Designing longitudinal studies in epidemiology

1. Designing longitudinal studiesin epidemiology Donna Spiegelman Professor of Epidemiologic Methods Departments of Epidemiology and Biostatistics stdls@channing.harvard.edu Xavier Basagana Doctoral StudentDepartment 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. 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. 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:

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: 712-720). 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 • points • 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 proc mixed; class id; model tics=grad age gradage/s ddfm=bw; Random intercept age/type=un subject=id;

22. Program optitxs.r makes it all possible

28. http://www.hsph.harvard.edu/faculty/spiegelman/software.html

29. http://www.hsph.harvard.edu/faculty/spiegelman/optitxs.html

35. Illustration of use of softwareoptitxs.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): -0.006 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 = 0.4206059

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?

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. 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.