Multivariable Analysis: A Primer for Readers of Medical Research

Most published medical research uses multivariable analysis. Unfortunately, many readers, especially those uncomfortable with mathematics, treat multivariable models as a black box, accepting the author's explanation of the results without independently assessing whether the models are correctly constructed or interpreted. However, multivariable models can be understood without undue concern for the underlying mathematics. I review the basics of multivariable analysis, including why multivariable models are used, what types exist, what assumptions underlie them, how they should be interpreted, and how they can be evaluated. What Is Multivariable Analysis? Multivariable analysis is a statistical tool for determining the unique contributions of various factors to a single event or outcome. For example, numerous factors are associated with the development of coronary heart disease, including smoking, obesity, sedentary lifestyle, diabetes, elevated cholesterol level, and hypertension. These factors are called risk factors, independent variables, or explanatory variables. Multivariable analysis allows us to determine the independent contribution of each of these risk factors to the development of coronary heart disease (called the outcome, the dependent variable, or the response variable). Why Is Multivariable Analysis Needed? In many clinical situations, experimental manipulation of study groups would be unfeasible, unethical, or impractical. In these circumstances, multivariable analysis can be used to assess the association between multiple risk factors and outcomes. For example, we cannot test whether smoking increases the likelihood of coronary heart disease by randomly assigning persons to groups who smoke and groups who do not smoke. Although bivariate analysis of longitudinal data demonstrates that smokers are more likely than nonsmokers to develop coronary heart disease, this is weak evidence of a causal association. Perhaps the only reason smokers are more likely to develop coronary heart disease is that they are more likely to be male, live in poverty, and have a sedentary lifestyle. In other words, the relationship between smoking and coronary artery disease may be confounded by these other variables. Confounding occurs when the apparent association between a risk factor and an outcome is affected by the relationship of a third variable to the risk factor and to the outcome; the third variable is a confounder. For a variable to be a confounder, the variable must be associated with the risk factor and causally related to the outcome (Figure 1). Male sex, poverty, and sedentary lifestyle could be confounders because they are associated with both smoking and coronary heart disease. With multivariable analysis, we can demonstrate that even after adjusting for male sex, poverty, and sedentary lifestyle, smoking has an independent relationship with coronary artery disease (Figure 2). Figure 1. Relationship among risk factor, confounder, and outcome. Figure 2. Multivariable association between four risk factors and coronary artery disease. A study of the association between periodontal disease and coronary heart disease illustrates how multivariable analysis can be used to identify confounders (2). Bivariate analysis demonstrates that persons with periodontitis have a markedly increased rate of coronary heart disease (relative hazard, 2.66 [95% CI, 2.34 to 3.03]). If this relationship were independent and causal, then interventions that would reduce periodontitis would decrease the occurrence of coronary heart disease. However, periodontitis is also associated with several factors known to be related to coronary heart disease, including older age, male sex, poverty, smoking, increased body mass index, and hypertension, raising the question of whether the association between periodontitis is due to confounding by these factors (Figure 3). With multivariable adjustment for these variables, sampling design, and sampling weights, the association between periodontitis and coronary heart disease weakens substantially: the relative hazard decreases to 1.21; the 95% CIs for the relative hazard (0.98 to 1.50) crosses 1.0; and the association between periodontitis and coronary artery disease is no longer statistically significant. Figure 3. Potential confounders of the relationship between periodontitis and coronary heart disease. Although one can theoretically distinguish independent associations from confounding, a variable may have both an independent effect on outcome and be a confounder of another variable's relationship to outcome. For example, poverty is a confounder of the relationship between smoking and coronary artery disease (poor people are more likely to smoke and to develop coronary artery disease), but poverty also has an independent effect on development of coronary artery disease (after adjustment for smoking, cholesterol level, and other known risk factors, poor persons are more likely to develop coronary artery disease). Multivariable analysis is not the only statistical method for eliminating confounding. Stratified analysis can also assess the effect of a risk factor on an outcome while holding other variables constant, thereby eliminating confounding. For example, the effect of periodontitis on coronary heart disease can be examined separately for men and women, which removes the effect of sex on the relationship between these diseases. If periodontitis is no longer significantly associated with coronary heart disease when men and women are looked at separately, then sex was confounding the relationship between the two. If periodontitis is still associated with coronary heart disease when men and women are assessed separately, then the effect of periodontitis on coronary heart disease is independent of sex. Stratification works well when there are only two or three confounders. However, when there are many potential confounders, stratifying for all of them will create literally hundreds of groups in which the investigators would need to determine the relationship between periodontitis and coronary heart disease. Because the sample sizes would be small, the estimates of risk would be unstable. Whether investigators use multivariable analysis or stratification, it is important to remember that they can only adjust for measured variables. Results may still be confounded by known and unknown unmeasured factors. What Types of Multivariable Analysis Are Commonly Used in Clinical Research? The three types of multivariable analysis that are commonly used in clinical research are multiple linear regression, multiple logistic regression, and proportional hazards (Cox) regression (Table). Linear regression is used with interval (also called continuous) outcomes (such as blood pressure). With interval variables, equally sized differences on all parts of the scale are equal. Blood pressure is an interval variable because the difference between a blood pressure of 140 and 143 mm Hg (3 mm Hg) is the same as the difference between a blood pressure of 150 and 153 mm Hg (3 mm Hg). Logistic regression is used with dichotomous outcomes (yes or no; for example, death). Proportional hazards regression is used when the outcome is the length of time to reach a discrete event (such as time from baseline visit to death). Table. Types of Multivariable Analysis How Is the Effect of an Individual Variable on Outcome Assessed in a Multivariable Analysis? The regression coefficient for each variable must be estimated by fitting the model to the data and adjusting for all other variables in the model. With logistic regression and proportional hazards regression, the coefficients have a special meaning. The antilogarithm of the coefficient equals the odds ratio (for logistic regression) and the relative hazard (for proportional hazards regression). The hazard is the probability that a person experiences an outcome in a short time interval, given that the person has survived to the beginning of the interval. When the outcome is uncommon (<15%), the odds ratio and relative hazard are reasonable estimates of the relative risk. For example, if the odds ratio or relative hazard for the association between smoking and fatal heart attacks is 3.0 (assuming that fatal heart attacks occurred in <15% of patients), then smoking roughly triples the risk for a fatal heart attack. If the odds ratio or relative hazard for the association between estrogen use and development of a pathologic fracture is 0.33, then persons who take estrogen have roughly a third of the risk for fracture as persons who do not take estrogen. When the outcome is common, the odds ratio remains a useful measure of association, but it does not approximate the relative risk. For example, a randomized trial of persons with bronchopulmonary aspergillosis showed better response to itraconazole (13 of 28 patients) than to placebo (5 of 27 patients) (3). The odds ratio is 4.7 [(13 27)/(15 5)], but the relative risk is only 3.0 [(13/28)/(5/32)]. With interval-independent variables, the coefficient and the resulting odds ratio or relative hazard can be misunderstood. For example, an observational study reported that the odds ratio for the effect of low-density lipoprotein cholesterol on coronary artery calcification was 1.01 (CI, 1.00 to 1.02) (4). This may seem like a trivial effect until you notice that the odds ratio of 1.01 is for each increase of 1 mg/dL of low-density lipoprotein cholesterol. An increase of 40 mg/dL of cholesterol would produce an odds ratio of 1.49 (1.01) 40. This example demonstrates that the size of the coefficient of an interval variable is entirely dependent on the units being used. With interval-independent variables, readers must also assess whether the model accurately captures the relationship between the variable and the outcome. Multivariable models assume that increases (or decreases) in an interval-independent variable will be associated with