Continuous outcome logistic regression for analyzing body mass index distributions

Introduction

Body mass index (BMI) is an anthropometric measure that is relatively easy to capture in epidemiological studies. Thus, it is widely used for describing underweight, overweight, and obesity^1,2. The most prominent standard BMI categories, underweight, normal weight, overweight, and obesity as defined by the World Health Organization [WHO, 3], are commonly applied to ensure comparability and reproducibility of statistical analyses across epidemiological studies^4,5. Such international standards are important for the communication of scientific results, for risk factor assessment and monitoring in populations, and for providing information to the general public. However, categorization of BMI values inevitably leads to information loss because an individual’s weight and height can be measured precisely using simple tools¹, but this precision is lost in statistical analyses by such an ad hoc categorization⁶. The most important problem, however, is the lack of comparability across studies that rely on different categorization schemes. Even more troublesome is the problem of comparability of studies and findings over time because the WHO categories can be expected to be updated to better reflect contemporary BMI distributions. Only roughly half of the studies published up to 2000 that used BMI as a risk factor for death used the WHO categories; the other half relied on a variety of different alternative schemes^4,5. The same problem occurs when the primary interest in a statistical analysis is the comparison of BMI distributions between different risk groups. In this latter situation, we advocate post hoc categorization of model outputs instead of ad hoc categorization of BMI measurements to better combine measurement precision, ease of communication, comparability, and reproducibility. Specifically, we propose that statistical analyses should be based on precise BMI measurements without ad hoc categorization, and then parameters and interesting contrasts thereof should then be categorized post hoc. Such results would be interpretable and universally comparable between studies using any type of category.

Methods

Empirical results

Comparison of estimated probabilities obtained from the four different likelihoods for model (4) showed that these probabilities were practically identical. For females and males of all smoking categories with baseline covariates, the estimated conditional BMI distribution evaluated at the WHO categories b ∈ {18.5, 25, 30} obtained from model (4) are given in Table 1. The model was fitted to BMI observations categorized according to the WHO and to a different categorization with intervals of two BMI units (Int 1). Furthermore, numeric intervals taking rounding error into account (Int 2) and “exact” BMI values were used to estimate model (4). The approximation of the likelihood by the density was very accurate, as the estimated probabilities obtained from models estimated from numeric intervals taking rounding error into account (Int 2) and “exact” BMI values were very close. Differences occurred in the third decimal place if at all. Slightly larger differences were observed between numeric intervals (Int 2) and intervals obtained by categorization Int 1. The more extreme WHO categorization led to the largest differences in these estimated probabilities, but the results were still practically identical.

In addition to a comparison of the estimated probabilities, we also compared the proportional log-odds ratios β among the four BMI likelihoods (Table 2) and did not find relevant differences. The approximation of the likelihood based on the density resulted in odds ratios numerically almost identical to those obtained from numeric intervals that take the rounding error into account (Int 2). The odds ratios obtained with intervals of Int 1 differed more, but were still negligible. This also applied to the marginally less accurate odds ratios obtained from models fitted to BMI values categorized according to WHO criteria. It should be noted that the lengths of the confidence intervals between the four different BMI likelihoods were in line, which indicated that not only the estimated parameters $\widehat{\beta }$ but also their estimated standard errors are comparable among the four approaches. The large sample size led to almost all odds ratios being significant. Age was associated with a shift towards larger BMI values, while higher alcohol intake was associated with marginally reduced BMI. Lower intake of fruits and vegetables as well as less physical activity also indicated a shift to higher BMI values. The BMI distributions of people with a higher education were shifted to the left compared to those of less well-educated people.

The BMI values of people of the German-speaking part of Switzerland were higher than those of the French- and Italian-speaking regions.

The estimated conditional BMI distribution for all combinations of smoking and sex were clearly non-symmetric, and the impacts of smoking and sex of the individual related to changes in the mean and higher moments (distribution functions in Figure 1 and density functions in Figure 2). The BMI distribution shifted towards larger BMI values from males who never smoked to male former smokers. In this case, only the mean was affected; the shape of the distribution was constant. The BMI distribution of females who never smoked and female former smokers was similar to those of males. The difference between the two sexes could not be described by a simple shift because the shapes of the two distributions clearly differed. In general, the association of smoking and BMI was less pronounced for females than for males. Compared to the associations of sex (Figure 1 and Figure 2), the smoking associations were much smaller. We quantified the odds ratios of the smoking association for both sexes for the BMI categories. Table 3 presents the same information as the distribution functions evaluated with the BMI categories (gray vertical lines in Figure 1) on the odds ratio scale in a condensed form. The odds of lower BMI evaluated at BMI ∈ {25, 30} for male former smokers were smaller than for males who never smoked. The odds ratios for underweight and normal weight (BMI ≤ 25) and for non-obesity (BMI ≤ 30) increased for both males and females.

