Anova lecture
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The document discusses analysis of variance (ANOVA) which is used to compare the means of three or more groups. It explains that ANOVA avoids the problems of multiple t-tests by providing an omnibus test of differences between groups. The key steps of ANOVA are outlined, including partitioning variation between and within groups to calculate an F-ratio. A large F value indicates more difference between groups than expected by chance alone.
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Anova lecture
- 1. ANOVA:ANOVA: Analysis of VariationAnalysis of Variation
- 2. ANOVA:ANOVA: Learning ObjectivesLearning Objectives • Underlying methodological principles ofUnderlying methodological principles of ANOVAANOVA • Statistical principles: Partitioning ofStatistical principles: Partitioning of variabilityvariability • Summary table for one-way ANOVASummary table for one-way ANOVA
- 3. tt-Test vs ANOVA:-Test vs ANOVA: The Case for Multiple GroupsThe Case for Multiple Groups • tt-tests can be used to compare mean differences for two-tests can be used to compare mean differences for two groupsgroups • Between Subject DesignsBetween Subject Designs • Within Subject DesignsWithin Subject Designs • Paired Samples DesignsPaired Samples Designs • The test allows us to make a judgment concerningThe test allows us to make a judgment concerning whether or not the observed differences are likely to havewhether or not the observed differences are likely to have occurred by chance.occurred by chance. • Although helpful the t-test has its problemsAlthough helpful the t-test has its problems
- 4. Multiple GroupsMultiple Groups • Two groups are often insufficientTwo groups are often insufficient • No difference between Therapy X and Control:No difference between Therapy X and Control: Are other therapies effective?Are other therapies effective? • .05mmg/l Alcohol does not decrease memory.05mmg/l Alcohol does not decrease memory ability relative to a control: What about otherability relative to a control: What about other doses?doses? • What if one control group is not enough?What if one control group is not enough?
- 5. Multiple Groups 2Multiple Groups 2 • With multiple groups we could make multipleWith multiple groups we could make multiple comparisons using t-testscomparisons using t-tests • ProblemProblem: We would expect some differences in: We would expect some differences in means to be significant by chance alonemeans to be significant by chance alone • How would we know which ones to trust?How would we know which ones to trust?
- 6. Comparing Multiple Groups:Comparing Multiple Groups: TheThe ““One-WayOne-Way”” DesignDesign • Independent variable (Independent variable (““factorfactor””):): • Dependent variable (Dependent variable (““measurementmeasurement””):): • Analysis: Variation between and within conditionsAnalysis: Variation between and within conditions ....... Alcohol Level (Dose in mg/kg) 0 (Control) 10 20
- 7. Analysis of Variance (F test):Analysis of Variance (F test): AdvantagesAdvantages • Provides an omnibus test & avoids multiple t-tests andProvides an omnibus test & avoids multiple t-tests and spurious significant resultsspurious significant results • it is more stable since it relies on all of the data ...it is more stable since it relies on all of the data ... • recall from your work on Std Error and T-testsrecall from your work on Std Error and T-tests • the smaller the sample the lessthe smaller the sample the less stable are the populationstable are the population parameter estimatesparameter estimates..
- 8. What does ANOVA do?What does ANOVA do? • Provides an F ratio that has an underlyingProvides an F ratio that has an underlying distribution which we use to determine statisticaldistribution which we use to determine statistical significance between groups (just like a t-test or asignificance between groups (just like a t-test or a z-test)z-test) • e.g., Take an experiment in which subjects aree.g., Take an experiment in which subjects are randomly allocated to 3 groupsrandomly allocated to 3 groups • The means and std deviations will all be different fromThe means and std deviations will all be different from each othereach other • We expect this because that is the nature of samplingWe expect this because that is the nature of sampling (as you know!)(as you know!)
- 9. The question is …The question is … are the groups more different thanare the groups more different than we would expect by chance?we would expect by chance?
- 10. How does ANOVA work?How does ANOVA work? • Instead of dealing with means as data points weInstead of dealing with means as data points we deal with variationdeal with variation • There is variation (variance) within groups (data)There is variation (variance) within groups (data) • There is variance between group means (ExptlThere is variance between group means (Exptl Effect)Effect) • If groups are equivalent then the varianceIf groups are equivalent then the variance between and within groups will be equal.between and within groups will be equal. • Expected variation is used to calculate statisticalExpected variation is used to calculate statistical significance in the same way that expectedsignificance in the same way that expected differences in means are used in t-tests or z-testsdifferences in means are used in t-tests or z-tests
- 11. The basic ANOVA situationThe basic ANOVA situation Two variables: 1 Categorical, 1 Quantitative Main Question: Do the (means of) the quantitative variables depend on which group (given by categorical variable) the individual is in? If categorical variable has only 2 values: • 2-sample t-test ANOVA allows for 3 or more groups
- 12. An example ANOVA situationAn example ANOVA situation Subjects: 25 patients with blisters Treatments: Treatment A, Treatment B, Placebo Measurement: # of days until blisters heal Data [and means]: • A: 5,6,6,7,7,8,9,10 [7.25] • B: 7,7,8,9,9,10,10,11 [8.875] • P: 7,9,9,10,10,10,11,12,13 [10.11] Are these differences significant?
