STAT 2120 Tim Keaton

ANalysis Of VAriance (ANOVA) ANOVA is a generalization of the comparison of two population means In ANOVA, we compare k population means where k >= 3 It is a terrible name because we are comparing means by analyzing variances

Fear of Heights Example People with a fear of heights were randomly assigned to one of three forms of psychology or a control condition. Each client in a treatment receives 10 sessions with a trained psycho-therapist. Clients in the control group receive no treatment. All clients are then taken to the 15 th floor of an office building and asked to stand next to a floor to ceiling window and rate their anxiety on a 20 point scale (20=strongest fear; 0 = no fear).

Anxiety Scores (Scale 1-20) ControlGestalt PsychoanalysisBehavior Modification

To use the Chapter 13 stuff we learned, we would have to use 6 different hypothesis tests.

Each of these six tests has its own P(Type I error) = . Therefore, with six tests, there is a higher probability than  that we will make a Type I error. When performing many tests, you are almost guaranteed to make at least one Type I error. That is, you are almost guaranteed to find something significant when nothing is. One-Way ANOVA performs one overall test with P(Type I error) = . That is, it ensures that P(Type I error) = .

ANOVA works by comparing the variation between treatments to the variation within treatments. If the variation between treatments is large compared to the variation within treatments, then we decide there is a significant difference between at least two treatment means. The results of ANOVA are stored in an ANOVA table (surprise!)

One-Way ANOVA Output Results for: ANXIETY.MTW One-way ANOVA: control, gestalt, psycho, behave Source DF SS MS F P Factor Error Total S = R-Sq = 36.06% R-Sq(adj) = 25.96%

One-Way ANOVA Output Results for: ANXIETY.MTW One-way ANOVA: control, gestalt, psycho, behave Source DF SS MS F P Factor Error Total S = R-Sq = 36.06% R-Sq(adj) = 25.96%

Look at the Tukey simultaneous confidence intervals for the difference between each pair of means If a confidence interval does not contain 0, then there is a significant difference between those two means. Pairwise Comparisons Recall the 6 different hypothesis tests. Now that we know at least two means are different, we can make these pairwise comparisons.

Pairwise Comparisons Recall the 6 different hypothesis tests. Now that we know at least two means are different, we can make these pairwise comparisons.

Informal Pairwise Comparisons Look at the individual confidence intervals for each mean. If two intervals do not overlap, then we may conclude that their population means differ. Individual 95% CIs For Mean Based on Pooled StDev Level N Mean StDev control ( * ) gestalt ( * ) psycho ( * ) behave ( * )

Tukey 95.0% Simultaneous Confidence Intervals Response Variable anxiety All Pairwise Comparisons among Levels of treatment treatment = behave subtracted from: treatment Lower Center Upper control ( * ) gestalt ( * ) psycho ( * ) treatment = control subtracted from: treatment Lower Center Upper gestalt ( * ) psycho ( * ) treatment = gestalt subtracted from: treatment Lower Center Upper psycho ( * ) Pairwise Comparisons

Tukey 95.0% Simultaneous Confidence Intervals Response Variable anxiety All Pairwise Comparisons among Levels of treatment treatment = behave subtracted from: treatment Lower Center Upper control ( * ) gestalt ( * ) psycho ( * ) treatment = control subtracted from: treatment Lower Center Upper gestalt ( * ) psycho ( * ) treatment = gestalt subtracted from: treatment Lower Center Upper psycho ( * ) Pairwise Comparisons

Anxiety Scores (Scale 1-20) ControlGestalt PsychoanalysisBehavior Modification

One-Way ANOVA Output Results for: ANXIETY.MTW One-way ANOVA: control, gestalt, psycho, behave Source DF SS MS F P Factor Error Total S = R-Sq = 10.75% R-Sq(adj) = 0.00%

Informal Pairwise Comparisons Look at the individual confidence intervals for each mean. If two intervals do not overlap, then we may conclude that their population means differ. Individual 95% CIs For Mean Based on Pooled StDev Level N Mean StDev control ( * ) gestalt ( * ) psycho ( * ) behave ( * )

Tukey 95.0% Simultaneous Confidence Intervals Response Variable anxiety All Pairwise Comparisons among Levels of treatment control subtracted from: Lower Center Upper gestalt ( * ) psycho ( * ) behave ( * ) gestalt subtracted from: Lower Center Upper psycho ( * ) behave ( * ) psycho subtracted from: Lower Center Upper behave ( * ) Pairwise Comparisons