Toolkit · 11 of 13

Dose-response

Outcome as a function of a continuous exposure, such as hospital volume.

What it is

Some questions are not about two groups but about a gradient: does mortality fall as hospital volume rises, and is the fall steady or does it level off? Each study then reports several exposure categories (for example, 1-10, 11-30 and more than 30 cases a year), each compared with the same reference category.

A study contributes a curve, not one number. Its estimates share the reference group, so they are correlated; the Greenland-Longnecker method rebuilds that correlation from the cases and totals in each category. Each category is represented by its midpoint.

The curves are pooled in two stages. A linear model gives the change in risk per unit of exposure; a restricted cubic spline lets the curve bend, and a Wald test of the spline terms says whether it departs from a straight line.

Worked exampleVolume and mortality, AAA repair

	Estimate	P or 95% CI
Slope per case a year	−0.0102	P = 0.011
Non-linearity	spline terms	P = 0.32
OR, 90th vs 10th percentile	0.65	0.44-0.96

Higher volume, lower mortalitya steady, roughly linear declineNo threshold can be claimedthe curve does not flatten within the observed range

6 studies, 202,212 patients, from one of my reviews.

When to use it

Use it when

The exposure is quantitative and has many levels: hospital or surgeon volume, drug dose, duration of treatment.
Each study reports at least 3 exposure categories, with cases and totals (or person-time) in each.
The question is about the shape of the relation, or about the effect at clinically meaningful levels of exposure.

Choose another tool when

Studies only compare high with low exposure: use pairwise binary outcomes, knowing that the cut-offs differ between studies.
The covariate is a study-level summary, such as the mean dose of each trial: use meta-regression.
Category sizes cannot be reported or derived: the covariance cannot be reconstructed, and the study cannot enter the dose-response model.

The template, parameter by parameter

Click a highlighted argument to see what it does and which values to use. Picking a value updates the code.

# dat: one row per category; reference row has logrr = 0
lin <- dosresmeta(logrr ~ dose, id = id, se = se,
                   = "cc", cases = cases, n = n, data = dat,
                   = "gl",  = "reml")

knots <- quantile(dat$dose,  = c(0.10, 0.50, 0.90))
spl <- dosresmeta(logrr ~ rcs(dose, knots), id = id, se = se,
                  type = "cc", cases = cases, n = n, data = dat,
                  covariance = "gl", method = "reml")
# test for non-linearity
waldtest(b = coef(spl), Sigma = vcov(spl), Terms = 2)

# curve: see section 4

The dose-response curve, parameter by parameter

These arguments change only how the result is shown, not the result itself. Click one to see the options.

grid <- data.frame(dose = seq(0,  = 110, by = 1))
pred <- cbind(grid, predict(spl, newdata = grid,
                             = 0, expo = TRUE))

ggplot(pred, aes(dose, pred)) +
  geom_ribbon(aes(ymin = ci.lb, ymax = ci.ub),
               = "grey90") +
  geom_line(linewidth = 1) +
  geom_hline(yintercept = 1, linetype = "dashed") +
  labs(x = "Hospital volume (cases/year)", y = "Odds ratio")

How to read the results

The shape

Look at the Wald test for non-linearity: a non-significant test favours the simpler linear model.
Show both curves, linear and spline, so the reader sees how similar they are.
Do not draw beyond the observed range of exposure.

The effect

Report the slope per clinically meaningful unit (per 10 cases a year, per 1 mg/kg).
Contrast fixed points, such as the 90th versus the 10th percentile, rather than arbitrary thresholds.
A minimum of the curve near the edge of the data is not an optimal threshold.

Heterogeneity

Dose-response models often show high heterogeneity: report it.
Explore it with meta-regression of the study-specific slopes.
Remember that categories and midpoints differ between studies, an extra source of error.

In my work

The main result of a published dose-response review, next to the calls behind it. Click the figure to enlarge it.

Hospital volume and overall mortality after elective AAA repair6 studies, 202,212 patients. Linear β = −0.0102 (P = 0.011); non-linearity P = 0.32; I² = 90.9%. From Ann Vasc Surg, 2026 ↗

lin <- dosresmeta(logrr ~ dose, ...)  # linear trend
spl <- dosresmeta(logrr ~ rcs(dose, knots),
                  ...)                # spline, 3 knots
waldtest(coef(spl), vcov(spl),
         Terms = 2)                   # P = 0.32

pred <- predict(spl, newdata = grid,
                xref = 0, expo = TRUE) # OR vs 0

References

Greenland S, Longnecker MP. Methods for trend estimation from summarized dose-response data, with applications to meta-analysis. Am J Epidemiol. 1992;135:1301-1309.
Orsini N, Li R, Wolk A, Khudyakov P, Spiegelman D. Meta-analysis for linear and nonlinear dose-response relations: examples, an evaluation of approximations, and software. Am J Epidemiol. 2012;175:66-73.
Crippa A, Orsini N. Multivariate dose-response meta-analysis: the dosresmeta R package. J Stat Softw. 2016;72(Code Snippet 1):1-15.
Hamling J, Lee P, Weitkunat R, Ambühl M. Facilitating meta-analyses by deriving relative effect and precision estimates for alternative comparisons from a set of estimates presented by exposure level or disease category. Stat Med. 2008;27:954-970.
Harrell FE. Regression Modeling Strategies. 2nd ed. Springer; 2015.