Cuckoo Birds Analysis of Variance I Cuckoo birds have a behavior in which they lay their eggs in other birds nests I The other birds then raise and care for the newly hatched cuckoos I Cuckoos return year after year to the same territory and lay their eggs in the nests of a particular host species I Furthermore cuckoos appear to mate only within their territory I Therefore geographical sub species are developed each with a dominant foster parent species I A general question is are the eggs of the different sub species adapted to a particular foster parent species I Specifically we can ask are the mean lengths of the cuckoo eggs the same in the different sub species Bret Larget Departments of Botany and of Statistics University of Wisconsin Madison Statistics 371 29th November 2005 I Analysis of variance ANOVA is a statistical procedure for analyzing data that may be treated as multiple independent samples with a single quantitative measurement for each sampled individual I ANOVA is a generalization of the methods we saw earlier in the course for two independent samples I The bucket of balls model is that we have I different buckets of balls each of which contains numbered balls 22 Display of Cuckoo Bird Egg Lengths I The populations means and standard deviations of the numbers in each bucket are i and i respectively for i 1 I 20 Analysis of Variance I In ANOVA we often assume that all of the population standard deviations are equal Here is a plot of egg lengths mm of cuckoo bird eggs categorized by the species of the host bird 21 23 24 25 I HedgeSparrow MeadowPipet PiedWagtail Robin birdSpecies TreePipet Wren A Dotplot of the Data Notation 24 25 This notation is used to describe calculations of variability within samples and variability among samples although for historical reasons of poor grammar the term between samples is more commonly used 23 yij 22 I 20 21 ni HedgeSparrow MeadowPipet PiedWagtail Robin TreePipet Wren the jth observation in the ith group the number of groups the ith sample size y i the mean of the ith sample I X ni the total number of observations n y The Big Picture I ANOVA is a statistical procedure where we test the null hypothesis that all population mean are equal versus the alternative hypothesis that they are not all equal I The test statistic is a ratio of the variability among sample means over the variability within sample means I I When this ratio is large this indicates evidence against the null hypothesis The test statistic will have a different form than what we have previously seen The null distribution is an F distribution named after Ronald Fisher I An ANOVA table is an accounting method for computing the test statistic I We introduce a lot of new notation on the way i 1 PI P n j i 1 j 1 yij n the grand mean Sums of Squares within Groups We measure variability by sums of squared deviations The sums of squares within groups or SS within is a combined measure of the variability within all groups SS within nj I X X i 1 j 1 I X i 1 yij y i 2 ni 1 si2 Notice that this measure of variability is a weighted sum of the sample variances where the weights are the degrees of freedom for each respective sample Degrees of Freedom Sums of Squares Between Among Means I The degrees of freedom within samples is simply the sum of degrees of freedom for each sample I This is equal to the total number of observations minus the number of groups I We measure variability by sums of squared deviations The sums of squares between groups or SS between is a measure of the variability among sample means SS between df within I X i 1 i 1 ni 1 I n I Mean Square Within I I X ni y i y 2 Notice that this measure of variability is a weighted sum of the deviations of the sample means from the grand mean weighted by sample size Degrees of Freedom In ANOVA a mean square will be the ratio of a sum of squares over the corresponding degrees of freedom MS within SS within df within n1 1 s12 nI 1 sI2 n I I In other words the mean square within is a weighted average of the sample variances where the weights are the degrees of freedom within each sample I The square root of the mean sqaure within is the estimate of the common variance for all the I populations p spooled MS within I The degrees of freedom between samples is simply the number of groups minus one df between I 1 Mean Square Between I Total Degrees of Freedom The mean square will be the ratio of a sum of squares over the corresponding degrees of freedom I I MS between I n 1 n I I 1 I This is the total sum of squares ni I X X I 1 j 1 I df total df within df between The F Statistic If we treated all observations as coming from a single population which would be the case if all population means were equal and all population standard deviations were equal as well then it would make sense to measure deviations from the grand mean SS total There is a similar decomposition SS between df between PI 2 i 1 ni y i y I 1 Total Sum of Squares I Similarly the total degrees of freedom would be n 1 The F statistic is the ratio of the mean square between over the mean square within F yij y 2 It turns out that the total sum of squares can be decomposed into the sum of squares within and the sum of squares between SS total SS within SS between I MS between MS within If the populations are normal the population means are all equal the standard deviations are all equal and all observations are independent then the F statistic has an F distribution with I 1 and n I degrees of freedom The F Statistic I I ANOVA Table for the Cuckoo Example An F distribution is positive and skewed right like the chi square distribution but it has two separate degrees of freedom the numerator degrees of freedom and the denominator degrees of freedom If X1 and X2 are independent 2 random variables with k1 and k2 degrees of freedom respectively then F I Each mean square is the ratio of the corresponding sum of squares and degrees of freedom I The square root of the mean square within is about 0 9 an estimate of I There is very strong evidence that all six population means are not equal There is very strong evidence that the mean length of cuckoo bird eggs is not the same for all subpopulations This is evidence for adaptation cuckoo birds that lay eggs of length similar to that of the host bird might have a selective advantage X1 k1 X2 k2 has an F …
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