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