Dose-response meta-analysis

What the method is for

Many questions in epidemiology are about amounts. How does the risk of a disease change with each additional serving of a food, each year of smoking, each unit of air pollution or each hour of physical activity? A dose-response meta-analysis answers such questions by combining studies that report risk across several levels of an exposure. Compared with a simple meta-analysis of the highest against the lowest exposure category, it uses all of the information, avoids the problem that categories and their cut-points differ between studies, and describes the shape of the relationship. A graded relation between exposure and risk is also one of the considerations that strengthens, though does not prove, a causal interpretation.

The studies are mostly cohort and case-control studies in which exposure is reported in categories, such as quartiles, with a relative risk or odds ratio for each category compared with the lowest, and often with the number of cases and the person-years or controls in each. The analysis fits a curve to each study's categories, accounting for the fact that the estimates share a common reference group, and then combines the study-specific curves. The general principles are in meta-analysis.

Data needed

For each study, the analysis needs, for at least three exposure categories, the exposure level in each category, the effect estimate and its interval for each category compared with the reference, and the distribution of cases and non-cases or person-years across the categories. The exposure level is usually a midpoint of the category, the reported mean or median in each, or an assumed value for open-ended top and bottom categories, and these assumptions should be stated. Units must be harmonized across studies, for example grams per day, converting from servings or other units when needed, which requires conventions that are written down in the protocol and tested in sensitivity analysis. Where the numbers of cases and non-cases are not all reported, they can often be estimated from the total and the effect estimates, with a method described in the literature, at some cost in precision.

The within-study step

The estimates for different categories in one study are not independent, because each is compared with the same reference group. Treating them as independent would give wrong standard errors. Greenland and Longnecker proposed a method that accounts for the correlation using the reported numbers of cases and non-cases, and it has become the standard. In essence, for each study it estimates the slope of the log relative risk against dose by generalized least squares, using a covariance matrix that reflects the shared reference. The result is a study-specific slope with a standard error, which is the change in the log relative risk per unit of exposure. For nonlinear models, the same approach gives study-specific coefficients for the terms of the curve and their covariance matrix.

Software for these steps is available in several packages, which implement the generalized least squares estimation and the pooling. The software and the options are reported.

Pooling across studies

The study-specific slopes are combined by a random-effects meta-analysis on the log scale, as in the page on random-effects meta-analysis. The result is a pooled slope with its interval, which is reported as a relative risk per unit of exposure, for example per 10 grams per day. A worked arithmetic shows the interpretation. If the pooled relative risk is 1.08 per 10 units, then, under a log-linear relationship, the relative risk for an increase of 20 units is 1.08 squared, which is 1.166, and for 50 units it is 1.08 to the fifth power, which is 1.47. That is, the effects multiply and not add, on the ratio scale. If the standard error of the log relative risk per 10 units were 0.012, the 95 percent interval for the relative risk would be 1.05 to 1.11. The figures are invented to illustrate the calculation.

The one-stage approach, which fits all studies in a single mixed model, is an alternative that can handle studies with few categories and complex structures, and it is increasingly used. The two-stage approach remains the most common and the simplest to report.

Nonlinear relations

A straight line is often too simple. Risk may rise steeply at low exposure and flatten, show a threshold below which exposure has no effect, or follow a U or J shape in which both low and high exposure carry risk, as with some nutrients. To describe such shapes, the dose is modeled with a flexible function. Restricted cubic splines, with knots at fixed percentiles of the exposure distribution, are the usual choice, since they are smooth, behave well at the extremes and are easily fitted. Fractional polynomials are an alternative. The coefficients are estimated within each study and pooled, and the curve is presented graphically with its confidence band, with the reference level chosen on a stated rationale. A test of nonlinearity compares the model with spline terms with the linear model.

The flexible curve is more informative than a line and also less stable, particularly in the tails, where data are sparse and the curve may wander. The analyst should show the distribution of the exposure in the contributing studies and be cautious about the range at which the curve is supported, and should not extrapolate beyond the doses observed.

