Regression

Logistic, linear and conditional logistic regression - and when to use robust standard errors

Published

July 2, 2026

Warning

Under development. More model types and diagnostics are coming.

Regression examines the relationship between an outcome and one or more explanatory variables (your exposure plus covariates), and is how you adjust for confounding in the analysis.

The outcome is also called the dependent variable and the explanatory variables the independent ones. With one explanatory variable it is a simple regression; with several, a multiple regression. Explanatory variables can be either continuous (e.g. age) or categorical (e.g. sex).

glm stands for generalized linear model: one function that, via the family argument, can fit several model types (here binomial = logistic). clogit() is a “conditional” version of logistic regression that accounts for the matching. gtsummary::tbl_regression() shows the result as a publication-ready table.

Each model also rests on some assumptions. They are noted with each model below, together with how to check them; if they do not hold, the estimates can be misleading.

Tip

New to regression? This page shows how to run the models in R - not the theory behind them. For a thorough introduction (model types, assumptions, interpretation) see the Epidemiologist R Handbook and R for Data Science. If it is the statistical theory behind regression you are missing, Learning Statistics with R (Navarro) is beginner-friendly. More resources: Learning resources.

Note

The code examples use generic path and variable names. Adapt them to your project. The packages used (gtsummary, survival, sandwich, lmtest) must be installed in your R environment on DST.

Starting point

library(dplyr)                         # %>% (pipe)
library(gtsummary)                     # tbl_regression() for clean output

df <- readRDS("path/to/analysis.rds")  # analysis-ready dataset

Logistic regression (binary outcome)

model <- glm(outcome ~ exposure + age + sex,   # outcome explained by exposure + covariates
             data = df, family = binomial)     # family = binomial -> logistic regression

model %>%                                # pass the model on to a table
  tbl_regression(exponentiate = TRUE)    # exponentiate = TRUE -> odds ratios (OR)
  • family = binomial makes it logistic regression.
  • exponentiate = TRUE shows odds ratios (OR) instead of log-odds coefficients.

Assumptions: the observations are independent (one row per person - otherwise use robust standard errors, see below), and continuous variables are linearly related to the log-odds. If linearity does not hold, you can let the variable enter as a smooth curve instead of a straight line - see Non-linear relationships (splines) below.

Linear regression (continuous outcome)

lm(bmi ~ exposure + age + sex, data = df) %>%  # linear model for a continuous outcome
  tbl_regression()                       # coefficients (no exponentiation)

Assumptions: a linear relationship, constant variance (homoscedasticity), roughly normal residuals and independent observations. Save the model to a variable and check with plot(model), which gives the classic residual plots; if they look skewed, a transformation of the outcome or a different model may be needed.

Matched case-control: conditional logistic regression

When you have matched in a nested case-control design (Case-control), the analysis must respect the matching with conditional logistic regression. Use clogit() with a strata() term for the matched set.

library(survival)                        # clogit()

clogit(case ~ exposure + age + sex + strata(match_id),  # strata() = the matched set
       data = cc) %>%                        # cc = your case-control dataset (from Phase 10b)
  tbl_regression(exponentiate = TRUE)    # -> odds ratios
  • strata(match_id) tells the model which rows belong to the same matched set (case.id from incidenceMatch() or Set from Epi::ccwc()).
  • case = 1 for a case, 0 for a control.

Assumptions: as for logistic regression (continuous variables linear with the log-odds; the matched sets independent of one another). The matching itself is handled by strata().

Interaction (effect modification)

Sometimes an exposure’s effect depends on a third variable: a drug may work more strongly in younger than older people, or a risk factor may hit men harder than women. This is called effect modification (or statistical interaction), and you examine it with a product term in the model.

