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Psych 200 Methods & Analysis Dr. Kulstad for Dr. Lindgren Fall 2008 Outline 1. Turn in homework 2. Stats in the news 3. ANOVA con’t 4. Bring to next class (Tuesday) -- COMPLETED ungraded pop quiz (available on Blackboard) as of end of class. -- List of statistical tests for group project 3. Components of the F-statistic Example A researcher is interested in testing the effectiveness of medication on reducing pain. She is interested in examining the effectiveness of Aspirin, Tylenol, and a placebo. She recruits subjects and randomly assigns them to one of the three groups. Participants perceptions on how much their pain is reduced is measured. What kind of design is this? What is the IV? DV? How many levels or conditions of the IV are there? What are the null & alternative hypotheses? Xbar Aspirin Tylenol Placebo 3 2 2 5 2 1 3 4 3 5 4 2 4 3 2 3. Components of the F-statistic • Aspirin MSwn: Basic formula – Add the squared deviations from the 3 mean for each condition & divide by df 5 Aspirin: (3-4)2 + (5-4)2+ (3-4)2 + (5-4)2 = 4 3 Tylenol: (2-3)2 + (4-3)2+ (2-3)2 + (4-3)2 = 4 5 Placebo: (2-2)2 + (1-2)2+ (3-2)2 + (2-2)2 = 2 – Sum of all of the squared deviations in Xbar 4 each condition: 4 + 4 + 2 = 10 – df: N –k = 12 - 3 = 9 Sum of Squares – SSwn = 10 = 1.111 within (SSwn) df 9 Mean Squares within (MSwn) Tylenol Placebo 2 2 2 1 4 3 4 2 3 2 df within (dfwn) 3. Components of the F-statistic • Components of ANOVAs are often arrayed in tables – Let’s fill in what we know so far with our sample data… Source Sum of Squares df Mean square 10 9 1.111 Between Within Total F 3. Components of the F-statistic • Mean square between groups (MSbn) – Estimate of variability of scores that occurs between levels/conditions in a factor • How much does each level mean deviate from overall mean of the experiment Way to measure how much the levels means differ from one another Takes into account error AND effect of treatment • • – – If null is true, MSbn is only estimating one population -- only estimating σ2error like MSwn If null is not true, MSbn is estimating error variance and treatment variance 3. Components of the F-statistic • Mean square between groups (MSbn) – Basic formula • Σn(Xbar - GM)2 – – – • Calculate the deviation of a condition mean from the grand mean & multiply by the number in that condition Repeat for each condition Add those values Divide by df: k-1 (k = # conditions) Grand mean (GM) = mean of all scores in the entire experiment 3. Components of the F-statistic Aspirin • Tylenol Placebo MSbw: Basic formula – Calculate the deviation of a condition mean 3 2 from the grand mean (mean of ALL scores) 5 2 & multiply by the number in that condition 3 4 – Repeat for each condition 5 4 – Add those values Xbar 4 3 Aspirin: 4* (4-3)2 = 4 Tylenol: 4 * (3-3)2 = 0 GM (Grand mean) Placebo: 4 * (2-3)2 = 4 – Sum for all conditions: 4 + 0 + 4 = 8 Sum of Squares – df: k-1 = 3 -1 = 2 between (SSbt) – SSbt = 8 = 4 df 2 Mean Squares between (MSbt) df between (dfwn) 2 1 3 2 2 3 3. Components of the F-statistic • Back to our ANOVA table Source Sum of Squares df Mean square Between 8 2 4 Within 10 9 1.111 F Sum of Squares total (SStot) = SSbt + SSwn = Σ(X- GM)2 df total (dftot) = dfbt + dfwn = N-1 Total – Now, we can complete it Fobt = MSbt /MSwn Source Sum of Squares df Mean square F Between 8 2 4 3.60 Within 10 9 1.111 Total 18 11 3. Components of the F-statistic • Logic of the F ratio (Fobt) – To conduct an ANOVA, we compare MSbn / MSwn • • If H0 is true – MSbn should = MSwn – NO treatment + Error = Error – Fobt = 1 If H0 is false – MSbn should be > MSwn – Treatment + Error vs. Error – Fobt > 1 – Does not mean we reject H0 every time Fobt > 1 » Effect has to be strong enough to rule out differences due to chance alone 3. Components of the F-statistic • F-Distribution – Sample distribution that shows values of F that occur when H0 is true Positively skewed Like t-distribution, F is a group of curves – – • – Dependent on df Unlike t-distribution, F, depends on 2 dfs • • – dfbw dfwn Need both to determine Fcrit • • (By hand) use chart in back of book (SPSS) if Fobt < α, reject