MARE 250 Dr. Jason Turner

Analysis of Variance(ANOVA)

Analysis of Variance (ANOVA)

Method for comparing the means of more than two populations

1-Sample t-test – 1R, 1F, 1 Level2-Sample t-test – 1R, 1F, 2 Levels1-Way ANOVA – 1R, 1F, >2 Levels

ANOVA

ANOVAResearch Question: Are there differences in the mean number of total urchins across locations (transects) at Onekahakaha?

ANOVAResearch Question: Are there differences in the mean number of total urchins across locations at Onekahakaha?

Null hypothesis:Ho: μ (urch shallow) = μ (urch middle) = μ (urch deep)

Ha: All means not equal

ANOVAWhy not run multiple T-test?

μ1 μ2 μ3

ANOVAWhy not run multiple T-test?

1. Number of t-tests increases with # of groupsbecomes cognitively difficult

2. ↑ Number of analyses = ↑ probability of committing Type I error

Probability of committing at least one type I error = experiment-wise error rate

μ1 μ2 μ3

Similarities T & ANOVA

A one-way analysis of variance (ANOVA) tests the hypothesis that the means of several populations are equal.

The method is an extension of the two-sample t-test, specifically for the case where the population variances are assumed to be equal.

“I pity the fools that think T and ANOVA are similar!”– Mr. T

Assumptions for One-Way ANOVA

Four assumptions for t-test hypothesis testing:1. Random Samples2. Independent Samples3. Normal Populations (or large samples)4. Variances (std. dev.) are equal

One-Way ANOVA

A one-way analysis of variance (ANOVA) tests the hypothesis that the means of several populations are equal

The null hypothesis for the test is that all population means (level means) are the same –

H0: μ1 = μ2 = μ3 The alternative hypothesis is that one or more population means differ from the others –

Ha: Not all means are equal

ANOVA

H0: μ1 = μ2 = μ3 Ha: Not all means are equal

One-way ANOVA: _ Urchins versus Location Source DF SS MS F PLocation 2 60.54 30.27 10.00 0.000Error 177 535.77 3.03Total 179 596.31

We reject the null that all means are equalAccept alternative that all means not equal

Is that all?

ANOVA

Allow you to determine the relations among all the means

Several methods: Tukey, Fisher’s LSD, Dunnett’s, Bonferroni, Scheffe, etc

Most focus on Tukey

Multiple Comparisons

3 ways to test:1) Confidence Intervals

- default on “older” Minitab versions - less intuitive than other methods2) Grouping Information - Just answers, no details - easy to interpret3) Simultaneous Tests - t-tests run after ANOVA - provides details; interpret like t-test

Multiple Comparisons

Tukey's methodTukey's method compares the means for each pair of factor levels using a family error rate to control the rate of type I error Results are presented as a set of confidence intervals for the difference between pairs of meansUse the intervals to determine whether the means are different:

If an interval does not contain zero, there is a statistically significant difference between the corresponding means

If the interval does contain zero, the difference between the means is not statistically significant

Tukey 95% Simultaneous Confidence Intervals

Tukey 95% Simultaneous Confidence Intervals

Deep vs. Middle = Not significantly differentDeep vs. Shallow = Significantly differentMiddle vs. Shallow = Significantly different

Tukey Grouping Information

Deep vs. Middle = Not significantly differentDeep vs. Shallow = Significantly differentMiddle vs. Shallow = Significantly different

Tukey Test (using GLM)

Deep vs. Middle = Not significantly differentDeep vs. Shallow = Significantly differentMiddle vs. Shallow = Significantly different

Non-Parametric Version of ANOVAIf samples are independent, similarly distributed dataUse nonparametric test regardless of normality or sample sizeIs based upon mean of ranks of the data – not the mean or variance (Like Mann-Whitney)If the variation in mean ranks is large – reject nullUses p-value like ANOVA

Last Resort/Not Resort –low sample size, “bad” data

Kruskal-Wallis

Non-Parametric Version of ANOVADoes not have multiple comparisons test (Tukey’s)

Will need to run separate “t-tests” (Mann-Whitney) to test for differences between individual “means”

Kruskal-Wallis

Non-Parametric Version of ANOVAKruskal-Wallis

When Do I Do the What Now?

If Data are normal –use ANOVA

Otherwise – use Kruskal-Wallis

“Well, whenever I'm confused, I just check my underwear. It holds the answer to all the important questions.” – Grandpa Simpson