# `*` expands to: exposure + sex + exposure:sex (the interaction term itself)
model <- glm(outcome ~ exposure * sex + age,        # tests whether the exposure's effect depends on sex
             data = df, family = binomial)

model %>%
  tbl_regression(exponentiate = TRUE)               # OR for each term, including the interaction term
  • exposure * sex means “both variables plus their interplay”. It is the interaction term exposure:sex that you read off.
  • If the interaction term’s OR sits close to 1 (and p is large), the exposure works the same in the two groups. If it differs clearly from 1, the effect depends on sex, and you should then report group-specific estimates (e.g. by running the model separately for each sex, or with emmeans / marginaleffects).
Note

Multiplicative vs. additive scale. A product term tests interaction on the multiplicative (ratio) scale: does the exposure change the odds ratio differently between groups? At the public-health level the additive scale (differences in absolute risk) is often more relevant, and it is measured with RERI (relative excess risk due to interaction). The two scales can give different answers, so be explicit about which you report. See Read more.

Warning

A subgroup analysis is not an interaction test. Running the model separately in two groups and seeing that one is “significant” and the other is not does not show that the effects differ. Only the interaction term (or a formal test of it) settles whether the difference is real.

Why not? (click)

Two reasons:

  1. The two tests answer a different question. A significance test within each group only asks: “is this group’s effect different from no effect (OR = 1)?” It never asks whether the two groups’ effects differ from each other - and only the latter is interaction.
  2. Significance depends on precision (group size), not just on the size of the effect. A smaller group gives a wider confidence interval and more easily lands “non-significant”, even when the point estimate is the same.

Example with an identical effect in both groups:

  • Men: OR = 1.5 (95% CI 1.1-2.0), p = 0.01 → “significant”
  • Women: OR = 1.5 (95% CI 0.9-2.5), p = 0.12 → “not significant”

The point estimate is the same (1.5); only the precision differs, because the women’s group is smaller. Concluding “the effect depends on sex” would be wrong. To compare the two effects you need the uncertainty of both estimates at once, and that is exactly what the interaction term does. (The phenomenon is known as “the difference between ‘significant’ and ‘not significant’ is not itself statistically significant”, Gelman & Stern, The American Statistician, 2006.)

Non-linear relationships (splines)

By default, the models above assume a continuous variable is related to the outcome in a straight line (on the model’s scale): each extra year of age changes the risk by the same amount, whether you go from 30 to 31 or from 70 to 71. That often fits poorly - the risk of many outcomes rises only slowly at young ages and steeply later. That is a curve, not a straight line.

A spline lets the variable enter as a smooth curve instead of a straight line, so the model finds the shape itself. You only decide how bendy the curve may be (via the number of degrees of freedom - how many “bends” the curve may have).

library(splines)                         # ns() = "natural spline" (a smooth curve)

glm(outcome ~ ns(age, df = 3) + exposure + sex,  # age as a smooth curve instead of a straight line
    data = df, family = binomial)
  • ns(age, df = 3) replaces the straight line for age with a smooth curve. df = 3 controls the bendiness: more degrees of freedom = a more flexible curve. 3-4 is a common starting point.
  • It works in all the models on these pages (logistic, linear, Cox, Poisson) - just wrap the continuous variable in ns(...).
  • Especially useful when the variable is your exposure (so you see the dose-response shape itself) or a strong confounder (so a too-rigid straight line does not leave residual confounding, i.e. confounding left after adjustment).
Note

You no longer read off a single number. A spline variable has no single odds ratio, because the effect changes across the values. Instead you plot the fitted curve (e.g. predicted risk against age) to show the relationship. The ggeffects or marginaleffects packages do this in a few lines.

