Analysis of Variance (ANOVA)

Transcription

1 Analysis of Variance (ANOVA) Used for comparing or more means an extension of the t test Independent Variable (factor) = categorical (qualita5ve) predictor should have at least levels, but can have many more! Dependent Variable = con5nuous (quan5ta5ve) outcome

2 ANOVA: From t to F In the independent samples t test, you use the t distribu@on to test the sta@s@cal hypothesis that there is no difference between two means But lets say we want to test whether there are differences between three or more means? You could use the t test a bunch to make all possible comparisons, but this is: o o o Tedious Boring Poor methodology to make so many comparisons

3 ANOVA Basics Type of Music (IV) & Aggression Scores (DV) Metallica Bob Marley Jus5n Bieber Subject 1: 7 Subject 6: Subject 11: 0 Subject : 4 Subject 7: 1 Subject 1: 0 Subject 3: 6 Subject 8: 3 Subject 13: 1 Subject 4: 6 Subject 9: Subject 14: Subject 5: 7 Subject 10: 0 Subject 15: 0 h\p:// h\p:// 8eLEW8g h\p:// In ANalysis Of VAriance (ANOVA), we compare the variability between groups to the variability within groups If our means are sta5s5cally different then o o The between group differences will be BIG The within group differences will be SMALL

4 ANOVA Basics Type of Music (IV) & Aggression Scores (DV) Metallica Bob Marley Jus5n Bieber Subject 1: 7 Subject 6: Subject 11: 0 Subject : 4 Subject 7: 1 Subject 1: 0 Subject 3: 6 Subject 8: 3 Subject 13: 1 Subject 4: 6 Subject 9: Subject 14: Subject 5: 7 Subject 10: 0 Subject 15: 0 _ X 1 = 6 Within Groups _ X = 1.6 Within Groups _ X 3 = 0.6 Within Groups Between Groups

5 Two Sources of Variability Within group differences (variance) Differences due to chance (random sampling error) Part of finding differences between groups is to show that there are NOT differences within groups We want members WITHIN each group to look alike This tells us that the mean for each group describes each individual well Between group differences (variance) Differences due to chance (random sampling error) AND treatment effects ANOVA Total Variation Among Scores Within-Groups Variation Variation due to chance. Between-Groups Variation Variation due to chance and treatment effect (if any existis).

6 Within- group variability Take Metallica, for example, to illustrate within- group variability (or error) Metallica _ X Subject 1: Subject : Subject 3: Subject 4: Subject 5: _ X - X

7 Group 1 Group Group Group 1 Group Group Group 1 Group Group 3 Group 1 Group Group

8 Hypothesis Tes5ng in ANOVA H 0 : no mean differences (μ 1 = μ = μ 3 ) H 1 : there are mean differences between the groups (H 0 is false) In order to sta@s@cally test for these differences we use the F- ra@o F = Between Group Variability Within Group Variability If F is large, what does this mean? If F is small, what does this mean?

9 A few new concepts q q q One-factor ANOVA: simplest type of ANOVA that tests for differences among populations means categorized by one independent variable Mean Square: estimate of variance within OR between groups. Calculated by dividing a sum of squares by its degrees of freedom (SS/ df) Ø Ø Sum of Squares (SS): (you know this!!) the sum of squared deviations of a set of scores about their mean Degrees of freedom (df): (you know this!!) the number of deviations free to vary in any sum of squares term, given one or more restrictions N à total # of participants in the study n à # of participants in a group k à # of groups Grand mean: every data values divided by the total sample size

10 Variability as a Picture F = Variability Between Groups Variability Within Groups F > 1 Between: effect Within: error

11 The F ra@o F = Between Group Variability Within Group Variability Mean squares between F = MS between MS wit hi n Mean squares within sum of squares between MS between = SS between df between degrees of freedom between sum of squares within MS wit hin = SS wit hin df wit hin degrees of freedom within

12 The F ra@o MS between = SS between df between SS between = n* (x group mean - x grand mean ). Note that n = # per group df between = # groups 1 df between = k 1 MS wit hin = SS wit hin df wit hin SS within = (x - x group mean ) df within = total # subjects # groups df within = N k

