One-way Analysis of Variance. Major Points. T-test. Ψ320 Ainsworth

Size: px

Start display at page:

Download "One-way Analysis of Variance. Major Points. T-test. Ψ320 Ainsworth"

Marsha Foster
6 years ago
Views:

1 One-way Analysis of Variance Ψ30 Ainsworth Major Points Problem with t-tests and multiple groups The logic behind ANOVA Calculations Multiple comparisons Assumptions of analysis of variance Effect Size for ANOVA T-test So far, we have made comparisons between a single group and population, -related samples and independent samples What if we want to compare more than groups? One solution: multiple t-tests 3

2 T-test With 3 groups, you would perform 3 t- tests Not so bad, but what if you had 10 groups? You would need 45 comparisons to analyze all pairs That s right 45!!! 4 The Danger of Multiple t-tests Each time you conduct a t-test on a single set of data, what is the probability of rejecting a true null hypothesis? Assume that H 0 is true. You are conducting 45 tests on the same set of data. How many rejections will you have? Roughly or 3 false rejections! So, multiple t-tests on the same set of data artificially inflate α 5 Summary: The Problems With Multiple t-tests Inefficient - too many comparisons when we have even modest numbers of groups. Imprecise - cannot discern patterns or trends of differences in subsets of groups. Inaccurate - multiple tests on the same set of data artificially inflate α What is needed: a single test for the overall difference among all means e.g. ANOVA 6

3 Logic of the Analysis of Variance Null hypothesis h 0 : Population means equal µ 1 = µ = µ 3 = µ 4 Alternative hypothesis: h 1 Not all population means equal. 7 Logic Create a measure of variability among group means MS BetweenGroups AKA s BetweenGroups Create a measure of variability within groups MS WithinGroups AKA s WithinGroups 8 Logic MS BetweenGroups /MS WithinGroups Ratio approximately 1 if null true Ratio significantly larger than 1 if null false approximately 1 can actually be as high as or 3, but not much higher 9

4 So, why is it called analysis of variance anyway? Aren t we interested in mean differences? Variance revisited Basic variance formula s ( ) X X SS = = n 1 df 10 Why is it called analysis of variance anyway? What if data comes from groups? We can have different sums of squares ( ij ) ( ij j ) j ( j ) SS = Y Y 1.. SS = Y Y. SS3 = n Y. Y.. Where i represents the individual and j represents the groups 11 Logic of ANOVA Grand Mean (Ungrouped Mean) X John s Score YGroup1 Y Group Y Group3 X-Axis 1

5 Sums of Squares The total variability can be partitioned into between groups variability and within groups variability. ( Yij Y.. ) = nj ( Y. j Y.. ) + ( Yij Y. j ) SSTotal = SSBetweenGroups + SSWithinGroups SST = SSBG + SSWG SS = SS + SS T Effect Error 13 Degrees of Freedom (df ) Number of observations free to vary df T = N -1 Variability of N observations df BG = g -1 Variability of g means df WG = g (n -1) or N - g n observations in each group = n -1 df times g groups df T = df BG + df WG 14 Mean Square (i.e. Variance) MS MS MS T BG WG = s = T = s = BG = s = WG ( Yij Y.. ) N 1 j (. j..) n Y Y g 1 ( Yij Y. j ) gn ( 1) 15

6 F-test MS WG contains random sampling variation among the participants MS BG also contains random sampling variation but it can also contain systematic (real) variation between the groups (either naturally occurring or manipulated) 16 F-test Grand Mean (Ungrouped Mean) Y Y Y X-Axis The F-test is a ratio of the MS BG /MS WG and if the group differences are just random the ratio will equal 1 (e.g. random/random) 17 F-test YGroup1 Y Group Y Group3 If there are real differences between the groups the difference will be larger than 1 and we can calculate the probability and hypothesis test 18

7 F distribution 1 F-ratio There is a separate F distribution for every df like t but we need both df bg and df wg to calculate the F CV from the F table E.3 for alpha =.05 and E.4 for alpha = Example A researcher is interested in knowing which brand of baby food babies prefer: Beechnut, Del Monte or Gerber. He randomly selects 15 babies and assigns each to try strained pees from one of the three brands Liking is measured by the number of spoonfuls the baby takes before getting upset (e.g. crying, screaming, throwing the food, etc.) 0 Hypothesis Testing 1. H o : µ Beechnut = µ Del Monte = µ Gerber. At least µs are different 3. α = More than groups ANOVA F 5. For F cv you need both df BG = 3 1 = and df WG = g (n - 1) = 3(5 1) = 1 Table E.3 F cv (,1) = 3.89, if F o > 3.89 reject the null hypothesis 1

