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Lecture Overview on ANOVA

Lecture Overview on ANOVA. Review hypothesis testing; inferential statistics z-test, t-test, independent & dependent t-test New Stuff Power – Ability to reject Ho ANOVA An alysis o f Va riance Done with 3 or more groups Playground Exercise Complete SPSS Example. Power.

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Lecture Overview on ANOVA

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  1. Lecture Overview on ANOVA • Review • hypothesis testing; inferential statistics • z-test, t-test, independent & dependent t-test • New Stuff • Power – Ability to reject Ho • ANOVA • Analysis of Variance • Done with 3 or more groups • Playground Exercise • Complete SPSS Example Dr. Sinn, PSYC301, The joy of 1-way ANOVA

  2. Power • Review: Hypothesis Testing Errors • Wrongly rejecting Ho: Chance of Type I error: α • Wrongly retaining Ho: Chance of Type II error: β • Power • Opposite of β • Power = 1- β • Ability to reject Ho (when Ho should be rejected). • Researchers want Power! • Want ability to reject Ho; Show you were right to suspect a difference. • Want to show IV affects your DV. Dr. Sinn, PSYC301, The joy of 1-way ANOVA

  3. α area (where we reject the Ho, and we shouldn’t) beyond tcritical under Ho β area (where we retain the Ho, and we shouldn’t) inside tcritical under Ha α α tc tc tc β Error Areas Ho: μ=55 Ha: μ>55 Dr. Sinn, PSYC301, The joy of 1-way ANOVA

  4. #1: Increase Treatment: Increase difference between groups (μ’s) H0: μ=55 H0: μ=55 Reality: μ=57 Reality: μ=72 β tc • #2: Decrease Sampling Error: Decrease differences within groups. tc tc tc β β β Increasing Power Dr. Sinn, PSYC301, The joy of 1-way ANOVA

  5. #1 Increase Treatment Effect (Increase BG differences) Rat study 0,3,or 6 mg 0,10,or 20 mg Therapy study 10 therapy sessions 1 therapy session #2 Decrease Sampling Error (Decrease WG differences) Rat study Different strains of rats Same strain of rat Rats allowed to eat freely Rats all unfed for 24 hours Therapy study Diff. types of Therapy Same type of Therapy Examples of increasing power Rat Study: IV:Caffeine Level DV:Amt. Food Found Therapy Study: IV:Therapy (drug, talk, drug+talk, or control) DV: Improvement] Dr. Sinn, PSYC301, The joy of 1-way ANOVA

  6. 1-Way ANOVA • ANOVA • Analysis of Variance • 1-way means 1 Independent Variable (IV) • Purpose: • ANOVA allows hypothesis testing with 3+ sample means • Imagine study on interventions to help frosh make friends • Three IV levels: Standard courses, interactive courses, clustered courses. • ANOVA uses F-test • Strategy: Compare variability within group to variability between groups. • F is ratio between two values: Dr. Sinn, PSYC301, The joy of 1-way ANOVA

  7. ANOVA Playground (Download from Website) Dr. Sinn, PSYC301, The joy of 1-way ANOVA

  8. Matching Exercise Dr. Sinn, PSYC301, The joy of 1-way ANOVA

  9. Draw Conclusions from Playground • What does a large F mean? • What two things will make F large? Dr. Sinn, PSYC301, The joy of 1-way ANOVA

  10. Partitioning Variance • Partition • fancy word for “divide up” • ANOVA partitions variance (MS means variance) • Types of variance • Total variance = MSWG + MSBG • MSWG= sampling error (background noise) • MSBG = sampling error + treatment (includes effect of Independent Variable) • If just error F tends toward 1.0 • If treatment effectF gets larger Dr. Sinn, PSYC301, The joy of 1-way ANOVA

  11. Example of 1-way ANOVA • Studying effect of caffeine on productivity • Does caffeine help or hurt? • IV: Level of Caffeine: 0, 10, 20 mg • DV: Number of Food Pellets Found Number of Food Pellets Found Dr. Sinn, PSYC301, The joy of 1-way ANOVA

