2. 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?

3. 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.

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

5. Tests to Compare Means Independent Variables and # of levels Independent SamplesRelated Samples One IV, 2 levelsIndependent t-testPaired-samples t-test One IV, 2 or more levelsANOVARepeated measures ANOVA Tow IVs, each with 2 or more levels Factorial ANOVARepeated measures factorial ANOVA Design of experiment

6. 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.

7. Results PillPlaceboNo Pill Sub 1576064 Sub 2717274 Sub 3757678 Sub 4939296 Mean747578

8. For now... Ignore that it is a repeated design PillPlaceboNo Pill Sub 1576064 Sub 2717274 Sub 3757678 Sub 4939296 Mean747578

9. PillPlaceboNo Pill Sub 1576064 Sub 2717274 Sub 3757678 Sub 4939296 Mean747578 Between Variability = low

10. PillPlaceboNo Pill Sub 1576064 Sub 2717274 Sub 3757678 Sub 4939296 Mean747578 Within Variability = high

12. Notice – the within variability of a group can be predicted by the other groups PillPlaceboNo Pill Sub 1576064 Sub 2717274 Sub 3757678 Sub 4939296 Mean747578

13. Notice – the within variability of a group can be predicted by the other groups PillPlaceboNo Pill Sub 1576064 Sub 2717274 Sub 3757678 Sub 4939296 Mean747578 Pill and Placebo r =.99; Pill and No Pill r =.99; Placebo and No Pill r =.99

14. PillPlaceboNo PillMean Sub 157606460.33 Sub 271727472.33 Sub 375767876.33 Sub 493929693.66 Mean747578 These scores are correlated because, in general, some subjects tend to do very well and others tended to do very poorly

15. 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.

16. PillPlaceboNo PillMean Sub 157606460.33 Sub 271727472.33 Sub 375767876.33 Sub 493929693.66 Mean74757875.66 Example

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

18. PillPlaceboNo PillMean Sub 157606460.33 Sub 271727472.33 Sub 375767876.33 Sub 493929693.66 Mean74757875.66 SS total = (57-75.66) 2 + (71-75.66) 2 +.... (96-75.66) 2 = 908 *What makes this value get larger?

19. PillPlaceboNo PillMean Sub 157606460.33 Sub 271727472.33 Sub 375767876.33 Sub 493929693.66 Mean74757875.66 SS total = (57-75.66) 2 + (71-75.66) 2 +.... (96-75.66) 2 = 908 *What makes this value get larger? *The variability of the scores!

20. 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)

21. PillPlaceboNo PillMean Sub 157606460.33 Sub 271727472.33 Sub 375767876.33 Sub 493929693.66 Mean74757875.66 SS Subjects = 3((60.33-75.66) 2 + (72.33-75.66) 2 +.... (93.66-75.66) 2) = 1712 *What makes this value get larger?

22. PillPlaceboNo PillMean Sub 157606460.33 Sub 271727472.33 Sub 375767876.33 Sub 493929693.66 Mean74757875.66 SS Subjects = 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

23. 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)

24. PillPlaceboNo PillMean Sub 157606460.33 Sub 271727472.33 Sub 375767876.33 Sub 493929693.66 Mean74757875.66 SS Treatment = 4((74-75.66) 2 + (75-75.66) 2 +(78-75.66) 2) = 34.66 *What makes this value get larger?

25. PillPlaceboNo PillMean Sub 157606460.33 Sub 271727472.33 Sub 375767876.33 Sub 493929693.66 Mean74757875.66 SS Treatment = 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

26. Sum of Squares Have a measure of how much all scores differ –SS Total Have a measure of how much this difference is due to subjects –SS Subjects Have a measure of how much this difference is due to the treatment condition –SS Treatment To compute error simply subtract!

27. Sum of Squares SS Error = SS Total - SS Subjects – SS Treatment 8.0 = 1754.66 - 1712.00 - 34.66

28. Compute df SourcedfSS Subjects1712.00 Treatment34.66 Error8.00 Total111754.66 df total = N -1

29. Compute df SourcedfSS Subjects31712.00 Treatment34.66 Error8.00 Total111754.66 df total = N -1 df subjects = n – 1

30. Compute df SourcedfSS Subjects31712.00 Treatment234.66 Error8.00 Total111754.66 df total = N -1 df subjects = n – 1 df treatment = k-1

31. Compute df SourcedfSS Subjects31712.00 Treatment234.66 Error68.00 Total111754.66 df total = N -1 df subjects = n – 1 df treatment = k-1 df error = (n-1)(k-1)

32. Compute MS SourcedfSSMS Subjects31712.00 Treatment234.6617.33 Error68.00 Total111754.66

33. Compute MS SourcedfSSMS Subjects31712.00 Treatment234.6617.33 Error68.001.33 Total111754.66

34. Compute F SourcedfSSMSF Subjects31712.00 Treatment234.6617.3313.00 Error68.001.33 Total111754.66

35. Test F for Significance SourcedfSSMSF Subjects31712.00 Treatment234.6617.3313.00 Error68.001.33 Total111754.66

36. Test F for Significance SourcedfSSMSF Subjects31712.00 Treatment234.6617.3313.00* Error68.001.33 Total111754.66 F(2,6) critical = 5.14

