1. AdvancedAnalysesAdvancedAnalyses

2. Example 11.1 Treatments Litter #Group AGroup BGroup C 111.813.69.2 212.014.49.6 310.712.88.6 411.113.08.5 512.113.49.8 mean11.5413.449.14 variance0.3730.3880.338 Total variance = 3.631 Five litters of mice of about the same age are selected. One member of each litter was randomly assigned to 1 of 3 diets, and their weight gain was recorded.

3. One-Way ANOVA to compare means

4. Randomized Complete Block Designs Sometimes experiments are set up where each observation in one of the treatments has something in common with one observation in each of the other treatments - each of these groups of related data is referred to as a block. The idea here is that some populations contain variance that makes the within groups variance very high. However, sometimes this variance can be grouped, or blocked out, in order to partition out this variance from the error variance.

5. Randomized Complete Block Designs calculate another source of variance – ‘block’ This design thus partitions some of the variance and degrees of freedom from the error variance into the block variance

6. Example 11.1 Treatments Litter #Group AGroup BGroup C Block 111.813.69.2 212.014.49.6 310.712.88.6 411.113.08.5 512.113.49.8 mean11.5413.449.14 variance0.3730.3880.338 Total variance = 3.631 Five litters of mice of about the same age are selected. One member of each litter was randomly assigned to 1 of 3 diets, and their weight gain was recorded.

7. Example 11.1 Generalized ANOVA Table Source of Variation Sum of Squares dfMSF Treatment SS Trt k-1SS Trt /k-1MS Trt /MS E Block SS B B-1SS B /B-1 Error SS E (k-1)(B-1)SS E / (k-1)(B-1) Total SS Tot N-1SS Tot /N-1 K = number of groups; N = total number of observations B = number of blocks

8. Let’s move to the SPSS demo:

9. Factorial Designs In many situations more than one factor, or treatment interacts to produce effects beyond the sum of the effects of the 2 acting alone. –In other words, factors may act synergistically or antagonistically. When some form of interaction is suspected, a two factor ANOVA, or a Factorial design, is appropriate. These are similar to a blocked design, except that the “block” now represents a factor in which we have an interest and each “block” x treatment cell consists of repeated observations.

10. Generalized ANOVA Table Source of Variation Sum of Squares dfMSF Category A SS Trt A A-1SS Trt /k-1MS Trt A /MS E Category B SS Trt B B-1SS B /B-1MS Trt B /MS E Interaction SS A x B (A-1)(B-1)MS A x B /MS E Error SS E (AxB)(n-1)SS E / (k-1)(B-1) Total SS Tot N-1SS Tot /N-1 A = number of levels in category A; B = number of levels in category B N = total number of observations, n = number of observations each category

11. DamageNo Damage No Yeast 00 1530 1245 2560 40100 Yeast 30115 15140 2075 2060 25110

12. No DamageDamage 20406080100 No Yeast Yeast

13. Example (Q 11.9) The possible influence of crowding and sex on plasma corticosterone in a highly inbred strain of rats was investigated using a factorial design. Sex (males, nongravid females, and gravid females) and crowding (low, moderate, and high) were used as the main treatment effects.

14. Example (Q 11.9) SexLowModerateHigh Males5121.5253 8122249 13119260 9130257 15114280 11129263 Nongravid Females12117.7219 19115222 15121218 20117220 11118223 18120225 Gravid Females37157289 42160273 50173280 35182291 40168205 36170296

15. SPSS Means Figure

16. At low crowding stress, gravid females have the greatest plasma corticosterone levels, followed by nongravid females, with males exhibiting the lowest levels. At high and moderate stress levels, gravid females still exhibit the greatest plasma corticosterone levels. However, at these two levels of stress, males have higher corticosterone levels than nongravid females. Low Medium High Males Females Gravid Females Corticosterone Concentration (X + 1 sd)

17. Nested ANOVAS BENCH FLAT PLANT Variation in Growth

18. Nested ANOVAS BENCH FLAT PLANT Variation in Growth 81 87 82 84 75 79 76 77 7567 68 69 75 66 73 72 76 77 75 74 72 74 73 80

19. For Nested ANOVA: - First, EDIT > OPTIONS > “Open syntax window @ startup” - see if it opens… if not reboot. NOW, from taskbar, there should be a new window…syntax Analyze > General Lin. Model dependent var = growth random effects = bench flat model = bench flat main effects Run Copy LOG box from output Paste in syntax window… change flat to flat(bench) Get cursor to top line, before UNIANOVA PRESS green ARROW to RUN!

20. UNIANOVA growth BY bench flat /RANDOM=bench flat /METHOD=SSTYPE(3) /INTERCEPT=INCLUDE /CRITERIA=ALPHA(0.05) /DESIGN=bench flat(bench). Tests of Between-Subjects Effects Dependent Variable:growth SourceType III Sum of SquaresdfMean SquareFSig. InterceptHypothesis136052.0421136052.042753.056.001 Error361.3332180.667 a benchHypothesis361.3332180.6675.510.099 Error98.375332.792 b flat(bench)Hypothesis98.375332.7923.754.030 Error157.250188.736 c a. MS(bench) b. MS(flat(bench)) c. MS(Error)

21. Mixed Model ANOVAS BENCH FLAT PLANT Variation in Growth Nitrogen Potassium

22. Mixed Model ANOVAS BENCH FLAT PLANT Variation in Growth 81 87 82 84 75 79 76 77 7567 68 69 75 73 66 72 76 77 75 74 80 74 72 73 Nitrogen Potassium

23. UNIANOVA growth BY fert bench flat /RANDOM=bench flat /METHOD=SSTYPE(3) /INTERCEPT=INCLUDE /CRITERIA=ALPHA(0.05) /DESIGN=bench fert flat(bench) fert*flat(bench). Run an ANOVA as before with the terms… copy LOG, and rewrite DESIGN line as follows:

24. EXPERIMENTAL DESIGN MATTERS – IT AFFECTS HOW YOU ANALYZE THE PATTERNS IN THE DATA