Stat 512 – Lecture 15 Two-way ANOVA (Ch. 12)

Last Time – Analysis of Variance (ANOVA) When: Want to compare two or more population/true treatment means  Have one qualtitative and one quantitative variable How: Minitab F-statistic and p-value  H 0 :    …  g H a : at least one  differs  F measures ratio of variation in observed group means to pooled measure of “chance” variability  Is test statistic too large?

Last Time – Analysis of Variance (ANOVA) Checking technical conditions (see Sec 12.8)  Normality: Samples well behaved or large sample sizes  Equal population variances: ratio of largest sample SD to smallest sample SD < 2  Independence: random samples or random assignment to treatment groups

Last Time – Analysis of Variance (ANOVA) Follow-up Analysis: Multiple comparison procedures  Look at all possible pairs, is the difference (in the population/true treatment) means statistically significant?

PP Conditions of each analysis Performing each analysis Vs t, number of variables

Example 2: Restaurant Spending How do different factors affect the size of the p-value?  when the population means are further apart, the p-value is usually smaller (more evidence they aren’t equal)  when the within group variability is larger, the p- value is larger (less evidence didn’t happen by chance)  when the sample sizes are larger, and there is a true difference between the population means, then the p-value is smaller

Example 4: Lifetimes of Notables One-way vs. One-way (stacked) Overall F = 9.37, p-value <.001  Strong evidence that the mean lifetime differs in at least one of these populations Multiple comparisons (99.81%)  Business leaders > Artists  Military leaders < Business  Philosophers < Business  Politicians < Business  Writers < Business  Military leaders < Historians  Writers < Historians  Scientists > Mil leaders  Social reformers > Mil leaders  Social reformers > Politicians  Writers < Scientists  Writers < Social reformers

Graphic writers military leaders politicians artists philosophers historians scientists social reformers business leaders 76.31

Example 5: Nature vs. Nuture How many observational units? How many variables? Want to run a “two-way ANOVA”  Two “factors”/explanatory variables  But “unbalanced” design → Minitab’s “General Linear Model”

Example 5: Nature vs. Nurture Interaction Plot

Example 5: Nature vs. Nurture How would you interpret this plot?

Independent Samples? Slightly different question from independent observations and independence variables (p. 226) Dependent samples occur when each observation in sample 1 matches with an observation from sample 2.  “Matched pairs” data  Examples: Same subjects, twins  Analysis: Look at the one sample of differences

Why care? Example 1 How do the “dependent samples” and the “independent samples” analyses compare? Why? By explaining some of the variation, the standard deviations are (slightly) smaller, allowing us to more easily see the “treatment” effect

Example 2: Marriage ages Statistically significant? Better way to compare husband’s ages to wife’s ages?  Directly compare each husband to his wife!  Let  = average age difference/couple and analyze the differences SD = 4.81 years

Example 2: Marriage Ages Use Minitab to carry out the ANOVA procedure  First ignore the coupling  Then treat the couple as another explanatory variable  Note the changes in SSE…

Example 1 Revisited See how the ANOVA analysis compares to the analysis of the differences in hippocampus volumes… F = = (3.23) 2 P-value =.006 Twins: p-value =.005  also a highly significant factor in hippocampus volumes

Example 3: Melting Properties Randomized comparative experiment to 3 treatment groups butterscotch studentsmilk chocolatemelting time semi-sweet

Example 3: Melting Properties How would you design a randomized block experiment to compare these 3 types?

For Tuesday Read 9.1, 9.2 Submit PP 13 in Blackboard, including final exam time commitment