Propensity scores are a statistical method for adjusting for baseline differences between study groups. The scores are based on the probability of a subject being in a particular group, conditional on that subject’s values on those independent variables though to influence group membership. Propensity scores with multivariable analysis produces a better adjustment for baseline differences than simply including potential confounders in a multivariable model predicting outcome. Propensity scores are also particularly helpful when outcomes are rare and the proportions of subjects in the independent groups are relatively equal. Another advantage of propensity scores is that they make no assumptions about the relationship between the individual confounders and outcome. The adequacy of a propensity score is judged by whether there is sufficient overlap between the groups and whether it balances the covariates.

There are four major ways you can use propensity scores: matching, weighting, stratified, as a covariate in a model predicting outcome.

The choice of multivariable model depends primarily on the type of outcome variable. Use multiple linear regression and analysis of variance for interval outcomes, multiple logistic regression and log-binomial regression with dichotomous outcomes, proportion odds regression with ordinal outcomes, multinomial logistic regression for nominal outcomes, proportional hazards analysis for time to outcome, Poisson regression and negative binomial regression for counts and for incidence rates. Each model has a different set of underlying assumptions. All of the models assume that there is only one observation of outcome for each subject.

Having determined the type of multivariable analysis to perform based on the outcome variable, one must next determine how to incorporate independent variables into the model. The important considerations are the type of independent variables you have: dichotomous, nominal, interval, or ordinal, and the relationship between the independent variable and the outcome; and the relationship between the independent variable and the outcome. Dichotomous independent variables can be used in any multivariable analysis. The other types of independent variables require special consideration. With interval variables, identifying non-linear associations is particularly important; when the association is nonlinear the variable should be transformed. The type of transformation will depend on the association with outcome. Splines are a sophisticated method of modeling complex relationships between an interval independent variable and the outcome.

Standard statistical analyses assume that each observation (subject) is independent. In other words, the outcomes of different subjects are not correlated. For example, in a longitudinal study, subjects may be assessed repeatedly. Subjects may also be enrolled in established groups or clusters such as families or physician practices. However, when this is not the case, multivariable models that incorporate correlated observations are needed. Common choices are generalized estimating equations and mixed-effects models. Generalized estimating equations are population-averaged models; they estimate the mean difference between the two groups. This is in contrast to mixed-effects models which estimate subject specific differences. Conditional logistic regression is useful with a dichotomous outcome that is measured repeatedly. The Andersen-Gill formation of the proportional hazards model is useful for censored data with outcomes that can over more than once to a subject over time.

Assessing the underlying assumptions of multiple models enables us to improve their fit. But it is a complicated process that is more art than science. The basic measure to assess the fit of models is residuals. Residuals are the difference between the observed and the estimated value. They can be thought of as the error in estimation. There are a number of possible transformations of the residuals for different multivariable procedures. For proportional hazards analysis it is important to test the proportionality assumption. This can be done using a log-minus-log survival plot, Schoenfeld’s residuals, division of time into discrete intervals, or time-dependent covariates.

An emulation (or target) trial uses observational data to simulate a trial. Because there is no actual randomization, multivariable methods need to adjust for differences between groups. However, emulation trials improve traditional observational studies by conducting all the same steps as a randomized trial with the exception of randomization. With an emulation trial, before conducting data analysis, specify research question eligibility criteria, determination of treatment groups, start of study and end of follow-up, outcome, and analysis plan. Active comparators can minimize indication bias. By setting eligibility, treatment assignment, and start of follow-up, emulation trials minimize immortal time bias.

Classification and regression trees (CART): a technique for separating subjects into distinct subgroups based on a dichotomous outcome. Its major advantage over multiple logistic regression—it more closely reflects how clinicians make decisions. Certain pieces of information take you down a particular diagnostic path for more information to prove/disprove you are on the right path. Most clinicians do not total up a weighted version of the information and make a decision.

Multivariable analysis is needed because most events, whether medical, politica, social, or personal, have multiple causes. And these causes are related to one another. Multivariable analysis enables us to determine the relative contributions of different causes to a single event or outcome.

Multivariable analysis enables us to identify and adjust for confounders. Confounders are associated with the risk factor and causally related to the outcome. Adjustment for confounders is key to distinguishing important etiologic risk factors from variables that only appear to be associated with outcomes due to their association with the true risk factor.

Stratification can also be used for identifying independent relationships between risk factors and outcomes but becomes too cumbersome when there are more than one or two possible confounders.

