Summary Case–parent trio studies concerned with children affected by a disease and their parents aim to detect single nucleotide polymorphisms (SNPs) showing a preferential transmission of alleles from the parents to their affected offspring. A popular statistical test for detecting such SNPs associated with disease in this study design is the genotypic transmission/disequilibrium test (gTDT) based on a conditional logistic regression model, which usually needs to be fitted by an iterative procedure. In this article, we derive exact closed-form solutions for the parameter estimates of the conditional logistic regression models when testing for an additive, a dominant, or a recessive effect of a SNP, and show that such analytic parameter estimates also exist when considering gene–environment interactions with binary environmental variables. Because the genetic model underlying the association between a SNP and a disease is typically unknown, it might further be beneficial to use the maximum over the gTDT statistics for the possible effects of a SNP as test statistic. We therefore propose a procedure enabling a fast computation of the test statistic and the permutation-based p-value of this MAX gTDT. All these methods are applied to whole-genome scans of the case–parent trios from the International Cleft Consortium. These applications show our procedures dramatically reduce the required computing time compared to the conventional iterative methods allowing, for example, the analysis of hundreds of thousands of SNPs in a few minutes instead of several hours.