Reading your result

For each variable the regression table gives an estimate, a confidence interval and a p-value:

  • The estimate is the effect measure. For logistic regression it is an odds ratio (OR), for log-binomial a relative risk (RR), and for Cox (Time-to-event) a hazard ratio (HR) - all are ratios, where 1 means no difference (above 1 = higher odds/risk/rate, below 1 = lower). For linear regression the estimate is instead a difference in means, where 0 means no difference. Example: OR = 1.5 → 50% higher odds; OR = 0.8 → 20% lower.
  • The 95% confidence interval is the range the true effect can plausibly lie in: narrow = precise, wide = much uncertainty. For a ratio: if the interval crosses 1, the effect is not statistically significant at the 5% level.
  • The p-value is the probability of seeing an effect at least as large as yours if there were truly none. A small p (typically < 0.05) argues against “no effect”, but does not tell you how large or important the effect is - the estimate and confidence interval do. In very large register datasets even tiny, unimportant differences often become “significant”, so always look at the size (the estimate), not just p.
Note

OR and HR are non-collapsible. Unlike a risk difference or relative risk, an adjusted odds ratio (and hazard ratio) can differ from the crude one even when there is no confounding, purely because you added covariates. So an adjusted OR is a conditional effect (within strata of the covariates), not the population-average effect, and “adjusted ≠ crude” is not by itself evidence of confounding. If you need a marginal (population-average) effect, standardization or IPTW gives one - see IP weighting.

Advanced: robust (clustered) standard errors - skip if each person appears only once

When the same person appears several times in the dataset - e.g. if the comparison cohort is matched with replacement, or a comparison person later becomes exposed (crossover, see Comparison cohort) - the rows are not independent. If you ignore this, the confidence intervals come out too narrow. The fix is clustered (robust) standard errors: a way of computing the uncertainty that accounts for rows from the same person being linked. (If each person appears only once, you don’t need this.)

library(sandwich)                        # vcovCL(): "sandwich" estimator of the variance
library(lmtest)                          # coeftest(): test the coefficients with a chosen variance

model <- glm(outcome ~ exposure + age + sex, data = df, family = binomial)

coeftest(model,                          # show the coefficients ...
         vcov = vcovCL(model, cluster = ~ pnr))  # ... with SEs clustered on person id
  • vcovCL(model, cluster = ~ pnr) is the cluster-robust covariance matrix (clustered on pnr). “Sandwich” only refers to the shape of the formula - the point is that it does not assume independent rows.
  • For a Cox model you do it with the cluster argument in coxph() instead (see Time-to-event).
Note

Mixed models: when groups in your data should each have their own level.

A cluster is a group of rows that belong together and are therefore more alike than two random rows - a group of rows, not necessarily of people. It can be several rows from the same person (e.g. repeated measurements over time), all patients at the same hospital, or siblings in the same family. So a person is a cluster only if they appear in several rows; if you have exactly one row per person, there is no person cluster. Rows in the same cluster are correlated because they share something, and that breaks the assumption of independent rows. You have two choices:

  • Cluster-robust standard errors (fold-out above): keep your model, but correct the confidence intervals so they account for the correlation. The effect estimate is unchanged; you still get one overall effect.
  • Mixed model (random effects): let each cluster have its own baseline level in the model (a random intercept) - e.g. each hospital gets its own baseline risk, instead of the model pretending all hospitals are the same. (To “model the cluster” means exactly that: building the groups’ differences into the model instead of ignoring them.) “Mixed” = it mixes fixed effects (the usual coefficients, the same for everyone, e.g. the exposure effect) with random effects (the variation between clusters).

Choose a mixed model when the clustering structure is itself of interest, when you have repeated measurements or many small clusters, or when you want to separate variation within and between clusters. In R: lme4 (glmer()/lmer()), and coxme for a Cox model. A thorough treatment is beyond this page.

Note

Remember: anything leaving DST must go through output control - no small cells, only aggregated results. See Phase 14 - Export and repatriation.

TipFurther information

Further depth in The Epidemiologist R Handbook:

On interaction / effect modification:

On non-linear relationships (splines):

  • Harrell, Regression Modeling Strategies, and the rms package - the standard reference on flexible modelling with splines.

On mediation (when you want to split an effect into a direct part and an indirect part that goes through an intermediate variable):

  • CMAverse - causal mediation analysis in R.
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