1. Psychometrics & Statistical Concepts PSYC 101 Dr. Gregg Fall, 2006

2. Vocabulary Item:1 question or task Scale:Set of items that measure a single trait or characteristic Test:Usually large set of items that measure one or several traits May consist of several scales or “subtests” (IQ; SAT; ACT)

3. Likert Scale Item with following response forms:Strongly AgreeAgreeDisagreeDisagree [ ] [ ] [ ] [ ] Strongly Agree [ ][ ] [ ] [ ] [ ] [ ] [ ] Disagree

4. Psychometrics: Test Design Theory-based strategy:Galton Prediction-based strategy:Binet

5. Psychometrics: Test Design Theory-based strategy: Create items based on theory “Some people are born with an urge to jump from high places.”

6. Psychometrics: Test Design Prediction-based strategy: 1. Identify criterion group (with trait) & group without trait. 2. Select items criterion group answers differently than non- criterion group.

7. Psychometrics: Designing an Accurate Test Reliability:Does test consistently measure what it measures? Validity:Does test measure what it aims to measure?

8. Reliability Does test consistently measure what it measures? Internal consistency Test-retest reliability

9. Validity Does test measure what it aims to measure? Convergent Validity:Correlations with other measures of same trait. Divergent Validity:Non- correlation with measures of different traits.

10. Need to Understand Correlation Regression Factor Analysis  Key concept: variance

11. Types of Variables

12. Nominal / Categorical: each value is distinct category [gender, blood type, city] Scale / Interval: linear measure, same interval between each value [age, weight, IQ, GPA, SAT, income] Ordinal: ranking, un-equal intervals between values [Likert scale, preference ranking]

13. Variables & Statistical Tests Variable TypeExampleCommon Stat Method Nominal by nominal Blood type by gender Chi-square Scale by nominalGPA by gender GPA by major T-test Analysis of Variance Scale by scaleWeight by height GPA by SAT Regression Correlation

14. Strength of association of scale measures r = -1 to 0 to +1 +1 perfect positive correlation -1 perfect negative correlation 0 no correlation Interpret r in terms of variance

15. Mean & Variance

16. Survey of Class n = 42 Height Mother’s height Mother’s education SAT Estimate IQ Well-being (7 pt. Likert) Weight Father’s education Family income G.P.A. Health (7pt Likert)

17. Frequency Table for:HEIGHT Valid Cum Value Label Value Frequency Percent Percent Percent 59.00 1 2.4 2.4 2.4 61.00 2 4.8 4.8 7.1 62.00 3 7.1 7.1 14.3 63.00 3 7.1 7.1 21.4 65.00 5 11.9 11.9 33.3 66.00 3 7.1 7.1 40.5 67.00 4 9.5 9.5 50.0 68.00 5 11.9 11.9 61.9 69.00 1 2.4 2.4 64.3 70.00 6 14.3 14.3 78.6 71.00 1 2.4 2.4 81.0 72.00 4 9.5 9.5 90.5 73.00 3 7.1 7.1 97.6 74.00 1 2.4 2.4 100.0 ------- ------- ------- Total 42 100.0 100.0 Valid cases 42 Missing cases 0

18. Frequency Table for:HEIGHT Valid Cum Value Label Value Frequency Percent Percent Percent 59.00 1 2.4 2.4 2.4 61.00 2 4.8 4.8 7.1 62.00 3 7.1 7.1 14.3 63.00 3 7.1 7.1 21.4 65.00 5 11.9 11.9 33.3 66.00 3 7.1 7.1 40.5 67.00 4 9.5 9.5 50.0 68.00 5 11.9 11.9 61.9 69.00 1 2.4 2.4 64.3 70.00 6 14.3 14.3 78.6 71.00 1 2.4 2.4 81.0 72.00 4 9.5 9.5 90.5 73.00 3 7.1 7.1 97.6 74.00 1 2.4 2.4 100.0 ------- ------- ------- Total 42 100.0 100.0 Valid cases 42 Missing cases 0 Descriptive Statistics for:HEIGHT Valid Variable Mean Std Dev Variance Range Minimum Maximum N HEIGHT 67.33 3.87 14.96 15.00 59.00 74.00 42 mean

