Where are we?

1. Where are we?

3. What we have covered: - How to write a primary research paper

4. What we have covered: - How to write a primary research paper - How to keep a research notebook

5. What we have covered: - How to write a primary research paper - How to keep a research notebook - Types of variables and scales

6. What we have covered: - How to write a primary research paper - How to keep a research notebook - Types of variables and scales - Types of distributions

7. What we have covered: - How to write a primary research paper - How to keep a research notebook - Types of variables and scales - Types of distributions - Attributes of the normal distribution - testing single values – “z” test (one sample t-test)

8. What we have covered: - How to write a primary research paper - How to keep a research notebook - Types of variables and scales - Types of distributions - Attributes of the normal distribution - Comparing means of two independent groups - t-test and MWU

9. What we have covered: - How to write a primary research paper - How to keep a research notebook - Types of variables and scales - Types of distributions - Attributes of the normal distribution - Comparing means of two independent groups - Experimental designs

10. What we have covered: - How to write a primary research paper - How to keep a research notebook - Types of variables and scales - Types of distributions - Attributes of the normal distribution - Comparing means of two independent groups - Experimental designs - Comparing > 2 groups, and multiple effects - ANOVA

11. Fig. 15.1

12. Differences Between Means • A. 1 Sample • one sample t-test (z test) • B. 2 samples • 1. Independent Groups • - t-test (parametric) • - MWU (non-parametric) • 2. Related Samples • - Paired t-test

13. 2. Related Samples - Paired t-test

14. 2. Related Samples - Paired t-test Suppose we wanted to assess the effect of a muscle-building supplement, and randomly assign people to two groups – placebo and experimental. People differ in many characteristics (ethnicity, sex, weight, diet, etc.), and so these have been randomized across groups. Effects due to these variables are part of the “within group variance” in the denominator.

15. 2. Related Samples - Paired t-test But if we give everyone the drug, and assess their performance before and after, then there is no within group variance between sample points – they are the same individual.

16. 2. Related Samples - Paired t-test We can look at the distribution of differences between before and after weights, and do a “z-test” asking the question: is “0” an unusual value for this sample? Or, how likely is it that “0” (no difference) is a part of this population of differences? If “0” is unlikely, then our population is different from zero; the difference between “before” and “after” is NOT zero – there IS an effect.

17. 2. Related Samples - Paired t-test Y1- 0 S/ n1

18. Person wt. Before wt. After Difference 1 150 156.5 1.5 2 155 156.3 1.3 3 158 159.6 1.6 4 160 161.4 1.4 5 163 164.5 1.5 6 167 166.8 -0.2 7 175 176.3 1.3 8 180 181.5 1.5 9 185 186.1 1.1 10 191 192.6 1.6 mean = 1.26 sd = 0.536 1.26 – 0 0.536 / 10 t = = 7.46

19. In a two-tailed test, we are asking if a value (or sample) IS DIFFERENT FROM a sample… (it can differ because it is LARGER or SMALLER.)

20. In a one-tailed test, we have a preconceived hypothesis about the direction of the effect. In our experiment here, “0” should be LOWER than the mean of our differences. So, our type one error can be “pooled” into one tail of the distribution.

21. So, our type one error can be “pooled” into one tail of the distribution. This means we use the t value in the table A.2 corresponding to p = 0.1 to test at the p = 0.05 level.

22. You need to look at the table, or to enter the correct test in SPSS

23. Differences Between Means • A. 1 Sample • one sample t-test (z test) • B. 2 samples • 1. Independent Groups • - t-test (parametric) • - MWU (non-parametric) • 2. Related Samples • - Paired t-test • - Sign Test (non-parametric)

24. 2. Related Samples - Paired t-test - Sign Test (non-parametric) For the matched pairs, you simply record whether one partner is greater (+) or less than (-) the other:

25. Person wt. Before wt. After Sign 1 150 156.5 + 2 155 156.3 + 3 158 159.6 + 4 160 161.4 + 5 163 164.5 + 6 167 166.8 - 7 175 176.3 + 8 180 181.5 + 9 185 186.1 + 10 191 192.6 +

26. Person wt. Before wt. After Sign 1 150 156.5 + 2 155 156.3 + 3 158 159.6 + 4 160 161.4 + 5 163 164.5 + 6 167 166.8 - 7 175 176.3 + 8 180 181.5 + 9 185 186.1 + 10 191 192.6 + Now, if we are testing the hypothesis of NO effect, then we would expect the “after” to be greater 1/2 the time (p = 0.5), and less than ½ the time (q = 0.5). Pairs that don’t differ are dropped from the analysis, with reduction in n.