  12. SPSS Data Entry IV DV Label levels of IV so output is easier to read. Dr. Sinn, PSYC301, The joy of 1-way ANOVA

  13. SPSS Analysis • Go to Analyze, Compare Means, & select One-way ANOVA Put DV here. Put IV here. Dr. Sinn, PSYC301, The joy of 1-way ANOVA

  14. SPSS Analysis, Part #2 Select this to get descriptive statistics like sample means & standard deviations. Alpha level still set to .05, just like it was with t-tests. Gives you a line graph of the sample means Conducts “after the fact” test to compare all pairs of sample means. Dr. Sinn, PSYC301, The joy of 1-way ANOVA

  15. SPSS Output Sample means from 3 groups, plus mean amount of food found overall. Source of Variation Table Dr. Sinn, PSYC301, The joy of 1-way ANOVA

  16. Where does F come from? • MSWG = SSWG/dfWG = Sum of Squares / degrees of freedom • MSBG = SSBG/dfBG = Sum of Squares / degrees of freedom • Degrees of freedom • dfWG: NT – K (Total # of subjects - # of groups) • dfBG: K-1(# of groups – 1) • dfTOTAL: NT – 1 (Total # of subjects – 1) • Expectations: • If I give you df and SS, you can calculate F • You don’t have to get any SS by hand. Dr. Sinn, PSYC301, The joy of 1-way ANOVA

  17. SPSS Output –Post Hoc Test No Sig. Diff. Between 0 & 10mg Rats at 20 mg found significantly more food than rats on 0 or 10 mg of caffeine. Dr. Sinn, PSYC301, The joy of 1-way ANOVA

  18. SPSS Output– Practical Significance • η2 (“eta squared”) • Effect size statistic – indicates % of variance explained • Measures impact of IV on DV • We can explain 68% of the variance in how much food a rat finds if we know the level of caffeine. Dr. Sinn, PSYC301, The joy of 1-way ANOVA

  19. Hypothesis Testing Steps • Comparison: cf. three sample means. • Hypothesis: Ho: μ1= μ2 = μ3 Ha: Not all μ’s equal • Set-up: α= .05 , dfbg= K-1= 2,dfwg= NT-K = 16-3=13, Fcrit = 3.80 • Fobt = 13.653 • Reject Ho. • The hypothesis was largely supported. Rats found sig. more food on 20mg of caffeine (M=4.33) than on 0mg (M=2.40) or 10mg (M=1.80), F(2,13) = 13.653, p <=.05. Caffeine has a large effect on food finding behavior, accounting for about 68% of the variance, η2 = .6775. Dr. Sinn, PSYC301, The joy of 1-way ANOVA

  20. df Between Groups F-table df Within Groups Dr. Sinn, PSYC301, The joy of 1-way ANOVA

  21. Lab #8: 1-way ANOVA • TV Problem: The hypothesis was supported. LightTV users provided more community service (M = 6.13) than did moderateusers (M = 4.00), who provided more than heavyusers (M = 1.75), F(2,21) = 15.963, p ≤ .05. TV accounts for about 60% of the variance in community service, η2 = .6032. Dr. Sinn, PSYC301, The joy of 1-way ANOVA

  22. Follow-up Questions • Q1: Variance within group? MSwg = 2.399 • Q2: Variance between groups? MSbg=38.292 • Q3: Replacing heavy scores with 4,5,4,5,6,5,4,3 would decrease the difference between groups because the heavy users would then difference less from the other groups. • Q4: Decreasing between group differences (decreasing treatment) would decrease F. Dr. Sinn, PSYC301, The joy of 1-way ANOVA

  23. Problem #2: Post Hoc Explanation Dr. Sinn, PSYC301, The joy of 1-way ANOVA

  24. Problem #2: Post Hoc Explanation Dr. Sinn, PSYC301, The joy of 1-way ANOVA

  25. Problem #2: • The hypothesis was supported. People commuting 0 minutes participated significantly more (M=3.4 hours) than people commuting 45 (M=1.2) or60 minutes (M=1.6), F (3,16) = 7.256, p≤.05. Commuting accounted for a large amount of variance in community involvement, η2 = .5764. Dr. Sinn, PSYC301, The joy of 1-way ANOVA

