One-way Between Groups Analysis of Variance Ψ320 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 Psy 320 -Cal State Northridge 2 T-test So far, we have made comparisons between a single group and population, 2-related samples and 2 independent samples What if we want to compare more than 2 groups? One solution: multiple t-tests Psy 320 -Cal State Northridge 3 1 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!!! Psy 320 -Cal State Northridge 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 is true. You are conducting 0 45 tests on the same set of data. How many rejections will you have? Roughly 2 or 3 false rejections! So, multiple t-tests on the same set of data artificially inflate α Psy 320 -Cal State Northridge 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 Psy 320 -Cal State Northridge 6 2 LOGIC OF THE ANALYSIS OF VARIANCE Psy 320 -Cal State Northridge 7 Logic of the Analysis of Variance Null hypothesis h : Population 0 means equal µ = µ =µ =µ 1 2 3 4 Alternative hypothesis: h 1 –Not all population means equal. Psy 320 -Cal State Northridge 8 Logic Create a measure of variability among group means –MS AKA s2 BetweenGroups BetweenGroups Create a measure of variability within groups –MS AKA s2 WithinGroups WithinGroups Psy 320 -Cal State Northridge 9 3 Logic MS /MS BetweenGroups WithinGroups –Ratio approximately 1 if null true –Ratio significantly larger than 1 if null false –“approximately 1” can actually be as high as 2 or 3, but not much higher Psy 320 -Cal State Northridge 10 “So, why is it called analysis of variance anyway?” Aren’t we interested in mean differences? Variance revisited –Basic variance formula ∑( )2 X − X SS s2 = i = n−1 df Psy 320 -Cal State Northridge 11 “Why is it called analysis of variance anyway?” What if data comes from groups? –We can have different sums of squares SS = ∑(Y −Y )2 1 i GM SS =∑(Y −Y )2 2 i j SS =∑n (Y −Y )2 3 j j GM Where i represents the individual, j represents the groups and GM represent the ungrouped (grand) mean Psy 320 -Cal State Northridge 12 4 Logic of ANOVA Grand Mean (Ungrouped Mean) X John’s Y Y Y Score Group1 Group2 Group3 X-Axis Psy 320 -Cal State Northridge 13 CALCULATIONS Psy 320 -Cal State Northridge 14 Sums of Squares The total variability can be partitioned into between groups variability and within groups variability. ∑(Y −Y )2 =∑n (Y −Y )2+∑(Y −Y )2 i GM j j GM i j SS =SS +SS Total BetweenGroups WithinGroups SS =SS +SS T BG WG SS =SS +SS T Effect Error Psy 320 -Cal State Northridge 15 5 Degrees of Freedom (df ) Number of “observations” free to vary –df = N-1 T •Variability of Nobservations –df = g-1 BG •Variability of gmeans –df = g (n-1) or N -g WG •nobservations in each group = n-1 df times ggroups –df = df + df T BG WG Psy 320 -Cal State Northridge 16 Mean Square (i.e. Variance) ∑(Y −Y )2 MS =s2 = i GM T T N−1 ∑ ( )2 n Y −Y MS =s2 = j j GM BG BG #groups−1 ∑( )2 Y −Y MS =s2 = i j WG WG #groups*(n−1) Psy 320 -Cal State Northridge 17 F-test MS contains random sampling WG variation among the participants MS also contains random sampling BG variation but it can also contain systematic (real) variation between the groups (either naturally occurring or manipulated) Psy 320 -Cal State Northridge 18 6 F-test Systematic BG Variance+Random BG Variance F = Ratio Random WS Variance And if no “real” difference exists between groups Random BG Variance F = ≈1 Ratio Random WS Variance Psy 320 -Cal State Northridge 19 F-test Grand Mean (Ungrouped Mean) Y Y Y X-Axis The F-test is a ratio of the MS /MS and BG WG if the group differences are just random the ratio will equal 1 (e.g. random/random) Psy 320 -Cal State Northridge 20 F-test Y Y Y Group1 Group2 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 Psy 320 -Cal State Northridge 21 7 F distribution Probability There is a separate F distribution for every df like t but we need both df and df to calculate bg wg the F from the F table D.3 for alpha = .05 and CV D.4 for alpha = .01 Psy 320 -Cal State Northridge 22 1-WAY BETWEEN GROUPS ANOVA EXAMPLE Psy 320 -Cal State Northridge 23 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 peas 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.) Psy 320 -Cal State Northridge 24 8 Hypothesis Testing 1. H : µ = µ = µ o Beechnut Del Monte Gerber 2. At least 2 µs are different 3. α= .05 4. More than 2 groups → ANOVA → F 5. For F you need both df = 3 – 1 = 2 cv BG and df = g (n - 1) = 3(5 – 1) = 12 WG Table D.3 F (2,12) = 3.89, if F > 3.89 cv o reject the null hypothesis Psy 320 -Cal State Northridge 25 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 Psy 320 -Cal State Northridge 26 Step 6 – Calculate F-test `Brand Baby Spoonfuls (Y) Group Means (Yij−Y..)2 (Yij−Y.j)2 nj(Y.j−Y..)2 1 3 Beechnut 234 444 4.6 5 8 6 7 0.445 1 Del Monte 789 486 6 520...471471391 440 [5 * (=6 0 -. 565.353 3)2] 10 5 1.777 1 11 9 7.113 0.36 Gerber 111234 1680 8.4 1023...174174197 250...571666 [5 * (8=. 42 1- .63.63 33)2] 15 9 7.113 0.36 Mean 6.333 Sum 71.335 34.4 36.93 Psy 320 -Cal State Northridge 27 9 ANOVA summary table and Step 7 Source SS df MS F BG 36.93 WG 34.4 Total 71.335 Remember –MS = SS/df –F = MS /MS BG WG Step 7 – Since ______ > 3.89, reject the null hypothesis Psy 320 -Cal State Northridge 28 Conclusions The F for groups is significant. –We would obtain an Fof this size, when H true, less than 5% of the time. 0 –The difference in group means cannot be explained by random error. –The baby food brands were rated differently by the sample of babies. Psy 320 -Cal State Northridge 29 ALTERNATIVE COMPUTATIONAL APPROACH Psy 320 -Cal State Northridge 30 10