of the standard deviation of outcomes at time _t_ to that in the previous period (see equation 2). For externally standardized outcomes, the scaling factor will be greater than one if the relative dispersion of the sample distribution to that of the reference population is growing over time. Consequently, in anything other than the internally standardized case, slope coefficients derived from equation (1) will not exclusively capture the speed of convergence toward the sample mean. In itself, this ambiguity motivates a search for alternative tests of this form of catch-up.

A second approach to identifying catch-up growth is often found in studies of disadvantaged populations (samples). In contrast to focussing on within-group convergence, Hirvonen (2014) recommends a shift of attention toward _α_ in equation (1). Least squares estimates for this coefficient imply: and where the latter estimate refers to externally-standardized outcomes. In this case, the same author suggests the simple difference captures between-group catch-up, defined as the extent of convergence of the sample mean toward that of the reference population. In the unconditional case, this is obtained directly from equation (5b) when _γ_ = 0 _∧ β_ = 1, which is equivalent to examining mean differences in the standardized outcome, ∆ _y_ ̄<sup>_∗_</sup> . Similarly, Cameron et al. (2005) argue that _α_ ˆ provides a meaningful measure of catch-up in stature from initial stunting. However, they recommend _β_ remains unconstrained in order to address potential regression to the mean effects (see further below; also Adair 1999).

# **2.2 Existing literature**

The above discussion distinguished between two forms of catch-up growth. Table 1 summarizes a selection of ten recent papers in both these traditions (published in the past 5 years), all of which refer to developing country contexts.<sup>4</sup> From this, four general comments merit note. First, existing studies have predominantly considered catch-up growth in relation to child height. One rationale is that low height-for-age is associated with stunting, which is known to adversely affect outcomes in adulthood (Almond et al. 2017, Hoddinott et al. 2013). However, low BMI-for-age values are associated with child wasting, which is a symptom of acute under-nutrition. Shrimpton et al. (2001) argue that adverse _in utero_ conditions tend to affect child birth weight more than length, but in contrast to height disadvantages, weight disadvantages can be addressed relatively more quickly in

> 4 The studies were identified from Google Scholar based on the combined search text: ‘ "catch-up growth" & children & "developing countries" ’. The search period was limited to 2012-2017; only primary empirical studies undertaken in developing countries were retained; and those concerned with catch-up growth either after specific interventions or associated with specific medical conditions were excluded. Papers were also excluded that have a primary focus on the determinants of variation in growth rates (e.g., Georgiadis et al. 2016; 2017). The final list (in the table) includes the most relevant peer-reviewed studies; but we recognize this is somewhat subjective and is likely to be selective.

5