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## STAT 2120 Tim Keaton

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**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 15th 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).**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) = a.**Therefore, with six tests, there is a higher probability than a 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) = a. That is, it ensures that P(Type I error) = a.**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 3 146.6 48.9 3.57 0.033 Error 19 260.0 13.7 Total 22 406.6 S = 3.699 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 3 146.6 48.9 3.57 0.033 Error 19 260.0 13.7 Total 22 406.6 S = 3.699 R-Sq = 36.06% R-Sq(adj) = 25.96%**Pairwise Comparisons**Recall the 6 different hypothesis tests. • 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. 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. Individual 95% CIs For Mean Based on Pooled StDev Level N Mean StDev +---------+---------+---------+--------- control 5 17.000 2.236 (---------*--------) gestalt 6 15.000 4.000 (--------*--------) psycho 6 14.000 4.000 (--------*--------) behave 6 10.000 4.000 (--------*--------) +---------+---------+---------+--------- 7.0 10.5 14.0 17.5 If two intervals do not overlap, then we may conclude that their population means differ.**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 0.696 7.000 13.30 (--------*--------) gestalt -1.011 5.000 11.01 (-------*--------) psycho -2.011 4.000 10.01 (--------*-------) ---+---------+---------+---------+--- -7.0 0.0 7.0 14.0 treatment = control subtracted from: treatment Lower Center Upper ---+---------+---------+---------+--- gestalt -8.304 -2.000 4.304 (--------*--------) psycho -9.304 -3.000 3.304 (--------*--------) ---+---------+---------+---------+--- -7.0 0.0 7.0 14.0 treatment = gestalt subtracted from: treatment Lower Center Upper ---+---------+---------+---------+--- psycho -7.011 -1.000 5.011 (--------*-------) ---+---------+---------+---------+--- -7.0 0.0 7.0 14.0**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 0.696 7.000 13.30 (--------*--------) gestalt -1.011 5.000 11.01 (-------*--------) psycho -2.011 4.000 10.01 (--------*-------) ---+---------+---------+---------+--- -7.0 0.0 7.0 14.0 treatment = control subtracted from: treatment Lower Center Upper ---+---------+---------+---------+--- gestalt -8.304 -2.000 4.304 (--------*--------) psycho -9.304 -3.000 3.304 (--------*--------) ---+---------+---------+---------+--- -7.0 0.0 7.0 14.0 treatment = gestalt subtracted from: treatment Lower Center Upper ---+---------+---------+---------+--- psycho -7.011 -1.000 5.011 (--------*-------) ---+---------+---------+---------+--- -7.0 0.0 7.0 14.0**One-Way ANOVA Output**Results for: ANXIETY.MTW One-way ANOVA: control, gestalt, psycho, behave Source DF SS MS F P Factor 3 25.3 8.4 0.76 0.529 Error 19 210.0 11.1 Total 22 235.3 S = 3.325 R-Sq = 10.75% R-Sq(adj) = 0.00%**Informal Pairwise Comparisons**Look at the individual confidence intervals for each mean. Individual 95% CIs For Mean Based on Pooled StDev Level N Mean StDev -----+---------+---------+---------+---- control 5 17.000 2.236 (-----------*-----------) gestalt 6 15.000 4.000 (----------*----------) psycho 6 14.000 4.000 (----------*----------) behave 6 15.000 2.449 (----------*----------) -----+---------+---------+---------+---- 12.5 15.0 17.5 20.0 If two intervals do not overlap, then we may conclude that their population means differ.**Pairwise Comparisons**Tukey 95.0% Simultaneous Confidence Intervals Response Variable anxiety All Pairwise Comparisons among Levels of treatment control subtracted from: Lower Center Upper --+---------+---------+---------+------- gestalt -7.665 -2.000 3.665 (-------------*-------------) psycho -8.665 -3.000 2.665 (--------------*-------------) behave -7.665 -2.000 3.665 (-------------*-------------) --+---------+---------+---------+------- -8.0 -4.0 0.0 4.0 gestalt subtracted from: Lower Center Upper --+---------+---------+---------+------- psycho -6.402 -1.000 4.402 (-------------*------------) behave -5.402 0.000 5.402 (-------------*-------------) --+---------+---------+---------+------- -8.0 -4.0 0.0 4.0 psycho subtracted from: Lower Center Upper --+---------+---------+---------+------- behave -4.402 1.000 6.402 (-------------*------------) --+---------+---------+---------+------- -8.0 -4.0 0.0 4.0