Categorizing continuous variables arises as an important task in statistical analysis, especially in analyzing dose-response relationships. Creating meaningful groups of the predictor variables regarding the outcome variable is desirable in many settings, especially if the form of the relationship is unknown. However it is not always obvious how many groups should be build and where the cutpoints should be placed. Usually more than one explanatory variable has to be included in the analysis, and therefore one has to apply an appropriate statistical model. For this purpose we need a simple approach to model the data without many requirements. Another important issue in statistical analysis and especially in toxicology studies is proving a dose response relationship: increasing response probability with increasing predictor variable. This theses deals with cases where categorization of numerical or categorical predictor variables results as an effect of the dose-response relationship.

Isotonic regression is an alternative proposal when one wishes to establish a dose-response relationship, categorize continuous variables and estimate threshold values. The only assumption for this approach is the monotonicity in the response variable. The isotonic regression summarizes the description of n observations to l categories (level sets or solution blocks) by automatically splitting the predictor in constant risk groups. The result is always a step function, and therefore the isotonic regression can be used to fit a changepoint model. The Pooled Adjacent Violators Algorithm (PAVA) is used to fit the data.

In relation to model fitting and testing, some problems arise when the response is binary, and in the present work the difficulties are highlighted and some proposals to solve them are given. Regarding isotonic regression and binary response, the isotonic test for trend, the reduced isotonic model, multidimensional isotonic models and methods to assess threshold limit values are discussed.

The isotonic framework provides a reliable test for trend which unlike other widely used tests (the Cochran-Armitage test for example) is independent of any monotonic transformation of the dose variable and does not assume a linear shape. However the proposed large sample approximation (a weighted chi-square distribution) does not hold when the overall response probability is less than 5\% and thus exact methods are proposed in order to assess the correct p-value. In a simulation study it has been shown that the isotonic likelihood ratio test is more powerful than the Cochran-Armitage test, the Wilcoxon test and the Iso-chi-squared test.

The model resulting from PAVA can become more parsimonious if the level sets which correspond to a non significant change for the response variable are eliminated. This model is called reduced isotonic regression. That can be accomplished by two means: a sequence of Fisher tests for the adjacent 2x2 tables or the application of a variation of a "closed testing" procedure. The correction for multiple comparisons is made for the first method by an a-priori estimation of the overall significance level in a permutation procedure. In the second method the control for the expense of the type I error is effected by the closure principal. To select between full isotonic and reduced model, a procedure based on parametric bootstrap is proposed. A simulation study proved that when the maximal coefficient of determination for the analyzed data set is at least 50% and the data can be represented by a step function, the reduced monotonic regression controls successfully the trade off between model complexity and goodness of fit.

When more than one predictor is to be taken into account an additive isotonic model can be applied. Alternatively, an isotonic-surfaces model is proposed. This can be estimated by an iterative version of the Pooled Adjacent Violators Algorithm. The result is a sequence of surfaces which is mo