Abstract By the so-called delta-method a test statistic is used which is a function of statistics with known variance-covariance structure. The standard deviation of the statistic is found by linearizing it. Significance is declared if the ratio between the test statistic and its estimated standard deviation surpasses c, where c is the 1-ε fractile of the normal distribution, and ε is asymptotically, the level of significance. Generalizing this method let η=(η1,..., ηv ) be the parameter in the model and H 0 a hypothesis that reduces the number of freely varying parameters to t. “Effects” are functions f(η) of η. They are “contrasts” relatively to H 0 if f(η)=0 for η;єH 0. Multiple comparison consists in looking for contrasts which are “present”, i.e. for which f(η) > 0. According to “The delta multiple comparison method” any contrast may be declared present if significance is obtained by using the delta method with the critical point c replaced by , where z is the (1-ε)-fractile of the chi-square distribution with w=v-t degrees of freedom. It is shown below that then the level still holds (asymptotically). It is also shown that this multiple comparison method is related to the likelihood ratio test for H 0 in a similar manner as the Scheffé method establishes a connection between modified Student testing and Fisher's analysis variance test. However, as discussed in chapter II below, our attitude to the null hypothesis is that we are not interested in the truth or the falsehood of it. The hypothesis is demoted to a tool which on the one hand is used to impose limitations on the possible comparisons to be undertaken in the statistical analysis. On the other hand the hypothesis defines the degrees of freedom, i.e. the critical point in the comparisons. Thus the hypothesis provides the tie between the comparisons which are desired and the actual shape of the decision criterions (see Sverdrup, 1975, 1977 a and 1977 b). Besides showing how to adjust the method to a certain level of significance, it is also shown how to study the performance function, i.e. the probability of stating the presence of certain effects for different values of η. The general result is given in Theorem 4 in chapter II below. The general theory is applied to multinomial situations where, of course, likelihood ratio testing may be replaced by chi-square goodness of fit testing. The term “general” must, however, be taken with a pinch of salt. The theory is general in the sense that the densities of the independent observations are only subject to general regularity conditions. The observations are groupwise identically distributed. Except for a review of Scheffe's theory of the linear-normal situations, the theory is asymptotic and the numbers of groups are kept constant. Thus frequency tables with small expected cell counts are not treated. I. Introduction. The basic statistical ideas: The general idea of contrasts and multiple comparison; Review of Scheffé's method; Outline of the general delta multiple comparison methods; The relationship to likelihood ratio testing; A comment on simultaneous confidence intervals; The multinormal model; Restrictive multinomial models; Statistics collected by the official Central statistical bureaus. II. General theory: The likelihood, assumptions and definitions; Properties of maximum likelihood estimates and likelihood ratios; Proofs of the lemmas; The case of Darmois-Koopman classes; The contrast analysis. III. Application to categorical data: Assumptions; The likelihood ratio and the chi-square goodness of fit statistics; Contrast analysis for categorical observations; Exact relationships in the case of framework, models and contrasts linear in the multinomial probabilities; Homogenity testing. References.