13 df within df between

14 Prac5ce Example Type of Music (IV) & Aggression Scores (DV) Metallica Bob Marley Jus5n Bieber Subject 1: 7 Subject 6: Subject 11: 0 Subject : 4 Subject 7: 1 Subject 1: 0 Subject 3: 6 Subject 8: 3 Subject 13: 1 Subject 4: 6 Subject 9: Subject 14: Subject 5: 7 Subject 10: 0 Subject 15: 0 A study compared the aggression of subjects after listening to one of three types of music. Determine the significance of the difference among groups, using an alpha of.05 DO NOT FORGET THE STEPS OF HYPOTHESIS TESTING! Define study question: Does music affect aggression? Identify statistical hypotheses Specify decision rule Calculate observed statistical test value Make a decision & interpret (IN ENGLISH PLEASE)

15 ANOVA Example Statistical hypotheses H 0 : μ metallica = μ marley = μ bieber H 1 : H 0 is false Decision rule α =.05 df between = 3-1 = df within = 15-3 = 1 Cri@cal F = 3.89 df between df within

16 Metallica Devia5on Devia5on Marley Devia5on Devia5on Bieber Devia5on Devia5on Σ = 30 _ X =6 Σ = 6 df between = k 1 = 3 1= df between devations + deviations + SS within = = = SS within df within df within = N k = 15 3 = 1 Σ = 8 _ X = 1.6 Σ = 5. deviations MS between MS Σ = 3 _ X =.6 Σ = 3. SS between = n (X GM ) GM = =. 73 = [(6.73) + (1.6.73) + (.6.73) ]*5 3 SS between SS between = [ ]*5 = 8.53 = within SS df = between between SS df within within = 8.53 = = 41.7 = 1.0

17 Don t forget to check your work! = SS total ( X GM ) SS total = ( 7.43) + (4.43) + (6.43) + (6.43) + (7.43) + (.43) + (1.43) + (3.43) + (.43) (0.43) (0.43) (0.43) (1.43) (.43) (0.43) SS total = SS = SS + total between SS within = = 96.93

18 ANOVA Summary Table Source df SS MS F. Between Groups Within Groups Total F = MS between MS wit hi n Observed statistical test value F (, 1) = 34.39, p <.05 Make a decision & interpret Reject H 0 because > 3.89 There is a significant difference in aggression depending on what type of music a person listens to

19 Now it is your turn to prac5ce Type of (IV) & Anxiety Scores (DV) A\ending Class Taking a Test Giving a Presenta5on A study compared the anxiety of college students after subjecting them to one of three situations: attending a class, taking a test, and giving a presentation. Determine the significance of the difference among groups, using an alpha of.05 Go through all steps of hypothesis testing, of course J Define study question Identify statistical hypotheses Specify decision rule Calculate observed statistical test value Make a decision & interpret (IN ENGLISH PLEASE)

20 Class Devia5on Devia5on Devia5on from from Mean Mean Sum= Sum=10 Devia5on Test Present Devia5on from Mean Devia5on Mean=6 Mean=10 Mean=10 Sum=8 SS SS SS SS SS SS between between between within within within = ( X = [(6 8.67) = 4.67 = = 0 GM ) * n + ( ) = devations + deviations + + ( ) deviations ]*4 F = MS MS GM = 3 between within = = 8.67 Reject H 0, 9.6>4.6 = 9.6 Source df SS MS F Between Groups Within Groups 9 0. Total There is a significant difference in anxiety depending on the classroom situa@on.

21 Back to our example which group is more aggressive? When an F is significant we know (with some degree of confidence) that there is a real difference between our means but we don t know which means are actually different if we have more than groups! Post hoc tests: make pair-wise comparisons after a significant F is obtained Several options: Bonferroni, Tukeys HSD, Scheffe, Lucky for you, this is something you won t have to learn about in depth until graduate stats! How big is our effect? Eta-squared: η = SS between /Ss total Proportion of variance that is explained by group differences Eta-squared for music & aggression: η = 8.53 / =.851 à 85.1% of aggression scores explained by music choice

22 Now it is your turn to prac5ce again Type of (IV) & Anxiety Scores (DV) A\ending Class Taking a Test Giving a Presenta5on What is the effect size (eta-squared) of the data you just analyzed? Is this something that you think is a big deal? Should programs be put into place to reduce anxiety in some situations? Why or why not?

23 Online Resources h\p:// and h\p://