Repeated Measures ANOVA Quantitative Methods in HPELS 440:210

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Transcript of Repeated Measures ANOVA Quantitative Methods in HPELS 440:210

  • Slide 1
  • Repeated Measures ANOVA Quantitative Methods in HPELS 440:210
  • Slide 2
  • Agenda Introduction The Repeated Measures ANOVA Hypothesis Tests with Repeated Measures ANOVA Post Hoc Analysis Instat Assumptions
  • Slide 3
  • Introduction Recall There are two possible scenarios when obtaining two sets of data for comparison: Independent samples: The data in the first sample is completely INDEPENDENT from the data in the second sample. Dependent/Related samples: The two sets of data are DEPENDENT on one another. There is a relationship between the two sets of data.
  • Slide 4
  • Introduction Three or more data sets? If the three or more sets of data are independent of one another Analysis of Variance (ANOVA) If the three or more sets of data are dependent on one another Repeated Measures ANOVA
  • Slide 5
  • Agenda Introduction The Repeated Measures ANOVA Hypothesis Tests with Repeated Measures ANOVA Post Hoc Analysis Instat Assumptions
  • Slide 6
  • Repeated Measures ANOVA Statistical Notation Recall for ANOVA: k = number of treatment conditions (levels) n x = number of samples per treatment level N = total number of samples N = kn if sample sizes are equal T x = X for any given treatment level G = T MS = mean square = variance
  • Slide 7
  • Repeated Measures ANOVA Additional Statistical Notation: P = total score for each subject (personal total) Example: If a subject was assessed three times and had scores of 3, 4, 5 P = 12
  • Slide 8
  • Repeated Measures ANOVA Formula Considerations Recall for ANOVA: SS between = T 2 /n G 2 /N SS within = SS inside each treatment SS total = SS within + SS between SS total = X 2 G 2 /N
  • Slide 9
  • ANOVA Formula Considerations: df total = N 1 df between = k 1 df within = (n 1) df within = df in each treatment
  • Slide 10
  • ANOVA Formula Considerations: MS between = s 2 between = SS between / df between MS within = s 2 within = SS within / df within F = MS between / MS within
  • Slide 11
  • Repeated Measures ANOVA New Formula Considerations: SS between SS between treatments = T 2 /n G 2 /N SS between subjects = P 2 /k G 2 /N SS within SS within treatments = SS inside each treatment SS error = SS within treatments SS between subjects
  • Slide 12
  • Repeated Measures ANOVA New Formula Considerations: df between df between treatments = k 1 df within df within treatments = N k df between subjects = n 1 df error = (N k) (n 1)
  • Slide 13
  • Repeated Measures ANOVA MS between treatments = SS between treatments / df between treatments MS error = SS error / df error F = MS between treatments / MS error
  • Slide 14
  • Repeated Samples Designs One-group pretest posttest (repeated measures) design: Perform pretest on all subjects Administer treatments followed by posttests Compare pretest to posttest scores and posttest to posttest scores OXO XO
  • Slide 15
  • Agenda Introduction The Repeated Measures ANOVA Hypothesis Tests with Repeated Measures ANOVA Post Hoc Analysis Instat Assumptions
  • Slide 16
  • Hypothesis Test: Repeated Measuers ANOVA Example 14.1 (p 457) Overview: Researchers are interested in a behavior modification technique on outbursts in unruly children Four students (n=4) are pretested on the # of outbursts during the course of one day Teachers begin using cost-response technique Students are posttested one week later, one month later and 6 months later
  • Slide 17
  • Hypothesis Test: ANOVA Questions: What is the experimental design? What is the independent variable/factor? How many levels are there? What is the dependent variable?
