1. Modeling Continuous Longitudinal Data

2. Introduction to continuous longitudinal data: Examples

3. Homeopathy vs. placebo in treating pain after surgery
Day of surgery
Mean pain assessments by visual analogue scales (VAS)
Days 1-7 after surgery
(morning and evening)
Copyright ©1995 BMJ Publishing Group Ltd.
Lokken, P. et al. BMJ 1995;310:

4. Divalproex vs. placebo for treating bipolar depression
Davis et al. “Divalproex in the treatment of bipolar depression: A placebo controlled study.” J Affective Disorders 85 (2005)

5. Randomized trial of in-field treatments of acute mountain sickness
Mean (SD) score of acute mountain sickness in subjects treated with simulated descent (One hour of treatment in the hyperbaric chamber) or dexamethasone.
Keller, H.-R. et al. BMJ 1995;310:
Copyright ©1995 BMJ Publishing Group Ltd.

6. Pint of milk vs. control on bone acquisition in adolescent females
Mean (SE) percentage increases in total body bone mineral and bone density over 18 months.
P values are for the differences between groups by repeated measures analysis of variance
Cadogan, J. et al. BMJ 1997;315:
Copyright ©1997 BMJ Publishing Group Ltd.

7. Counseling vs. control on smoking in pregnancy
Copyright ©2000 BMJ Publishing Group Ltd.
Hovell, M. F et al. BMJ 2000;321:

8. Longitudinal data: broad form
id time1 time2 time3 time4
Hypothetical data from Twisk, chapter 3, page 26, table 3.4
Jos W. R. Twisk. Applied Longitudinal Data Analysis for Epidemiology: A Practical Guide. Cambridge University Press, 2003.

9. Longitudinal data: Long form
id time score
id time score
Hypothetical data from Twisk, chapter 3, page 26, table 3.4

10. Converting data from broad to long in SAS…
data long;
set broad;
time=1; score=time1; output;
time=2; score=time2; output;
time=3; score=time3; output;
time=4; score=time4; output;
run;

11. Profile plots (use long form)
The plot tells a lot!

12. Mean response plot

13. Superimposed…

14. smoothed

15. smoothed

16. Superimposed…

17. Two groups (e.g., treatment placebo)
id group time1 time2 time3 time4 A A A B B B
Hypothetical data from Twisk, chapter 3, page 40, table 3.7

18. Profile plots by group B A

19. Mean plots by group B A

20. Possible questions…
Overall, are there significant differences between time points?
From plots: looks like some differences (time3 and 4 look different)
Overall, are there significant changes from baseline?
From plots: at time3 or time4 maybe
Do the two groups differ at any time points?
From plots: certainly at baseline; some difference everywhere
Do the two groups differ in their responses over time?**
From plots: their response profile looks similar over time, though A and B are closer by the end.

21. Statistical analysis strategies
Strategy 1: ANCOVA on the final measurement, adjusting for baseline differences (end-point analysis)
Strategy 2: repeated-measures ANOVA “Univariate” approach
Strategy 3: “Multivariate” ANOVA approach
Strategy 4: GEE
Strategy 5: Mixed Models
Strategy 6: Modeling change
Traditional approaches: this week
Newer approaches: next week
In two/three weeks

22. Comparison of traditional and new methods
FROM:
Ralitza Gueorguieva, PhD; John H. Krystal, MD Move Over ANOVA : Progress in Analyzing Repeated-Measures Data and Its Reflection in Papers Published in the Archives of General Psychiatry. Arch Gen Psychiatry. 2004;61:

23. Things to consider: 1. Spacing of time intervals 3. Missing Data
Repeated-measures ANOVA and MANOVA require that all subjects measured at same time intervals—our plots above assumed this too!
MANOVA weights all time intervals evenly (as if evenly spaced)
2. Assumptions of the model
ALL strategies assume normally distributed outcome and homogeneity of variances
But all strategies are robust against this assumption, especially if data set is >30
**Univariate repeated-measures ANOVA assumes sphericity, or compound symmetry
3. Missing Data
All traditional analyses require imputation of missing data
(also need to know: does the SAS PROC require long or broad form of data?)

