Question

For the bird egg length data set, conduct an appropriate test to determine if bird egg length differs among species. Assume that before you conducted this test you hypothesized about 3 contrasts a priori. These were that (1) Meadow Pipits, Wagtails, and Robins would be different than the other 3 bird species, (2) Hedge Sparrows would differ from Wrens, and (3) Tree Pipits would be different than all other birds. Use the approach we outlined in class to evaluate these a priori contrasts.

The approach outlined in class is as follows:

A priori contrasts

First, we must decide how many and which planned comparisons to make. Technically, we can make as many as we would like, but many statisticians recommend that our planned contrasts be orthogonal to one another to ensure independence of results (i.e., that each contrast tests an independent relationship among the means). This way our P-values for each contrast are not correlated with one another. If there are k groups, then, at most, there can be k-1 orthogonal contrasts (although we can create the k-1 contrasts in multiple ways). We use an approach similar to the one outlined above for the Scheffe’s test, in that we generate coefficients for each of the means in the contrast. The rules for building contrasts and assigning coefficients are presented by Gotelli and Ellison 2004 (pp. 339-341):

1. The sum of the coefficients for any contrast must equal 0 2. Sets of means averaged together have the same coefficient 3. Means not included in a contrast have a coefficient of 0 4. A maximum of k-1 orthogonal contrasts are possible 5. All of the pair-wise cross products must sum to 0

Rules 4 and 5 apply only when we want to limit our comparisons to orthogonal contrasts. If we chose to test non-orthogonal contrasts, we must adjust our alpha (α) level since the non
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independence of our tests will inflate our probability of making a Type I error. These types of adjustments to our alpha level are collectively referred to as Bonferroni adjustments and there are several types. The simplest is the Bonferroni method which sets alpha = α/k, where k = the number of tests performed.

More powerful options For the Holm-Bonferroni method (Holm 1979): 1) start by ordering the P-values from smallest to largest 2) compare your smallest P-value against 0.05/k 3) If your smallest observed P-value is smaller, you reject the null, and go to the next smallest P-value, which you compare to 0.05/k-1 4) If you reject that null, you move to your next smallest P-value and compare it to 0.05/k-2 5) You continue in this manner until you fail to reject a null hypothesis, at which point all remaining P-values would be nonsignificant.

Hedge Sparrow = 20.85, 21.65, 22.05, 22.85, 23.05, 23.05, 23.05, 23.05, 23.45, 23.85, 23.85, 23.85, 24.05, 25.05

Meadow Pipit = 19.65, 20.05, 20.65, 20.85, 21.65, 21.65, 21.65, 21.85, 21.85, 21.85, 22.05, 22.05, 22.05, 22.05

Pied Wagtail = 21.05, 21.05, 21.85, 21.85, 21.85, 22.05, 22.45, 22.65, 23.05, 23.05, 23.25, 23.45, 24.05, 24.85

Robin = 21.85, 22.05, 22.05, 22.25, 22.45, 22.45, 22.65, 23.05, 23.05, 23.05, 23.05, 23.05, 23.25, 23.85

Tree Pipit = 21.05, 21.85, 22.05, 22.45, 22.65, 23.25, 23.25, 23.45, 23.45, 23.65, 23.85, 24.05, 24.05, 24.05

Wren = 19.85, 20.05, 20.25, 20.85, 20.85, 20.85, 21.05, 21.05, 21.25, 21.45, 22.05, 22.05, 22.05, 22.25

Answer 1