  26. Follow-up Questions • Q1: Variance within group? MSwg = .650 • Q2: Variance between groups? MSbg=4.717 • Q3: Replacing 30 minute commuting scores with 1,4,1,4,3 would increase the within group variability. • Q4: Increasing sampling error would decrease F. Dr. Sinn, PSYC301, The joy of 1-way ANOVA

  27. Study: Does alcohol affect reaction time? Identify the treatment effect in this case. Explain how sampling error might arise. μna=?? μ2b=?? μ4b=?? Population Means Review Partitioning 14 23 26 Sample Means Dr. Sinn, PSYC301, The joy of 1-way ANOVA

  28. One-Way ANOVA Part 2!! Dr. Sinn, PSYC301, The joy of 1-way ANOVA

  29. Study: Does alcohol affect reaction time? What accounts for variability within groups? What accounts for variability between groups? What’s the Formula for F? Review Partitioning Dr. Sinn, PSYC301, The joy of 1-way ANOVA

  30. Study: Does alcohol affect reaction time? If the alcohol content of the beers is not held constant, what happens to F? increases decreases neither Review Partitioning • If the alcohol content of the beers is not held constant, what happens? • error increases • error decreases • treatment effect increases • treatment effect decreases Dr. Sinn, PSYC301, The joy of 1-way ANOVA

  31. Hypothesis Testing Steps • Comparison: cf. three sample means. • Hypothesis: Ho: μ1= μ2 = μ3 Ha: Not all μ’s equal • Set-up: α= .05 , dfbg=K-1=3-1=2,dfwg=NT-K=12-3=9, Fcrit = 4.26 • now do one-way ANOVA on SPSS Dr. Sinn, PSYC301, The joy of 1-way ANOVA

  32. SPSS Output - Charts Dr. Sinn, PSYC301, The joy of 1-way ANOVA

  33. SPSS Output - Graphs Dr. Sinn, PSYC301, The joy of 1-way ANOVA

  34. Hypothesis Testing Steps • Comparison: cf. three sample means. • Hypothesis: Ho: μ1= μ2 = μ3 Ha: Not all μ’s equal • Set-up: α= .05 , dfbg=K-1=3-1=2,dfwg=NT-K=12-3=9, Fcrit = 4.26 • Fobt = 2.633 • Retain Ho. • The hypothesis was not supported. The reaction times following no alcohol (M=13.75), two beers (M=22.50), and four beers (M=26.25) did not differ significantly, F(2,9) = 2.633, n.s.. Dr. Sinn, PSYC301, The joy of 1-way ANOVA

  35. Bet. Group Varib: L M H MSbg: _______ With. Group Varib: L M H MSwg: _______ Numb. of Words Recalled: Dataset A Dr. Sinn, PSYC301, The joy of 1-way ANOVA

  36. Bet. Group Varib: L M H MSbg: _______ With. Group Varib: L M H MSwg: _______ Numb. of Words Recalled: Dataset B Dr. Sinn, PSYC301, The joy of 1-way ANOVA

  37. Bet. Group Varib: L M H MSbg: _______ With. Group Varib: L M H MSwg: _______ Numb. of Words Recalled: Dataset C Dr. Sinn, PSYC301, The joy of 1-way ANOVA

  38. Bet. Group Varib: L M H MSbg: _______ With. Group Varib: L M H MSwg: _______ Numb. of Words Recalled: Dataset D Dr. Sinn, PSYC301, The joy of 1-way ANOVA

  39. Bet. Group Varib: L M H MSbg: _______ With. Group Varib: L M H MSwg: _______ Numb. of Words Recalled: Dataset E Dr. Sinn, PSYC301, The joy of 1-way ANOVA

  40. Bet. Group Varib: L M H MSbg: _______ With. Group Varib: L M H MSwg: _______ Numb. of Words Recalled: Dataset F Dr. Sinn, PSYC301, The joy of 1-way ANOVA

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