8 Step 6 Calculate F-test Start with Sum of Squares (SS) We need: SS T SS BG SS WG Then, use the SS and df to compute mean squares and F `Brand Baby Spoonfuls (Y) Group Means ( Y Y ) ( Y Y ) n ( Y Y ) Beechnut Del Monte Gerber Step 6 Calculate F-test ij.. ij. j j. j ( ) = (4-4.6) = [5 * ( ) ] = [5 * ( ) ] = [5 * ( ) ] = 1.36 Mean Sum ANOVA summary table and Step 7 Source SS df MS F BG WG Total Remember MS = SS/df F = MS BG /MS WG Step 7 Since > 3.89, reject the null hypothesis 4

9 Conclusions The F for groups is significant. We would obtain an F of this size, when H 0 true, less than 5% of the time. The difference in group means cannot be explained by random error. The baby food brands were rated differently by the sample of babies. 5 Alternative Analysis computational approach to SS Equations ( Y ) T SS Y Y N ( ) aj T SSBG = n N ( a ) = T = = j SSWG Y n Under each part of the equations, you divide by the number of scores it took to get the number in the numerator N 6 Computational Approach Example Brand Baby Spoonfuls (Y) Sum Sum Sum 4 Beechnut Del Monte Gerber Total Sum Y Squared T 95 SST = Y = 673 = N 15 ( ) aj T SSBG = = = n N 5 15 = = ( a ) j SSWG = Y = n = 673 = = Note: You get the same SS using this method

10 Unequal Sample Sizes With one-way, no particular problem Multiply mean deviations by appropriate n i as you go The problem is more complex with more complex designs, as shown in next chapter. Equal samples only simplify the equation because when n 1 = n = = n g (...) = (...) n Y Y n Y Y j j j 8 Multiple Comparisons Significant F only shows that not all groups are equal We want to know what groups are different. Such procedures are designed to control familywise error rate. Familywise error rate defined Contrast with per comparison error rate 9 More on Error Rates Most tests reduce significance level (α) for each t test. The more tests we run the more likely we are to make Type I error. Good reason to hold down number of tests 30

11 Fisher s LSD Procedure Requires significant overall F, or no tests Run standard t tests between pairs of groups. Often we replace s pooled with MS error from overall analysis It is really just a pooled error term, but with more degrees of freedom (pooled across all treatment groups) 31 Fisher s LSD Procedure Comparing Beechnut and Gerber * MSWG MSWG s = + = + XBeechnut XGerber nbeechnut ngerber 5 5 X X t = = = Beechnut Gerber * sxbeechnut XGerber t cv (5+5-=8) α =.05 = Since 3.55 > 1.860, the groups are significantly different. 3 Bonferroni t Test Run t tests between pairs of groups, as usual Hold down number of t tests Reject if t exceeds critical value in Bonferroni table Works by using a more strict value of α for each comparison 33

12 Bonferroni t Critical value of α for each test set at.05/c, where c = number of tests run Assuming familywise α =.05 e. g. with 3 tests, each t must be significant at.05/3 =.0167 level. With computer printout, just make sure calculated probability <.05/c 34 Assumptions for Analysis of Variance Assume: Observations normally distributed within each population Population variances are equal Homogeneity of variance or homoscedasticity Observations are independent 35 Assumptions Analysis of variance is generally robust to first two A robust test is one that is not greatly affected by violations of assumptions. 36

13 Magnitude of Effect Eta squared (η ) Easy to calculate Somewhat biased on the high side Formula SSBG η = SS Total Percent of variation in the data that can be attributed to treatment differences 37 Magnitude of Effect Omega squared (ω ) Much less biased than η Not as intuitive We adjust both numerator and denominator with MS error Formula SSBG ( k 1) MS ω = SS + MS T WG WG 38 η and ω for Foa, et al. SSBG η = = =.50 SST SSBG ( k 1) MSWG (.867) ω = = =.40 SS + MS T WG η =.50: 50% of variability in symptoms can be accounted for by treatment ω =.40: This is a less biased estimate, and note that it is 5% smaller. 39

14 Other Measures of Effect Size We can use the same kinds of measures we talked about with t tests (e.g. d and γ) Usually makes most sense to talk about groups at a time or effect size between the largest and smallest groups, etc. And there are methods for converting η to d and vice versa 40

10/31/2012. One-Way ANOVA F-test

10/31/2012. One-Way ANOVA F-test PSY 511: Advanced Statistics for Psychological and Behavioral Research 1 1. Situation/hypotheses 2. Test statistic 3.Distribution 4. Assumptions One-Way ANOVA F-test One factor J>2 independent samples