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Psychology 10

Psychology 10. Analysis of Psychological Data April 21, 2014. The plan for today. Another example of ANOVA with post hoc comparisons. Introducing two-way ANOVA. Main effects and interactions. Computing and interpreting two-way ANOVA. Working with incomplete ANOVA tables.

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Psychology 10

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  1. Psychology 10 Analysis of Psychological Data April 21, 2014

  2. The plan for today • Another example of ANOVA with post hoc comparisons. • Introducing two-way ANOVA. • Main effects and interactions. • Computing and interpreting two-way ANOVA. • Working with incomplete ANOVA tables.

  3. Another ANOVA example • A data set from the Data and Story Library (Google DASL to find it) contains heights in inches for a large group of singers. • The singers are classified as Sopranos, Altos, Tenors, or Basses. • H0 : mSoprano= mAlto= mTenor= mBass. • We will conduct the test using an alpha level of .05.

  4. Singers example (cont.) • Sopranos: T = 2251, S X2 = 144889, n = 35. • Altos: T = 2271, S X2 = 147621, n = 35. • Tenors: T = 2456, S X2 = 172658, n = 35. • Basses: T = 2481, S X2 = 176051, n = 35. • All: G = 9459, S X2 = 641219, N = 140.

  5. ANOVA Table Source SS df MS F -------------------------------- Voice 1244.821 3 Within -------------------- Total

  6. ANOVA Table Source SS df MS F -------------------------------- Voice 1244.821 3 Within 883.600 136 -------------------- Total

  7. ANOVA Table Source SS df MS F -------------------------------- Voice 1244.821 3 Within 883.600 136 -------------------- Total 2128.421 139

  8. ANOVA Table Source SS df MS F ------------------------------ Voice 1244.821 3 414.940 Within 883.600 136 6.497 -------------------- Total 2128.421 139

  9. ANOVA Table Source SS df MS F -------------------------------- Voice 1244.821 3 414.940 63.87 Within 883.600 136 6.497 -------------------- Total 2128.421 139

  10. Singers example (cont.) • Our F was 63.87. • From the table, the critical value is 2.68. • We reject the null hypothesis and conclude that there is significant evidence of differences among the population mean heights for the four different vocal ranges.

  11. Singers example (cont.) • But where are those differences? • Tukey’s HSD: • q = 3.68. • HSD = 3.68 √(6.4970588 / 35) = 1.59. • Our means are 64.3, 64.9, 70.2, and 70.9 inches. • Sopranos differ from tenors; sopranos differ from basses; altos differ from tenors; altos differ from basses. • There is no evidence that sopranos differ from altos or that tenors differ from basses.

  12. Eysenck ANOVA Table Source SS df MS F ----------------------------------- Between 351.52 4 87.88 9.09 Within 435.30 45 9.67 --------------------- Total 786.82 49

  13. Tukey’s HSD for Eysenck • From the table, q = 4.04. • HSD = 4.04 √(9.67/10) = 3.97. • Means for the Counting, Rhyming, Adjective, Imagery, and Intentional Learning groups were 7, 6.9, 11, 13.4, and 12. • The Counting and Rhyming means are significantly different from the other three.

  14. Two-way ANOVA • (Note: two-way and two-factor ANOVA are the same thing.) • Introducing two-way ANOVA through an example. • The truth about the Eysenck data: • Eysenck was also interested in how memory is related to age. • We have been looking at results for a group of old people.

  15. Two-way ANOVA questions • Is there an effect of level of processing? • Is there an effect of age? • Does the effect of level of processing vary depending on age?

  16. Graphics for two-way ANOVA • Parallel bar plots. • Interaction plots. • Technically, interaction plots should be used only when the variable on the X axis is interval or ratio level. • That guideline is frequently ignored to good effect.

  17. Two-way ANOVA calculations • Main effect for first factor: pretend the second factor isn’t there, calculate as if you were doing a one-way ANOVA with more observations per cell. • Main effect for second factor: pretend that the first factor isn’t there, calculate as if you were doing a one-way ANOVA.

  18. Two-way ANOVA calculations (cont.) • Interaction: calculate as if you were doing a one-way ANOVA across all cells of the two-way design; then subtract the sums of squares for the main effects. • Within groups: calculate a sum of squares within each cell, add up across cells. • Total: calculate as in one-way ANOVA.

