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Remember. You just invented a “magic math pill” that will increase test scores. On the day of the first test you give the pill to 4 subjects. When these same subjects take the second test they do not get a pill Did the pill increase their test scores?. What if.

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  1. Remember • You just invented a “magic math pill” that will increase test scores. • On the day of the first test you give the pill to 4 subjects. When these same subjects take the second test they do not get a pill • Did the pill increase their test scores?

  2. What if. . . • You just invented a “magic math pill” that will increase test scores. • On the day of the first test you give a full pill to 4 subjects. When these same subjects take the second test they get a placebo. When these same subjects that the third test they get no pill.

  3. Note • You have more than 2 groups • You have a repeated measures design • You need to conduct a Repeated Measures ANOVA

  4. Tests to Compare Means

  5. What if. . . • You just invented a “magic math pill” that will increase test scores. • On the day of the first test you give a full pill to 4 subjects. When these same subjects take the second test they get a placebo. When these same subjects that the third test they get no pill.

  6. Results

  7. For now . . . Ignore that it is a repeated design

  8. Between Variability = low

  9. Within Variability = high

  10. Notice – the within variability of a group can be predicted by the other groups

  11. Notice – the within variability of a group can be predicted by the other groups Pill and Placebo r = .99; Pill and No Pill r = .99; Placebo and No Pill r = .99

  12. These scores are correlated because, in general, some subjects tend to do very well and others tended to do very poorly

  13. Repeated ANOVA • Some of the variability of the scores within a group occurs due to the mean differences between subjects. • Want to calculate and then discard the variability that comes from the differences between the subjects.

  14. Example

  15. Sum of Squares • SS Total • The total deviation in the observed scores • Computed the same way as before

  16. SStotal = (57-75.66)2+ (71-75.66)2+ . . . . (96-75.66)2 = 908 *What makes this value get larger?

  17. SStotal = (57-75.66)2+ (71-75.66)2+ . . . . (96-75.66)2 = 908 *What makes this value get larger? *The variability of the scores!

  18. Sum of Squares • SS Subjects • Represents the SS deviations of the subject means around the grand mean • Its multiplied by k to give an estimate of the population variance (Central limit theorem)

  19. SSSubjects = 3((60.33-75.66)2+ (72.33-75.66)2+ . . . . (93.66-75.66)2) = 1712 *What makes this value get larger?

  20. SSSubjects = 3((60.33-75.66)2+ (72.33-75.66)2+ . . . . (93.66-75.66)2) = 1712 *What makes this value get larger? *Differences between subjects

  21. Sum of Squares • SS Treatment • Represents the SS deviations of the treatment means around the grand mean • Its multiplied by n to give an estimate of the population variance (Central limit theorem)

  22. SSTreatment = 4((74-75.66)2+ (75-75.66)2+(78-75.66)2) = 34.66 *What makes this value get larger?

  23. SSTreatment = 4((74-75.66)2+ (75-75.66)2+(78-75.66)2) = 34.66 *What makes this value get larger? *Differences between treatment groups

  24. Sum of Squares • Have a measure of how much all scores differ • SSTotal • Have a measure of how much this difference is due to subjects • SSSubjects • Have a measure of how much this difference is due to the treatment condition • SSTreatment • To compute error simply subtract!

  25. Sum of Squares • SSError = SSTotal - SSSubjects – SSTreatment 8.0 = 1754.66 - 1712.00 - 34.66

  26. Compute df df total = N -1

  27. Compute df df total = N -1 df subjects = n – 1

  28. Compute df df total = N -1 df subjects = n – 1 df treatment = k-1

  29. Compute df df total = N -1 df subjects = n – 1 df treatment = k-1 df error = (n-1)(k-1)

  30. Compute MS

  31. Compute MS

  32. Compute F

  33. Test F for Significance

  34. Test F for Significance F(2,6) critical = 5.14

  35. Additional tests Can investigate the meaning of the F value by computing t-tests and Fisher’s LSD

  36. Remember

  37. Pill vs. Placebo

  38. Pill vs. Placebo t=1.23

  39. Pill vs. Placebo t=1.23 t (6) critical = 2.447

  40. Pill vs. Placebo t=1.23 Pill vs. No Pill t =4.98* t (6) critical = 2.447

  41. Pill vs. Placebo t=1.23 Pill vs. No Pill t =4.98* Placebo vs. No Pill t =3.70* t (6) critical = 2.447

  42. Practice • You wonder if the statistic tests are of all equal difficulty. To investigate this you examine the scores 4 students got on the three different tests. Examine this question and (if there is a difference) determine which tests are significantly different.

  43. SPSS Homework – Bonus 1) Determine if practice had an effect on test scores. 2) Examine if there is a linear trend with practice on test scores.

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