1. Review: The Logic Underlying ANOVA The possible pair-wise comparisons: X 11 X 12. X 1n X 21 X 22. X 2n Sample 1Sample 2 means: X 31 X 32. X 3n Sample 3

2. Review: The Logic Underlying ANOVA There are k samples with which to estimate population variance X 11 X 12. X 1n X 21 X 22. X 2n Sample 1Sample 2 X 31 X 32. X 3n Sample 3

3. Review: The Logic Underlying ANOVA There are k samples with which to estimate population variance X 11 X 12. X 1n X 21 X 22. X 2n Sample 1Sample 2 X 31 X 32. X 3n Sample 3

4. Review: The Logic Underlying ANOVA There are k samples with which to estimate population variance X 11 X 12. X 1n X 21 X 22. X 2n Sample 1Sample 2 X 31 X 32. X 3n Sample 3

5. Review: The Logic Underlying ANOVA The average of these variance estimates is called the “Mean Square Error” or “Mean Square Within”

6. Review: The Logic Underlying ANOVA There are k means with which to estimate the population variance X 11 X 12. X 1n X 21 X 22. X 2n Sample 1Sample 2 X 31 X 32. X 3n Sample 3

7. Review: The Logic Underlying ANOVA This estimate of population variance based on sample means is called Mean Square Effect or Mean Square Between

8. The F Statistic MS error is based on deviation scores within each sample but… MS effect is based on deviations between samples MS effect would overestimate the population variance when there is some effect of the treatment pushing the means of the different samples apart

9. The F Statistic We compare MS effect against MS error by constructing a statistic called F

10. The F Statistic F is the ratio of MS effect to MS error

11. The F Statistic If the hull hypothesis: is true then we would expect: except for random sampling variation

12. The F Statistic F is the ratio of MS effect to MS error If the null hypothesis is true then F should equal 1.0

13. ANOVA is scalable You can create a single F for any number of samples

14. ANOVA is scalable You can create a single F for any number of samples It is also possible to examine more than one independent variable using a multi- way ANOVA –Factors are the categories of independent variables –Levels are the variables within each factor

15. ANOVA is scalable A two-way ANOVA: 4 levels of factor 1 X1X2XnX1X2Xn X1X2XnX1X2Xn X1X2XnX1X2Xn X1X2XnX1X2Xn X1X2XnX1X2Xn X1X2XnX1X2Xn X1X2XnX1X2Xn X1X2XnX1X2Xn X1X2XnX1X2Xn X1X2XnX1X2Xn X1X2XnX1X2Xn X1X2XnX1X2Xn 3 levels of factor 2 1 2 34 1 2 3

16. Main Effects and Interactions There are two types of findings with multi-way ANOVA: Main Effects and Interactions –For example a main effect of Factor 1 indicates that the means under the various levels of Factor 1 were different (at least one was different)

17. Main Effects and Interactions 4 levels of factor 1 X1X2XnX1X2Xn X1X2XnX1X2Xn X1X2XnX1X2Xn X1X2XnX1X2Xn X1X2XnX1X2Xn X1X2XnX1X2Xn X1X2XnX1X2Xn X1X2XnX1X2Xn X1X2XnX1X2Xn X1X2XnX1X2Xn X1X2XnX1X2Xn X1X2XnX1X2Xn 3 levels of factor 2 1 2 34 1 2 3

18. Main Effects and Interactions 4 levels of factor 1 X1X2XnX1X2Xn X1X2XnX1X2Xn X1X2XnX1X2Xn X1X2XnX1X2Xn X1X2XnX1X2Xn X1X2XnX1X2Xn X1X2XnX1X2Xn X1X2XnX1X2Xn X1X2XnX1X2Xn X1X2XnX1X2Xn X1X2XnX1X2Xn X1X2XnX1X2Xn 3 levels of factor 2 1 2 34 1 2 3

19. Main Effects and Interactions 4 levels of factor 1 X1X2XnX1X2Xn X1X2XnX1X2Xn X1X2XnX1X2Xn X1X2XnX1X2Xn X1X2XnX1X2Xn X1X2XnX1X2Xn X1X2XnX1X2Xn X1X2XnX1X2Xn X1X2XnX1X2Xn X1X2XnX1X2Xn X1X2XnX1X2Xn X1X2XnX1X2Xn 3 levels of factor 2 1 2 34 1 2 3

20. Main Effects and Interactions 4 levels of factor 1 X1X2XnX1X2Xn X1X2XnX1X2Xn X1X2XnX1X2Xn X1X2XnX1X2Xn X1X2XnX1X2Xn X1X2XnX1X2Xn X1X2XnX1X2Xn X1X2XnX1X2Xn X1X2XnX1X2Xn X1X2XnX1X2Xn X1X2XnX1X2Xn X1X2XnX1X2Xn 3 levels of factor 2 1 2 34 1 2 3

21. Main Effects and Interactions A main effect of Factor 1 Factor 1 1 2 3 4 Levels of Factor 2 1 2 3 dependent variable means of each sample