Figure 1. Conditional distribution of BMI.

For each combination of smoking and sex, the conditional distribution function of BMI ℙ(BMI ≤ b | smk, sex, x) corresponding to model (4) was evaluated for baseline covariates x at all possible BMI values b. Red, female BMI distributions; blue, male BMI distributions; solid lines, BMI distributions of active smokers; dashed lines, never smoked; gray vertical lines, WHO categories 18.5, 25, 30. The model was fitted using “exact” BMI values.

Figure 2. Conditional distribution of BMI.

For each combination of smoking and sex, the conditional density of BMI corresponding to model (4) was evaluated for baseline covariates x at all possible BMI values b. Red, female BMI distributions; blue, male BMI distributions; solid lines, BMI distributions of active smokers; dashed lines, never smoked; gray vertical lines, WHO categories 18.5, 25, 30. The model was fitted using “exact” BMI values.

For current smokers, the odds ratio patterns that depended on BMI differed between males and females. All smoking levels were associated with larger odds of being underweight for females and had a U-shaped pattern. For males, this association was reversed and had an inverted U-shaped pattern. In the center of the BMI distribution (BMI ≤ 25), the odds ratios were much closer to 1 for both sexes. The odds ratios for non-obesity (BMI ≤ 30) for females indicated a trend towards smaller BMI values for current smokers. Except for heavy smokers, this effect was also found for males.

Discussion

Our study showed that it was possible to analyze and compare BMI distributions in terms of standard parameters without the need of ad hoc categorization. Continuous BMI logistic regression, which avoided ad hoc categorization of BMI values, led to deeper insights into the impact of sex of the individuals and smoking status on the continuous BMI distribution. The model results were insensitive to BMI measurement scales or categorization schemes and matched previously reported findings on the impact of smoking and sex of the individuals on BMI. It was obvious from the conditional BMI densities (Figure 2) that more restrictive models, e.g., a conditional normal distribution with or without sex- and smoking-specific variance²², would describe the BMI distributions less accurately. The corresponding BMI-dependent odds ratios derived from continuous BMI logistic regression (Table 3) also indicated that a model that assumed proportional and thus BMI-independent odds would not be appropriate because odds ratios varied substantially as BMI cut-off points increased.

We used a parsimonious approach in defining covariate parameters and described the impact of the covariates on the BMI distribution as being linear on the log-odds scale. We therefore assumed that the covariate parameters would be the same in all binary or polytomous logistic regression models regardless of the ad hoc categorization applied. This corresponds to the proportional odds assumption in polytomous logistic regression models. In principle, this assumption could be relaxed by allowing BMI-dependent regression coefficients β (b), as in multinomial regression. Similar outcome-varying parameters are called time-varying parameters in survival analysis and distribution regression in econometrics^23,24 and are a special case of conditional transformation models¹⁴. Ongoing research²⁵ suggests that the assumption of a constant and sex-independent age effect for BMI is oversimplistic and conditional transformation models^14,15, allowing BMI distributions to vary smoothly with age, might provide additional insights.

From a practical point of view, one advantage of continuous outcome logistic regression is the possibility of evaluating the likelihood of BMI values obtained at different measurement scales or using different categorization schemes. This aspect allows the same model to be fitted to data obtained at different scales, and thus allows models from studies using different BMI measurement scales to be compared. The narrower the interval representing the BMI value for a particular individual, the more information is contributed by this individual to the likelihood. In contrast to the common procedure of downscaling all analyses by ad hoc categorization of BMI measurements to the ubiquitous WHO categories^4,5, we propose to fit the same or even joint continuous BMI model to all studies by maximizing a likelihood with measurement-scale specific contributions. In subject-level meta analyses, these likelihood contributions are a mixture of exact, interval, or category-based BMI measurement scales. The likelihood can also be extended to incorporate study-specific left and right truncation when only individuals with BMI values in a pre-defined range are enrolled.

Our findings on the association between smoking and BMI are consistent with the results of previous studies. It has been shown that former smoking is associated with being overweight as well as obesity, especially for males^8,9,11,12,26. Other studies have also observed a positive association of male heavy smokers with obesity, although the association was non-significant when male heavy smokers were compared with males who never smoked^8,9. By contrast, light and moderate smoking was associated with lower BMI values^8,9. In general, current smoking is associated with lower BMI values^12,27,28. These findings are consistent with previous findings on the effect of smoking on body weight^29,30.

Waiving the need for ad hoc categorization and thus also for agreement on standard categories that define the parameters in models for BMI distributions makes reported scientific results less dependent on these standard categories, and most importantly, less dependent on the WHO criteria. Considering that BMI distributions are subject to change at the population level over time², insistence on the application of standards defined decades ago leads to an increasing discrepancy between models and data. Continuous BMI logistic regression is an attempt to narrow this gap.