19. Variance   x i - Mean ) 2 Variance = s 2 = ----------------------- N Standard Deviation = s =  variance

20. Frequency Table for:WEIGHT Valid Cum Value Label Value Frequency Percent Percent Percent 115.00 1 2.4 2.4 2.4 120.00 1 2.4 2.4 4.8 124.00 1 2.4 2.4 7.1 125.00 4 9.5 9.5 16.7 128.00 1 2.4 2.4 19.0 130.00 6 14.3 14.3 33.3 135.00 4 9.5 9.5 42.9 136.00 1 2.4 2.4 45.2 140.00 3 7.1 7.1 52.4 145.00 2 4.8 4.8 57.1 150.00 3 7.1 7.1 64.3 155.00 2 4.8 4.8 69.0 160.00 6 14.3 14.3 83.3 165.00 2 4.8 4.8 88.1 170.00 1 2.4 2.4 90.5 185.00 1 2.4 2.4 92.9 190.00 2 4.8 4.8 97.6 210.00 1 2.4 2.4 100.0 ------- ------- ------- Total 42 100.0 100.0 Valid cases 42 Missing cases 0 Descriptive Statistics for:WEIGHT Valid Variable Mean Std Dev Variance Range Minimum Maximum N WEIGHT 146.38 21.30 453.80 95.00 115.00 210.00 42 mean

21. Relationship of weight & height: Regression Analysis

23. “Least Squares” Regression Line Dependent = ( B ) (Independent) + constant weight = ( B ) ( height ) + constant

24. Regression line

25. Regression:WEIGHTonHEIGHT Multiple R.59254 R Square.35110 Adjusted R Square.33488 Standard Error 17.37332 Analysis of Variance DF Sum of Squares Mean Square Regression 1 6532.61322 6532.61322 Residual 40 12073.29154 301.83229 F = 21.64319 Signif F =.0000 ------------------ Variables in the Equation ------------------ Variable B SE B Beta T Sig T HEIGHT 3.263587.701511.592541 4.652.0000 (Constant) -73.367236 47.311093 -1.551 [ Equation:Weight = 3.3 ( height ) - 73 ]

26. Regression line W = 3.3 H - 73

27. Strength of Relationship “Goodness of Fit”: Correlation How well does the regression line “fit” the data?

29. Frequency Table for:WEIGHT Valid Cum Value Label Value Frequency Percent Percent Percent 115.00 1 2.4 2.4 2.4 120.00 1 2.4 2.4 4.8 124.00 1 2.4 2.4 7.1 125.00 4 9.5 9.5 16.7 128.00 1 2.4 2.4 19.0 130.00 6 14.3 14.3 33.3 135.00 4 9.5 9.5 42.9 136.00 1 2.4 2.4 45.2 140.00 3 7.1 7.1 52.4 145.00 2 4.8 4.8 57.1 150.00 3 7.1 7.1 64.3 155.00 2 4.8 4.8 69.0 160.00 6 14.3 14.3 83.3 165.00 2 4.8 4.8 88.1 170.00 1 2.4 2.4 90.5 185.00 1 2.4 2.4 92.9 190.00 2 4.8 4.8 97.6 210.00 1 2.4 2.4 100.0 ------- ------- ------- Total 42 100.0 100.0 Valid cases 42 Missing cases 0 Descriptive Statistics for:WEIGHT Valid Variable Mean Std Dev Variance Range Minimum Maximum N WEIGHT 146.38 21.30 453.80 95.00 115.00 210.00 42 mean

30. Regression line mean

31. Correlation: “Goodness of Fit” Variance (average sum of squared distances from mean) = 454 “Least squares” (average sum of squared distances from regression line) = 295 454 – 295 = 159159 / 454 =.35 Variance is reduced 35% by calculating from regression line

32. r 2 = % of variance in WEIGHT “explained” by HEIGHT Correlation coefficient = r

33. Correlation:HEIGHTwith WEIGHT HEIGHT WEIGHT HEIGHT 1.0000.5925 ( 42) ( 42) P=. P=.000 WEIGHT.5925 1.0000 ( 42) ( 42) P=.000 P=.