24. Compound symmetry Compound symmetry requires :
The variances of the outcome variable must be the same at each time point
(b) The correlation between repeated measurements are equal, regardless of the time interval between measurements.

25. (a) Variances at each time points (visually)
Does variance look equal across time points??
--Looks like most variability at time1 and least at time4…

26. (a) Variances at each time points (numerically)
id time1 time2 time3 time4
Variance:

27. (b) Correlation (covariance) across time points
time time time time4
time
time
time
time
Certainly do NOT have equal correlations!
Time1 and time2 are highly correlated, but time1 and time3 are inversely correlated!

28. Compound symmetry would look like…
time time time time4
time
time
time
time

29. Missing Data
Very important to fill in missing data! Otherwise, you have to throw out the whole observation.
With missing data, changes in the mean over time may just reflect drop-out pattern; you cannot compare time point 1 with 50 people to time point 2 with 35 people!
We will implement classic “last observation carried forward” strategy for simplicity
Other more complicated imputation strategies may be more appropriate

30. LOCF Subject HRSD 1 HRSD 2 HRSD 3 HRSD 4 Subject 1 20 13 Subject 2 2119
Subject 3 18 10 6
Subject 4 30 25 23

31. Last Observation Carried Forward
LOCF
Last Observation Carried Forward
Subject
HRSD 1
HRSD 2
HRSD 3
HRSD 4
Subject 1 20 13
Subject 2 21 19
Subject 3 18 10 6
Subject 4 30 25 23

32. Strategy 1: End-point analysis
Removes repeated measures problem by considering only a single time point (the final one).
Ignores intermediate data completely
Asks whether or not the two group means differ at the final time point, adjusting for differences at baseline (using ANCOVA).
proc glm data=broad;
class group;
model time4 = time1 group;
run;
Comparing groups at every follow-up time point in this way would hugely increase your type I error.

33. Strategy 1: End-point analysis
Sum of
Source DF Squares Mean Square F Value Pr > F
Model
Error
Corrected Total
R-Square Coeff Var Root MSE time4 Mean
Source DF Type I SS Mean Square F Value Pr > F
time
group
group time4 LSMEAN Pr > |t| A B

34. Strategy 1: End-point analysis
Sum of
Source DF Squares Mean Square F Value Pr > F
Model
Error
Corrected Total
R-Square Coeff Var Root MSE time4 Mean
Source DF Type I SS Mean Square F Value Pr > F
time
group
Least-squares means of the two groups at time4, adjusted for baseline differences (not significantly different)
group time4 LSMEAN Pr > |t| A B

35. From end-point analysis…
Overall, are there significant differences between time points?
Can’t say
Overall, are there significant changes from baseline?
Do the two groups differ at any time points?
They don’t differ at time4
Do the two groups differ in their responses over time?

36. Strategy 2: univariate repeated measures ANOVA (rANOVA)
Just good-old regular ANOVA, but accounting for between subject differences

37. BUT first… Naive analysis
Run ANOVA on long form of data, ignoring correlations within subjects (also ignoring group for now):
proc anova data=long;
class time;
model score= time ;
run;
Compares means from each time point as if they were independent samples. (analogous to using a two-sample t-test when a paired t-test is appropriate). Results in loss of power!

38. One-way ANOVA (naïve) id time1 time2 time3 time4 MEAN 1 31 29 15 26
MEAN:
Within time
Between times

39. One-way ANOVA results Twisk: Output 3.3 The ANOVA Procedure
Dependent Variable: score
Sum of
Source DF Squares Mean Square F Value Pr > F
Model
Error
Corrected Total
Source DF Anova SS Mean Square F Value Pr > F
time
Twisk: Output 3.3

40. Univariate repeated-measures ANOVA
Explain away some error variability by accounting for differences between subjects:
-SSE was
-This will be reduced by variability between subjects
proc glm data=broad;
model time1-time4=;
repeated time;
run; quit;

