Introduction to the Analysis of Variance (ANOVA) Computing One-Way Independent Measures (Between Subjects) ANOVAs
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1 Introduction to the Analysis of Variance (ANOVA) Computing One-Way Independent Measures (Between Subjects) ANOVAs
2 The Analysis of Variance (ANOVA) The analysis of variance (ANOVA) is a statistical technique for testing for differences between the means of multiple (more than two) groups It is probably the most prevalent statistical technique used in psychological research. The ANOVA is a flexible technique that can be used with a variety of different research designs. In today s lecture, I will explain the logic behind the ANOVA and introduce the one-way between groups ANOVA, which is an ANOVA in which the groups are defined along only one independent (or quasi-independent) variable
3 The Analysis of Variance The purpose of ANOVA is much the same as the t tests presented in the preceding lectures Are the mean differences obtained for sample data sufficiently large for us to conclude that there are mean differences between the populations from which the samples were obtained The difference between ANOVA and the t tests is that ANOVA can be used in situations where there are two or more means being compared, whereas the t tests are limited to situations where only two means are involved.
4 The Problem of Multiple Comparisons The ANOVA is necessary to protect researchers from an excessive experimentwise error rate in situations where a study is comparing more than two population means. Experimentwise error rate: the probability of making at least one Type I error across mutliple comparisons These situations would require a series of several t tests to evaluate all of the mean differences. (Remember, a t test can compare only two means at a time) So? Why not just use multiple t-tests?
5 The Problem of Multiple Comparisons Why not just use multiple t-tests? Although each t test can be evaluated using a specific α-level (risk of Type I error), the α-levels accumulate over a series of tests so that the final familywise α-level can be quite large Example: For 5 levels of the independent variable, there are 10 possible pairwise comparisons between group means: {1,2},{1,3},{1,4},{1,5},{2,3},{2,4},{2,5},{3,4},{3,5},{4,5}
6 The Problem of Multiple Comparisons Assume H 0 is true and α=0.05. Then the probability of accepting H 0 in a single pairwise comparison is: P accept single pairwise H However, we have to make 10 such comparisons. Using the multiplicative law of probability (remember that?), and assuming independent pairwise tests, the probability of correctly retaining the null in all 10 comparisons is: accept all P H Therefore, experiment 1 P accept all H We now have a 40% overall chance of making a Type I error! 0
7 Intro to ANOVA
8 Null and Alternative Hypotheses in ANOVAs The omnibus null hypothesis is the null hypothesis in the ANOVA: that the population means of all groups being compared are equal i.e., for three groups, H 0 : μ 1 = μ 2 = μ 3 Alternative Hypothesis: at least one population mean is different from the others.
9 Assumptions of the ANOVA Normality of Scores I.e., we assume that the scores in all of our group populations are normally distributed Since this is important primarily for the sampling distribution of the mean, the ANOVA is fairly robust to violations of this assumption, especially if the sample sizes are reasonably large Homogeneity of variances We assume that each population of scores has the same variance E.g., [error variance] ANOVA is fairly robust to violations of this assumption Independence of observations E.g., given the population parameters, knowing one person s score tells you nothing about another person s score. Violations of this assumption can have serious implications for an analysis.
10 Alternative Hypothesis: µ 1, µ 2, and µ 3 are not all equal Instructor 1 Instructor 2 Instructor 3 Populations (µ,σ unknown) Samples
11 Omnibus Null Hypothesis: µ 1 = µ 2 = µ 3
12 The Logic of the Analysis of Variance The test statistic for ANOVA is an F-ratio, which is a ratio of two estimates of the population variance. F variance including any treatment effects variance without any treatment effects MS MS between within In the context of ANOVA, these variance estimates are called mean squares, or MS values The numerator, MS between, estimates variance using the sample means of different treatment groups The denominator, MS within (or MS error ), estimates variance using the sample variances within each treatment group
13 The Logic of the Analysis of Variance Total Variance Between Treatments Variance Measures differences caused by: Systematic treatment effects Sampling & other nonsystematic errors Within Treatments Variance Measures differences caused by: Sampling & other nonsystematic errors
14 The Logic of the ANOVA Regardless of whether or not the null hypothesis is true, the assumption of homogeneity of variances implies that all population variances are equal Thus, as we did for the independent-samples t-test, we can estimate this shared population variance by taking the average of the sample variances (the pooled variance),, s s s n s Avg s1 s s ˆwithi p (assuming n 1 = n 2 = n 3 )