Appendix: Computational details

The intercept functions α(b)_smk:sex for each combination of smoking and sex were estimated as smooth and monotonically increasing functions of b. The constraints expit(r(∞ | smk, sex, x)) = 1 and expit(r(0 | smk, sex, x)) = 0 restrict the BMI distribution on the positive numbers. For each of the ten strata given by the five smoking categories and two categories of sex, an intercept function was defined by six increasing parameters of a Bernstein polynomial³¹ of order five. This choice ensures smoothness and monotonicity and allows flexible intercept functions and thus regression functions r and conditional BMI distributions to be described by model (4). The monotonicity constrained on the intercept functions renders the addition of smoothing penalty terms to the likelihood unnecessary, because the effective number of parameters is less than the order of the Bernstein polynomial [see 15,32, for numerical experiments with varying numbers of parameters]. Simple maximum-likelihood estimation was performed for all model parameters simultaneously. When the likelihood was evaluated for BMI values in WHO categories, the sex- and smoking-specific intercept function was parameterized in terms of the step-function α(b) (see Formula (3)) defined for the proportional odds model. All computations were performed using R version 3.4.2³³. The mlt package^32,34 was used to estimate continuous outcome logistic regression models. The underlying statistical theory is described in 15.

A blueprint for the estimation of conditional BMI logistic regression using the mlt package in R, assuming the data are available in a data frame sgb with variables bmi (the numeric BMI values), smoking, and sex (smoking and sex as factors), as well as age and alcohol (numeric age and alcohol intake) with optional sampling weights weights, is

### attach mlt package
library("mlt")
### compute support of BMI distribution
bmis <- quantile(sgb$bmi,
    prob = c(.01, .99), na.rm = TRUE)
vBMI <- numeric_var("bmi",
    bounds = c(0, Inf),
    support = bmis, add = c(-5, 5))
### set-up increasing Bernstein polynomial
bBMI <- Bernstein_basis(vBMI, order = 5,
                           ui = "increasing")
### set-up dummy encodings for smoking
### and sex
bSMK <- as.basis(~ smoking - 1, data = sgb)
bSEX <- as.basis(~ sex - 1, data = sgb)
### specify the model with strata sex
### and smoking and covariates age
### and alcohol
mod <- ctm(bBMI,
    interacting = b(sm = bSMK, sex = bSEX),
    shifting = ~ age + alcohol, data = sgb,
    todistr = "Logistic")
### fit model to data with weighted #
### ‘exact’ likelihood
fmod <- mlt(mod, data = sgb, scale = TRUE,
              weights = sgb$weights)
### plot conditional BMI distribution for
### 18 year-old never-smoking non-drinking
### female
nsf18 <- data.frame(
     sex = factor(c("Female", "Male"))[1],
     smoking = factor(c("Never", "Former",
          "Light", "Medium", "Heavy"))[1],
     age = 18, alcohol = 0)
plot(fmod, newdata = nsf18,
     type = "distribution")

Continuous outcome logistic regression, as a model for a continuous conditional distribution implemented in mlt, has a very strong connection to the Cox proportional hazards model, which describes the conditional continuous distribution of a survival time outcome with fully parameterized log-cumulative hazard function^15,32. A Cox model for the conditional BMI distribution could be written as (see 35)

                         cloglog(ℙ(BMI ≤ b | smk, sex, x)) = r(b | smk, sex, x).

In this case, the logistic link in (1) was replaced by the complementary log-log link. In the absence of covariates x, the results obtained from our continuous BMI logistic regression model and a Cox model stratified by sex and smoking would not be affected by this change, because for each combination of sex and smoking, a corresponding equivalent intercept function α(b)_smk:sex (the sex- and smoking-specific log-cumulative hazard in the stratified Cox model) can be found on both the logit and cloglog scales. However, the interpretation of β changes from proportional log-odds ratios to proportional log-hazard ratios. In contrast to the partial likelihood of Cox models that treat the intercept functions as nuisance parameters, the likelihood for continuous outcome logistic regression is evaluated for fully parameterized intercept functions and all model parameters are estimated by maximum likelihood [similar to 36]. The corresponding monotonicity constraint allows smooth conditional distribution functions to be estimated without adding smoothing parameters to the likelihood^15,32.

Data availability

Data from the Swiss Health Survey 2012 can be obtained from the Swiss Federal Statistics Office (Email: sgb12@bfs.admin.ch). Data is available for scientific research projects, and a data protection application form must be submitted. More information can be found here http://www.bfs.admin.ch/bfs/de/home/statistiken/gesundheit/erhebungenSupplementary