In setting up your model, include those variables, in addition to the risk factor or group assignment, that have been theorized or shown in prior research to be confounders or those that empirically are associated with the risk factor and the outcome in bivariate analysis.

Exclude variables that are on the intervening pathway between the risk factor and outcome, those that are extraneous because they are not on the causal pathway, redundant variables, and variables with a lot of missing data.

Sample size calculation for multivariable analysis is complicated but statistical programs exist to help you to calculate it. Missing data on independent variables can compromise your multivariable analysis. Several methods exist to compensate for missing independent data including deleting cases, using indicator variables to represent missing data and estimating the value of missing cases. Methods also exist for estimating missing outcome data using other data you have about the subject and multiple imputation.

A published report should include a sufficient explanation of the statistical methods so that someone with access to the original data could reproduce the reported results. Generally, it is best to divide the methods section of your paper into how subjects were enrolled (Subjects), what interventions were used or how data were acquired (Procedures), how the variables were coded (Measures), and how the data were analyzed (Statistical analysis). Unless there is no missing data, it is important to report the n for each analysis.

What results to report in your paper will vary based on your research question, your analysis, and the style of the journal. In general, for multiple linear regression models, report the regression coefficients, the standard errors of the coefficients, and the statistical significance levels of the coefficients. For logistic regression, report the odds ratio and the 95% confidence interval; for proportional hazards regression, report the relative hazard and the 95% confidence interval.

A valid model is one whether the inferences drawn from it are true. Many factors can threaten the validity of a model including imprecise or inaccurate measurements, bias in study design or in sampling, and misspecification of the model itself.

A key way to validate a model is to replicate the findings with new data. The best method of replication is collecting new data. However, when that is not possible, it is possible to perform a replicate by dividing the sample using a split-group, jackknife, or bootstrap method. Of these 3 methods, split-group is the strongest but requires a dataset large enough to split your sample. A bootstrap is the weakest method of replication, but produces more valid confidence intervals than a simple model.

Multivariable techniques produce two major kinds of information: Information about how well the model (all the independent variables together) fit the data and information about the relationship of each of the independent variables to the outcome (with adjustment for all other independent variables in the analysis). Common measures of the strength of the relationship between an independent variable and the outcome are odds ratio, relative hazard, and relative risk. Adjusting for multiple comparisons is challenging; most important, is to decide ahead of time whether there will be adjustments of multiple comparisons. A common convention is to not adjust the primary outcome, but to adjust secondary outcomes for multiple comparisons.

The strength of multivariable analysis is its ability to determine how multiple independent variables, which are related to one another, are related to an outcome. However, if two variables are so closely correlated that if you know the value for one variable you know the value of the other, multivariable analysis cannot separately assess the impact of these two variables on outcome. This problem is called multicollinearity.

To assess whether there is multicollinearity, investigators should first run a correlation matrix. However, the matrix only tells you the relationship between any two independent variables. Harder to detect is whether a combination of variables accounts for another variable’s value. Two related measure of muticollinearity are tolerance and the reciprocal of tolerance: the variation inflation factor. If you have variables that are highly related, you can omit one or more of the variables, use an “and/or” clause or create a scale.

Sensitivity analysis tests how robust the results are to changes in the underlying assumptions of your analysis. In other words, if you made plausible changes in your assumptions, would you still draw the same conclusions? The changes could be a more restrictive or inclusive sample, a different way to measure your variables, a different way for handling missing data, or a change of a different feature of your analysis. With sensitivity analysis you cannot lose. If you vary the assumptions of your analysis and you get the same result, you will have more confidence in the conclusions of your study. Conversely, if plausible changes in your assumptions lead to a different conclusion, you will have learned something important. A common assumption tested in sensitivity analysis is that there are no unmeasured confounders, which can be tested with E values or falsification analysis. Other common assumptions tested are that losses to follow-up are random, that the sample is unbiased, that there is the correct exposure period and follow-up period, that there is a biased predictor or outcome, or that the model is misspecified.

Multivariable analysis is used for four major types of studies: observational studies of etiology, randomized and nonrandomized intervention studies, studies of diagnosis, and studies of prognosis.

For observational studies, whether etiologic or intervention, the most important reason to do multivariable analysis is to eliminate confounding, since in observational studies the groups are not randomly assigned. With randomized studies, multiple analysis is used to adjust for baseline differences that occurred by chance, to identify other independent predictors of outcome besides the randomized group, and x.

With studies of diagnosis, multivariable analysis is used to identify the best combination of diagnostic information to determine whether a person has a particular disease. Multivariable analysis can also be used to predict the prognosis of a group of patients with a particular set of known prognostic factors.