15 The Logic of the ANOVA However, if all the population means are equal (under H 0 ), then we have a second way to estimate the population variance we can estimate the population variance using the variance of the sample means Recall that the Central Limit Theorem tells us how to compute the variance of sample means from the population variance: 2 2 M n We can rearrange this formula to solve for the population variance given the variance of sample means: n 2 2 M
16 The Logic of the ANOVA Of course, we don t have the variance of sample means either. However, we can estimate it by computing the variance of our three group means s Var M, M, M 2 2 ˆM M Plugging this into the previous equation, our second estimate of the population variance is ns 2 2 ˆbetween M
17 The Logic of the ANOVA We now have two estimates of the population variance: An estimate computed from the sample variances, which should estimate the population variance regardless of whether H 0 is true ˆwith,, in s p Avg s1 s2 s 3 A second estimate computed from the sample means, which only estimates the population variance if H 0 is true 2 2 ˆ between nsm nvar M1, M 2, M 3
18 The Logic of the ANOVA The F-ratio used as the test statistic for the ANOVA is simply the ratio between these two estimates of the population variance F 2 MS ˆ 1,, between nvar M M M between MS ˆ within within Avg s1, s2, s 3 If H 0 is true, then these two estimates should be equal (on average) In this case, the ratio should be 1.0 However, if H 0 is false, then the estimate in the numerator (which is based on the variability of sample means) will include the treatment effect in addition to differences in sample means expected by chance In this case, the ratio should be greater than
19 The F distribution reject H 0 retain H 0
20 Populations Samples
21 The Logic of the ANOVA Sample 1 Sample 2 Sample 3 n = 20 n = 20 n = 20 M = 65.4 M = M = 71.2 s 2 = s 2 = s 2 = M sum M 2 ˆ s 2 2 within p s1 s2 s ns 2 2 ˆbetween M SS n df M M M T SS M nm M n k 3 M M k = ˆbetween 215 F 2 ˆ within
22 Computations for the ANOVA In computing the terms required for the F-statistic, we won t explicitly compute any sample variances or standard deviations Instead, in all intermediate steps, we ll deal exclusively with sums of squared deviations (SS) and means of squared deviations (MS) Computing the F-statistic using sample standard deviations or variances gets you the same answer, but requires more calculations
23 Computing the F-statistic F df MS between between, dfwithin MSwithin where, MS between SS df between between and MS within SS df within within
24 Computations for the ANOVA: Preliminaries x x 2 Start by computing and for each group, then compute: Grand total: The overall total, computed over all scores in all groups (samples) x T Total sum of squared scores: The sum of squared scores computed over all scores in all groups x k i n j x ij k n 2 2 T xij i j
25 Computations for the ANOVA: SS terms Intro to ANOVA SS total : The sum of squared deviations of all observations from the grand mean or 2 x 2 T 2 SStotal x MT xt N (conceptual) (computational) SS SS SS total between within Not strictly needed for computing the F ratio, but it makes computing the needed SS terms much easier
26 Computations for the ANOVA: SS terms Intro to ANOVA Total Variance SS df SS SS total between within df df total between within Between Treatments Variance Within Treatments Variance SS df SS SS between total within df df between total within SS df SS SS within total between df df within total between
27 Computations for the ANOVA: SS terms Intro to ANOVA SS between : The sum of squared deviations of the sample means from the grand mean multiplied by the number of observations k 2 SS n M M between i i T i or SS between SS total SS within SS within (SS error ): The sum of squared deviations within each sample k SSwithin SS j SS1 SS2... SS k j or SS SS SS within total between
28 Computations for the ANOVA: df terms df total = N-1 : degrees of freedom associated with SS total N is the total number of scores Intro to ANOVA df between = k-1 : degrees of freedom associated with SS between k is the number of groups (samples) df within (or df error )= df total -df between = N-k : degrees of freedom associated with SS within Can also be computed as: 1 2 k 1 2 df df... df n 1 n 1... n 1 k
29 Computing the F-statistic MS MS between within SS df SS df between between within within F df between between, dfwithin MSwithin MS
30 The One-Way ANOVA: Steps 1. State Hypotheses 2. Compute F-ratio statistic: F df between between, dfwithin MSwithin MS For data in which I give you raw scores, you will have to compute: Sample means SS total, SS between, & SS within df total, df between, & df within 3. Use F-ratio distribution table to find critical F-value representing rejection region 4. Make a decision: does the F-statistic for your sample fall into the rejection region?