Heterogeneity

Between-study heterogeneity is examined as in other meta-analyses, with the added source of differences in how exposure is measured and categorized. Heterogeneity in the slope is explored through subgroup analysis and meta-regression by study characteristics such as design, sex, region, duration of follow-up, adjustment for particular confounders and exposure assessment method. The assumptions needed for exposure harmonization, such as converting units or assigning midpoints, may themselves be sources of heterogeneity, and they are tested by sensitivity analyses that vary them. Where the number of studies is small, the exploration is limited, as described under meta-regression.

Bias, confounding and measurement error

Dose-response meta-analyses of observational studies share all of the weaknesses of the underlying studies and add a few. Confounding is the central one. Exposure is correlated with other behaviors and characteristics, and adjustment for the measured ones is imperfect. A pooled slope from studies adjusted for different sets of confounders is hard to interpret, and subgroups by adjustment level or restriction to the most adjusted estimates are used. Measurement error in the exposure, especially self-reported diet and activity, typically attenuates associations and distorts the shape of curves. Reverse causation can arise when disease affects exposure, as when illness leads to weight loss or to changes in diet, and is addressed by excluding early follow-up. Publication bias is assessed with funnel plots and tests on the study-specific slopes when there are enough studies.

The conclusion should respect these limits. An association that is graded, consistent and biologically plausible is suggestive, and does not by itself establish that changing exposure would change the outcome. The reasoning is classically summarized in the considerations for causal inference set out by Austin Bradford Hill, which remain a useful checklist and not a test.

Reference level and presentation

The curve needs a reference level, the exposure at which the relative risk is defined as one. The lowest observed exposure, a clinically meaningful level such as none, or a median is chosen, and the choice changes the appearance of the curve without changing the underlying relationship, so it is stated and justified. The standard figure shows the pooled curve with its confidence band against exposure, with a rug or histogram of the exposure distribution beneath it so that readers see where the data are. A table gives the relative risk at selected doses, such as the quartiles, with intervals. Showing both the linear and the nonlinear fits, and the test of nonlinearity, helps readers judge whether the extra flexibility is warranted. A curve presented without the data distribution invites over-reading of the tails.

Other outcome types

The method was developed for relative risks and odds ratios, and it has been extended. For continuous outcomes reported as differences in means across exposure categories, a corresponding approach estimates the slope of the mean difference with dose, accounting for the correlation between categories that share a reference. For survival outcomes, hazard ratios across categories are handled in the same way as relative risks. For binary outcomes with rare events, odds ratios approximate relative risks, and for common outcomes the distinction should be kept in mind. In each case the choice of scale follows the outcome and the data, and the interpretation of the slope, as a ratio per unit or a difference per unit, is stated.

Conducting a review

1. Frame the exposure and outcome

   The exposure, its unit, the outcome and the population are defined, with the dose range of interest.

2. Search and select

   Observational studies reporting risk across at least three exposure levels are eligible, and the criteria for categories and doses are stated.

3. Extract and harmonize

   Doses, cases, non-cases or person-years, and risk estimates are extracted for each category, units are harmonized, and assumptions are recorded.

4. Model

   Study-specific slopes or spline coefficients are estimated and pooled with random effects, and nonlinearity is tested.

5. Explore, test and report

   Heterogeneity and bias are examined, sensitivity analyses vary the assumptions, and the review is reported with PRISMA 2020 and MOOSE.

Limitations

Dose-response meta-analysis relies on observational data that are subject to confounding and measurement error. The harmonization of exposure units and category midpoints requires assumptions that are unavoidable and add uncertainty. Nonlinear curves are unstable where data are sparse, and extrapolation outside the observed range is unreliable. Studies with few exposure categories contribute little to the shape. The pooled slope applies to the range of exposure in the studies and not necessarily to individuals. And the finding of a dose-response relation does not show causation. The results are not advice about the care of individuals, or dietary or other recommendations for a patient.

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