### Article

## Using information from a linear mixed effects regression model analysis in the planning of a clinical trial

### Search Medline for

### Authors

Published: | September 8, 2005 |
---|

### Outline

### Text

#### Introduction and problem

Age - related macular degeneration (AMD) is the most common cause for the loss of central vision associated with aging. It is a complex disorder caused by many factors. Beside choroidal neovascularization and pigment epithelial detachment, geographic atrophy (GA) is a frequent cause of severe destruction of sharp, central vision in patients with AMD.

At present, there are no treatment options available for patients with GA. Beside surgical interventions, prophylactic measures with antioxidants as well as zinc and statin [Ref. 1], [Ref. 2] are investigated. Especially, studies on the prophylactic potential of drugs will be of interest in public health because of an effective application of such drugs in large sections of the population. To design such studies, it is helpful to possess a detailed knowledge on the natural course and its variability.

We propose a linear mixed effects regression model to describe main aspects of the natural course of GA. The parameter estimates from the model fit are used to plan the sample size of a randomized controlled clinical trial to prove treatment options for an effective reduction of GA growth.

#### Material and methods

We use data on the natural course of GA from 114 patients (178 eyes) recruited in the multi-center, longitudinal, observational FAM-Study (*
F
*undus

*utofluorescence in A*

__A__*D) for an investigation of the GA growth process. The size of GA is measured in fundus autofluorescence images generated by scanning laser ophthalmoscopy. A formal description of the GA growth process is made by using the linear mixed effects regression model LMEM [Ref. 3], which reflects main aspects of the natural course, and is methodologically simple.*

__M__Mixed effects regression models are useful for the analysis of longitudinal data [Ref. 4], [Ref. 5] . Beside estimators of fixed effects (i.e. influence of the whole study population), estimators of variance components (i.e. random effects and residual variance) are the most important results of fitting a mixed effects regression model. The estimators of model parameters from the LMEM model are used to plan the sample size for two different scenarios of a controlled interventional trial.

All considerations are based on the comparison of two parallel groups with the primary endpoint „size of GA“. This endpoint is assumed as normally distributed with mean μ_{1} and standard deviation σ_{1} in the intervention group, and with mean μ_{2} and standard deviation σ_{2} in the control group. Assuming equal sample size for both groups, the following formula can be used to determine the group size [Ref. 6], which is needed to detect a mean difference between yearly growth in the intervention and in the control group of Δ= μ_{1} - μ_{2} on a significance level of a with a power of 1-β:

n_{group} = [z_{1-α/2} + z_{1-β}]^{2 }× [σ²_{1} + σ²_{2}] / Δ^{2} .

The numbers z_{1-α/2} resp. z_{1-β} are the (1-α/2)-quantile resp. (1-β)-quantile of N(0; 1). Two major points in practice consist in making assumptions about the variances (σ²_{1} and σ²_{2}) and the clinically relevant difference Δ. The latter has large influence on the group size. An estimation of Δ is derived from the model estimator of the populations average slope in the mixed effects regression model. Planning data for the control group give the natural history of the patients in the FAM study: The estimation of σ²_{2} is derived from the estimators of model parameters. Getting a similar estimation for the variance in the intervention group (σ²_{1}) is not so straightforward because such information does not exist. To solve this problem we make assumptions, which are plausible on a heuristic ground.

All scenarios we discuss are based on estimations, which are derived from a regression analysis. An aspect consists here in an adequate consideration of the estimator’s precision of the model parameter. To address this problem both estimators and upper limits of the confidence intervals are used for sample size calculation.

#### Results and discussion

In the first scenario we discuss the situation that only one eye per patient is under study. This would be the case if only patients with unilateral GA are under study or an *eye-specific treatment* (e.g. photodynamic therapy) is investigated. A sample size between 410 and 690 patients seems to be realistic for this situation.

In the second scenario, a certain percentage of patients will contribute two eyes to the study. This situation occurs if a *patient-specific* *treatment* is investigated, e.g. prophylactic measures with antioxidant drugs. In this situation, the total sample size is between 294 and 592 patients.

Such high numbers of patients leads to the inevitable conclusion to carry out the study as a multi-center project.

#### Acknowledgement

Our project is supported by DFG grants: SPP AMD 1088, MA 1723/1-1.

### References

- 1.
- Age-Related Eye Disease Study Research Group. A randomized, placebo-controlled, clinical trial of high-dose supplementation with vitamins C and E, beta carotene, and zinc for age-related macular degeneration and vision loss: AREDS report no. 8. Arch Ophthalmol 2001;119: 1417-1436
- 2.
- G McGwin Jr, Owsley C, Curcio CA, Crain RJ. The association between statin use and age related maculopathy. Br J Ophthalmol 2003;87: 1121-1125.
- 3.
- Dreyhaupt J, Mansmann U, Pritsch M, Dolar-Szczasny J, Bindewald A, Holz FG. Modelling the natural history of geographic atrophy in patients with age-related macular degeneration. Ophthalmic epidemiology. (submitted)
- 4.
- Pinheiro JC, Bates DM. Mixed-effects models in S and S-plus. Berlin, Heidelberg, New York: Springer; 2000
- 5.
- Fitzmaurice GM, Laird NM, Ware JH. Applied Longitudinal Analysis. New Jersey: John Wiley & Sons; 2004
- 6.
- Bock J. Bestimmung des Stichprobenumfangs für biologische Experimente und kontrollierte klinische Studien. München: Oldenbourg; 1998