Common pitfalls in statistical analysis: Logistic regression (2024)

Abstract

Logistic regression analysis is a statistical technique to evaluate the relationship between various predictor variables (either categorical or continuous) and an outcome which is binary (dichotomous). In this article, we discuss logistic regression analysis and the limitations of this technique.

Keywords: Biostatistics, logistic models, regression analysis

INTRODUCTION

In a previous article in this series,[1] we discussed linear regression analysis which estimates the relationship of an outcome (dependent) variable on a continuous scale with continuous predictor (independent) variables. In this article, we look at logistic regression, which examines the relationship of a binary (or dichotomous) outcome (e.g., alive/dead, success/failure, yes/no) with one or more predictors which may be either categorical or continuous.

Let us consider the example of a hypothetical study to compare two treatments, variceal ligation and sclerotherapy, in patients with esophageal varices [Table 1a]. A simple (univariate) analysis reveals odds ratio (OR) for death in the sclerotherapy arm of 2.05, as compared to the ligation arm. This means that a person receiving sclerotherapy is nearly twice as likely to die than a patient receiving ligation (please note that these are odds and not actual risks – for more on this, please refer to our article on odds and risk).[2]

We, however, know that factors other than the choice of treatment may also influence the risk of death. These could include age, gender, concurrent beta-blocker therapy, and presence of other illnesses, among others. Let us look at the effect of beta-blocker therapy on death by constructing a 2 × 2 table [Table 1b]; this reveals an OR for death in the “no beta-blocker” arm of 4.1 as compared to the “beta-blocker” arm. Similarly, we can determine the association of death with other predictors, such as gender, age, and presence of other illnesses. Each of these analyses assesses the association of the dichotomous outcome variable - death - with one predictor factor; these are known as univariate analyses and give us unadjusted ORs.

However, often, we are interested in finding out whether there is any confounding between various predictors, for example, did equal proportion of patients in ligation and sclerotherapy arms receive beta-blockers? This can be done by stratifying the data and making separate tables for two levels of the likely confounder, for example, beta-blocker and no beta-blocker [Table 1c]. From this analysis, it is obvious that ligation increased the risk of death among those receiving beta-blockers but reduced this risk among those not receiving beta-blockers. Therefore, it would be inappropriate for us to look at the effect of ligation versus sclerotherapy without accounting for beta-blocker administration. Similarly, we may wish to know whether the age of patients, a continuous variable, was different in the two treatment arms and whether this difference could have influenced the association between treatment and mortality. In fact, in real life, we are interested in assessing the concurrent effect of several predictor factors (both continuous and categorical) on a single dichotomous outcome. This is done using “multivariable logistic regression” – a technique that allows us to study the simultaneous effect of multiple factors on a dichotomous outcome.

HOW DOES MULTIPLE LOGISTIC REGRESSION WORK?

The statistical program first calculates the baseline odds of having the outcome versus not having the outcome without using any predictor. This gives us the constant (also known as the intercept). Then, the chosen independent (input/predictor) variables are entered into the model, and a regression coefficient (known also as “beta”) and “P” value for each of these are calculated. Thus, the software returns the results in a form somewhat like [Table 2a]. The “P” value indicates whether the particular variable contributes significantly to the occurrence of the outcome or not. These results can also be expressed as an equation [Table 2b], which includes the constant term and the regression coefficient for each variable, which has been found to be significant (usually using P < 0.05). The equation provides a model which can be used to predict the probability of an event happening for a particular individual, given his/her profile of predictor factors.

The coefficients represent the logarithmic form (using the natural base represented by “e”) of odds associated with each factor and are somewhat difficult to interpret by themselves. The software tools often also automatically calculate antilogs (exponentials; as shown in the last column of Table 2a) of the coefficients; these provide adjusted ORs (aOR) for having the outcome of interest, given that a particular exposure is present, while adjusting for the effect of other predictor factors. These aORs can be used to provide an alternative representation of the model [Table 2c].

For categorical predictors, the aOR is with respect to a reference category (exposure absent). For example, the aOR for treatment gives the chance of death in the sclerotherapy group as compared to the ligation group, i.e., patients receiving sclerotherapy are 1.4 times likely to die than those receiving ligation, after adjusting for age, gender, and presence of other illnesses. For gender, it refers to the odds of death in women versus men (i.e., women are 0.25 times [one-fourth] as likely to die as males, after adjusting for type of treatment, age, and presence of other illnesses). Most software tools allow the user to choose the reference category. For example, for gender, one could choose “female” as the reference category – in that case, the result would provide the odds of death in men as compared to women. The results would obviously be different in that case – with software returning the aOR for gender of 4 (= 1/0.25), i.e., men are four times more likely to die than women after adjusting for other factors. If a factor has more than two categories (e.g., nonsmoker, ex-smoker, current smoker), then separate ORs are calculated for each of the other categories relative to a particular reference category (ex-smoker vs. nonsmoker; current smoker vs. nonsmoker).

For continuous predictors (e.g., age), the aOR represents the increase in odds of the outcome of interest with every one unit increase in the input variable. In the example above, increase in age by each one year increases the odds of death by 6% (OR of 1.06). This increase is multiplicative; for instance, an increase of age by 3 years would lead to an increase in odds of death by 1.06 × 1.06 × 1.06 (or [1.06]³).

As discussed in our previous article on odds and risk,[2] standard errors and hence confidence intervals can be calculated for each of these aORs. Many software programs do this automatically and include these values in the results table. Further, these softwares also provide an estimate of the goodness-of-fit for the regression model (i.e., how well the model predicts the outcome) and how much of the variability in the outcome can be explained by each predictor.

STATISTICAL TECHNIQUES FOR REGRESSION MODELS

Various methods have been proposed for entering variables into a multivariate logistic regression model. In the “Enter” method (which is the default option on many statistical programs), all the input variables are entered simultaneously. Alternative methods include “forward stepwise” regression (where various factors are introduced one by one, beginning with the strongest, and stopping when addition of the next factor does not significantly improve prediction), “backward stepwise” (where all the factors are initially introduced and then various factors are withdrawn one by one, till the overall prediction does not deteriorate), or bidirectional (a mix of the forward and backward methods).

CAUTIONS AND PITFALLS

SUGGESTED READING

Antwi et al. developed and validated a prediction model for gestational hypertension (GH).[5] They first compared groups of women with and without GH, using the independent t-test for continuous variables and the Chi-square test for categorical variables (univariate analyses). Predictors that were found to be related to GH (P ≤ 0.20) were then entered into a multivariable logistic regression model, using stepwise backward selection. The final model with aORs for the various predictors is shown in Table 3. Readers may like to read this paper as a practical example.

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