## Analysis of proportions responding

Another widely used approach to evaluating immune response is through the proportion of individuals responding in some specified manner. For a single group of subjects, a meaningful analysis is simply the point estimate of the proportion responding and its confidence interval. The following methods are for comparing two or more groups, or for evaluating the effects of covariates.

### Commonly used methods

Chi-square and Fisher's exact test Both these methods compare proportions responding in a specified manner to immune stimuli. Fisher's exact test is used alternatively to the chi-square.

Logistic regression Logistic regression differs from simple or multiple regression in that the response variable is discrete and typically dichotomous rather than continuous, and also in that the distribution of the errors is binomial rather than normal. Thus, this method is adaptable to the study of binary immune response status (e.g. positive or negative). It may be used to assess the association between immune response and one or more explanatory factors, or it may be employed for classifying subjects as does discriminant analysis (see below).

Proportional hazards (Cox) regression This method, a type of survival or time-to-event analysis, has been used to evaluate the effect of covariates on time to seroconversion. Survival methods are especially useful when individual subjects in a study are not followed for equal periods of time, or when there is censoring (which occurs when an individual is lost to follow-up during the study). In order for this analysis to be informative about serological data, blood samples must be drawn at numerous time points. However, it may not be possible to know precisely when seroconversion actually occurs - only when it is discovered through a laboratory test.

Discriminant analysis This technique has been used to assess the effects of covariates on seroconversion status in response to vaccination. The basic purpose of discriminant analysis is to classify an individual with specified covariate characteristics into one of two or more population groups. In serological analysis, these groups are usually only two, responders and nonresponders. The linear discriminant function is appropriate when the explanatory variables in each comparison group are approximately multivari-ately normally distributed with equal variances/ covariances. If explanatory factors are qualitative rather than continuous, then logistic regression, which does not require normality, may be preferable.

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