Background
Dichotomisation of continuous outcomes is common in epidemiology. For example, certain conditions of interest are defined by a continuous variable over or below some threshold, such as, hyperglycaemia is determined by either pre-meal blood glucose (BG) exceeding 7.78 mmol/L or random BG exceeding 10 mmol/L [
1,
2]. The nature of the outcome determines the statistical approach taken to analyse the data. For example, linear model and logit model (or logistic regression model) are commonly performed on continuous and binary outcomes respectively. Hence, for a continuous outcome where its binary representation is also widely used, studies have reported findings from both linear and logit models for outcomes with dual representations [
3‐
5].
However, estimates from dichotomised outcomes have large variances [
6,
7] and low power [
6‐
10]. Despite of these disadvantages, there are practical reasons for justifying dichotomisation, such as: (a) following practices used in previous research, (b) simplifying analyses or presentation of results, (c) addressing skewed variable, and (d) using clinically significant thresholds [
9,
10]. To address the problems associated with dichotomising continuous outcomes, approaches that use the analytical results from continuous outcomes to infer the association between an exposure and the dichotomised outcome have been proposed.
The proposed approaches transform the estimates obtained from continuous outcomes into familiar measures of association for binary outcomes, such as, risk differences, odds ratios (ORs), and relative risks (RRs). The estimates are obtained from applying least squares [
11], method of moments [
12‐
14], maximum likelihood [
15,
16], or Bayesian [
17,
18] estimation method to continuous outcomes. The simplest transformation multiplies a scaling factor to the estimates from the linear model to obtain log-OR [
11] but it assumes the errors have a logistic distribution. The Bayesian method allows the distribution of the error to be unspecified [
18]. An alternative approach uses the dichotomised marginal means from the linear model for continuous outcomes [
12,
13,
15] to obtain the measures of association for binary outcomes and the skew normal distribution [
14] has been considered to address potential skewness in the continuous outcomes.
The marginal mean approach estimates probabilities of different exposure levels by assuming an individual having a confounding profiling that corresponds to the mean values of the confounders [
19]. When one of the confounders is binary, an individual having a binary confounder that equals to its mean value does not exist in the real world setting. However, when the outcome is continuous, the marginal mean is equivalent to the overall mean of the population in the linear regression model because the model has an identity link. Given that the computation of probabilities for binary outcomes involve non-linear link function (e.g., logit link), marginal standardisation is commonly used to generate probabilities and RRs from the logit model for making inference on the overall study population [
19]. Interestingly, when the study of continuous outcomes does not require adjustment for confounders with regression model, for example, in randomised controlled trials, the marginal mean method for estimating one-sample risk and RR from two-sample [
12,
15] could be equivalent to the marginal standardisation method under certain assumptions (see Additional file
1 Section 1 for the details).
In this paper, we leverage on the marginal standardisation method for binary outcomes to estimate the RR of dichotomised outcomes using the linear regression model with adjustment for confounders. Extending the marginal standardisation method from binary to dichotomised outcomes becomes apparent when we realise that the logit (or probit) model assumes an underlying latent variable that corresponds to a linear model with standard logistic (or normal) error [
20‐
22], and when the latent variable exceeds some threshold (i.e., dichotomised latent variable) it can be modelled using the logit (or probit) model. As both the logit and probit models are commonly used to model binary outcomes, we extend the marginal standardisation method to linear model to make inference on dichotomised outcomes in two scenarios: logistic and normal error distributions. We compare the marginal standardisation method between regression models for continuous (i.e., linear model) and dichotomised (i.e., logit or probit model) outcomes by comparing the estimates generated from these regression models. Among approaches that avoid dichotomisation, we compare RR estimates that used the marginal standardisation approach for linear model with those from the distributional approach proposed by Sauzet et al. [
16] that estimates RR from the marginal mean obtained from the same linear model. We assess the statistical properties of the estimates and diagnostic tests using simulated data. We also apply the various approaches to evaluate the effect of an intervention that aims to improve inpatient management of hyperglycaemia in a pilot study where multiple thresholds were used to represent varying levels of severity in hyperglycaemia.
Discussion
Dichotomisation of continuous outcome is a common and appealing analytical approach especially when the dichotomisation process is also practiced in the clinical setting. However, the use of dichotomisation has been greatly criticised for non-negligible loss of power and increased variability in the estimate [
6‐
10]. In particular, the magnitude of loss in power is greater when the threshold value is distant from the mean or median [
7,
9], which corresponds to the scenario where OR approximates RR. To avoid the drawbacks of dichotomisation and facilitate the interpretation of binary representations of continuous outcomes from linear models, we proposed to transform estimates from linear models to RR through marginal standardisation. We evaluated the performance of our proposed approach, and compared it with the dichotomized and distributional approaches using both simulated and real datasets.