- 13. Informal InvestigationInformal Investigation Graphical investigation: • side-by-side box plots • multiple histograms Whether the differences between the groups are significant depends on • the difference in the means • the standard deviations of each group • the sample sizes ANOVA determines P-value from the F statistic
- 14. Side by Side BoxplotsSide by Side Boxplots PBA 13 12 11 10 9 8 7 6 5 treatment days
- 15. What does ANOVA do?What does ANOVA do? At its simplest (there are extensions) ANOVA tests theAt its simplest (there are extensions) ANOVA tests the following hypotheses:following hypotheses: HH00: The means of all the groups are equal.: The means of all the groups are equal. HHaa: Not all the means are equal: Not all the means are equal doesndoesn’’t say how or which ones differ.t say how or which ones differ. Can follow up withCan follow up with ““multiple comparisonsmultiple comparisons”” Note: we usually refer to the sub-populations asNote: we usually refer to the sub-populations as ““groupsgroups”” when doing ANOVA.when doing ANOVA.
- 16. Assumptions of ANOVAAssumptions of ANOVA • each group is approximately normaleach group is approximately normal • check this by looking at histograms or usecheck this by looking at histograms or use assumptionsassumptions • can handle some non-normality, but notcan handle some non-normality, but not severe outlierssevere outliers • standard deviations of each group arestandard deviations of each group are approximately equalapproximately equal • rule of thumb: ratio of largest to smallestrule of thumb: ratio of largest to smallest sample st. dev. must be less than 2:1sample st. dev. must be less than 2:1
- 17. Normality CheckNormality Check We should check for normality using: • assumptions about population • histograms for each group With such small data sets, there really isn’t a really good way to check normality from data, but we make the common assumption that physical measurements of people tend to be normally distributed.
- 18. Notation for ANOVANotation for ANOVA • n = number of individuals all together • k = number of groups • = mean for entire data set is Group i has • ni = # of individuals in group i • xij = value for individual j in group i • = mean for group i • si = standard deviation for group i ix x
- 19. How ANOVA works (outline)How ANOVA works (outline) ANOVA measures two sources of variation in the data and compares their relative sizes • variation BETWEEN groups • for each data value look at the difference between its group mean and the overall mean • variation WITHIN groups • for each data value we look at the difference between that value and the mean of its group
- 20. The ANOVA F-statistic is a ratio of the Between Group Variaton divided by the Within Group Variation: F = Between Within = MSB MSW A large F is evidence against H0, since it indicates that there is more difference between groups than within groups.
- 21. How are these computationsHow are these computations made?made? We want to measure the amount of variation due to BETWEEN group variation and WITHIN group variation For each data value, we calculate its contribution to: • BETWEEN group variation: • WITHIN group variation: xi − x( ) 2 2 )( iij xx −
- 22. An even smaller exampleAn even smaller example Suppose we have three groups • Group 1: 5.3, 6.0, 6.7 • Group 2: 5.5, 6.2, 6.4, 5.7 • Group 3: 7.5, 7.2, 7.9 We get the following statistics: SUMMARY Groups Count Sum Average Variance Column1 3 18 6 0.49 Column2 4 23.8 5.95 0.176667 Column3 3 22.6 7.533333 0.123333
- 23. Analysis of Variance for days Source DF SS MS F P treatment 2 34.74 17.37 6.45 0.006 Error 22 59.26 2.69 Total 24 94.00 ANOVA OutputANOVA Output 1 less than # of groups # of data values - # of groups (equals df for each group added together) 1 less than # of individuals (just like other situations)
- 24. ANOVA OutputANOVA Output MSB = SSB / DFB MSW = SSW / DFW Analysis of Variance for days Source DF SS MS F P treatment 2 34.74 17.37 6.45 0.006 Error 22 59.26 2.69 Total 24 94.00 F = MSB / MSW P-value comes from F(DFB,DFW) (P-values for the F statistic are in Table E)
- 25. So How big is F?So How big is F? Since F is Mean Square Between / Mean Square Within = MSB / MSW A large value of F indicates relatively more difference between groups than within groups (evidence against H0) To get the P-value, we compare to F(I-1,n-I)-distribution • I-1 degrees of freedom in numerator (# groups -1) • n - I degrees of freedom in denominator (rest of df)
- 26. WhereWhere’’s the Difference?s the Difference? Analysis of Variance for days Source DF SS MS F P treatmen 2 34.74 17.37 6.45 0.006 Error 22 59.26 2.69 Total 24 94.00 Individual 95% CIs For Mean Based on Pooled StDev Level N Mean StDev ----------+---------+---------+------ A 8 7.250 1.669 (-------*-------) B 8 8.875 1.458 (-------*-------) P 9 10.111 1.764 (------*-------) ----------+---------+---------+------ Pooled StDev = 1.641 7.5 9.0 10.5 Once ANOVA indicates that the groups do not all appear to have the same means, what do we do? Clearest difference: P is worse than A (CI’s don’t overlap)