Characterize a dose-response relation between a quantitative exposure (e.g. fluoride intake) and a continuous response (e.g. neurodevelopmental outcomes) based on multiple studies

Research question:

• Is the any relation between the exposure and the continuous response?

• What is the shape of the relation?

Many meta-analyses summarize/contrast results for high vs. no/low exposure levels

Major limitations:

• Intermediate exposure values are excluded from the analysis

• Exposure contrasts may vary across studies

A dose-response approach is more informative and uses the whole information

The outcome variable may change across studies because of diversity in subjects characteristics

The effect size will be based on differences. Because the response variable is measured on different scales

Standardized mean differences: \[d_{ij}^* = \frac{ \bar{Y}_{ij} - \bar{Y}_{i0}}{s_{p_i}}\]
\(\bar{Y}_{ij}\) is the mean response in the i-th study for the j-th exposure level
\(\bar{Y}_{i0}\) is the referent exposure value in i-th study
\(s_{p_i}\) is the pooled standard deviation in i-th study

Standardized mean differences are used as summary statistics in meta-analysis when the studies all assess the same outcome but measure it in a variety of ways (e.g. example, all studies measure impairments in learning and memory but they use different tests). In this circumstance it is necessary to standardize the results of the studies to a uniform scale before they can be combined. The standardized mean difference expresses the size of the intervention effect in each study relative to the variability observed in that study.

handbook.cochrane.org

In each single study \[{d}_i^* = f\left( {x}_i , {\theta}_i \right) + {\varepsilon}_i \]

\(f\) is the dose-response model (specifies how the effect size varies according to the dose values)

Different alternatives: linear, quadratic, …

(Restricted cubic) splines
Smooth functions that can take almost any shape

The study-specific curves are combined using meta-analysis

\[{\bar \theta} = \frac{\sum_{i = 1}^I {\theta}_i w_i}{\sum_{i = 1}^I w_i}\]

\(w_i = (v_i + \tau^2)^{-1}\)
\(v_i\) is the variance of the dose-response coefficients
\(\tau^2\) represents the heterogeneity