When comparing marginal standardisation method that avoided and did not avoid dichotomisation, our simulation results suggested β
1 and RR estimates from linear, probit and logit models were generally unbiased when applied appropriately, but probit and logit models had larger SEs and smaller power than those estimates from linear models. The improvement in precision and power of estimates from continuous outcomes were also reported in other studies [
11,
12,
16,
36]. Although β
1 estimates from mis-specified binary models had somewhat pronounced biases when effect size was large, past studies have found that small sample size or extremely common (or rare) binary outcomes could lead to biased estimates from logit model, which were negligible when compared to the magnitude of the SEs [
37]. The OR approximates to the RR when baseline risk is low (i.e., threshold at 92.5-th percentile), however, the reduction in precision and power of OR estimates from logistic linear model to logit model was also more pronounced when baseline risk is low.
Often, model diagnostics are performed to assess model misspecification [
38]. Our simulation results suggested that the Lilliefors corrected KS test and KS test had better power in assessing the distributional assumptions of the error terms for linear models when compared to Pregibon link tests for binary models. Our findings were consistent with a previous study that reported low power when differentiating probit and logit links in binary models [
39]. These findings were expected as probit and logit links are known to be similar [
40].
From the real data analyses, in general, we found the estimates from normal linear and probit models to be similar, and this phenomenon was also observed in estimates from logistic linear and logit models. However, estimates from probit and logit models had wider confidence intervals and fewer significant findings than those from linear models. These observations were consistent with findings from our simulation and the literature [
11,
12,
15]. Although we had used the log-transformed outcome in the real dataset analyses,
\( \hat{\mathrm{RR}}\mathrm{s} \) obtained from this transformation is equivalent to that obtained from the original outcome without transformation, which corresponds to a multiplicative regression model with log-normal [
41] or log-logistic error distribution [
42]. When comparing approaches that avoided dichotomisation within normal linear model in the real data analysis, we found the marginal standardisation and marginal mean methods had similar
\( \hat{\mathrm{RR}}\mathrm{s} \). From the Jensen’s inequality [
43], the estimated probability obtained via marginal mean method could differ from that obtained via marginal standardisation method because the link function is not linear. However, if the range of the linear predictor values is in a neighbourhood where a linear function can be used to approximate the link function,
\( \hat{\mathrm{RR}}\mathrm{s} \) from the two methods can be similar. Although these two methods gave similar results in our real data analysis, the marginal standardisation method is more commonly used in epidemiology studies [
19,
25‐
27,
44] when compared with the marginal means method, and can be viewed as a special case of G-computation method in the causal inference literature [
19], and can be generalised to binomial models with other link functions [
26].
When faced with multiple threshold values to dichotomise the continuous outcome, the conventional approach would apply probit or logit model to the dichotomised outcome at each threshold value resulting in a multiple testing problem when assessing whether β1 (or RR) equals to 0 (or 1). In the real dataset analyses, after applying Bonferroni correction to account for the multiple testing problem with binary models, only RR defined at threshold value corresponding to 20 mmol/L was significant while β1s for all threshold values were not significant. However, with linear models, we avoided the multiple testing problem for β1. We first assessed whether β1 returned a significant finding before proceeding to identify the threshold value where RR ≠ 1. Linear models in the real dataset analyses returned significant findings for β1 and the follow-up analyses for RR, in general, returned significant results at various threshold values after accounting for multiple testing. In particular, applying marginal standardisation method to normal linear model to obtain \( \hat{\mathrm{RR}}\mathrm{s} \) provided stronger evidence of reduction in hyperglycaemia risk after intervention although \( \hat{\mathrm{RR}}\mathrm{s} \) were similar across methods. For future studies involving multiple thresholds, one could mimic the ANOVA testing procedure by starting with a test for β1 from the linear model before proceeding to perform post-hoc tests for RR at each threshold value with marginal standardisation or mean method.
Our proposed method has some limitations. We did not consider dichotomisation based on two thresholds, e.g., I(τ
1 <
Yi < τ
2) or I(
Yi < τ
1,
Yi > τ
2), but our proposed approach can be extended to this scenario and it is beyond the scope of this paper. Although we have presented an application to binary exposure, our approach can be applied to categorical or continuous exposure as well [
11,
16,
36,
45]. The marginal standardisation and mean methods in this paper were used to estimate RR only, but both methods can also be used to estimate absolute risk reduction, RR reduction and number needed to treat pending on the research question [
44,
46].
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