The standard χ2 test used in limiting dilution assays is insufficient for estimating the goodness-of-fit to the single-hit Poisson model

Abstract

Limiting dilution analysis is a common technique that is used in immunology to estimate accurately the frequency of cells possessing a wide variety of functional activities, such as growth, cytotoxicity and production of lymphokins. In the literature, most experiments are fit well by the single-hit Poisson model (SHPM), which assumes that only one cell of one defined cell subset is necessary for a positive response. This is somewhat surprising since other models such as multi-hit or multi-target models that involve the interaction of one or more cells from one or more cell subpopulations for generating or inhibiting a positive response are conceivable. Since the validity of the SHPM is usually investigated by performing a standard χ 2 test, based on the number of observed and expected positive and negative responses, we questioned here the efficiency of this test in comparison with other validity tests for the SHPM, the log likelihood test derived by Cox, and the modified Weibull plot tests, the principles of which are entirely different from that of the standard χ 2 test. We used the following theoretical approach. First, we generated artificial data corresponding to multi-hit and multi-target models. Second, considering that these data were derived from real experiments, we calculated the frequency of the desired cell subset according to the SHPM using the maximum likelihood method. Then, the goodness-of-fit of these data with the SHPM was evaluated. The log likelihood test and the modified Weibull plot tests rejected the SHPM hypothesis, while the standard χ 2 test did not. Thus, the standard χ 2 test is unable to discrimenate sensitively between the SHPM and more complicated (non-single-hit) Poisson models. We concluded that the results of limiting dilution studies published thus far must be evaluated with caution. The statistical tests presented here should be routinely applied for each limiting dilution experiment.