Research Article: An algorithm for computing profile likelihood based pointwise confidence intervals for nonlinear dose-response models

Date Published: January 25, 2019

Publisher: Public Library of Science

Author(s): Xiaowei Ren, Jielai Xia, Christian Ritz.


This study was inspired by the need to estimate pointwise confidence intervals (CIs) for a nonlinear dose-response model from a dose-finding clinical trial. Profile likelihood based CI for a nonlinear dose response model is often recommended. However, it is still not commonly used in dose-finding studies because it cannot generally be calculated explicitly. Most previous research has mainly focused on the performance of the profile likelihood based CI method compared with other common approaches. However, there are still no reports on computing profile likelihood based pointwise CIs for an entire dose-response curve. Based on a previous dose-finding trial with binary-response data, this present study proposed to calculate profile likelihood based pointwise CIs by using the bisection method with proper calculation order for doses in the curve plus crude search when the expected response is close to a boundary. The convergence could be improved by applying appropriate starting values for the optimization procedure with straightforward programming techniques. The algorithm worked well in most cases based on the simulation study and it can be applied more generally to other dose-response models, and other type of data like normally distributed response. It would indeed be great to be able to use profile likelihood approaches more routinely when constructing pointwise CIs for nonlinear dose-response models.

Partial Text

In dose-finding studies, together with parameter estimates from the fitted dose-response model, the pointwise confidence intervals (CIs) of the expected response for doses on the entire curve that characterize the uncertainty of the fitted model are always needed to provide essential information for identifying the optimal dose (s). To compute CIs for parameters, the Wald-type CI is the most commonly used approximate CI mainly because of its intuitive appeal and computational ease. However, the use of this approach in nonlinear dose-response models and binomial inferences has been discouraged because of its woeful performance [1–5]. The profile likelihood approach is one of the recommended methods for generating CIs for parameters from a nonlinear dose-response model [3–5]. Compared with Wald-type CI, the profile likelihood based CI generally has a better coverage, can avoid aberrations such as limits outside [0,1], and takes monotonicity into account. Furthermore, it even performs better than the bootstrap method (nonparametric percentile method) in some situations, such as when boundaries are used in the estimation of parameters in a nonlinear model [3–5].

The data that motivated this present study and that were used to illustrate the proposed analysis were extracted from trial NCT02131662 on The trial was a phase 2 dose-finding study that was randomized and placebo-controlled with a total of four active doses (0.5 mg, 1 mg, 2 mg, and 4 mg) and placebo. The primary variable was assumed to be binomially distributed; the number of responders (n)/total number of participants (N) in each group from the lowest dose (0 mg, placebo) group to the highest dose (4 mg) group were as follows n/N = 1/58, 18/60, 34/61, 33/61, and 36/60. The dose-response model (4-parameter logistic model) was applied to investigate the dose-response relationship, which was assumed to be a monotonically increasing function in dose.

To obtain the pointwise CIs, a grid of doses ranging from the lowest dose to the highest dose (0–4 mg) was defined. A total of 41 equidistant points, d* (0–4 mg, by 0.1 mg), were defined in our example. The grid of values can be extended to obtain greater accuracy, if necessary. In addition, we reparameterized the model to encompass our target parameter which is the expected response p* in our case. We replaced Emax with a function of p* in model (1),
thus making the model contain p*f(d,p0,p*,d*,δ)=p0+(p*−p0){1+e[(ED50−d*)/δ]}/{1+e[(ED50−d)/δ]}(2)
This is in principle the usual approach for estimating confidence intervals of a certain estimate [3,4,8]. After the reparameterization, the profile likelihood versus the expected response can be obtained in the model [8], and subsequently to obtain the CI of p* for each specific dose d* in the defined grid like other parameters.

To test the algorithm proposed in different sample sizes and to evaluate the performance of the profile likelihood approach in different doses for the entire curve, a simulation study was performed. The simulation study was based on the example discussed in the motivation section with the following doses (0 mg, 0.5 mg, 1 mg, 2 mg, and 4 mg) in a simulated 4-parameter logistic model. The coefficients were chosen to be the ML estimates based on our example (p0 = 0.15%, Emax = 56.9%, ED50 = 0.49 mg, δ = 0.14). Sample sizes of 20, 50, 70, and 100 per dose-group were generated. For each sample size, we simulated 1,000 datasets. For each set of data, the profile likelihood, normal-based Wald, and nonparametric percentile bootstrap pointwise 95% CIs were calculated. For the nonparametric percentile bootstrap, 1,000 resamples were generated. Simulation study was written in SAS (version 9.2).

In this study, we have addressed the computation of profile likelihood based pointwise CIs for a nonlinear 4-parameter logistic model with binomially distributed data based on a phase 2 dose-finding trial. An algorithm, the bisection method with proper calculation order for doses in the curve, plus crude search when the expected response was close to a boundary, was proposed. The starting values for a specific fitting with the ML estimates of parameters from previous fitting and the log likelihood value for each fitting can be taken from the standard output of the SAS procedure with straightforward programming techniques, with no extra efforts. The non-convergence can be minimized significantly with this proposed approach.




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