1. Analysis of variance (3)

2. Normality Check Frequency histogram (Skewness & Kurtosis) Probability plot, K-S test Normality Check Frequency histogram (Skewness & Kurtosis) Probability plot, K-S test Descriptive statistics Measurements (data) Measurements (data) Mean, SD, SEM, 95% confidence interval YES Check the Homogeneity of Variance Data transformation NO Data transformation NO Median, range, Q1 and Q3 Non-Parametric Test(s) For 2 samples: Mann- Whitney For 2-paired samples: Wilcoxon For >2 samples: Kruskal-Wallis Sheirer-Ray-Hare Non-Parametric Test(s) For 2 samples: Mann- Whitney For 2-paired samples: Wilcoxon For >2 samples: Kruskal-Wallis Sheirer-Ray-Hare Parametric Tests Student’s t tests for 2 samples; ANOVA for  2 samples; post hoc tests for multiple comparison of means Parametric Tests Student’s t tests for 2 samples; ANOVA for  2 samples; post hoc tests for multiple comparison of means YES Multi-way ANOVA (Ch 14) Nested ANOVA (Ch 15) e.g. F max test Friedman p. 263-265 Non-parametric Repeated-measures ANOVA Univariate ANOVA Only one dependent variable Non-parametric 2-way ANOVA with replication

3. Multi-way ANOVA Multi-way ANOVA Effects of sex and water temperature on the oxygen consumption rate of three species of inter-tidal crabs

4. Multi-way ANOVA e.g. We would like to investigate the effects of sex and temperature (10, 20, 30  C) on oxygen consumption rate (OC; mg O 2 /hr/individual) of three different species of crabs of similar size Dependent variable = OC 3-Factors = Species (3 levels), Sex (2 levels) and Temperature (3 levels) 4 Replicates per group (balanced design), thus Total N = 3 x 2 x 3 x 4 = 72

5. The oxygen consumption rate (mg O2/hr/individual) of the crabs

6. Species A Species C Species B

7. Data input in SPSS Column 1 (Species): 1, 2, 3 Column 2 (Temp): 1, 2, 3 Column 3 (Sex): 1, 2 Column 4 (OC): dependent variable Model or effects in hypothesis: –Species –Temp –Sex –Species  Temp –Species  Sex –Temp  Sex –Species  Temp  Sex

8. Computer Output There is no critical values in Table B4 for d.f. = 54, so the values for the next lower d.f. = 50 were utilized. How to obtain the DF?

9. Computer Output In conclusion, the oxygen consumption rates (OCR) of the three species are not the same; and OCR increase with temperate. Furthermore, OCR of a species is dependent on temperature and sex as indicated by the significant interactions.

10. Species A Species C Species B Remember to double check your ANOVA results against the figures!

11. Number of hypotheses potentially testable in ANOVA

12. Exercises – Please try to do these exercises at home or on this coming Computer lab Chapter 14 (Zar 1999) Page 301-2: Questions 14.4, 14.5 and 14.6

13. Nested (Hierarchical) ANOVA Chapter 15 In some experimental designs, we may have –some levels of one factor occurring in combination with the levels of one or more other factors, and other distinctly different levels occurring in combination with others. –e.g. testing the influence of drugs (3 types) on the blood cholesterol level in women while the drugs are produced by different sources (2 company)

14. Nested (Hierarchical) ANOVA e.g. testing the influence of drugs (3 types) on the blood cholesterol level in women while the drugs are originated from different sources –Each drug obtained from 2 sources but the 2 sources were different for all the drugs –Two factors: drug type and drug source –Nested design: with one factor (drug source) being nested within the major factor (drug type)

15. The nested factor is typically random (as this example) This example may be considered to be a kind of one-way ANOVA, however, a different denominator (i.e. not the error MS), subgroups MS, must be used to calculate the F value for the main factor (i.e. drug type). Drug XDrug YDrug Z Source A B C D E F Test Ho: Same blood cholesterol concentrations

16. C = (1270) 2 /12 = 134408.33 Total SS = 134480 - 134408.33 = 71.67 Among all subgroups SS = 134471 - 134408.33 = 62.67 Error SS = Total SS – among all subgroups SS = 71.67 – 62.67 = 9.00 Groups SS = 134469.5 – 134408.33 = 61.17 Subgroups SS = among all subgroups SS – groups SS = 62.67 =61.17 = 1.50

17. a = 3 [3 drugs] b = 2 [2 sources] Total DF = N - 1 = 12 – 1 = 11 Among all subgroups DF = ab –1 = (3)(2) – 1 = 5 Error DF = Total – among all subgroups = 11 –5 = 6 Groups SS = 3 – 1 = 2 Subgroups SS = a(b-1) = 3(2-1) = 3 Using subgroups MS as the denominator In conclusion, (1) there is no significant difference among the drug sources in affecting blood cholesterol concentrations (F 0.05(1), 3, 6 = 4.76, P > 0.5) ; and (2) there is significant difference in cholesterol concentrations owing to the three drugs (F 0.05(1), 2, 3 = 9.55, P < 0.01).

18. Example 15.2 (p. 309; Zar 1999) ANOVA with a random-effects factor (blood collection) nested within the two-factor crossed experimental design Effects of sex and hormone treatment on plasma calcium concentrations (mg/ 100 ml) of birds: For each of the 4 combinations of sex and hormone treatment (a = 2 & b =2), there are 5 animals (c = 5), from each of which three blood collections are taken (n = 3). N = abcn = 60

19. Example 15.2 (p. 309, Zar, 1999) ANOVA with a random-effects factor (blood collection) nested within the two-factor crossed experimental design

20. Exercise In an experiment on cholesterol level in blood of mice two levels of fat intake are fixed by the researchers and coded as level 0 and 1. For each level of intake, there are three populations of mice in separate cages and from each of these individual mice are selected at random for blood testing.

21. Other example