1. Spiders on Mazurian lake islands: Wigry –Mikołajki, Nidzkie, Bełdany) Lecture 2 Analysis of variance Photo: Wigierski Park Narodowe Photo: Ruciane.net Araneus diadematus Salticidae Photo: Eurospiders.com

2. Spider species richness on Mazurian lake islands Does species richness differ with respect to the degree of disturbance? If we use the same test several times with the same data we have to apply a Bonferroni correction. Single test n independent tests

3. Spider species richness on Mazurian lake islands sH2sH2 sM2sM2 sL2sL2 sP2sP2 sT2sT2 If there would be no difference between the sites the average within variance s Within 2 should equal the variance between the sites s Between 2. One way analysis of variance Sir Ronald Aylmer Fisher (1890-1962) s Between 2 We test for significance using the F-test of Fisher with k-1 (Between) and n-k (Within) degrees of freedom. n-1 = n-k + k-1 df Total df Within df Between

5. Welch test The Levene test compares the group variances using the F distribution. Variances shouldn’t differ too much (shouldn’t be heteroskedastic)!!! The Tuckey test compares simultaneously the means of all combinations of groups. It’s a t-test corrected for multiple comparisons (similar to a Bonferroni correction)

7. We include the effect of island complex (Wigry – Nidzkie, Bełdany, Mikołaiki) There must be at least two data for each combination of groups. We use a simple two way ANOVA Main effectsSecondary effects

8. The significance levels have to be divided by the number of tests (Bonferroni correction) Spider species richness does not significantly depend on island complex and degree of disturbance.

9. Correcting for covariates: Anaysis of covariance Instead of using the raw data we use the residuals. These are the area corrected species numbers. The conmparison of within group residuals and between group residuals gives our F-statistic.

10. Disturbance does not significantly influence area corrected species richness SS total = SS between + SS error Within group residuals Total residuals We need four regression equations: one from all data points and three within groups.

11. Repetitive designs In medical research we test patients before and after medical treatment to infer the influence of the therapy. We have to divide the total variance (SS total ) in a part that contains the variance between patients (SS between ) and within the patient (SS within ). The latter can be divided in a part that comes from the treatment (SStreat) and the error (SS error ) Medical treatment Before After SS within SS between

12. Before – after analysis in environmental protection In the case of unequal variances between groups it is save to use the conservative ANOVA with (n-1) df error and only one df Effect in the final F-test. df treat = k-1 df Error = (n-1)(k-1)

13. Bivariate comparisons in environmental protection The outlier would disturb direct comparisons of species richness Due to possible differences in island areas between the two island complexes we have to use the residuals. A direct t-test on raw data would be erroneous.

14. Permutation testing Observed P(t) Upper 2.5% confidence limit. 10000 randomizations of observed values gives a null distribution of t-values and associated probability levels with which we compare the observed t. This gives the probability level for our t-test.

15. Bivariate comparisons using ANOVA t and F tests can both be used for pair wise comparisons.

16. Repeated measures Species richness of ground living Hymenoptera in a beech forest Photo Tim Murray Photo Simon van Noort

17. Advices for using ANOVA:  You need a specific hypothesis about your variables. In particular, designs with more than one predicator level (multifactorial designs) have to be stated clearly.  ANOVA is a hypothesis testing method. Pattern seeking will in many cases lead to erroneous results.  Predicator variables should really measure different things, they should not correlate too highly with each other  The general assumptions of the GLM should be fulfilled. In particular predicators should be additive. The distribution of errors should be normal.  It is often better to use log-transformed values  In monofactorial designs where only one predicator variable is tested it is often preferable to use the non-parametric alternatives to ANOVA, the Kruskal Wallis test. The latter test does not rely on the GLM assumptions but is nearly as powerful as the classical ANOVA.  Another non-parametric alternative for multifactorial designs is to use ranked dependent variables. You loose information but become less dependent on the GLM assumptions.  ANOVA as the simplest multivariate technique is quite robust against violations of its assumptions.