### Introduction

### Methods

_{ij}= X

_{i}+ β

_{1}(t

_{ij}) + β

_{2}u

_{ij}+ β

_{3}× (t

_{ij}− u

_{ij}) I(u

_{ij}> 0) + b

_{1i}+ b

_{2i}t

_{ij}+ e

_{ij}

_{ij}is the observed eGFR of subject i at time t

_{ij}; X

_{i}is the ideal unobserved GFR for subject i at baseline; u

_{ij}is the amount of time up to time t

_{ij}that the subject was on treatment; I(u

_{ij}> 0) is 1 if u

_{ij}> 0, and 0 otherwise. β

_{1}, β

_{2}, and β

_{3}are fixed effects terms. β

_{1}(•) is a function that describes the trajectory in the placebo group. When there is a constant rate of change, this term can be replaced by β

_{1}× t

_{ij}. β

_{2}is the effect of the treatment on the chronic slope and β

_{3}is the carryover effect; b

_{1i}and b

_{2i}are random effects assumed to be normally distributed with a mean of zero. In order to illustrate the difference between t

_{ij}and u

_{ij}, assume patient i is assigned to the treatment in period 1 and that the duration of treatment is T. Then, u

_{ij}= T whenever t

_{ij}> T. The residual error terms in the model, e

_{ij}, are assumed to be mutually independent and normally distributed as N(0,σ

^{2}); they are also assumed to be independent of the random effects. No acute effect is used in the model, because it is assumed that all of the measurements were made while the patient was off of treatment.

^{2}annually, while that during experimental treatment was −3, for a chronic treatment effect of 1. The exception was in the case of a carryover effect in the crossover design (explained below).

### Analysis

### Asymptotic relative efficiency

### Type 1 error rate

_{2}= β

_{3}= 0. In the first scenario, a constant rate of change over time was assumed (decline of 4); that is β

_{1}(t

_{ij}) = –4 t

_{ij}. In the second scenario, the natural history of the rate of change was assumed to decline slightly over time. This rate of change was defined by a decline of 4.0 annually in period 1 and 3.5 annually in period 2, as follows: β

_{1}(t

_{ij}) = –4 t

_{ij}when t

_{ij}≤ 2 and β

_{1}(t

_{ij}) = –8 – 3.5 (t

_{ij}– 2) when t

_{ij}> 2. The third scenario assumed an increasing rate of change over time. This rate of change was defined by a decline of 4.0 annually in period 1 and 4.5 annually in period 2, as follows: β

_{1}(t

_{ij}) = –4 t

_{ij}when t

_{ij}≤ 2 and β

_{1}(t

_{ij}) = –8 – 4.5 (t

_{ij}– 2) when t

_{ij}> 2. The targeted type 1 error rate used for the hypothesis tests was the conventional one sided 0.025.