31 The One-Way ANOVA: Textbook Example Intro to ANOVA A psychologist wants to determine whether having a job interferes with student academic performance. She measures academic performance using students GPAs. She selects a sample of 30 students. Of these students,10 did not work, 10 worked part-time, and 10 worked full-time during the previous semester Conduct an ANOVA at a.05 level of significance testing the hypothesis that having a job interferes with student performance
32 Work Status No Work Part-Time Full-Time M 1 =3.43 M 2 =3.3 M 3 =2.98 M T =3.24 n 1 =10 n 2 =10 n 3 =10 N =30 xt x T Set up a summary ANOVA table: Source df SS MS F Between Within (error) Total 1. Compute degrees of freedom df df df total between within N 1 29 k 1 2 N k 27
33 Work Status No Work Part-Time Full-Time M 1 =3.43 M 2 =3.3 M 3 =2.98 M T =3.24 n 1 =10 n 2 =10 n 3 =10 N =30 xt x T Set up a summary ANOVA table: Source df SS MS F Between 2 Within (error) 27 Total Compute SS total SS total x 2 T x N T 2 2
34 Work Status No Work Part-Time Full-Time M 1 =3.43 M 2 =3.3 M 3 =2.98 M T =3.24 n 1 =10 n 2 =10 n 3 =10 N =30 xt x T Set up a summary ANOVA table: Source df SS MS F Between 2 Within (error) 27 Total Compute SS between (or SS within ) directly SS n M M between T
35 Work Status No Work Part-Time Full-Time M 1 =3.43 M 2 =3.3 M 3 =2.98 M T =3.24 n 1 =10 n 2 =10 n 3 =10 N =30 xt x T Set up a summary ANOVA table: Source df SS MS F Between Within (error) 27 Total Compute the missing SS value (SS between or SS within ) via subtraction: SS SS SS within total between
36 Work Status No Work Part-Time Full-Time M 1 =3.43 M 2 =3.3 M 3 =2.98 M T =3.24 n 1 =10 n 2 =10 n 3 =10 N =30 xt x T Set up a summary ANOVA table: Source df SS MS F Between Within (error) Total Compute the MS values needed to compute the F ratio: MS between SSbetween df 2 between MS within SSwithin df 27 within
37 Work Status No Work Part-Time Full-Time M 1 =3.43 M 2 =3.3 M 3 =2.98 M T =3.24 n 1 =10 n 2 =10 n 3 =10 N =30 xt x T Set up a summary ANOVA table: Source df SS MS F Between Within (error) Total Compute the F ratio: F df F between, df error 2, MS MS between error
38 F table for α=0.05 reject H 0 df error df numerator
39 F table for α=0.05 reject H 0 df error df numerator
40 Work Status No Work Part-Time Full-Time M 1 =3.43 M 2 =3.3 M 3 =2.98 M T =3.24 n 1 =10 n 2 =10 n 3 =10 N =30 SS = 5.19 T Set up a summary ANOVA table: Source df SS MS F Between Within (error) Total Compare computed F statistic with F crit and make a decision F crit ; reject H 0 Conclusion: Having a job does significantly interfere with academic performance
41 The One-way ANOVA: Example 2 Return to our running example: Do test scores vary as a function of the instructor? x 1 : sample scores from Dr. M s class x 2 : sample scores from Dr. K s class x 3 : sample scores from Dr. A s class Null Hypothesis H 0 : µ 1 = µ 2 = µ 3 Research Hypothesis H 1 : one of the population means is different Do we accept or reject the null hypothesis? Assume α = 0.05
42 Instructor Dr. M Dr. K Dr. A n 1 =7 n 2 =10 n 3 =8 N =25 M 1 =71.43 M 2 =64.20 M 3 =67.50 M T =67.28 SS 1 =89.71 SS 2 =91.60 SS 3 =68.00 SS T = Set up a summary ANOVA table: Source df SS MS F Between Within (error) Total Compute degrees of freedom df df df total between within N 1 24 k 1 2 N k 22
43 Instructor Dr. M Dr. K Dr. A n 1 =7 n 2 =10 n 3 =8 N =25 M 1 =71.43 M 2 =64.20 M 3 =67.50 M T =67.28 SS 1 =89.71 SS 2 =91.60 SS 3 =68.00 SS T = Set up a summary ANOVA table: Source df SS MS F Between 2 Within (error) 22 Total Compute SS within (or SS between ) directly (This time, we ll compute SS within ) SSwithin SS
44 Instructor Dr. M Dr. K Dr. A n 1 =7 n 2 =10 n 3 =8 N =25 M 1 =71.43 M 2 =64.20 M 3 =67.50 M T =67.28 SS 1 =89.71 SS 2 =91.60 SS 3 =68.00 SS T = Set up a summary ANOVA table: Source df SS MS F Between 2 Within (error) Total Compute the missing SS value (SS between or SS within ) via subtraction: SS SS SS between total within
45 Instructor Dr. M Dr. K Dr. A n 1 =7 n 2 =10 n 3 =8 N =25 M 1 =71.43 M 2 =64.20 M 3 =67.50 M T =67.28 SS 1 =89.71 SS 2 =91.60 SS 3 =68.00 SS T = Set up a summary ANOVA table: Source df SS MS F Between Within (error) Total Compute the MS values needed to compute the F ratio: MS between SSbetween df 2 between MS within SSwithin df 22 within
46 Instructor Dr. M Dr. K Dr. A n 1 =7 n 2 =10 n 3 =8 N =25 M 1 =71.43 M 2 =64.20 M 3 =67.50 M T =67.28 SS 1 =89.71 SS 2 =91.60 SS 3 =68.00 SS T = Set up a summary ANOVA table: Source df SS MS F Between Within (error) Total Compute the F ratio: F df F between, df error 2, MS MS between error