  • Slide 18
  • Step 1: State Hypotheses Non-Directional H 0 : pre = 1week = 1month = 6months H 1 : At least one mean is different than the others Step 2: Set Criteria Alpha ( ) = 0.05 Critical Value: Use F Distribution Table Appendix B.4 (p 693) Information Needed: df between treatments = k 1 = 4 1 = 3 df error = (N-k)-(n-1) = (16-4)-(4-1) = 9 Table B.4 (p 693) Critical value = 3.86
  • Slide 19
  • Step 3: Collect Data and Calculate Statistic Total Sum of Squares SS total = X 2 G 2 /N SS total = 222 44 2 /20 SS total = 222 - 121 SS total = 101 Sum of Squares Between each Treatment SS between treatment = T 2 /n G 2 /N SS between treatment = 26 2 /4+8 2 /4+6 2 /4+4 2 /4 44 2 /20 SS between treatment = (169+16+9+4) - 121 SS between treatment = 77 Sum of Squares Within each Treatment SS within = SS inside each treatment SS within = 11+2+9+2 SS within = 24 Sum of Squares Between each Subject SS between subjects = P 2 /k G 2 /N SS between subjects = (12 2 /4+6 2 /4+10 2 /4+16 2 /4) - 44 2 /16 SS between subjects = (36+9+25+64) 121 SS between subjects = 13 Sum of Squares Error SS error = SS within treatments SS between subjects SS error = 24 - 13 SS within = 11 Raw data can be found in Table14.3 (p 457)
  • Slide 20
  • Step 3: Collect Data and Calculate Statistic Mean Square Between each Treatment MS between treatment = SS between treatment / df between treatment MS between treatment = 77 / 3 MS between = 25.67 Mean Square Error MS errorn = SS error / df error MS error = 11 / 9 MS within = 1.22 F-Ratio F = MS between treatment / MS error F = 25.67 / 1.22 F = 21.04 Step 4: Make Decision
  • Slide 21
  • Agenda Introduction Repeated Measures ANOVA Hypothesis Tests with Repeated Measures ANOVA Post Hoc Analysis Instat Assumptions
  • Slide 22
  • Post Hoc Analysis What ANOVA tells us: Rejection of the H 0 tells you that there is a high PROBABILITY that AT LEAST ONE difference exists SOMEWHERE What ANOVA doesnt tell us: Where the differences lie Post hoc analysis is needed to determine which mean(s) is(are) different
  • Slide 23
  • Post Hoc Analysis Post Hoc Tests: Additional hypothesis tests performed after a significant ANOVA test to determine where the differences lie. Post hoc analysis IS NOT PERFORMED unless the initial ANOVA H 0 was rejected!
  • Slide 24
  • Post Hoc Analysis Type I Error Type I error: Rejection of a true H 0 Pairwise comparisons: Multiple post hoc tests comparing the means of all pairwise combinations Problem: Each post hoc hypothesis test has chance of type I error As multiple tests are performed, the chance of type I error accumulates Experimentwise alpha level: Overall probability of type I error that accumulates over a series of pairwise post hoc hypothesis tests How is this accumulation of type I error controlled?
  • Slide 25
  • Two Methods Bonferonni or Dunns Method: Perform multiple t-tests of desired comparisons or contrasts Make decision relative to / # of tests This reduction of alpha will control for the inflation of type I error Specific post hoc tests: Note: There are many different post hoc tests that can be used Our book only covers two (Tukey and Scheffe)
  • Slide 26
  • Repeated Measures ANOVA Bonferronni/Dunns method is appropriate with following consideration: Use related-samples T-tests Tukeys and Scheffe is appropriate with following considerations: Replace MS within with MS error in all formulas Replace df within with df error in all formulas Note: Statisticians are not in agreement with post hoc analysis for Repeated Measures ANOVA
  • Slide 27
  • Agenda Introduction The Repeated Measures ANOVA Hypothesis Tests with Repeated Measures ANOVA Post Hoc Analysis Instat Assumptions
  • Slide 28
  • Instat Label three columns as follows: Block: This groups your data by each subject. Example: If you conducted a pretest and 2 posttests (3 total) on 5 subjects, the block column will look like: 1 1 1 2 2 2 3 3 3 4 4 4 5 5 5 Treatment: This tells you which treatment level/condition occurred for each data point. Example: If each subject (n=5) received three treatments, the treatment column will look like: 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 Response: The data for each subject and treatment condition
  • Slide 29
  • Instat Convert the Block and Treatment columns into factors: Choose Manage Choose Column Properties Choose Factor Select the appropriate column to be converted Indicate the number of levels in the factor Example: Block (5 levels, n = 5), Treatment (3 levels, k = 3) Click OK
  • Slide 30
  • Instat Choose Statistics Choose Analysis of Variance Choose General Response variable: Choose the Response variable Treatment factor: Choose the Treatment variable Blocking factor: Choose the Block variable Click OK. Interpret the p-value!!!
  • Slide 31
  • Instat Post hoc analysis: Perform multiple related samples t-Tests with the Bonferonni/Dunn correction method
  • Slide 32
  • Reporting ANOVA Results Information to include: Value of the F statistic Degrees of freedom: Between treatments: k 1 Error: (N k) (n 1) p-value Examples: A significant treatment effect was observed (F(3, 9) = 21.03, p = 0.002)
  • Slide 33
  • Reporting ANOVA Results An ANOVA summary table is often included SourceSSdfMS Between77325.67F = 21.03 Within Treatments2412 Between subjects133 Error1191.22 Total10115
  • Slide 34
  • Agenda Introduction The Analysis of Variance (ANOVA)