27. Person wt. Before wt. After Sign 1 150 156.5 + 2 155 156.3 + 3 158 159.6 + 4 160 161.4 + 5 163 164.5 + 6 167 166.8 - 7 175 176.3 + 8 180 181.5 + 9 185 186.1 + 10 191 192.6 + Now, if we are testing the hypothesis of NO effect, then we would expect the “after” to be greater 1/2 the time (p = 0.5), and less than ½ the time (q = 0.5). So, this reduces to calculating the probability of a particular BINOMIAL OUTCOME, OR SOMETHING MORE EXTREME. Pairs that don’t differ are dropped from the analysis, with reduction in n.

28. So, if p = 0.5 and q = 0.5 (no effect), what would be the probability of having at least 9/10 individuals show a weight gain just by chance? p(9) = (p9) (q1) = 0.009760 p(10) = (p10) (q0) = 0.0009760 10! (9!) (1!) 10! (10!) (0!) 0.010736 This is the one-tailed probability of seeing at least 9/10 showing a weight gain (directional).

29. 2. Related Samples - Paired t-test - Sign Test (non-parametric) - Wilcoxon Signed-ranks test

30. 2. Related Samples - Paired t-test - Sign Test (non-parametric) - Wilcoxon Signed-ranks test In the sign test, the magnitude of the difference doesn’t matter. You could have 5 big positive diffs and 4 very small diffs, and it would still be 5 + and 4 - .

31. 2. Related Samples - Paired t-test - Sign Test (non-parametric) - Wilcoxon Signed-ranks test In the sign test, the magnitude of the difference doesn’t matter. You could have 5 big positive diffs and 4 very small diffs, and it would still be 5 + and 4 - . The signed-ranks test takes the magnitude of the difference into account.

32. Example 9.5 AGGRESSION Female w/o kittens w kittens Diff. Rank 1 3 7 4 4 2 2 8 6 6 3 5 4 -1 1.5(-) 4 6 9 3 3 5 5 10 5 5 6 1 9 8 7 7 8 9 1 1.5

33. Example 9.5 AGGRESSION Female w/o kittens w kittens Diff. Rank 1 3 7 4 4 2 2 8 6 6 3 5 4 -1 1.5(-) 4 6 9 3 3 5 5 10 5 5 6 1 9 8 7 7 8 9 1 1.5 Sum ranks with positive and with negative values: Negative = 1.5 Positive = 26.5 T = lower value = 1.5

34. Example 9.5 AGGRESSION Female w/o kittens w kittens Diff. Rank 1 3 7 4 4 2 2 8 6 6 3 5 4 -1 1.5(-) 4 6 9 3 3 5 5 10 5 5 6 1 9 8 7 7 8 9 1 1.5 Sum ranks with positive and with negative values: Negative = 1.5 Positive = 26.5 T = lower value = 1.5

35. N 0.05 6 0 7 2 8 4 9 6 10 8 11 11 Compare SMALLER value to critical value, at n for number of paired samples. Reject Ho if calculated value is SMALLER THAN critical value, as our is here (1.5 < 2).

36. Fig. 15.1

37. Differences Between Means • A. 1 Sample • B. 2 Samples • C. >2 Samples • 1. 1 factor • - One way ANOVA • - Kruskal-Wallis

38. Consider 4 groups that are not normally distributed, with 10 values each. Rank all values across categories. Sum ranks for categories: 1 2 3 4 162.5 208.5 316.5 132.5 H = 14.273 Use Chi-square distribution,k-1 df.