Most well-known regression models focus on the estimation of the conditional mean given a set of explanatory variables. Higher moments of the distribution function are usually assumed as constant. This typically implies strict assumptions such as homoscedasticity or symmetry. In contrast, in the flexible model class of conditional transformation models (CTMs), the whole conditional distribution function is modelled directly. Thereby, higher moments of the conditional distribution (ie variance, kurtosis, and skewness) are allowed to depend on explanatory variables. CTMs include linear transformation models (eg, proportional hazards and proportional odds models) as a special case, which display a powerful tool for data analysis in various fields. To provide a broad literature overview, the development of linear transformation models over the past twenty years is summarised in this thesis. Frequently used regression models are reviewed from the perspective of transformation models to clarify their common model basis. The methodological emphasis of this thesis is the introduction of conditionally linear transformation models (CLTMs), which constitute an important special case of CTMs, and the extension of CTMs to censored response variables. In the suggested parametrisations of CLTMs, the influence of the explanatory variables on the first two moments of the distribution function is interpretable, and closer insights into model structure can be gained. For some low-parametrised CLTMs, a likelihood-based estimation approach is presented that can be easily extended to any type of censoring. This approach is especially appealing because the associated maximum likelihood theory comes for free. Alternatively, CLTMs can be estimated based on regularised optimisation using component-wise boosting. This estimation approach is not restricted to low-parametrised CLTMs and can be used for the whole cascade of CLTMs. For applications especially in survival analysis, the target function is extended to right-censored responses by including inverse probability of censoring weights.

The superiority of C (L) TMs in comparison to less flexible standard regression models was shown in two simulation studies. Moreover, two applications of C (L) TMs in biostatistics have been selected for this thesis. The influence of ultrasound measurements on the future birth weight for newborns from the Perinatal Database Erlangen, Germany, has been analysed using low-parametrised likelihood-based CLTMs. Thereby, the birth weights were allowed to follow some arbitrary distribution with fetus-specific means and variances. Additionally, more flexible C (L) TMs have been used to estimate fetus-specific prediction intervals for the future birth weight. In survival analysis, the estimation of patient-specific survivor functions that are conditional on a set of patient characteristics is of special interest. The consideration of CTMs is advisable because the conditional survivor function