In this article, we use the effect of breastfeeding on intergenerational change in intelligence as a case study of regression to the mean in latent change score models. The effect of breastfeeding on intelligence is controversial. Earlier studies using observational data have found that breastfed children have higher intelligence compared to those not breastfed [1,2,3,4]. A risk for confounding is apparent, as breastfeeding mothers tend to be more intelligent than non-breastfeeding mothers or mothers who breastfeed only for a short period of time [4,5,6] and because intelligence is strongly hereditary [7,8,9,10]. A positive association between breastfeeding and child intelligence has indeed been shown to be less than completely attenuated when adjusting for maternal intelligence [2, 4, 5, 11]. However, since intelligence is measured with imperfect reliability, breastfeeding mothers may tend to have higher true intelligence than non-breastfeeding mothers even if they have the same measured intelligence. Hence, the remaining adjusted association between breastfeeding and child intelligence could be due to residual confounding. This example may illustrate the susceptibility of latent change score models to regression to the mean, a phenomenon which is well-understood in theory [12], but nevertheless often overlooked in practice.

Galton coined the term regression toward mediocrity to describe the phenomenon that tall parents tended to have tall offspring, but not quite as tall as themselves, while short parents tended to have short offspring, but not quite as short as themselves [13]. This phenomenon, nowadays usually called regression to the mean, occurs because extreme outcomes usually require an extreme combination of causative factors, and the probability is higher for some combination of causative factors that results in a less extreme outcome. So, even if an offspring has partly inherited their parent’s genome that increases the likelihood for tall/short stature, they may not experience the same extreme combination of other factors, such as nutrition, activity levels, and medical conditions, and this tends to result in a less extreme stature. An important feature of regression to the mean is that it has an effect backward as well as forward in time. Tall offspring can be expected to have tall parents, but not quite as tall as themselves, while short offspring can be expected to have short parents, but not quite as short as themselves.

The heights of parents and offspring in Galton’s example above are likely to have been measured with very high reliability. However, if we have an outcome Y that is measured with less than perfect reliability and a predictor X that has an association with the true value on Y, we can expect an association between X and observed change in Y between two measurements when adjusting for initial value on Y, even if no true change in Y has taken place. The reason for this spurious association is that with a positive (negative) association between X and the true value on Y, given the same initial value on Y those with a high value on X will tend to have a higher (lower) value on true Y and, consequently, a more positive (negative) residual in the measurement of Y compared with those with a lower value on X. And as residuals and measurement errors tend to regress toward a mean value of zero, those with a high value on X will tend to experience a more positive (negative) change in Y to a subsequent measurement compared with those with the same initial value on Y but with a lower value on X. The effect of X on the change score in Y is less susceptible to this fallacy when not adjusting for the initial value on Y [14,15,16,17].

Confounding refers to a phenomenon where two variables X and Y are associated without having any effect on each other because both of them are associated with a third variable Z. In attempting to evaluate whether X and Y are independently associated, it is common to estimate the association while adjusting for an indicator of Z. However, it is far from certain that such adjustment will eliminate the problem completely and some degree of residual confounding may remain. Residual confounding is increased by higher true degree of confounding, higher reliability in the measurements of X and Y, lower reliability in the measurement of Z, and larger sample size [18,19,20,21,22].

Latent change score modeling is a form of structural equation modeling for analyzing change in an outcome between measurements [23,24,25]. The use of latent change score modeling rather than traditional regression models has been recommended for analyzing change over time [24]. However, similarly to simpler regression models, latent change score models can be susceptible to the influence of regression to the mean if regressing the latent change score factor on the initial value on the outcome variable in addition to the predictor. For example, studies employing latent change score modeling have demonstrated what seems to be spurious effects of vocabulary on change in matrix reasoning scores, and vice versa, and of intelligence on change in academic achievement, and vice versa [26, 27]. Therefore, we have recommended to verify effects shown in latent change score models by analyses where the latent change score is not regressed on the initial value on the outcome variable [27].

Thus, we aimed to investigate the association between breastfeeding and child intelligence using two latent change score models susceptible to regression to the mean either on the mother’s or the child’s intelligence, in order to evaluate whether these approaches would diverge. If breastfeeding has a true causal effect on child intelligence, a positive association is predicted between breastfeeding and the latent intergenerational change score in intelligence, from mother to child, both when adjusting and when not adjusting for maternal intelligence. Moreover, a negative association is predicted between breastfeeding and backward intergenerational change in intelligence, from child to mother, when conditioning on child intelligence. This negative association would indicate that given the same intelligence, breastfed children tend to have mothers with lower intelligence and have, consequently, experienced a more positive intergenerational change in intelligence compared with non-breastfed children. Additionally, we simulated data with similar descriptive characteristics as in the empirical data and without any independent effect of breastfeeding on child intelligence, in order to test whether spurious associations would appear when it is known that no true effect is present.