HR (bpm)

1. Effects of posture 84 80 HR (bpm) 76 72 68 64 lying sitting standing Does posture affect HR?

2. Analysis of variance (ANOVA): Based on the null hypothesis that all 3 samples are drawn from the same population, so that the variance of each sample will be an estimate of the same (population) variance (). Thus, according to the null hypothesis, the variance within samples or the variance between samples are both estimates of the population variance  The “within samples” variance is determined by calculating, for each sample, the sum of squares (SS) of differences between each value and the sample mean, adding these up and dividing by the number of degrees of freedom. The “between samples” variance is determined by (1) calculating the mean of all values for all samples (the “grand mean”), (2) in each sample, replace each individual value by the mean of that sample; (3) then calculating the sum of squares (SS) of differences between each new replaced value and the grand mean, adding these up and dividing by the number of degrees of freedom.

3. “within samples” variance “between samples” variance 70.56 77.64 81.03 70.56 77.64 81.03 Grand mean = 76.41 54.49 768.40 1233.01 4336.97 (SS) 3322.89 3252.31 Total SS = 3322.89 + 3252.31 + 4336.97 = 10912.17 Total SS = 1233.01 + 54.49 + 768.40= 2055.90 No. of degrees of freedom = (36 -1) + (36 -1) + (36 -1) = 105 No. of degrees of freedom = (3-1) = 2 Variance estimate = 10912.17/105 = 103.93 Variance estimate = 2055.87/2 = 1027.95

4. Thus, the estimate of the population variance based on the variations between sample means (1027.95) is much greater than the estimate based on variations of individual values about their own sample means (103.93). If the null hypothesis were correct, one would expect the two estimates to be similar. The degree of discrepancy is called F, and is calculated as the ratio: between sample variance estimate/ within sample variance estimate In this case, F = 1029.95/103.93 = 9.89 Assuming the null hypothesis is true, what is the probability of obtaining By chance a value of F ≥ 9.89 on (2,105) degrees of freedom? Answer: P = 0.00012 Therefore the null hypothesis is rejected

5. ANOVA tests can be performed on Excel. The results will look like the following:

6. In the data set above, the gender of the subject was also recorded, as follows: Effects of posture 92 girls 88 boys 84 80 76 72 68 64 lying sitting standing Does posture affect HR? Is there a difference between males and females? 3) Are the effects of posture and gender on HR additive or interactive?

7. In this case, there are two factors that affect the variable (posture and gender). To answer the previous questions, an ANOVA (two factor with replication) is required. The results are as follows:

8. Imagine that we have a slightly different set of data, where the differences between the genders are more exaggerated, as follows: Effects of posture 92 girls 88 boys 84 80 76 72 68 64 lying sitting standing Does posture affect HR? Is there a difference between males and females? 3) Are the effects of posture and gender on HR additive or interactive?

9. The results are as follows. Note that the P-values are less

10. Imagine a quite different data set, as follows: Effects of posture 92 girls 88 boys 84 80 76 72 68 64 lying sitting standing Does posture affect HR? Is there a difference between males and females? 3) Are the effects of posture and gender on HR additive or interactive?

11. The results are as follows: