What is health? Contrary to simplistic notions of its being defined as the absence of disease, it is now increasingly understood that most outcomes of public health significance are continuous in nature (Keyes and Galea, 2016). This applies to both physical and mental health outcomes (Plomin et al., 2009; Keyes, 2002). The use of binary endpoints, while having utility in clinical applications, should not hinder investigation of the influences of health outcomes which are ultimately continuous. Further, analysing the determinants of health using continuous rather than binary outcomes is beneficial both practically (with more statistical power and less information loss) and substantively (greater aetiological understanding). Indeed, those at high risk of a developing an illness may comprise a minority of those who ultimately succumb (Rose, 2001).

Studies into the effect on continuous outcomes of exposures, be they risk factors in observational studies or interventions in randomised trials, typically focus on mean differences in the outcome, using linear regression. However linear regression assumes homoscedasticity, that is that the variability of the outcome is unrelated to the exposure, and often this is not the case. It is possible to extend regression analysis to model the variability as well as the mean, and this has benefits in terms of not only the model’s fit but also its interpretation. If for example the intervention in a trial can be shown to reduce variability in the outcome, this could reasonably be viewed as evidence of intervention success (Subramanian et al., 2018) independent of the intervention’s effect on the mean. Treatment for refractive vision errors—glasses, contact lenses, and/or corrective surgery—seeks to improve vision by shifting individuals towards a specified standard (e.g. 20/20 vision) (Vitale et al., 2006). Successful treatments alter the mean refraction, but they are even more successful if they also reduce the substantial variability in refraction arising from the mix of short- and long-sighted individuals.

Similarly, obesity interventions aim to reduce body mass index (BMI) and shift treated individuals from overweight (25–30 kg/m²), obese ( > 30 kg/m²), or severely obese ( > 45 kg/m²) to the normal range (20–25 kg/m²). However, here the effect of the intervention on variability is often to increase it. Even if not formally tested, visual comparisons of outcome distributions of some influential trials suggest that weight loss interventions increase rather than reduce BMI variability, (Truby et al., 2006) presumably since they are effective in some but not all participants.

Understanding if and how risk factors influence variability in health outcomes has aetiological significance, consistent with the goal of epidemiological science to understand the distribution of health (Porta, 2008). Risk factors could feasibly affect outcome variability yet not affect the mean—for example, one study found that breastfeeding was not related to mean childhood BMI, yet was related to lower childhood BMI variability (Beyerlein et al., 2008b). Similarly, sex may affect variability and/or average levels of an outcome—for instance, males may have greater variability than females in some cognitive traits (Hyde, 2014) and brain structures (Wierenga et al., 2022).

Identifying associations between risk factors and outcome variability may also be useful to identify the absence or presence of heterogeneity in susceptibility to interventions or risk factors and thus aid aetiological understanding. Indeed, the finding that substantial increases in mean BMI in recent decades have been matched by increases in BMI variability indicates that there may be differential susceptibility to the obesogenic environment (Flegal and Troiano, 2000; Johnson et al., 2015). In the context of randomised controlled trials, the finding of variability in treatment effects between individuals has been used to justify individualised approaches to treatment (personalised medicine). Reflecting the challenges of empirically testing this, however, five separate meta-analyses have tested heterogeneity in response to antidepressant therapy; despite using the same dataset, different methods and divergent conclusions were drawn (Luedtke and Kessler, 2021).

Another advantage of modelling varability arises in common situations where the outcome under study is non-linearly related to other outcomes of interest. For instance, BMI influences mortality and morbidity rates, but the relationship between BMI and mortality is thought to be J-shaped ( Bhaskaran et al., 2018) compared with those in the normal range, mortality risks are greater for those who are under- or overweight. In this case, the total effect of an intervention to reduce BMI on these wider outcomes is not fully captured by its average BMI effect. Rather, understanding the total distributional effect on BMI is required.

Figure 1 shows three hypothetical scenarios for an intervention to affect the distribution of an outcome. In the first case (Panel A), the intervention has an impact that is consistent across the population: all individuals are affected and to the same extent. In the second case (Panel B), the intervention has the same mean impact, but variability is also increased: some are positively affected, others negatively. In the third case (Panel C), the mean is again increased, but so is skewness. There is heterogeneity in response, with some seeing more positive responses than others. The policy implications may be different in each case. In the second and third scenarios, efforts could be directed to identify those who are (more) positively impacted, so as to increase the net benefit or cost-effectiveness of the intervention. Indeed, in a choice between interventions, an intervention generating lower expected benefits but smaller variability in outcomes may be chosen, in so far as reducing inequalities is seen as a policy goal in itself.

Figure 1

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Recent studies in biological (Sun et al., 2020; Nakagawa et al., 2014), environmental (Pitt et al., 2020), and economic science (Hohberg et al., 2020; Silbersdorff and Schneider, 2019; Silbersdorff et al., 2018) have begun to examine how risk factors relate to the distribution of the outcome of interest. However, there have been few epidemiological applications of this approach to date; (Beyerlein et al., 2008a) and fewer still that provide explanations for such findings, which are essential if such methods are to have utility. Indeed, one recent study which investigated the association between mental health symptoms and lower income explicitly avoided interpretation of its findings on variability, focusing instead on issues relating to the application of such methods (Silbersdorff and Schneider, 2019).

Regression methods that allow variability to be modelled are uncommon. One particular method, Generalised Additive Models for Location, Scale and Shape (GAMLSS) (Rigby and Stasinopoulos, 2005) has become the standard for constructing growth reference centiles, (Cole et al., 2009) where the aim is to model the outcome’s distribution as a function of age. It defines the distribution in terms of distribution moments, i.e. the mean, variance, and optionally skewness and kurtosis. This allows for factors influencing the higher moments to be identified in just the same way as for the mean, and it provides a simple and elegant interface for modelling variability in epidemiology.

Another arguably underutilised (Beyerlein, 2014) and related statistical approach to investigating risk factors for continuous outcomes is quantile regression. Recent epidemiological studies using this method have found that risk factors for higher BMI—particularly lower social class and physical inactivity—have sizably larger effect sizes at higher BMI centiles (Bann et al., 2020; Green and Rowe, 2020). This has potentially important policy implications—risk factors which have larger effects amongst those at highest health risk are likely to have a more favourable effect on population health than alternatives which do not (Bann et al., 2020). However, the reason for this phenomenon is not yet understood—it is likely to be logically consistent with results of GAMLSS analyses in which risk factors influence outcome means, variability and/or skewness.

In this paper, we provide a worked example of the use and interpretation of GAMLSS. Accompanying this is an online tutorial and full replication syntax for running GAMLSS in R (https://osf.io/5tvz6/). We investigate whether and how several established risk factors—sex, childhood socioeconomic circumstances, and physical inactivity (Stringhini et al., 2017)—relate to differences in outcome mean and variability. We choose two different continuous outcomes, an indicator of adiposity (body mass index, BMI) and mental wellbeing. These are two weakly correlated health outcomes, each of independent importance to population health. Each risk factor-outcome combination is the subject of previous (separate) literature which focuses largely on mean differences only. For instance, low socioeconomic position in childhood has been repeatedly related to higher BMI (Bann et al., 2018; Senese et al., 2009) and worse mental wellbeing in adulthood; (Wood et al., 2021; Simanek et al., 2021; Wood et al., 2017) greater physical activity has notable likely bi-directional links with lower BMI ; (Jakicic et al., 2019) and higher wellbeing; (Black et al., 2015; Choi et al., 2019; Pinto Pereira et al., 2014) while males and females seemingly have similar mean BMI and wellbeing, (Wood et al., 2017) this may mask differences in variability or skewness, as suggested in the sizable sex differences in overweight and obesity rates (Conolly et al., 2017).

The further investigation of differences in variability and skewness in these outcomes is therefore arguably of substantive interest, providing further motivation to the tutorial content. We highlight the contribution of GAMLSS by contrasting results with the more commonly used linear regression and (less commonly used) quantile regression models.

A total of 6007 participants had valid data for BMI and all risk factors, and 7104 for WEMWBS. Mean BMI was 28.4 (SD = 5.5), and mean WEMWBS 49.2 (8.3). Higher BMI was weakly associated with lower wellbeing (r = –0.07, p < 0.01). BMI was moderately right-skewed (Figure 2, left panel) and WEMWBS left-skewed (Figure 2, right panel). Visual and descriptive comparisons of the BMI and wellbeing distributions by risk factor suggest that differences in the outcome mean and variability are not always in the same direction.

Figure 2

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GAMLSS results for the binary risk factors are shown in Tables 1 and 2, with the results using the extra risk factor categories inSupplementary file 1. Associations were similar in the unadjusted and mutually adjusted analyses, so the former are described below.

Comparison with quantile regression findings

For BMI, the associations of lower social class and physical inactivity were stronger at upper quantiles (Table 3; e.g., manual social class had 3.7 (0.6) higher BMI at the the median, and 4.9 (0.7) at the 75th); estimates at higher centiles were also estimated less precisely than at lower centiles (larger SE). In contrast sex differences were present at lower centiles but absent at the 75th centile. These findings corresponded with those from GAMLSS using BCCG, with all BMI centiles plotted by risk factor group (Figure 3). This comparison highlights the utility of GAMLSS—risk factor differences in the mean, variability, and skewness can each be quantified and thus visually depicted.

Table 3

Outcome	Risk factor	25th centile	50th centile	75th centile
BMI @ Age 46	Male vs female	6.8 (0.5)	4.5 (0.6)	–0.8 (0.7)
	Father’s Class	3.7 (0.6)	3.7 (0.6)	4.9 (0.7)
	Exercise Level	1 (0.7)	3 (0.7)	4.3 (0.8)
WEMWBS @ Age 42	Sex	0 (0.7)	0 (0.5)	0 (0.3)
	Father’s Class	–4.5 (0.7)	–4 (0.5)	–1.8 (0.3)
	Exercise Level	–6.9 (0.5)	–6.1 (0.5)	–1.8 (0.5)

1. Note: results show the percentage difference (log-transformed x 100) in BMI or mental wellbeing (WEMWEBS; standard errors in parenthesis) at different centiles of the outcome distribution; estimates are mutually adjusted.

Figure 3

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For WEMWBS, the associations of lower social class and physical inactivity were also stronger at lower quantiles (Table 3), yet had larger standard errors. Sex was not associated with WEMWBS at any centile. These findings corresponded with those from GAMLSS (Figure 4).

Figure 4

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Using an underutilised analytical approach (GAMLSS), we present empirical evidence to support the idea that risk factors can relate to sizable differences in outcome variability, and even outcome skewness, in addition to differences in the outcome mean. Females had higher variability in BMI and mental wellbeing than males; lower social class and physical inactivity were each associated with higher variability in both BMI and mental wellbeing, despite having different directions of association with the mean (higher BMI yet lower mental wellbeing).

Our findings add to an emerging literature which has investigated associations between risk factors and outcome variability. Studies (Sun et al., 2020; Nakagawa et al., 2014; Pitt et al., 2020; Hohberg et al., 2020; Silbersdorff and Schneider, 2019; Silbersdorff et al., 2018; Beyerlein et al., 2008a) have reported that risk factors associated with higher means are also associated with higher outcome variability. For example, (Beyerlein et al., 2008a) found that multiple risk factors for high childhood BMI (such as more frequent television viewing and greater rapid infant weight gain) were related to both higher mean BMI and greater variability in BMI. However, previous studies have not utilised multiple outcomes or nationally representative samples, and have not systematically considered explanations for such findings or their implications.

Our findings help to reconcile findings from GAMLSS with those using quantile regression (Beyerlein et al., 2008a; Bann et al., 2020; Green and Rowe, 2020) which have reported stronger effect sizes for BMI risk factors at higher BMI centiles. This finding is both consistent with and helps explain the GAMLSS findings. For instance, lower social class and physical inactivity are related to higher BMI mean and variability, yet less BMI skewness; the net result is higher effect estimates at upper centiles which are less precisely estimated, as seen in quantile regression. While both analytical approaches have merit, GAMLSS has a number of attractive features for use in aetiological research: it enables each distribution moment to be separately investigated, and uses predetermined distribution families which enable computation of sparsely distributed variables.

