Repeated Measures/Longitudinal Analysis -Bob Feehan 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. 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/ 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) 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. 2. 3. 4. College Students Anxiety (Null Week1 = Week2 = Week3) High School Students Anxiety (Null Week1 = Week2 = Week3) High School vs College Overall (Null High School = College Overall) High School vs College Trend (Null No Interaction or “Parallel Lines”) 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) Minitab Tricks – “Stacked” Data 7 8 9 10 11 12 Minitab Tricks - “Stacked” Data Minitab Tricks - “Stacked” Data Minitab Tricks – “Subset” Data Minitab Tricks – “Subset” Data Data ANOVA ANOVA General Linear Mode: • Responses: Response • Model: Week Subject • Random Factors: 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! College Student Results Results for: College Students Main Effects Plot for Response Fitted Means General Linear Model: Response versus Week, Subject 9 Factor Type Levels Values Week fixed 3 Week 1, Week 2, Week 3 Subject random 6 1, 2, 3, 4, 5, 6 8 7 Analysis of Variance for Response, using Adjusted SS for Tests Source Week Subject Week*Subject Error Total DF 2 5 10 0 17 Seq SS 148.0000 1.3333 6.6667 * 156.0000 Adj SS 148.0000 1.3333 6.6667 * Adj MS F P 74.0000 111.00 0.000 0.2667 0.40 0.838 0.6667 ** * ** Denominator of F-test is zero or undefined. 1. 2. 3. 4. Mean 6 5 4 3 2 1 Week 1 Week 2 Week College Students Anxiety (Null Week1 = Week2 = Week3) High School Students Anxiety (Null Week1 = Week2 = Week3) High School vs College Overall (Null High School = College Overall) High School vs College Trend (Null No Interaction or “Parallel Lines”) Week 3 High School Student Results Results for: High School Main Effects Plot for Response Fitted Means General Linear Model: Response versus Week, Subject 7.5 Factor Type Levels Values Week fixed 3 Week 1, Week 2, Week 3 Subject random 6 1, 2, 3, 4, 5, 6 7.0 6.5 Analysis of Variance for Response, using Adjusted SS for Tests Source Week Subject Week*Subject Error Total DF Seq SS 2 44.3333 5 0.5000 10 7.6667 0 * 17 52.5000 Adj SS 44.3333 0.5000 7.6667 * Adj MS F P 22.1667 28.91 0.000 0.1000 0.13 0.982 0.7667 ** * ** Denominator of F-test is zero or undefined. 1. 2. 3. 4. Mean 6.0 5.5 5.0 4.5 4.0 3.5 Week 1 Week 2 Week College Students Anxiety (Null Week1 = Week2 = Week3) High School Students Anxiety (Null Week1 = Week2 = Week3) High School vs College Overall (Null High School = College Overall) High School vs College Trend (Null No Interaction or “Parallel Lines”) Week 3 Combined Analysis 1. 2. 3. 4. College Students Anxiety (Null Week1 = Week2 = Week3) High School Students Anxiety (Null Week1 = Week2 = Week3) High School vs College Overall (Null High School = College Overall) High School vs College Trend (Null No Interaction or “Parallel Lines”) High School / College Comparisons -Problems? 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!) 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: Response • Model: Week Year Subject Week*Year Week*Subject Year*Subject • Random Factors: Subject Combined Analysis Main Effects Plot for Response Fitted Means Week 8 7 General Linear Model: Response versus Week, Year, Subject Type fixed fixed random Levels 3 2 6 6 Values Week 1, Week 2, Week 3 Freshman, Senior 1, 2, 3, 4, 5, 6 Mean Factor Week Year Subject 3 Adj SS 175.1667 2.2500 1.2500 17.1667 8.8333 0.5833 5.5000 Adj MS 87.5833 2.2500 0.2500 8.5833 0.8833 0.1167 0.5500 F 99.15 19.29 0.56 15.61 1.61 0.21 2 P 0.000 0.007 0.744 x 0.001 0.234 0.950 Week 1 Week 2 Week 3 Freshman Senior Interaction Plot for Response Fitted Means 9 Year Freshman Senior 8 7 6 Mean Seq SS 175.1667 2.2500 1.2500 17.1667 8.8333 0.5833 5.5000 210.7500 5 4 Analysis of Variance for Response, using Adjusted SS for Tests Source DF Week 2 Year 1 Subject 5 Week*Year 2 Week*Subject10 Year*Subject 5 Error 10 Total 35 Year 5 4 3 2 1 Week 1 Week 2 Week Week 3 Combined Analysis Setting up the ANOVA GLM with Nested Factors: (Reminder – Subjects are nested within School) ANOVA General Linear Mode: • Responses: Response • Model: School Subject(School) Week School*Week Note: No Subject*Week interactions as • Random Factors: Subject School*Week included Subject*Week Combined Analysis General Linear Model: Response versus School, Week, Subject Factor School Subject(School) Week Type fixed random fixed Levels 2 12 3 Main Effects Plot for Response Fitted Means Values College, High School 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 Week 1, Week 2, Week 3 Week 8 School 7 Source School Subject(School) Week School*Week Error Total DF 1 10 2 2 20 35 Seq SS 2.250 1.833 175.167 17.167 14.333 210.750 Adj SS Adj MS 2.250 2.250 1.833 0.183 175.167 87.583 17.167 8.583 14.333 0.717 F 12.27 0.26 122.21 11.98 Mean 6 Analysis of Variance for Response, using Adjusted SS for Tests P 0.006 0.984 0.000 0.000 5 4 3 2 Week 1 Week 2 Week 3 College High School Interaction Plot for Response College Students Anxiety (Null Week1 = Week2 = Week3) High School Students Anxiety (Null Week1 = Week2 = Week3) High School vs College Overall (Null High School = College Overall) High School vs College Trend (Null No Interaction or “Parallel Lines”) 1. 2. 3. 4. College Students Anxiety (Iffy) High School Students Anxiety (Iffy) High School vs College Overall (P <0.001, Means differ - College less) 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) Fitted Means 9 School College High School 8 7 6 Mean 1. 2. 3. 4. 5 4 3 2 1 Week 1 Week 2 Week Week 3 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 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. 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