Quantile estimation from grouped data: the cell midpoint

Data are often collected in histogram form, especially in the context of computer simulation. While requiring less memory and computation than saving all observations, the grouping of observations in the histogram cells complicates statistical estimation of parameters of interest. In this paper the mean and variance of the cell midpoint estimator of the p^th quantile are analyzed in terms of distribution, cell width, and sample size. Three idiosyncrasies of using cell midpoints to estimate quantiles are illustrated. The results tend to run counter to previously published results for grouped data.     

Trajectories of low-income mothers’ and fathers’ engagement in learning activities and child socioemotional skills in middle childhood

Using Fragile Families and Child Wellbeing data (N = 3259), the current study examined Black and Latinx mothers’ and fathers’ trajectories of engagement in learning activities (e.g., storytelling) from infancy to age 5, and whether those trajectories predicted socioemotional skills at age 9, predictors of the trajectories (poverty, mother-father nonresidence, temperament, race/ethnicity), and moderators of the trajectories. Mothers’ and fathers’ learning activities decreased significantly over time as children got older. Higher rates of decline in fathers’ engagement in learning activities over time significantly predicted lower socioemotional skills. Mother-father nonresidence during infancy was associated significantly with higher rates of decline in mothers’ and fathers’ learning activities. Difficult temperament moderated the association between fathers’ trajectories and child outcomes.