34. r =.59 r 2 =.35 HEIGHT “explains” 35% of variance in WEIGHT

35. Heretibility % of variance in measures of a trait (such as height or IQ) that is “attributable to” genes

36. Multiple Regression Problem: relationship of weight and calorie consumption Both weight and calorie consumption related to height Need to “control for” height

37. Regression line mean Multiple Regression

38. Form of relationship --regression line: Weight = 3.3 ( height ) - 73 Each inch of height “adds” 3.3 pounds of weight Strength of relationship -- correlation: r =.59r 2 =.35% Height “explains” 35% of variance in weight

39. Statistical Significance

40. What is the probability that the relationship observed in the sample does not exist in the universe from which the sample was drawn? What are the chances that the sample could be a “quirky” one, which doesn’t reflect the real state of affairs in the larger world?

41. If the probably of having drawn a “quirky,” non-representative sample is less than 5 in 100, the finding from the sample can be said to be statistically significant. p <.05

42. Stat Sig of Height–Weight Correlation ( sample n = 42 ) In sample, r =.59 What are chances a sample with r =.59 could come from a population in which there is NO correlation between height and weight?

43. Statistical Significance Need to know: distribution of possible samples of 42 from population in which height and weight are NOT correlated: Sampling Distribution Is probability of drawing a sample in which r =.59 less than.05? r =.59p <.001

44. Distinguish Between Relationship -- slope of regression line Strength of the relationship – “goodness of fit” -- % of variance explained Statistical significancep <.05

45. Regression line

47. Height and Weight Relationship (regression line) Weight = 3.3 Height - 73 Strength of relationship (correlation) r =.59r 2 =.35 35% variance “explained” Statistical significance( p <.05 ) p <.001

48. Factor Analysis Charles Spearman

49. Believed IQ inherited Eugenics advocate Created factor analysis: Showed intercorrelation among Binet’s sub-tests Two-factor theory: g + s-s

50. Survey of Class n = 42 Height Mother’s height Mother’s education SAT Estimate IQ Well-being (7 pt. Likert) Weight Father’s education Family income G.P.A. Health (7pt Likert) How many pieces of cherry pie could you eat if you had to?

51. HeightFather Height Mother Height WeightPie Pieces Father Educ Mother Educ G.P.A.S.A.T.I.Q.IncomeHealthHappy Height 1.0.36*.57***.59**.57***.20.05.04.21.25-.09.06.10 F Height 1.0.30.05.16.23.08.25.38*.37*-.04-.40*-.01 M Height 1.0.19.29.08.003.05.001.09-.23-.10.03 Weight 1.0.54***-.06-.10-.02.04.05-.07.16-.09 Pie 1.0.16.19.03.25.35*.03.21-.02 F Educ 1.0.62***-.21-.02.10.29-.32*-.06 M Educ 1.0-.07.06.23.30.005.22 G.P.A. 1.0.63***.51***-.19.13.10 S.A.T. 1.0.67***-.22.15.28 I.Q. 1.0-.14.25.19 Income 1.0-.15-.23 Health 1.0.36* Happy 1.0

52. WeightPie PiecesG.P.A.S.A.T.I.Q.HealthHappy Height.59**.57***.04.21.25.06.10 Weight.54***-.06-.10.05.16-.09 Pie Pieces.03.25.35*-21-.02 G.P.A..63***.51***.13.10 S.A.T..67***.15.28 I.Q..25.19 Health.36*

53. WeightPie Pieces G.P.A.S.A.T.I.Q.HealthHappy Height.59**.57***.04.21.25.06.10 Weight.54***-.06-.10.05.16-.09 Pie Pieces.03.25.35*-21-.02 G.P.A..63***.51***.13.10 S.A.T..67***.15.28 I.Q..25.19 Health.36*

54. Three Factors “Size” “Smarts” “Good Life”

56. Lewis Thurstone Invented factor rotation technique Found 7 factors – “Primary Mental Abilities”

58. Thurstone: Primary Mental Abilities Verbal comprehension Word fluency Computational ability Spatial visualization Associative memory Perceptual speed Reasoning

59. Theories of Intelligence Single Galton Cattell Goddard Terman Spearman Herrnstein & Muray Multiple Binet Thurstone Gardner

60. David Wechsler Developed W.A.I.S. (Wechsler Adult Intelligence Scale)

61. W. A. I. S.