  19. Notation for two-way ANOVA calculations • For first main effect (factor A), TAi = total of scores in level i of factor A, nAi = number of scores in level i of factor A. • For the second main effect (factor B), TBj = total of scores in level j of factor B, nBj = number of scores in level j of factor B. • T and n without subscripts denote totals and sample sizes within cells of the crossed factors.

  20. Sums of Squares

  21. The Eysenck example (Strategy) • Totals for levels of processing are 135, 145, 258, 310, and 313. • The sample sizes are n=20 for each level. • The grand total is G=1161. • The overall N = 100. • SS = 1352/20 + 1452/20 + 2582/20 + 3102/20 + 3132/20 - 11612/100 = 1514.94.

  22. The ANOVA table Source SS df MS F ---------------------------------- Strategy 1514.94 4 Age Interaction Within ----------------------- Total

  23. The Eysenck example (Age) • Totals for levels of age are 503 and 658. • The sample sizes are n=50 for each level. • The grand total is still G=1161. • The overall N = 100. • SS = 5032/50 + 6582/50 - 11612/100 = 240.25.

  24. The ANOVA table Source SS df MS F ---------------------------------- Strategy 1514.94 4 Age 240.25 1 Interaction Within ----------------------- Total

  25. The Eysenck example (Interaction) • Totals for cells are 70, 69, 110, 134, 120, 65, 76, 148, 176, 193. • The sample sizes are n=10 for each cell. • The grand total is still G=1161. • The overall N = 100. • SS = 702/10 + 692/10 + 1102/10 + 1342/10 + 1202/10 + 652/10 + 762/10 + 1482/10 + 1762/10 + 1932/10 + - 11612/100 = 1945.49.

  26. The Eysenck example (Interaction, cont.) • But part of that 1945.49 has already been attributed to strategy and age main effects. • SSA×B=1945.49 – 1514.94 – 240.25 = 190.3. • df for the interaction is the product of the df’s for the main effects: 4 × 1 = 4.

  27. The ANOVA table Source SS df MS F ---------------------------------- Strategy 1514.94 4 Age 240.25 1 Interaction 190.30 4 Within ----------------------- Total

  28. The Eysenck example (Within) • The total sum of squares is the same as in one-way ANOVA: SST=SX2 – G2/N. • SST = 16147 – 11612/100 = 2667.79. • SSW = SST - SSA - SSB – SSA×B = 2667.79 – 1514.94 – 240.25 – 190.30 = 722.30. • dfW = N – (number of cells) = 90.

  29. The ANOVA table Source SS df MS F ---------------------------------- Strategy 1514.94 4 Age 240.25 1 Interaction 190.30 4 Within 722.30 90 ----------------------- Total 2667.79 99

  30. The ANOVA table Source SS df MS F ---------------------------------- Strategy 1514.94 4 378.735 47.19 Age 240.25 1 240.250 29.94 Interaction 190.30 4 47.575 5.93 Within 722.30 90 ----------------------- Total 2667.79 99

  31. Interpreting the ANOVA • We have three F statistics, one for each hypothesis. • For tests with 4 and 90 df, the critical value from the table is 2.48. • For the test with 1 and 90 df, the critical value is 3.96. • We reject all three null hypotheses.

  32. Interpreting the ANOVA (cont.) • The rejection of the interaction null hypothesis indicates that we have shown that the effect of level of processing differs depending on which age group we consider.

  33. Interpreting the ANOVA (cont.) • The rejection of the strategy null hypothesis indicates that if we ignore which age group people are in, memory differs according to what level of processing they employed.

  34. Interpreting the ANOVA (cont.) • The rejection of the age null hypothesis indicates that if we ignore which strategy people used, memory differs for old and young people.

  35. Incomplete ANOVA tables Source SS df MS F ----------------------------------------------------------- Strategy 47.19 Age Interaction 190.30 Within 8.026 ------------------------------------- Total 2667.79

  36. In class example • Two TAs (one male, one female). • Two genders of student (male and female). • Dependent variable = midterm score.

  37. Example (cont.) • Male TA, male student: T = 583.5. • Male TA, female student: T = 532. • Female TA, male student: T = 582.5. • Female TA, female student: T = 590.5. • n = 7. • S X2 = 193391.8.

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