1. Designing longitudinal studies in epidemiology
Donna Spiegelman
Professor of Epidemiologic Methods
Departments of Epidemiology and Biostatistics
Xavier Basagana
Doctoral Student Department of Biostatistics,
Harvard School of Public Health

2. Background
We develop methods for the design of longitudinal studies for the most common scenarios in epidemiology
There already exist some formulas for power and sample size calculations in this context.
All prior work has been developed for clinical trials applications

3. Based on clinical trials:
Background
Based on clinical trials:
Some are based on test statistics that are not valid or less efficient in an observational context, where (e.g. ANCOVA).

4. Based on clinical trials:
Background
Based on clinical trials:
In clinical trials:
The time measure of interest is time from randomization  everyone starts at the same time. We consider situations where, for example, age is the time variable of interest, and subjects do not start at the same age.
Time-invariant exposures
Exposure (treatment) prevalence is 50% by design

5. Xavier Basagaña’s Thesis
Derive study design formulas based on tests that are valid and efficient for observational studies, for two reasonable alternative hypotheses.
Comprehensively assess the effect of all parameters on power and sample size.
Extend the formulas to a context where not all subjects enter the study at the same time.
Extend formulas to the case of time-varying covariates, and compare it to the time-invariant covariates case.

6. Xavier Basagaña’s Thesis
Derive the optimal combination of number of subjects (n) and number of repeated measures (r+1) when subject to a cost constraint.
Create a computer program to perform design computations. Intuitive parameterization and easy to use.

7. Notation and Preliminary Results

8. Constant Mean Difference (CMD).
We study two alternative hypotheses:
Constant Mean Difference (CMD).

9. Linearly Divergent Differences (LDD)

10. Intuitive parameterization of the alternative hypothesis
the mean response at baseline (or at the mean initial time) in the unexposed group, where
the percent difference between exposed and unexposed groups at baseline (or at the mean initial time), where

11. Intuitive parameterization of the alternative hypothesis (2)
: the percent change from baseline (or from the mean initial time) to end of follow-up (or to the mean final time) in the unexposed group, where
When is not fixed, is defined at time s instead of at time
: the percent difference between the change from baseline (or from the mean initial time) to end of follow-up (or mean final time) in the exposed group and the unexposed group, where
When , will be defined as the percent change from baseline (or from the mean initial time) to the end of follow-up (or to the mean final time) in the exposed group, i.e.

12. Notation & Preliminary Results
We consider studies where the interval between visits (s) is fixed but the duration of the study is free (e.g. participants may respond to questionnaires every two years)
Increasing r involves increasing the duration of the study
We also consider studies where the duration of the study, , is fixed, but the interval between visits is free (e.g. the study is 5 years long)
Increasing r involves increasing the frequency of the measurements, s
 = s r.

13. Notation & Preliminary Results
Model
The generalized least squares (GLS) estimator of B is
Power formula

14. Notation & Preliminary Results
Let lm be the (l,m)th element of -1
Assuming that the time distribution is independent of exposure group.
Then, under CMD
Under LDD

15. Correlation structures
We consider three common correlation structures:
Compound symmetry (CS).

16. Correlation structures
Damped Exponential (DEX)
 = 0: CS
 = 0.3: CS
 = 1: AR(1)

17. Correlation structures
Random intercepts and slopes (RS).
Reparameterizing:
is the reliability coefficient at baseline
is the slope reliability at the end of follow-up ( =0 is CS; =1 all variation in slopes is between subjects).
With this correlation structure, the variance of the response changes with time, i.e. this correlation structure gives a heteroscedastic model.

18. Example
Goal is to investigate the effect of indicators of socioeconomic status and post-menopausal hormone use on cognitive function (CMD) and cognitive decline (LDD)
“Pilot study” by Lee S, Kawachi I, Berkman LF, Grodstein F (“Education, other socioeconomic indicators, and cognitive function. Am J Epidemiol 2003; 157: ). Will denote as Grodstein.
Design questions include power of the published study to detect effects of specified magnitude, the number and timing of additional tests in order to obtain a study with the desired power to detect effects of specified magnitude, and the optimal number of participants and measurements needed in a de novo study of these issues

19. Example
At baseline and at one time subsequently, six cognitive tests were administered to 15,654 participants in the Nurses’ Health Study
Outcome: Telephone Interview for Cognitive Status (TICS)
00=32.7 (4);
Implies model
= 1 point/10 years of age

20. Example Exposure: Graduate school degree vs. not (GRAD)
Corr(GRAD, age)=-0.01
points
Exposure: Post-menopausal hormone use (CURRHORM)
Corr(CURRHORM, age)=-0.06
Time: age (years) is the best choice, not questionnaire cycle or calendar year of test
The mean age was 74 and V(t0)4.