Why are risk factors associated with differences in outcome variability? There are multiple possible explanations. First, risk factors may not be sufficient for an outcome to occur but rather only have a causal effect in the presence of other factors, for instance as posited in models such as the stress-diathesis model of mental health (Zuckerman, 1999). Such additional factors could also operate as effect modifiers which increase the strength of the risk factor. Factors such as genetic propensity to weight gain may for example modify the effect on weight gain of exposure to adverse socioeconomic circumstances (Tyrrell et al., 2017). Other environmental factors could operate similarly—such that the association between lower social class and higher BMI is weaker amongst those living in a local environment which is less ‘obesogenic’ (i.e. less conducive to physical inactivity and lower energy intake) (Drewnowski et al., 2007; Stafford et al., 2007). The net result of such divergent effects would be increased variability since the effects would range from zero to the upper bound of the effect. This explanation may also apply to mental wellbeing, given evidence for the myriad environmental (Ludwig et al., 2012; Wood et al., 2021) and genetic determinants (Luciano et al., 2018; de Moor et al., 2015) which could modify the effects observed in the current study.

Alternatively, between-person differences in confounding and/or measurement error may also lead to risk factors being associated with outcome variability. For example, in the present study physical activity was measured via a single item capturing reported activity of a moderate-vigorous intensity for at least 30 min per day; this is an imperfect reflection of the underlying exposure which may have a causal effect (e.g. total energy expenditure [across all intensities of activity] in the case of adiposity; (Bann et al., 2014) or time spent in specific activities conducive to wellbeing in the case of mental wellbeing [Black et al., 2015]). The net result would be higher variability in those reporting higher physical activity levels. A related issue is the extent to which the exposure captures the same ‘dose’ across participants in a given study. The physical activity measure used here counted the number of days that bouts of activity lasted at least 30 min; this likely reflects substantial variability in the level of exercise actually undertaken, thus leading to greater differences in outcome variability. This could partly explain the associations of lower social class with greater outcome variability, since social class is one dimension of socioeconomic position, such that there may be substantial between-person variation in other dimensions (e.g. parental education, income and/or wealth [Moulton et al., 2021; Galobardes et al., 2006]) which may each influence outcomes, leading to greater variability.

The study highlights the fact that analyses by GAMLSS and quantile regression lead to similar results at the selected quantiles of the outcome distribution—see Figures 3 and 4. However GAMLSS, by analysing the whole distribution, can in some cases provide more efficient estimates of the quantiles. Compare for example the standard errors of the median as obtained by the BCCG distribution (Tables 2 and 3) and quantile regression (Table 3); the GAMLSS standard errors are smaller.

All data are available to download from the UK Data Archive: https://beta.ukdataservice.ac.uk/datacatalogue/series/series?id=200001.

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Author details

1. David Bann

   Centre for Longitudinal Studies, Social Research Institute, University College London, London, United Kingdom

   Contribution

   Conceptualization, Data curation, Formal analysis, Funding acquisition, Investigation, Methodology, Resources, Visualization, Writing - original draft, Writing - review and editing

   For correspondence

   david.bann@ucl.ac.uk

   Competing interests

   No competing interests declared

   "This ORCID iD identifies the author of this article:" 0000-0002-6454-626X

2. Liam Wright

   Centre for Longitudinal Studies, Social Research Institute, University College London, London, United Kingdom

   Contribution

   Formal analysis, Investigation, Methodology, Resources, Software, Visualization, Writing - review and editing

   Competing interests

   No competing interests declared

3. Tim J Cole

   Great Ormond Street Institute of Child Health, University College London, London, United Kingdom

   Contribution

   Conceptualization, Investigation, Methodology, Visualization, Writing - review and editing

   Competing interests

   No competing interests declared

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