1. Repeated Measures/Longitudinal Analysis Bob Feehan -Bob Feehan

2. What are you talking about? Repeated Measures – Measurements that are taken at two or more points in time on the same set of experimental units. (i.e. subjects) Longitudinal Data – Longitudinal data are a common form of repeated measures in which measurements are recorded on individual subjects over a period of time.

3. Example Researchers want to see if high school students and college students have different levels of anxiety as they progress through the semester. They measure the anxiety of 12 participants three times: Week 1, Week 2, and Week 3. Participants are either high school students, or college students. Anxiety is rated on a scale of 1-10, with 10 being “high anxiety” and 1 being “low anxiety”. Repeated Measurement? Anxiety Longitudinal Measurement? 3 Weeks Any other comparisons? High School vs College Overall Rate (interaction) -http://statisticslectures.com/topics/factorialtwomixed/http://statisticslectures.com/topics/factorialtwomixed/

4. Clarifications Repeated Measures/Longitudinal Design: – Need at least one Factor with two Levels. – The Levels have to be dependent upon the Factor Example Continued … – Factor: Subjects (12) – Levels: 3 (measured each week from SAME person)

5. Example Researchers want to see if high school students and college students have different levels of anxiety as they progress through the semester. They measure the anxiety of 12 participants three times: Week 1, Week 2, and Week 3. Participants are either high school students, or college students. Anxiety is rated on a scale of 1-10, with 10 being “high anxiety” and 1 being “low anxiety”. Before we even take any Data: What is our Hypothesis going in? (CRITICAL!!!) 1.College Students Anxiety (Null  Week1 = Week2 = Week3) 2.High School Students Anxiety (Null  Week1 = Week2 = Week3) 3.High School vs College Overall (Null  High School = College Overall) 4.High School vs College Trend (Null  No Interaction or “Parallel Lines”)

6. Data Why not just do multiple Paired/Independent T – Tests? Takes Time (Time is precious) Only can look at one Factor at a time. (ie Week) Factor can only be two levels (ie no repeated measures > 2) Cannot look at over-all interactions Why use ANOVA? Saves time Can look at multiple Factors Factors can have multiple levels Can look at differences between separate groups (ie College/High school)

7. Minitab Tricks – “Stacked” Data 7 8 9 10 11 12

8. Minitab Tricks - “Stacked” Data

10. Minitab Tricks – “Subset” Data

12. Data ANOVA ANOVA General Linear Mode: Responses: Model: Random Factors: Response Week Subject Subject *Note: Without Subjects as Random our N of 6 would be N of 18. It would count each measurement of a subject as INDEPENDENT!

13. College Student Results Results for: College Students General Linear Model: Response versus Week, Subject Factor Type Levels Values Week fixed 3 Week 1, Week 2, Week 3 Subject random 6 1, 2, 3, 4, 5, 6 Analysis of Variance for Response, using Adjusted SS for Tests Source DF Seq SS Adj SS Adj MS F P Week 2 148.0000 148.0000 74.0000 111.00 0.000 Subject 5 1.3333 1.3333 0.2667 0.40 0.838 Week*Subject 10 6.6667 6.6667 0.6667 ** Error 0 * * * Total 17 156.0000 ** Denominator of F-test is zero or undefined. 1.College Students Anxiety (Null  Week1 = Week2 = Week3) 2.High School Students Anxiety (Null  Week1 = Week2 = Week3) 3.High School vs College Overall (Null  High School = College Overall) 4.High School vs College Trend (Null  No Interaction or “Parallel Lines”)

14. High School Student Results Results for: High School General Linear Model: Response versus Week, Subject Factor Type Levels Values Week fixed 3 Week 1, Week 2, Week 3 Subject random 6 1, 2, 3, 4, 5, 6 Analysis of Variance for Response, using Adjusted SS for Tests Source DF Seq SS Adj SS Adj MS F P Week 2 44.3333 44.3333 22.1667 28.91 0.000 Subject 5 0.5000 0.5000 0.1000 0.13 0.982 Week*Subject 10 7.6667 7.6667 0.7667 ** Error 0 * * * Total 17 52.5000 ** Denominator of F-test is zero or undefined. 1.College Students Anxiety (Null  Week1 = Week2 = Week3) 2.High School Students Anxiety (Null  Week1 = Week2 = Week3) 3.High School vs College Overall (Null  High School = College Overall) 4.High School vs College Trend (Null  No Interaction or “Parallel Lines”)

15. Combined Analysis 1.College Students Anxiety (Null  Week1 = Week2 = Week3) 2.High School Students Anxiety (Null  Week1 = Week2 = Week3) 3.High School vs College Overall (Null  High School = College Overall) 4.High School vs College Trend (Null  No Interaction or “Parallel Lines”) High School / College Comparisons -Problems?