41. rANOVA Between subjects id time1 time2 time3 time4 MEAN
MEAN:

42. rANOVA results Between time variability Unexplained variability
Idea of G-G and H-F corrections, analogous to pooled vs. unpooled variance ttest: if we have to estimate more things because variances/covariances aren’t equal, then we lose some degrees of freedom and p-value increases.
rANOVA results
Repeated measures p-value = .0752
After G-G correction for non-sphericity=.1311
(H-F correction gives .1114)
The GLM Procedure
Repeated Measures Analysis of Variance
Univariate Tests of Hypotheses for Within Subject Effects
Adj Pr > F
Source DF Type III SS Mean Square F Value Pr > F G - G H - F
time
Error(time)
Greenhouse-Geisser Epsilon
Huynh-Feldt Epsilon
Between time variability
Unexplained variability
These epsilons should be 1.0 if sphericity holds. Sphericity assumption appears violated.

43. With two groups: Naive analysis
Run ANOVA on long form of data, ignoring correlations within subjects:
proc anova data=long;
class time;
model score= time group group*time;
run;
As if there are 8 independent samples: 2 groups at each time point.

44. Two-way ANOVA (naïve) grp time1 time2 time3 time4 MEAN A 31 29 15 26B B B
MEAN:
Within time
Overall mean=27
Between groups
Within time
Recall: SST= ; group by time= =26.79

45. Results: Naïve analysis
The ANOVA Procedure
Dependent Variable: score
Sum of
Source DF Squares Mean Square F Value Pr > F
Model
Error
Corrected Total
Source DF Anova SS Mean Square F Value Pr > F
time
group
time*group

46. Univariate repeated-measures ANOVA
Reduce error variability by between subject differences:
-SSE was
-This will be reduced by variability between subjects
proc glm data=broad;
class group;
model time1-time4= group;
repeated time;
run; quit;

47. rANOVA grp time1 time2 time3 time4 MEAN A 31 29 15 26 25.25B B B
MEAN:
Between subjects in each group
Overall mean=27
Between subjects in each group

48. rANOVA results (two groups)
Usually of less interest!
The GLM Procedure
Repeated Measures Analysis of Variance
Tests of Hypotheses for Between Subjects Effects
Source DF Type III SS Mean Square F Value Pr > F
group
Error
What we care about!
The GLM Procedure
Repeated Measures Analysis of Variance
Univariate Tests of Hypotheses for Within Subject Effects
Adj Pr > F
Source DF Type III SS Mean Square F Value Pr > F G - G H - F
time
time*group
Error(time)
Greenhouse-Geisser Epsilon
Huynh-Feldt Epsilon
No apparent difference in responses over time between the groups.

49. From rANOVA analysis…
Overall, are there significant differences between time points?
No, Time not statistically significant (p=.1743, G-G)
Overall, are there significant changes from baseline?
No, Time not statistically significant
Do the two groups differ at any time points?
No, Group not statistically significant (p=.1408)
Do the two groups differ in their responses over time?**
No, not even close; Group*Time (p-value>.60)

50. Strategy 3: rMANOVA Multivariate: More than one dependent variable
Multivariate Approach to repeated measures--Treats response variable as a multivariate response vector.
Not just for repeated measures, but appropriate for other situations with multiple dependent variables.

51. Analogous to paired t-test
Recall: paired t-test:
Paired t-test compares the difference values between two time points to their standard error.
MANOVA is just a paired t-test where the outcome variable is a vector of difference rather than a single difference:
Where T is the number of time points:
Called: Hotelling's Trace

52. T-1 differences id group diff1 diff2 diff3 1 A -2 -14 11 2 A 4 -8 12B B B
Note: weights all differences equally, so hard to interpret if time intervals are unevenly spaced.
Note: assumes differences follow a multivariate normal distribution + multivariate homogeneity of variances assumption

53. On same output as rANOVA
proc glm data=broad;
model time1-time4=;
repeated time;
run; quit;
Null hypothesis: diff1=0, diff2=0, diff3=0

