There are so many possible confounding variables and different ways this can happen. They seem to address these somewhat in the paper. (see below)
Earlier Mother's Age at Menarche Predicts Rapid Infancy Growth and Childhood Obesity
Early menarche tends to be preceded by rapid infancy weight gain and is associated with increased childhood and adult obesity risk. As age at menarche is a heritable trait, we hypothesised that age at menarche in the mother may in turn predict her children's early growth and obesity risk.
Methods and Findings
We tested associations between mother's age at menarche, mother's adult body size and obesity risk, and her children's growth and obesity risk in 6,009 children from the UK population-based Avon Longitudinal Study of Parents and Children (ALSPAC) birth cohort who had growth and fat mass at age 9 y measured by dual-energy X-ray absorptiometry. A subgroup of 914 children also had detailed infancy and childhood growth data. In the mothers, earlier menarche was associated with shorter adult height (by 0.64 cm/y), increased weight (0.92 kg/y), and body mass index (BMI, 0.51 kg/m2/y; all p <>2/y), and fat mass index (0.22 kg/m2/year; all p <0.001). Children in the earliest mother's menarche quintile (≤11 y) were more obese than the oldest quintile (≥15 y) (OR, 2.15, 95% CI 1.46 to 3.17; p < 0.001, adjusted for mother's education and BMI). In the subgroup, children in the earliest quintile showed faster gains in weight (p < 0.001) and height (p < 0.001) only from birth to 2 y, but not from 2 to 9 y (p = 0.3–0.8).
Earlier age at menarche may be a transgenerational marker of a faster growth tempo, characterised by rapid weight gain and growth, particularly during infancy, and leading to taller childhood stature, but likely earlier maturation and therefore shorter adult stature. This growth pattern confers increased childhood and adult obesity risks.