domain, the Sis are uncorrelated with each other, as are the eis crmda.KU.edu 16

17. Latent Variable Variants • a) X1= T+ S1+ e1 • b) X2 = T+ S2 + e2 • c) X3= T+ S3+ e3 • X1-X3: are multiple indicators of the same construct • T : common 'true' score across indicators • S1-S3: item-specific, yet reliable, component • e1-e3 : random error or noise. • Assume: • Ssand esare normally distributed (with mean of zero) and uncorrelated with each other • Across all items in a domain, the Ssare uncorrelated with each other, as are the es crmda.KU.edu 17

18. Empirical Pros • Psychometric Characteristics of Parcels • (vs. Items) • Higher reliability, communality, & ratio of common-to-unique factor variance • Lower likelihood of distributional violations • More, Smaller, and more-equal intervals Never Seldom Often Always Happy 1 2 3 4 Glad 1 2 3 4 Mean 1 1.5 2 2.5 3 3.5 4 Sum 2 3 4 5 6 7 8 crmda.KU.edu 18

19. More Empirical Pros • Model Estimation and Fit with Parcels • (vs. Items) • Fewer parameter estimates • Lower indicator-to-subject ratio • Reduces sources of parsimony error (population misfit of a model) • Lower likelihood of correlated residuals & dual factor loading • Reduces sources of sampling error • Makes large models tractable/estimable crmda.KU.edu 19

20. More Empirical Pros Variance of / 2 = = = ] = T T Construct T + Specific S1 S2 E1 E2 ¼ of their original size! Error 2 crmda.KU.edu 20

21. Construct = Common Variance of Indicators crmda.KU.edu 21

22. Construct = Common Variance of Indicators crmda.KU.edu 22

23. Empirical Cautions T T T Construct + M M S1 S2 Specific E1 E2 ¼ of their original size! Error 2 crmda.KU.edu 23

24. Construct = Common Variance of Indicators crmda.KU.edu 24

25. Construct = Common Variance of Indicators crmda.KU.edu 25

26. y21 Positive 1 Negative 2 y11 1* 1* y22 11 21 31 42 52 62 Great &Glad Cheerful & Good Happy & Super Terrible &Sad Down & Blue Unhappy & Bad q66 q11 q22 q55 q33 q44 Three is Ideal crmda.KU.edu 26

27. Three is Ideal Matrix Algebra Formula: Σ = Λ Ψ Λ´ + Θ Down & Blue Great & Glad Cheerful & Good Happy & Super Terrible & Sad Unhappy & Bad P1 11y11 11+q11 P2 11y112121y11 21+q22 P3 11y113121y113131y11 31+q33 N1 11y214221y214231y214242y22 42+q44 N211y215221y215231y215242y225252y22 52+q55 N311y216221y216231y216242y2262 52y226262y22 62+q66 • Cross-construct item associations (in box) estimated only via Ψ21– the latent constructs’ correlation. • Degrees of freedom only arise from between construct relations crmda.KU.edu 27

28. Empirical Cons • Multidimensionality • Constructs and relationships can be hard to interpret if done improperly • Model misspecification • Can get improved model fit, regardless of whether model is correctly specified • Increased Type II error rate if question is about the items • Parcel-allocation variability • Solutions depend on the parcel allocation combination (Sterba & McCallum, 2010; Sterba, in press) crmda.KU.edu 28

29. Psychometric Issues • Principles of Aggregation (e.g., Rushton et al.) • Any one item is less representative than the average of many items (selection rationale) • Aggregating items yields greater precision • Law of Large Numbers • More is better, yielding more precise estimates of parameters (and a person’s true score) • Normalizing tendency crmda.KU.edu 29

30. Construct Space with Centroid crmda.KU.edu

31. Potential Indicators of the Construct crmda.KU.edu

32. Selecting Six (Three Pairs) crmda.KU.edu

33. … take the mean crmda.KU.edu

34. … and find the centroid crmda.KU.edu

35. Selecting Six (Three Pairs) crmda.KU.edu

36. … take the mean crmda.KU.edu

37. … find the centroid crmda.KU.edu

38. How about 3 sets of 3? crmda.KU.edu

39. … taking the means crmda.KU.edu

40. … yields more reliable & accurate indicators crmda.KU.edu

41. Building Parcels • Theory – Know thy S and the nature of your items. • Random assignment of items to parcels (e.g., fMRI) • Use Sterba’s calculator to find an allocation method that is in the middle of the distribution. • Balancing technique • Combine items with higher loadings with items having smaller loadings [Reverse serpentine pattern] • Using a priori designs (e.g., CAMI) • Develop new tests or measures with parcels as the goal for use in research crmda.KU.edu 41

42. Techniques: Multidimensional Case • Example: ‘Intelligence’ ~ Spatial, Verbal, Numerical • Domain Representative Parcels • Has mixed item content from various dimensions • Parcel consists of: 1 Spatial item, 1 Verbal item, • and 1 Numerical item • Facet Representative Parcels • Internally consistent, each parcel is a ‘facet’ or • singular dimension of the construct • Parcel consists of: 3 Spatial items • Recommended method crmda.KU.edu 42

43. Domain Representative Parcels + + = S V N Parcel #1 + + = S V N Parcel #2 + + = S V N Parcel #3 SpatialVerbalNumerical crmda.KU.edu 43

44. Domain Representative crmda.KU.edu 44

45. Domain Representative Intellective Ability, Spatial Ability, Verbal Ability, Numerical Ability But which facet is driving the correlation among constructs? crmda.KU.edu 45

46. Facet Representative Parcels + + = S S S Parcel: Spatial + + = V V V Parcel: Verbal + + = N N N Parcel: Numerical crmda.KU.edu 46

47. Facet Representative crmda.KU.edu 47

48. Facet Representative Intellective Ability Diagram depicts smaller communalities (amount of shared variance) crmda.KU.edu 48

49. Facet Representative Parcels + + = + + = + + = A more realistic case with higher communalities crmda.KU.edu 49

50. Facet Representative crmda.KU.edu 50