54. Results (time only)
MANOVA Test Criteria and Exact F Statistics for the Hypothesis of no time Effect
H = Type III SSCP Matrix for time
E = Error SSCP Matrix
S=1 M= N=0.5
Statistic Value F Value Num DF Den DF Pr > F
Wilks' Lambda
Pillai's Trace
Hotelling-Lawley Trace
Roy's Greatest Root
4 separate F-statistics (slightly different versions of MANOVA statistic)
all give the same answer: change over time is not significant
compare to rANOVA results: G-G time p-value=.13
Use Wilks’ Lambda in general.
Use Pillai’s Trace for small sample sizes (when assumptions of model are violated)

55. On same output as rANOVA
proc glm data=broad;
class group;
model time1-time4= group;
repeated time;
run; quit;

56. Results (two groups) No differences between times.
The GLM Procedure Repeated Measures Analysis of Variance
MANOVA Test Criteria and Exact F Statistics for the Hypothesis of no time Effect
Statistic Value F Value Num DF Den DF Pr > F
Wilks' Lambda
Pillai's Trace
Hotelling-Lawley Trace
Roy's Greatest Root
MANOVA Test Criteria and Exact F Statistics for the Hypothesis of no time*group Effect
Wilks' Lambda
Pillai's Trace
Hotelling-Lawley Trace
Roy's Greatest Root
No differences between times.
No differences in change over time between the groups (compare to G-G time*group p-value=.6954)

57. From rMANOVA analysis…
Overall, are there significant differences between time points?
No, Time not statistically significant (p=.3287)
Overall, are there significant changes from baseline?
No, Time not statistically significant
Do the two groups differ at any time points?
Can’t say (never looked at raw scores, only difference values)
Do the two groups differ in their responses over time?**
No, not even close; Group*Time (p-value=.89)

58. Can also test for the shape of the response profile…
proc glm data=broad;
class group;
model time1-time4= group;
repeated time 3 polynomial /summary ;
run; quit;

59. linear quadratic cubic The GLM Procedure
Repeated Measures Analysis of Variance
Analysis of Variance of Contrast Variables
time_N represents the nth degree polynomial contrast for time
Contrast Variable: time_1
Source DF Type III SS Mean Square F Value Pr > F
Mean
group
Error
Contrast Variable: time_2
Mean
group
Error
Contrast Variable: time_3
Mean
group
Error
linear
quadratic
cubic

60. Can also get successive paired t-tests
proc glm data=broad;
class group;
model time1-time4= group;
repeated time profile /summary ;
run; quit;
**Not adjusted for multiple comparisons!

61. Time1 vs. time2 Time2 vs. time3 Time3 vs. time4
Repeated Measures Analysis of Variance
Analysis of Variance of Contrast Variables
time_N represents the nth successive difference in time
Contrast Variable: time_1
Source DF Type III SS Mean Square F Value Pr > F
Mean
group
Error
Contrast Variable: time_2
Mean
group
Error
Contrast Variable: time_3
Mean
group
Error
Time1 vs. time2
Time2 vs. time3
Time3 vs. time4

62. Univariate vs. multivariate
If compound symmetry assumption is met, univariate approach has more power (more degrees of freedom).
But, if compound symmetry is not met, then type I error is increased

63. Summary: rANOVA and rMANOVA
Require imputation of missing data
rANOVA requires compound symmetry (though there are corrections for this)
Require subjects measured at same time points
But, easy to implement and interpret

64. Practice: rANOVA and rMANOVA
What effects do you expect to be statistically significant?
Time?
Group?
Time*group?
Within-subjects effects, but no between-subjects effects.
Time is significant.
Group*time is significant.
Group is not significant.

65. Practice: rANOVA and rMANOVA
Between group effects; no within subject effects:
Time is not significant.
Group*time is not significant.
Group IS significant.

66. Practice: rANOVA and rMANOVA
Some within-group effects, no between-group effect.
Time is significant.
Group is not significant.
Time*group is not significant.

67. References
Jos W. R. Twisk. Applied Longitudinal Data Analysis for Epidemiology: A Practical Guide. Cambridge University Press, 2003.