21. Example
The estimated covariance parameters were
SAS code to fit the LDD model with CS covariance
proc mixed;
class id;
model tics=grad age gradage/s;
random id;
SAS code to fit the LDD model with RS covariance
model tics=grad age gradage/s ddfm=bw;
Random intercept age/type=un subject=id; CS RS
 or
0.27
0.26
0.04
-0.14

22. Program optitxs.r makes it all possible

35. Illustration of use of software optitxs.r
We’ll calculate the power of the Grodstein’s published study to detect the observed 70% difference in rates of decline between those with more than high school vs. others
Recall that 6.2% of NHS had more than high school; there was a –0.3% decline in cognitive function per year

36. > long.power()
Press <Esc> to quit
Constant mean difference (CMD) or Linearly divergent difference (LDD)? ldd
The alternative is LDD.
Enter the total sample size (N): 15000
Enter the number of post-baseline measures (r>0): 1
Enter the time between repeated measures (s): 2
Enter the exposure prevalence (pe) (0<=pe<=1): 0.062
Enter the variance of the time variable at baseline, V(t0)
(enter 0 if all participants begin at the same time): 4
Enter the correlation between the time variable at baseline and exposure, rho[e,t0]
(enter 0 if all participants begin at the same time): -0.01
Will you specify the alternative hypothesis on the absolute (beta coefficient) scale (1)
or the relative (percent) scale (2)? 2
The alternative hypothesis will be specified on the relative (percent) change scale.

37. Enter mean response at baseline among unexposed (mu00): 32.7
Enter the percent change from baseline to end of follow-up among unexposed (p2)
(e.g. enter 0.10 for a 10% change):
Enter the percent difference between the change from baseline to
end of follow-up in the exposed group and the unexposed group (p3) (e.g. enter 0.10 for a 10% difference): 0.7
Which covariance matrix are you assuming: compound symmetry (1),
damped exponential (2) or random slopes (3)? 2
You are assuming DEX covariance
Enter the residual variance of the response given the assumed model covariates (sigma2): 12
Enter the correlation between two measures of the same subject separated by one unit (rho): 0.3
Enter the damping coefficient (theta): 0.10
Power =

38. Power of current study
To detect the observed 70% difference in cognitive decline by GRAD
CS: 44%
RS: 35%
DEX : 42%
To detect a hypothesized ±10% difference in cognitive decline by current hormone use
CS & DEX: 7%
RS: 6%

39. How many additional measurements are needed when tests are administered every 2 years how many more years of follow-up are needed...
To detect the observed 70% difference in cognitive decline by GRAD with 90% power?
CS, DEX , RS: 3 post-baseline measurements =6
one more 5 year grant cycle
To detect a hypothesized ± 20% difference in cognitive decline by current hormone use with 90% power?
CS, DEX : 6 post-baseline measurements =12
More than two 5 year grant cycles
N=15,000 for these calculations

40. How many more measurements should be taken in four (1 NIH grant cycle) and eight years of follow-up (two NIH grant cycles)...
To detect the observed
70% difference in cognitive
decline by GRAD with 90%
power?
To detect a hypothesized
± 20% difference in cognitive
decline by current hormone
use with 90% power?
Duration of follow-up
4 years
8 years CS 8 1
DEX 10 RS
Duration of follow-up
4 years
8 years CS
>50 11
DEX 17 RS 13

41. Optimize (N,r) in a new study of cognitive decline
Assume
4 years of follow-up (1 NIH grant cycle);
cost of recruitment and baseline measurements are twice that of subsequent measurements
GRAD:
(N,r)=(26,795; 1) CS
=(26,930;1) DEX
=(28,945;1) RS
CURRHORM:
(N,r)=(97,662; 1) CS
=(98,155; 1) DEX
=(105,470;1) RS

42. Re: Constant Mean Difference (CMD)
Conclusions
Re: Constant Mean Difference (CMD)

43. Conclusions
CMD:
If all observations have the same cost, one would not take repeated measures.
If subsequent measures are cheaper, one would take no repeated measures or just a small number if the correlation between measures is large.
If deviations from CS exist, it is advisable to take more repeated measures.
Power increases as and as
Power increases as Var( ) goes to 0

44. Conclusions
LDD:
If the follow-up period is not fixed, choose the maximum length of follow-up possible (except when RS is assumed).
If the follow-up period fixed, one would take more than one repeated measure only when the subsequent measures are more than five times cheaper. When there are departures from CS, values of  around 10 or 20 are needed to justify taking 3 or 4 measures.
Power increases as , as , as slope reliability goes to 0, as Var( ) increases, and as the correlation between and exposure goes to 0

45. Conclusions
LDD:
The optimal (N,r) and the resulting power can strongly depend on the correlation structure. Combinations that are optimal for one correlation may be bad for another.
All these decisions are based on power considerations alone. There might be other reasons to take repeated measures.
Sensitivity analysis. Our program.

46. Future work Develop formulas for time-varying exposure.
Include dropout
For sample size calculations, simply inflate the sample size by a factor of 1/(1-f).
However, dropout can alter the relationship between N and r.