Registers are increasingly important sources of data to be analyzed. Examples include registers of congenital abnormalities, supermarket purchases, or traffic violations. In such registers, records are created when a relevant event is observed, and they contain the features characterizing the event. Understanding the structure of associations among the features is of primary interest. However, the registers often do not contain cases in which no feature is present and therefore, standard multiplicative or log-linear models may not be applicable. The number of babies born without any anomaly is known, and the register can be completed. Supermarket visits without any purchase do occur, although their number is unknown. But traffic violations when no rule is violated make no sense, and models should not imply an estimate for their number. For cases, when quasi models do not provide a good fit, the talk presents hierarchical multiplicative models built on restricting the values of non-homogeneous odds ratios and describes their relationship with quasi log-linear models, using concepts of algebraic statistics. This is joint work with Anna Klimova.