Experimental Design and Data Analysis Part 2 Assump@ons for Parametric Tests t- test and ANOVA Independence Variance Normality t-test Yes Yes Yes ANOVA Yes Yes Yes Lecture 7 AEC 460 Assume homogeneity of variances What s up with blocking, why should we do it, and what s a residual?!?! Examples of when you would use a block or a paired t- test, quick and dirty Environmental change in the sampling area, increases the within group varia@on, that may limit our ability to detect significant differences in our response variables Inter@dal Gradient of Exposure Edge effects along an eleva@on gradient ANOVAs Comparison of mul@ple means, normal distribu@on, equal varia@on F test for difference between any of the treatment means Then do a mul@ple comparison tests (LSD, least significant difference) 1
Nudibranchs in inter@dal regions with and without predators Preda@on on nudibranchs in the inter@dal community There is no difference in nudibranch numbers in reefs with or without predators Predator exclusion Experiment with more than two comparisons Have to compare more than two means Experiment on the effect of excluding predators of reef organisms What are we tes@ng (F Test) H0 = ucontrol = ucagecontrol = ucage Should you do three t- tests? No errors are compounded Be4er to use another technique that will avoid compounding error F = Treatment MS/Residual MS 2
NOTE: 3 Treatments (Cage, Control, Cage Control) This is a single Factor 6 Replicates in each treatment Here s how it works Each treatment (1-3) has associated mean and variance ANOVA par@@ons the variance into components TOTAL variance = Variance between + Variance within 3
Within Group Means: 13.67 8.67 10.33 Overall Mean = 10.9 NOTE: 3 Treatments (Cage, Control, Cage Control) 6 Replicates in each treatment One- way ANOVA General form of the t- test, can have more than 2 samples One- way ANOVA General form of the t- test, can have more than 2 samples A B C Ho: All samples the same Ha: At least one sample different Ho DATA A B A C B C Ha A C B 4
One- way ANOVA Just like t- test, compares differences between samples to differences within samples Mean squares: T- test sta@s@c (t) A B C Difference between means Standard error within sample MS= Sum of squares df Analogous to variance ANOVA sta@s@c (F) MS between groups MS within group ANOVA tables Variance: S 2 = Σ (x i x ) 2 n- 1 Sum of squared differences Treatment (between groups) Error (within groups) df SS MS F p df (X) SSX SSX MSX Look df (X)} MSE up! df (E) SSE SSE df (E) Total df (T) SST } 5
ANOVA in ANUTSHELL If the samples have been drawn from the same normally distributed popula@on, they should have equal means and variances The variance between groups = the variance within groups If NOT the case, then the samples were drawn from popula@ons with different means and/or variances How is it computed: Analysis of Variance Compute the variance between groups Divide by the variance within groups Logic: if there is no treatment effect, the variance between groups should be the same as the variance within groups As a result this would be the case: variance between variance within = 1 Deep Dark Secret WITHIN GROUPS is the same as ERROR Don t be confused by your sta@s@cal soeware output!! 6
between How much greater does your F- value need to be for you to say one of the groups is different? LOOK UP Cri@cal Value in your F- table! variation between > variation within 3.68 9.41 7
Assump_ons of ANOVA ANOVA is a parametric test which assumes that the data analyzed: Are con@nuous, interval data comprising a whole popula@on or sampled randomly from a popula@on. Has a normal distribu@on. Moderate departure from the normal distribu@on does not unduly disturb the outcome of ANOVA, especially as sample sizes increase. Highly skewed datasets result in inaccurate conclusions. The groups are independent of each other. The variances in the two groups should be similar. 8