Dose Response Models for Multiple Endpoints: A Simulation Study

Farkad Ezzet (1), Eyas Raddad (2)

(1) Aycer Pharma Consulting, Chatham, NJ, USA, (2) Chorus, Eli Lilly and Company, Lilly Corporate Center, Indianapolis, USA

Objectives: There are potential advantages in modeling multiple endpoints simultaneously in a single model fit: 1) perform tests on shared parameters between endpoints and 2) ability to establish models when one or more endpoints are less frequently observed. Using simulation we aim to explore the performance of a bivariate dose response model.

Methods: 

Consider Y_ij,k to be the response for k^th endpoint for the  i^th subject when treated with the j^th dose,

Y_ij,k = f_k(q_ij) + { d_1k + f_k(q_ij)^d2k } e_ij,k,  k=1,2,…,M

The parameters d₁ and d₂ alongside e_ij,k (assumed N(0, s_k²) ) describe an additive and  power error model, with s_k = g_k s₁, where, g_k is the fold difference in standard deviation of the for k^th endpoint. Data set is created by stacking observed responses as a single Y variable with a flag identifying the M endpoints. Consider 2 endpoints following an inhibitory and a stimulatory E_max models:

Y_ij,1 = placebo₁ e^h1i  [  1 – I_max { Dose_ij / (Dose_ij + ID₅₀) }  ] 

Y_ij,2 = placebo₂ e^h2i   [  E_max { Dose_ij / (Dose_ij + ED₅₀) }  ]

A data set (N=24) from a 5-period X-over was simulated. Subjects received placebo and 4 doses of 0.1, 1, 10 and 100 mg. The two endpoints are observed once after each treatment. Parameter values were, placebo₁=50, placebo₂=5, I_max=0.8, E_max =1, ID₅₀=ED₅₀=1, h₁=h₂=0.01, s₁=s₂=0.05, d₁ =1.5, and d₂ =1. Simulated data for the two endpoints were fitted simultaneously and independently. A cross-endpoint hybrid metric such as ID₈₀/ED₂₀ is of interest. Metric distribution was compared using the two methods based on simulated data repeated 1000 times using Splus.

Results: Modeling multiple responses was feasible under a general model and error structure. For the bivariate case discussed above, model estimates were in close agreement with the true values. Additional results will report comparisons with independent models under various data settings, including unequal endpoint sample size.

Conclusions: Modeling multiple endpoints in a single-model fit offers a means of estimating and comparing drug effects on multiple endpoints.

Reference: PAGE 23 () Abstr 3084 [www.page-meeting.org/?abstract=3084]

Poster: Methodology - Covariate/Variability Models

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