47 F table for α=0.05 reject H 0 df error df numerator
48 F table for α=0.05 reject H 0 df error df numerator
49 Instructor Dr. M Dr. K Dr. A n 1 =7 n 2 =10 n 3 =8 N =25 M 1 =71.43 M 2 =64.20 M 3 =67.50 M T =67.28 SS 1 =89.71 SS 2 =91.60 SS 3 =68.00 SS T = Set up a summary ANOVA table: Source df SS MS F Between Within (error) Total Compare computed F statistic with F crit and make a decision F crit ; reject H 0 Conclusion: Student test scores do vary across instructors
50 Effect Size for the One-Way ANOVA For ANOVAs, effect sizes are usually indicated using the R 2 -family measure eta-squared (η 2 ) R 2 -family measures indicate the effect size in terms of proportion of variance accounted for by the treatment effect(s) For our example: 2 variability explained by treatment effect R total variability SS SS between total 0.46
51 Post-hoc Tests for Multiple Comparisons Intro to ANOVA Rejecting H 0 only tells us that the omnibus null hypothesis (that all sample means are equal) is false However, we are often interested in knowing which particular means differ from each other Evaluating differences (usually pairwise) beyond the omnibus null hypothesis requires post-hoc testing
52 Post-hoc Tests The challenge in constructing a post-hoc multiple comparison test is keeping the experimentwise α low while maximizing the power of the test Power refers to the ability of a statistical test to pick up true differences between population means Researchers use many different post-hoc tests tailored to particular families of comparisons. Most of these tests are based on the t-test We will cover two such tests: Fisher s LSD (protected t-test) The Bonferroni procedure
53 Fisher s Least Significant Difference (LSD) Test Fisher s LSD (protected t) test was the first proposed method for post-hoc pairwise comparisons It is nearly identical to the independent measures t-test. The only differences are that the denominator uses MS error in place of pooled variance and uses df error as the degrees of freedom for the t-statistic 1 2 t df error MS n M error 1 M MS n The t is protected in that the omnibus null hypothesis must be rejected for this test to be valid The test is fairly liberal, producing higher than intended experimentwise α for post-hoc tests involving more than 3 pairwise comparisons error 2
54 The Bonferroni Procedure The Bonferroni procedure simply adjusts the pairwise alpha for a group of comparisons to ensure that, in the worst case scenario, the experimentwise alpha will never exceed 0.05 The worst case occurs when the rejection of H 0 under different pairwise comparisons is mutually exclusive In this case, via the additive rule, the probability of falsely rejecting H 0 in k comparisons is α 1 + +α k = kα
55 The Bonferroni Procedure Thus, the Bonferroni procedure requires that you divide the pairwise alpha by the number of comparisons. For example, if you wanted to make 10 pairwise comparisons at a desired experimentwise α of 0.05, you would choose the rejection region using a pairwise criterion of α/10 =0.005 The Bonferroni procedure is a very conservative test. It is guaranteed to keep the experimentwise Type I error rate below α but is more likely to lead to Type II errors (acceptance of H 0 when it is false). The formula for the Bonferroni procedure is exactly like that for Fisher s LSD test.
56 Post hoc tests: Example (Fisher s LSD) Intro to ANOVA ANOVA Summary Table Source df SS MS F Between Within (error) Total M M M n n n Let s do all possible comparisons: {1,2},{1,3},{2,3} t-statistic for Fisher s LSD test when comparing {A,B}: First, note that the denominator is the same for all comparisons: t df error MS n M A error A M B MS n error B t 27 M M M M M M A B A B A B
57 Post hoc tests: Example (Fisher s LSD) Intro to ANOVA M M M Let s do all possible comparisons: {1,2},{1,3},{2,3} Now we simply apply this formula to all comparisons: {1,2} {1,3} {2,3} t M M t M M t M M
58 Post hoc tests: Example (Bonferroni) M M M Let s do all possible comparisons: {1,2},{1,3},{2,3} We have three comparisons, so the Bonferroni correction to α would be # comparisons 3 {1,2} {1,3} {2,3} t M M t M M t M M
59 t-distribution Table α t One-tailed test α/2 α/2 -t t Two-tailed test Level of significance for one-tailed test Level of significance for two-tailed test df
60 Post hoc tests: Example M M M Let s do all possible comparisons: {1,2},{1,3},{2,3} Now we simply apply this formula to all comparisons: {1,2} {1,3} {2,3} Fisher s LSD: t M M , retain H 0 t M M , reject H , retain H0 t M M Bonferroni: , retain H , reject H , retain H0
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