38. Additional tests SourcedfSSMSF Subjects31712.00 Treatment234.6617.3313.00* Error68.001.33 Total111754.66 Can investigate the meaning of the F value by computing t-tests and Fisher’s LSD

39. Remember

40. PillPlaceboNo PillMean 74757875.66

41. PillPlaceboNo PillMean 74757875.66 Pill vs. Placebo

42. PillPlaceboNo PillMean 74757875.66 Pill vs. Placebo t=1.23

43. PillPlaceboNo PillMean 74757875.66 Pill vs. Placebo t=1.23 t (6) critical = 2.447

44. PillPlaceboNo PillMean 74757875.66 Pill vs. Placebo t=1.23 Pill vs. No Pill t =4.98* t (6) critical = 2.447

45. PillPlaceboNo PillMean 74757875.66 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

46. 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.

47. Test 1Test 2Test 3 Sub 1607078 Sub 2787685 Sub 3649089 Sub 4778194

50. 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.

52. Why is this important? Requirement Understand research articles Do research for yourself Real world

53. The Three Goals of this Course 1) Teach a new way of thinking 2) Teach “factoids”

54. Mean

55. r =

56. What you have learned! Describing and Exploring Data / The Normal Distribution Scales of measurement –Populations vs. Samples Learned how to organize scores of one variable using: –frequency distributions –graphs

57. What you have learned! Measures of central tendency –Mean –Median –Mode Variability –Range –IQR –Standard Deviation –Variance

58. What you have learned! –Z Scores –Find the percentile of a give score –Find the score for a given percentile

59. What you have learned! Sampling Distributions & Hypothesis Testing –Is this quarter fair? –Sampling distribution CLT –The probability of a given score occurring

60. What you have learned! Basic Concepts of Probability –Joint probabilities –Conditional probabilities –Different ways events can occur Permutations Combinations –The probability of winning the lottery –Binomial Distributions Probability of winning the next 4 out of 10 games of Blingoo

61. What you have learned! Categorical Data and Chi-Square –Chi square as a measure of independence Phi coefficient –Chi square as a measure of goodness of fit

62. What you have learned! Hypothesis Testing Applied to Means –One Sample t-tests –Two Sample t-tests Equal N Unequal N Dependent samples

63. What you have learned! Correlation and Regression –Correlation –Regression

64. What you have learned! Alternative Correlational Techniques –Pearson Formulas Point-Biserial Phi Coefficent Spearman’s rho –Non-Pearson Formulas Kendall’s Tau

65. What you have learned! Multiple Regression –Multiple Regression Causal Models Standardized vs. unstandarized Multiple R Semipartical correlations –Common applications Mediator Models Moderator Mordels

66. What you have learned! Simple Analysis of Variance –ANOVA –Computation of ANOVA –Logic of ANOVA Variance Expected Mean Square Sum of Squares

67. What you have learned! Multiple Comparisons Among Treatment Means –What to do with an omnibus ANOVA Multiple t-tests Linear Contrasts Orthogonal Contrasts Trend Analysis –Controlling for Type I errors Bonferroni t Fisher Least Significance Difference Studentized Range Statistic Dunnett’s Test

68. What you have learned! Factorial Analysis of Variance –Factorial ANOVA –Computation and logic of Factorial ANOVA –Interpreting Results Main Effects Interactions

69. What you have learned! Factorial Analysis of Variance and Repeated Measures –Factorial ANOVA –Computation and logic of Factorial ANOVA –Interpreting Results Main Effects Interactions –Repeated measures ANOVA

70. The Three Goals of this Course 1) Teach a new way of thinking 2) Teach “factoids” 3) Self-confidence in statistics

71. CRN 33515.0

73. Four Step When Solving a Problem 1) Read the problem 2) Decide what statistical test to use 3) Perform that procedure 4) Write an interpretation of the results

74. Four Step When Solving a Problem 1) Read the problem1) Read the problem 2) Decide what statistical test to use 3) Perform that procedure3) Perform that procedure 4) Write an interpretation of the results4) Write an interpretation of the results

75. Four Step When Solving a Problem 1) Read the problem 2) Decide what statistical test to use2) Decide what statistical test to use 3) Perform that procedure 4) Write an interpretation of the results

76. How do you know when to use what? If you are given a word problem, would you know which statistic you should use?

77. Example An investigator wants to predict a male adult’s height from his length at birth. He obtains records of both measures from a sample of males.

79. Example An investigator wants to predict a male adult’s height from his length at birth. He obtains records of both measures from a sample of males. Use regression

80. Type of Data Qualitative One categorical variable Goodness of Fit Chi Square Two categorical variables Independence Chi Square Quantitative Differences One Group One sample t-test Two Groups Independent Groups Two-sample t-test Dependent Groups Dependent t-test Multiple Groups Independent Groups One IV One-way ANOVA Two IVs Factorial ANOVA Dependent Groups Repeated mmeasures ANOVA Relationships One predictor Continuous measurement Degree of Relationship Pearson Correlation Prediction Regression Ranks Spearmn’s r or Kendell’s Tau Two predictors Multiple Regression