H0, 3. Components of the F-statistic • Back to our data – – – – Source Sum of Squares df Mean square F Between 8 2 4 3.60 9 1.111 α = .05 Within 10 Total 18 Fcrit = 4.26 The exact p-value for Fobt = .071 What do we decide? • – Fail to reject H0 How do we write that up? • Fobt > 1 &11not rejecting H0?Why? Due to sample size (which determines df), can’t rule out that differences observed are simply due to chance – i.e., treatment effects aren’t “extreme”enough to rule out chance. A one-way analysis of variance was conducted to examine the effect of three drugs on pain relief. There were no significant differences in pain relief, F(2,9) = 3.60, p > .05. Therefore, there was no evidence that Tylenol or Aspirin worked better than a placebo. 3. Components of the F-statistic • What if data and results were different? Xbar – – – – – Aspirin Tylenol Placebo 3 2 2 5 2 3 Source Sum of Squares df Mean square F 1 Between 10.167 2 5.083 5.229 4 2 Within 8.750 9 .972 5 4 2 Total 18 11 4 3 1.75 α = .05 Fcrit = 4.26 The exact p-value for Fobt = .031 What do we decide? What trouble do we run into? 4. Performing Post hoc Comparisons • Procedures to identify significant differences in condition/level/group means – – – Variety of procedures Some are more/less conservative Book covers 2 • Fisher’s protected t-test – • Used if groups have unequal n’s Tukey’s HSD multiple comparisons test – Used if groups have equal n’s 4. Performing Post hoc Comparisons • Fisher’s protected t-test – tobt = (Xbar1 – Xbar2) [ MSwn (1/n1 + 1/n2)]1/2 Same as independent samples t-test BUT use MSwn in lieu of s2pool Use this formula for every pair of means in experiment Use t table in book & find tcrit (use two-tailed values) – – – • – df is dfwn Remember not used for equal n’s 4. Performing Post hoc Comparisons • Tukey’s HSD (Honestly Significant Difference) – – Used with equal n’s Computes the minimum difference between means that is required for them to differ significantly Steps – • Identify qk (table in book) – • • • Need k (α, # levels, and dfwn) Multiply qk by (MSwn/n)½ Determine the differences between all the means Compare each difference to the HSD difference – – If difference is > than HSD, means differ significantly If difference is < HSD, means do not differ significantly 4. Performing Post hoc Comparisons • Which post hoc test would we run? Xbar – – – – Aspirin Tylenol Placebo 3 2 2 5 2 3 Source Sum of Squares df Mean square F 1 Between 10.167 2 5.083 5.229 4 2 Within 8.750 9 .972 5 4 2 Total 18 11 4 3 1.75 α = .05 Determine which means differ significantly & write up results using APA style & GST qk =3.95 Aspirin & Placebo differ significantly Tukey’s HSD (SPSS) Multiple Comparisons pain Tukey HSD 95% Confidence Interval Mean Difference (I-J) (I) drug (J) drug 1 2 1.00000 .69722 .365 -.9466 2.9466 3 2.25000* .69722 .025 .3034 4.1966 1 -1.00000 .69722 .365 -2.9466 .9466 3 1.25000 .69722 .226 -.6966 3.1966 1 -2.25000* .69722 .025 -4.1966 -.3034 2 -1.25000 .69722 .226 -3.1966 .6966 2 3 Std. Error *. The mean difference is significant at the 0.05 level. Sig. Lower Bound Upper Bound 5. Summary of Steps • Initial steps – • Computations – – – • • • Determine null & alternative hypotheses, choose alpha, check assumptions, collect data Sums of squares dfs Mean squares Identify Fcrit If you reject the null, perform the appropriate post hoc tests Interpret & write up your results 6. Describing the relationship • Can also calculate confidence intervals – – – • Upper bound: [( MSwn /n)1/2 * +tcrit]+ Xbar Lower bound: [( MSwn /n)1/2 * -tcrit]+ Xbar Calculated for every significant condition/mean/group Can also calculate the effect size (eta squared or η2) – – New correlation coefficient Akin to a squared correlation coefficient • – Remember squared correlation coefficient tells you proportion of variance of accounted for Formula: SSbn / SStot 7. Power • • F statistic (Fobt) : MSbn / MSwn Maximize power through – Good design • Strong manipulation – • Added control (lessen variability and error) – • Decreases denominator Larger n’s – – – – Increase numerator Increases dfwn Minimizes MSwn Decreases size of Fcrit All of the above increase Fobt