1. 1 G89.2228 Lect 11a G89.2228 Lecture 11a Example: Comparing variances ANOVA table ANOVA linear model ANOVA assumptions Data transformations Effect sizes and power

2. 2 G89.2228 Lect 11a Comparing 3 or more means: Examples Okasaki reported that Asian-American students had higher depression than Anglo students –Suppose she had wanted to compare second generation "Anglo" students of Italian, Irish, British and Jewish heritage Henderson-King and Nisbett constructed three experimental social experiences –Angry disruption –Disappointed disruption –Minimal disruption Under the null hypothesis in these cases, all k means vary only due to sampling fluctuations

3. 3 G89.2228 Lect 11a Implications of Null Hypothesis when k>2 Suppose also that n subjects are sampled in each of the k groups. Under H0 –All groups estimate a common grand mean –Group means,, differ due to sampling variation –Var( )=  2 /n –Var( ) can be studied empirically with –Note that under H0, the variation of individual Y values,  2, can be estimated using n*S 2  Y. Regardless of H0, an independent estimate of  2 can be obtained from the variation within groups.

4. 4 G89.2228 Lect 11a Analyzing Variance Under H0, both and estimate the same variance term,  2 (what Howell calls,  2 e ). The ratio of MSB/MSE will be approximately unity on the average, if H0 is true. Fisher described the distribution of the variance ratio under H0. It is called the central F distribution on (k-1) and k(n-1) degrees of freedom.

5. 5 G89.2228 Lect 11a The ANOVA Table The ANOVA table shows how the variability of the study can be partitioned. The degrees of freedom and the total Sums of Squares (SS) can be neatly broken down into Between Groups and Within Groups. The MS column shows how the SS and df terms are used to estimate the variance. The F statistic is the ratio of MS terms, and it is distributed with degrees of freedom given by the df of the MS components.

6. 6 G89.2228 Lect 11a A Numerical Example of the Null Hypothesis The data below are modeled after Table 11.2 in Howell, except that all scores are sampled from a normal distribution with mean 10 and variance 16

7. 7 G89.2228 Lect 11a Modeling group differences The alternative to the null hypothesis is that the groups have different means. Suppose that each group comes from different populations with means, It is often useful to write these group- specific means as deviations from some grand mean:  j reflects how much different µ j is from µ, the grand mean. The groups usually represent some FIXED comparisons. Then Later we consider RANDOM effects. Without expectations we write

8. 8 G89.2228 Lect 11a ANOVA assumptions The assumptions behind this ANOVA analysis (simple, one-way ANOVA) include: –Normally distributed observations in each cell of the design –Independent observations –Equal variances in each cell The rule of thumb is that ANOVA tolerates some violations of its assumptions. It can handle unequal sample sizes in each cell (with the calculations described in Howell), unequal variance or non-normality, but not multiple, strong violations Independence does matter!

9. 9 G89.2228 Lect 11a Data transformations Sometimes, your data violate an assumption of the ANOVA analysis but that is ameliorated by transforming your data: –arcsin transformation for probabilities (note: some suggest values of 0 and 1 be treated specially) –log transformation for strong positive skew (e.g. salaries) –square root for Poisson (e.g., counts of events that are plausibly independent) –reciprocal –Winsorizing (robust statistics)

10. 10 G89.2228 Lect 11a Effect sizes for ANOVA Eta-squared is the proportion of the total sums of squares that is attributable to the treatment groups (like a proportion of variance accounted for). As the number of groups k increases,  2 increases. It is a biased estimate of the underlying proportion. Omega-squared (  2 ) is a relatively unbiased estimate of the proportion of total variance that is attributable to the variance of  j. Slightly different forms are used when the groups are construed as a random sample of treatments rather than a fixed set of treatments.

11. 11 G89.2228 Lect 11a Effect sizes for ANOVA - 2 Cohen suggests using the square root of the expected ratio of  2   to  2 e. He calls this f (Howell calls it ). It is one half the size of the effect d for t- tests. One then computes and then uses the power tables (for the noncentral F distribution indexed by the degrees of freedom) to calculate power Cohen categorizes the effect sizes as –small: f=.1 –medium: f=.25 –large: f=.4