Statistical analysis of highly skewed immune response data

Abstract

This paper considers methods of statistical analysis for highly skewed immune response data. Observations from population studies of immunological variables are rarely normally distributed between individuals; typically the distribution shows extreme levels of skewness. In some situations, skewness remains considerable even after transforming the data. Using resampling techniques, applied to several actual datasets of ELISA assay data, we consider the robustness of normal parametric methods, e.g. t tests and linear regression. Despite the skewness of the transformed data, we demonstrate that such methods are quite robust depending on the number of observations, type of analysis and severity of skewness. We also illustrate how bootstrap resampling can be used to provide a valid alternative method of analysis that can be used either for checking normal parametric analysis or as a direct method of analysis. We illustrate this combined approach by analysing real data to test for association between human serum antibodies to malaria merozoite surface proteins, MSP1 and MSP2, and resistance to clinical malaria, and confirm the protective effect of antibodies to MSP1 and demonstrated a similar protective effect for some antibodies to MSP2.

Introduction

Population surveys of naturally acquired immune responses can be used to obtain quantitative and qualitative data for a number of immunological variables which can be related to clinical variables such as susceptibility to disease, disease progression or prognosis and individual variables such as age, sex, previous medical history, etc. From our own work, and the published data of many other research groups, we have observed that immune response data collected in such studies are often distributed, between individuals, in a very uneven or asymmetrical pattern. A rapid search of the published literature has revealed a number of serological datasets for a variety of different parasitic organisms, all of which showed some degree of skewness (Nutman et al., 1985; Gabra et al., 1986; Perlmann et al., 1989; Tenter et al., 1991; Deplazes et al., 1992; Helbeig et al., 1993; Van Gelder et al., 1993; Paranhos-Bacalla et al., 1994; Ferrari et al., 1995); additional examples can be found in a recent review by Muller et al. (1995). The degree of skewness appears to depend, to some extent, on the type of antigen used in the assay with the responses to defined single antigens, typically represented by recombinant proteins or synthetic peptides, being highly positively skewed and responses to crude parasite extracts, which are mixtures of many different antigens, tending to be less skewed. For example, severely skewed data was obtained for purified antigens of Echinococcus granulosus (Helbeig et al., 1993) and Plasmodium falciparum (Gabra et al., 1986; Perlmann et al., 1989) and moderately skewed data for defined Toxoplasma gondii (Tenter et al., 1991; Van Gelder et al., 1993) and Trypanosoma cruzi (Paranhos-Bacalla et al., 1994) antigens. In contrast, serological responses to crude extracts of a number of helminth parasites, including Echinococcus and Schistosoma, are only slightly skewed (Nutman et al., 1985; Deplazes et al., 1992; Ferrari et al., 1995).

A recent paper (Bennett and Riley, 1992) confirmed that skewness across a range of datasets is more the norm than the exception and that there is no consensus among immunologists about how such data should be analysed. Skewness may be so pronounced that the application of standard analysis can be rather problematic. One common approach to analysis is to ignore the skewness in the data and apply normal parametric methods such as t tests, analysis of variance (or covariance) or standard linear regression, with or without applying a transformation (usually logarithmic) to the data to reduce or remove skewness. As will be confirmed in this paper, transforming immune response data does not always normalize the distribution and may only have a minor impact on reducing the level of skewness (Bennett and Riley, 1992). At first sight, using normal parametric methods on skewed data may be expected to be invalid: confidence intervals intended to provide 95% probability of coverage (of the population value) may actually only attain a lower level of coverage and perhaps give misleading `statistically significant' results. However, because of the `Central Limit Theorem' (CLT) of statistics¹ (Ross, 1976) normal methods may still provide approximately valid confidence intervals (or equivalently, hypothesis test p values) provided that, (a) the sample size is large enough, and (b) the distribution is not too severely skewed. A second approach is to use standard non-parametric methods, e.g. Mann-Whitney or Kruskal-Wallis significance tests. However, these methods do not make full use of the available quantitative data and are limited when a multivariate analysis (e.g. multiple regression) is required which is often the case with epidemiological data. One aim of this paper is to investigate the range of values relating to (a) and (b) which will permit the use of normal parametric methods for sero-epidemiologic studies. A third approach to the analysis of immune response data is to avoid analysing the original quantitative data altogether but instead to analyse corresponding binary data after categorising each individual as a `responder' or a `non-responder', depending on whether the measured response is above or below some estimated cut-off value. Standard statistical methods such as the chi-squared test or logistic regression can then be validly applied. Although this approach may make biological sense, it is not ideal. Firstly, defining a cut-off value when, from plots of the data, there may be no obvious division of individuals into two groups, is potentially unjustified and a rather arbitrary process. Results from such analysis may also be sensitive to the particular choice of cut-off, especially for small studies. In addition, it has been shown previously (Bennett and Riley, 1992), that statistical power will be decreased when qualitative rather than quantitative data are analysed.

A relatively new approach to constructing confidence intervals, without making any parametric assumptions about the shape of the underlying population of data, is the bootstrap resampling method, first introduced by Efron (1979). A very readable account of the method is provided by Efron and Tibshirami (1993). The bootstrap method involves Monte Carlo sampling and is computationally intensive. It differs from traditional formula-driven methods in that the original data are resampled, many times, to simulate sampling variability and to provide an approximation for the unknown sampling distribution of the statistic of interest. This sampling distribution is used to provide estimates of standard error and to estimate confidence intervals. Traditional normal methods are based on the assumption that the underlying data distribution is normal so that the sampling variability of the statistic of interest is also normal and so predictable. The bootstrap method does not require assumptions about the underlying population of data, and therefore, intuitively, it is appropriate for analysing immune response data regardless of, for example, severe skewness. The bootstrap is a non-parametric method of analysis and provides a `gold standard' against which other analyses can be compared.

This paper aims to: (i) consider the robustness and limitations of normal parametric methods for the analysis of highly skewed immune response data, and (ii) to explore the use of the bootstrap method as a tool for checking normal analysis and also as a direct method of analysis.

We shall use as examples datasets obtained from enzyme-linked immunosorbent assays (ELISA) of human sera tested for the presence of antibodies to the malaria parasite, Plasmodium falciparum. The data obtained are examined in relation to both clinical and demographic covariates in order to test for any statistical association between immune responsiveness and malarial disease.