Summary: Growth curve analysis is beneficial in longitudinal studies, where the pattern of response variables measured repeatedly over time is of interest, yet unknown. In this article, we propose generalized growth curve models under a polynomial regression framework and offer a complete process that identifies the parsimonious growth curves for different groups of interest, as well as compares the curves. A higher order of a polynomial degree generally provides more flexible regression, yet it may suffer from the complicated and overfitted model in practice. Therefore, we employ the model selection procedure that chooses the optimal degree of a polynomial consistently. Consideration of a quadratic inference function [A. Qu et al., Biometrika 87, No. 4, 823–836 (2000; Zbl 1028.62045)] for estimation on regression parameters is addressed and estimation efficiency is improved by incorporating the within-subject correlation commonly existing in longitudinal data. In biomedical studies, it is of particular interest to compare multiple treatments and provide an effective one. We further conduct the hypothesis test that assesses the equality of the growth curves through an asymptotic chi-square test statistic. The proposed methodology is employed on a randomized controlled longitudinal dataset on depression. The effectiveness of our procedure is also confirmed with simulation studies.