A Researcher's Guide to Multiple Regression for Dose-Response and Covariate Analysis
- Aug 17
- 7 min read
Most bench scientists reach for a t-test or ANOVA by reflex, and for a clean two-group comparison that's the right call. But a lot of real experiments aren't clean two-group comparisons. Your treatment effect might depend on passage number. Your two groups might differ at baseline. Your predictor might be a continuous dose rather than a category. In each of those cases the reflex test either throws away information or gives you an answer that quietly builds in the confounder you were worried about.
Multiple regression handles all three. It lets you model an outcome as a function of several predictors at once, so you can ask what your treatment does while accounting for the other things that vary. And once you understand it, a fact that surprises most people becomes clear: ANOVA and ANCOVA are not separate techniques, they're special cases of the same underlying model.
This guide covers when multiple regression earns its place in biomedical work, how to read its coefficients honestly, the two covariate mistakes that invalidate the analysis, and the important exception where regression is the wrong tool for dose-response.
The principle: one model, several predictors
A simple linear regression fits a line: outcome = intercept + slope × predictor. Multiple regression extends this to several predictors at once:
Y = β₀ + β₁X₁ + β₂X₂ + ... + error
Each coefficient (β) estimates how much Y changes per one-unit increase in that predictor while the other predictors are held constant. That last clause is the entire value proposition. It's what lets you separate the effect of your treatment from the effect of age, batch, baseline value, or passage number, provided those things are in the model.
Note: this is also why ANOVA, ANCOVA, and multiple regression give identical results when set up equivalently. They're all the general linear model wearing different labels. ANCOVA is simply multiple regression with a categorical predictor of interest plus continuous covariates you want to control for. Knowing this saves you from treating them as a menu of unrelated tests.
When multiple regression beats a t-test or ANOVA
Reach for it in these situations:
Your predictor is continuous. Comparing three arbitrary dose groups by ANOVA throws away the dose information itself. Regression uses the actual concentrations and estimates a slope, which is both more powerful and more informative.
Your groups differ at baseline. If treated and control cells started at different confluence, comparing endpoints alone confounds treatment with starting point. Adding baseline as a covariate estimates the treatment effect adjusted for it. This is the classic ANCOVA use, and it typically increases power as well as reducing bias.
You need to account for nuisance variation. Passage number, plate, batch, operator, and run day all inject variance. Including them as covariates removes that variance from your error term, making a real treatment effect easier to detect.
You want to test an interaction. Regression lets you ask whether the treatment effect depends on another variable (does the drug work differently at high vs low baseline expression?), which no simple group comparison can answer.
Tip: in a properly randomized experiment, randomization already breaks the link between treatment and confounders, so adjusting for covariates isn't needed to remove bias. It can still be worth including strong prognostic covariates to reduce error variance and gain power.
The important exception: dose-response is usually nonlinear
Here's where a guide to regression has to warn you off regression, because this trips up a lot of pharmacology work.
Biological dose-response almost never follows a straight line. At very low concentrations, nearly nothing happens. Through a middle range, response climbs steeply. At high concentrations, receptors saturate and the response plateaus. Plotted against log dose, that's an S-shaped sigmoidal curve, not a line, and fitting a linear model to it will mis-estimate potency.
The standard tool is nonlinear regression with the four-parameter logistic (4PL) model, also called the Hill equation:
Y = Bottom + (Top − Bottom) / (1 + (X / IC50)^HillSlope)
Its four parameters are all directly interpretable: the bottom plateau, the top plateau, the IC50/EC50 (the concentration giving half-maximal response), and the Hill slope (curve steepness). EC50 and IC50 are the same quantity, differing only in whether the curve rises or falls.
Practical rules that follow from the model:
Fit and plot on a logarithmic dose axis. The sigmoid only appears on a log scale.
Design your dose grid to bracket both plateaus. If your concentrations don't reach the top and bottom, your IC50 is an extrapolation, not an estimate.
Report IC50 with a confidence interval, never as a bare point estimate.
Use the 5PL variant when the curve is visibly asymmetric.
Caution: the 4PL is nonlinear regression, not multiple linear regression. Where multiple regression belongs in dose-response work is alongside it, when you need to compare curves between conditions or adjust a potency comparison for a covariate.
How to read multiple regression coefficients without fooling yourself
Coefficients are easy to over-read. Three habits keep the interpretation honest.
