Nonlinear response surface in the study of interaction analysis of three combination drugs

Few articles have been written on analyzing three-way interactions between drugs. It may seem to be quite straightforward to extend a statistical method from two-drugs to three-drugs. However, there may exist more complex nonlinear response surface of the interaction index (II) with more complex local synergy and/or local antagonism interspersed in different regions of drug combinations in a three-drug study, compared in a two-drug study. In addition, it is not possible to obtain a four-dimensional (4D) response surface plot for a three-drug study. We propose an analysis procedure to construct the dose combination regions of interest (say, the synergistic areas with II≤0.9). First, use the model robust regression method (MRR), a semiparametric method, to fit the entire response surface of the II, which allows to fit a complex response surface with local synergy/antagonism. Second, we run a modified genetic algorithm (MGA), a stochastic optimization method, many times with different random seeds, to allow to collect as many feasible points as possible that satisfy the estimated values of II≤0.9. Last, all these feasible points are used to construct the approximate dose regions of interest in a 3D. A case study with three anti-cancer drugs in an in vitro experiment is employed to illustrate how to find the dose regions of interest.

A semiparametric approach to analysing dose-response data

In the analysis of a quantal dose-response experiment with grouped data, the most commonly used parametric procedure is logistic regression, commonly referred to as 'logit analysis'. The adequacy of the fit by the logistic regression curve is tested using the chi-square lack-of-fit test. If the lack-of-fit test is not significant, then the logistic model is assumed to be adequate and estimation of effective doses and confidence intervals on the effective doses can be made. When the tolerance distribution of the dose-response data is not known and cannot be assumed by the user, one can use non-parametric methods, such as kernel regression or local linear regression, to estimate the dose-response curve, effective doses and confidence intervals. This research proposes another alternative based on semi-parametric regression to analysing quantal dose-response data called model-robust quantal regression (MRQR). MRQR linearly combines the parametric and non-parametric predictions with the use of a mixing parameter. MRQR uses logistic regression as the parametric portion of the model and local linear regression as the non-parametric portion of the model. Our research has shown that the MRQR procedure can improve the fit of the dose-response curve by producing narrower confidence intervals for predictions while providing improved precision of estimates of the effective doses with respect to either logistic or local linear regression results.

A note on the small sample behavior of logistic regression in a bioassay setting

The logistic regression procedure is a popular statistical method used when analyzing quantal dose-response data. However, logistic regression results based on a poorly designed experiment can be seriously compromised. Our results indicate that depending on the spacing of the doses, the number of doses, and the number of replications at each dose, the user can get very misleading results, including ineffective lack-of-fit tests and severely biased coefficient estimates along with biased estimates of response. In addition, variance formulas based on asymptotic theory may be completely inappropriate. Simulation results are used to support these statements.

An improved data analysis method for interleukin 2 microassay

Development of the interleukin 2(IL 2) microassay, coupled with the use of highly purified or recombinant factors has allowed a detailed examination of the mechanism of action of this important biological response modifier. However, probit analysis of the microassay data does not allow inherent error of the system to be approximated nor can units of activity be assessed for significance. A computer program was developed to analyze the validity of each regression line and to generate 95% confidence intervals around each line. This program employs analysis of variance, linear regression analysis and the parallel line assay to fix confidence intervals for each IL 2 unit value. The use of recombinant IL 2 as an immunomodulator in clinical settings warrants a more precise statistical method to evaluate normal fluctuations of this factor than currently in use. The development of such a method is presented here.

Robust analysis of covariance

The simple analysis of covariance situation with two groups and one concomitant variable is considered. The parameters of this model with outliers present are estimated by the methods of at least squares and M-estimation. By use of simulation, several forms of M-estimators are compared with the least squares method. In terms of their efficiencies the tests on the equality of slopes and of adjusted means in the presence of outliers are examined under the null hypothesis by studying the behavior of t-like statistics based on the least squares and M-estimates. An example illustrates the techniques discussed.