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Psychometrics & Statistical Concepts

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

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Psychometrics & Statistical Concepts

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  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 Strongly Agree Agree Disagree Disagree [ ] [ ] [ ] [ ] Strongly 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. Types of Variables 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

  14. Correlation • 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. 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) Survey of Classn = 42

  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 = s2 = ----------------------- 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

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

  23. Regression line

  24. Regression: WEIGHT on HEIGHT 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 ]

  25. Regression line W = 3.3 H - 73

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

  27. 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

  28. Regression line mean

  29. 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 = 159 159 / 454 = .35 • Variance is reduced 35% by calculating from regression line

  30. Correlation coefficient = r r2 = % of variance in WEIGHT “explained” by HEIGHT

  31. Correlation: HEIGHT with WEIGHT HEIGHT WEIGHT HEIGHT 1.0000 .5925 ( 42) ( 42) P= . P= .000 WEIGHT .5925 1.0000 ( 42) ( 42) P= .000 P= .

  32. r = .59 r2 = .35 HEIGHT “explains” 35% of variance in WEIGHT

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

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

  35. Multiple Regression Regression line mean

  36. 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 = .59 r2 = .35% Height “explains” 35% of variance in weight

  37. Statistical Significance

  38. Statistical Significance 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?

  39. 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

  40. 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?

  41. 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 = .59 p < .001

  42. Distinguish Between • Relationship -- slope of regression line • Strength of the relationship – “goodness of fit” -- % of variance explained • Statistical significance p < .05

  43. Regression line

  44. Height and Weight • Relationship (regression line) Weight = 3.3 Height - 73 • Strength of relationship (correlation) r = .59 r2 = .35 35% variance “explained” • Statistical significance( p < .05 ) p < .001

  45. Factor Analysis Charles Spearman

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

  47. 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? Survey of Classn = 42

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