22. Main Effects and Interactions There are two types of findings with multi-way ANOVA: Main Effects and Interactions –For example a main effect of Factor 1 indicates that the means under the various levels of Factor 1 were different (at least one was different) –A main effect of Factor 2 indicates that the means under the various levels of Factor 2 were different

23. Main Effects and Interactions 4 levels of factor 1 X1X2XnX1X2Xn X1X2XnX1X2Xn X1X2XnX1X2Xn X1X2XnX1X2Xn X1X2XnX1X2Xn X1X2XnX1X2Xn X1X2XnX1X2Xn X1X2XnX1X2Xn X1X2XnX1X2Xn X1X2XnX1X2Xn X1X2XnX1X2Xn X1X2XnX1X2Xn 3 levels of factor 2 1 2 34 1 2 3

24. Main Effects and Interactions 4 levels of factor 1 X1X2XnX1X2Xn X1X2XnX1X2Xn X1X2XnX1X2Xn X1X2XnX1X2Xn X1X2XnX1X2Xn X1X2XnX1X2Xn X1X2XnX1X2Xn X1X2XnX1X2Xn X1X2XnX1X2Xn X1X2XnX1X2Xn X1X2XnX1X2Xn X1X2XnX1X2Xn 3 levels of factor 2 1 2 34 1 2 3

25. Main Effects and Interactions 4 levels of factor 1 X1X2XnX1X2Xn X1X2XnX1X2Xn X1X2XnX1X2Xn X1X2XnX1X2Xn X1X2XnX1X2Xn X1X2XnX1X2Xn X1X2XnX1X2Xn X1X2XnX1X2Xn X1X2XnX1X2Xn X1X2XnX1X2Xn X1X2XnX1X2Xn X1X2XnX1X2Xn 3 levels of factor 2 1 2 34 1 2 3

26. Main Effects and Interactions A main effect of Factor 2 Factor 1 1 2 3 4 Levels of Factor 2 1 2 3 dependent variable

27. Main Effects and Interactions There are two types of findings with multi-way ANOVA: Main Effects and Interactions –For example a main effect of Factor 1 means that the means under the various levels of Factor 1 were different (at least one was different) –A main effect of Factor 2 means that the means under the various levels of Factor 2 were different –An interaction means that there was an effect of one factor but the effect is different for different levels of the other factor

28. Main Effects and Interactions An Interaction Factor 1 1 2 3 4 Levels of Factor 2 1 2 3 dependent variable

29. Correlation We often measure two or more different parameters of a single object

30. Correlation This creates two or more sets of measurements

31. Correlation These sets of measurements can be related to each other –Large values in one set correspond to large values in the other set –Small values in one set correspond to small values in the other set

32. Correlation examples: –height and weight –smoking and lung cancer –SES and longevity

33. Correlation We call the relationship between two sets of numbers the correlation

34. Correlation Measure heights and weights of 6 people PersonHeightWeight a5’4120 b5’10140 c5’2100 d5’1110 e5’6140 f5’8150

35. Correlation Height vs. Weight 5’5’25’45’65’85’10 100110120130140150 Weight Height

36. Correlation Height vs. Weight 5’5’25’45’65’85’10 100110120130140150 a a Weight Height

37. Correlation Height vs. Weight 5’5’25’45’65’85’10 100110120130140150 a a b b Weight Height

38. Correlation Height vs. Weight 5’5’25’45’65’85’10 100110120130140150 a a b b, e c c d d ef f Weight Height

39. Correlation Notice that small values on one scale pair up with small values on the other 5’5’25’45’65’85’10 100110120130140150 a a b b, e c c d d ef f Weight Height

40. Correlation Scatter Plot shows the relationship on a single graph Like two number lines perpendicular to each other 5’5’25’45’65’85’10 100110120130140150 a a b b, e c c d d ef f Think of this as the y-axis Think of this as the x-axis

41. Correlation Scatter Plot shows the relationship on a single graph 5’5’25’45’65’85’10 a b cdef 100 110 120 130 140 150 a b, e c d f Weight Height * * * * * *

42. Correlation The relationship here is like a straight line We call this linear correlation * * * * * *

43. Various Kinds of Linear Correlation Strong Positive

44. Various Kinds of Linear Correlation Weak Positive

45. Various Kinds of Linear Correlation Strong Negative

46. Various Kinds of Linear Correlation No (or very weak) Correlation y values are random with respect to x values

47. Various Kinds of Linear Correlation No Linear Correlation

48. Correlation Enables Prediction Strong correlations mean that we can predict a y value given an x value…this is called regression Accuracy of our prediction depends on strength of the correlation

49. Spurious Correlation Sometimes two measures (called variables) both correlate with some other unknown variable (sometimes called a lurking variable) and consequently correlate with each other This does not mean that they are causally related! e.g. use of cigarette lighters positively correlated with incidence of lung cancer

50. Next Time: measuring correlations