16. Combined Analysis “Crossed” Factors vs “Nested” Factors for arbitrary Factors “A” & “B” Nested: Factor "A" is nested within another factor "B" if the levels or values of "A" are different for every level or value of "B". Crossed: Two factors A and B are crossed if every level of A occurs with every level of B. Our Factors: Subjects, School, & Week Crossed? School & Week Subjects & Week Nested? Subject is nested within School ie. Each subject has a measurement in High School or College not High school and College Therefore; any comparisons between them are independent (Not paired!)

17. Combined Analysis Setting up the ANOVA GLM with only Crossed Factors: (Pretend “Highschool” = Freshman year of College & “College” = Senior year) ANOVA General Linear Mode: Responses: Model: Random Factors: Response Week Year Subject Week*Year Week*Subject Year*Subject Subject

18. Combined Analysis General Linear Model: Response versus Week, Year, Subject Factor Type LevelsValues Week fixed 3 Week 1, Week 2, Week 3 Year fixed 2 Freshman, Senior Subject random 6 1, 2, 3, 4, 5, 6 Analysis of Variance for Response, using Adjusted SS for Tests Source DF Seq SS Adj SS Adj MS F P Week 2 175.1667 175.1667 87.5833 99.15 0.000 Year 1 2.2500 2.2500 2.2500 19.29 0.007 Subject 5 1.2500 1.2500 0.2500 0.56 0.744 x Week*Year 2 17.1667 17.1667 8.5833 15.61 0.001 Week*Subject10 8.8333 8.8333 0.8833 1.61 0.234 Year*Subject 5 0.5833 0.5833 0.1167 0.21 0.950 Error 10 5.5000 5.5000 0.5500 Total 35 210.7500

19. Combined Analysis Setting up the ANOVA GLM with Nested Factors: (Reminder – Subjects are nested within School) ANOVA General Linear Mode: Responses: Model: Random Factors: Response School Subject(School) Week School*Week Subject Note: No Subject*Week interactions as School*Week included Subject*Week

20. General Linear Model: Response versus School, Week, Subject Factor Type Levels Values School fixed 2 College, High School Subject(School) random12 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 Week fixed 3 Week 1, Week 2, Week 3 Analysis of Variance for Response, using Adjusted SS for Tests Source DF Seq SS Adj SS Adj MS F P School 1 2.250 2.250 2.250 12.27 0.006 Subject(School) 10 1.833 1.833 0.183 0.26 0.984 Week 2 175.167175.167 87.583 122.21 0.000 School*Week 2 17.167 17.167 8.583 11.98 0.000 Error 20 14.333 14.333 0.717 Total 35 210.750 Combined Analysis 1.College Students Anxiety (Null  Week1 = Week2 = Week3) 2.High School Students Anxiety (Null  Week1 = Week2 = Week3) 3.High School vs College Overall (Null  High School = College Overall) 4.High School vs College Trend (Null  No Interaction or “Parallel Lines”) 1.College Students Anxiety (Iffy) 2.High School Students Anxiety (Iffy) 3.High School vs College Overall (P <0.001, Means differ - College less) 4.High School vs College Trend (P <0.001, Rate at which Anxiety changes varies dependent on the week. High schoolers became less anxious as Time went on and college students more anxious)

21. My Data 20 Minute Body cooling procedure where measurements (etc. HR, BP, Skin Temperature) are taking at baseline and then ever 2 minutes during cooling all the way to 20 minutes. 11 Total measurements during the cooling procedure. Two Group (Younger and Older) Two Infusions (Saline and Vitamin C) on Both Older and Younger Two “Timepoints” (Pre Infusion and Post Infusion) on each injection day. 20 Subjects total (10 Older and 10 Younger) Summary: -Each subject comes for two visits. One visit is Saline, the other is Vitamin C injection -Each visit subjects puts on cold suit and is cooled twice. Once before the infusion and once after -Measurements are taking Before cooling (baseline) and then ever 2 minute increments up to 20 minutes. -Subjects are splits into two groups, Younger and Older

22. My Data Crossed Factors at total analysis: Infusion (Saline/Vit C), Timepoint (Pre/Post), Cooling (BL + every 2 minutes), & Subjects (1-20) Nested Factors at Total Analysis: Subjects and Group (Subjects are nested within Groups because Subjects have either a Young or Old attached to it, not both. Repeated Measures and Time: Each factor takes a repeated measure but the only longitudinal design in the 20 min cooling that has more then one non random level (it has 11). Subjects do not count as they are considered random.

23. SBP and HR Hypotheses Hypotheses on Systolic BP Change due to cooling while adding Vitamin C: 1.Young and Older Saline Days should NOT differ (accept Null hypothesis)* 2.Young and Older VitC days could Differ (reject Null Hypothesis)* 3.We can use Change in SBP as a standardization for different starting points 4.Older’s Change in SBP will be blunted compared to Younger’s* Hypotheses on HR changes due to cooling while adding Vitamin C: 1.Young and Older Saline Days should NOT differ (accept Null hypothesis)* 2.Young and Older VitC days should NOT differ (accept Null hypothesis)* 3.We can use Change in HR as a standardization for different starting points 4.Older’s Change is HR should not change from Younger’s* *Old published Data supports it