"Holding others constant" is a statement about the model, not reality. A coefficient tells you the association with Y per unit of X within this model, given these covariates. If a relevant confounder wasn't measured, no amount of adjustment removes its bias, and the coefficient absorbs it. Regression cannot fix unmeasured confounding, which is why observational estimates deserve more caution than experimental ones.
Coefficients change when you add correlated predictors, and that's expected. Adding a covariate correlated with both your predictor and the outcome will shrink your predictor's coefficient relative to the unadjusted version. That's the adjustment working, not a bug. It's good practice to report both the unadjusted and adjusted estimates so readers can see how much the adjustment moved things.
Report the effect size and interval, not just the p-value. A coefficient is itself an effect size, in the units of your outcome per unit predictor. Give it with a confidence interval. A statistically significant slope of trivial magnitude is the same trap covered in our piece on statistical versus biological significance.
Two covariate mistakes that invalidate a multiple regression
Most regression failures in biomedical papers come from choosing covariates badly rather than from the mathematics.
Mistake 1: adjusting for a mediator. If your treatment works through a variable, controlling for that variable removes part of the very effect you're trying to measure. This is overcontrol bias, and it can shrink a real effect toward zero or reverse its sign. Covariates should be things that could confound the treatment-outcome relationship, not steps on the causal path between them. Decide this from the biology before you fit anything.
Mistake 2: ignoring the homogeneity-of-slopes assumption. ANCOVA assumes the covariate's effect on the outcome is the same in every group, meaning the regression lines are parallel. This is the most critical and most frequently overlooked assumption in covariate analysis. If the slopes genuinely differ, an adjusted mean is meaningless, because there's no single treatment effect to report, it depends on the covariate value. Test it by fitting the group × covariate interaction; if the interaction is significant, report the interaction rather than an adjusted mean.
Beyond covariate choice, check the standard assumptions: a linear relationship between covariate and outcome, independent observations (see our piece on replicates for why this one bites in cell culture), roughly normal residuals, and constant residual variance. Plot your residuals, always. Describing your design in plain language and having the model specification checked before you commit is a cheap safeguard (Sophie's analytics will walk through covariate choice and flag assumption violations in the output).
The takeaway
Multiple regression is the general tool your familiar tests are special cases of, and it earns its keep whenever your predictor is continuous, your groups differ at baseline, nuisance variation is eating your power, or you need to test an interaction. Read its coefficients as model-conditional associations rather than proven causal effects, report them with intervals alongside unadjusted estimates, and choose covariates from the biology, never adjusting for a mediator and always checking whether the slopes are actually parallel. And when your predictor is dose, remember that biology saturates: fit a 4PL curve on a log axis with a dose grid that reaches both plateaus, and report your IC50 with a confidence interval. The right model is the one that matches the shape of the thing you measured.
FAQ
When should I use multiple regression instead of ANOVA? Use multiple regression when your predictor is continuous (a dose rather than groups), when you need to adjust for covariates such as baseline values or batch, or when you want to test an interaction. ANOVA is a special case of the same general linear model, so with equivalent setups they give identical results; regression is simply the more flexible framing.
What is the difference between ANCOVA and multiple regression? Essentially none mathematically. ANCOVA is multiple regression with a categorical predictor of interest plus one or more continuous covariates you want to control for. The labels reflect different traditions and emphases (group comparison versus prediction), not different models.
Can I use linear regression for a dose-response curve? Usually not. Biological responses saturate, producing a sigmoidal curve against log dose, so a linear fit mis-estimates potency. Use nonlinear regression with the four-parameter logistic (Hill) model, plot on a log dose axis, and design doses that bracket both the top and bottom plateaus.
What does "holding other variables constant" actually mean? It means the coefficient estimates the association between that predictor and the outcome within the fitted model, given the covariates included. It is not a guarantee of causality, and it cannot account for confounders you never measured.
Why did my coefficient change when I added a covariate? That's expected when the new covariate is correlated with both your predictor and the outcome; the adjusted estimate separates their contributions. Report both the unadjusted and adjusted coefficients so readers can see the size of the adjustment.
Should I control for every variable I measured? No. Adjusting for a mediator (a variable on the causal path from treatment to outcome) causes overcontrol bias and can erase a real effect. Choose covariates that plausibly confound the relationship, based on the biology, and keep the set as small as the question allows.
