Derive a dose-response curve from published results of multiple studies on the relation between a quantitative exposure (e.g. number of cigarettes, diet) and the occurrence of a health outcome (e.g. mortality risk, incidence of a disease)

Research questions:

• Is there any association? What is the shape of the association?
  
• What are the exposure values associated with the best or worst outcome?
  
• What are the factors that can influence the dose–response shape?

• Several fields of application (oncology, environmental and occupational health, nutrition epidemiology, endocrinology, and general internal medicine)

• Many leading medical and epidemiological journals (JAMA, Lancet, Stroke, Gastroenterology, American J of Medicine, American J of Clinical Nutrition, American J Epidemiology, International J Epidemiology, International J of Cancer, Statistics in Medicine)

• Global health organizations and foundations (World Cancer Research Fund, American Institute for Cancer Research, International Agency for Research on Cancer

• Measures of public health impact (burden of mortality, attributable fractions)

• Donna Spiegelman, Professor at Harvard T.H. Chan School of Public Health, Boston

• Stefan Leucht, Professor at Technische Universität München, Munich

• James Woodcock, Senior Research Associate at University of Cambridge School of Clinical Medicine, Cambridge

• Emir Veledar, Center for Healthcare Advancement and Outcomes, Baptist Health South Florida, Miami

• Marco Vinceti, member of the European Food Safety Authority, Panel on Dietetics Products

• Kristina Thayer, Deputy Director for Analysis, Division of the Search Results National Institute of Environmental Health Sciences

If any, what is the dose-response association between coffee consumption and all-cause mortality?

# devtools::install_github("alecri/dosresmeta")
library(dosresmeta)
coffee <- get(data(coffee_mort))
head(coffee, n = 10)
##    id           author year type dose cases    n   logrr    se gender   area
## 1   1   LeGrady et al. 1987   ci  0.5    57  249  0.0000 0.000      M    USA
## 2   1   LeGrady et al. 1987   ci  2.5   136  655 -0.2877 0.139      M    USA
## 3   1   LeGrady et al. 1987   ci  4.5   144  619 -0.1744 0.137      M    USA
## 4   1   LeGrady et al. 1987   ci  6.5   115  387  0.0862 0.140      M    USA
## 5   2 Rosengren et al. 1991   ci  0.0    17  192  0.0000 0.000      M Europe
## 6   2 Rosengren et al. 1991   ci  1.5    88 1121 -0.1204 0.253      M Europe
## 7   2 Rosengren et al. 1991   ci  3.5   155 2464 -0.3418 0.244      M Europe
## 8   2 Rosengren et al. 1991   ci  5.5   155 1986 -0.1262 0.244      M Europe
## 9   2 Rosengren et al. 1991   ci  7.5    39  575 -0.2665 0.278      M Europe
## 10  2 Rosengren et al. 1991   ci  9.5    24  427 -0.5108 0.280      M Europe

The results are presented in a tabular way

##     wtdiffcat    dose cases   n    rr    lb    ub logrr    se
## 1   [-7,-1.5) -2.5353     3  97  1.00    NA    NA 0.000    NA
## 2 [-1.5,-0.5) -1.0162     8 100  2.53 0.633  10.1 0.929 0.707
## 3  [-0.5,0.5) -0.0182     9  96  2.98 0.764  11.6 1.093 0.695
## 4     [0.5,3)  1.3870    24  71 13.05 3.584  47.5 2.569 0.660
## 5       [3,5)  3.4612     4   6 82.16 9.173 735.8 4.409 1.119

• Observations are not independent (share reference category)

• The RR in the referent is 1

Log-linear model:

\[ y = X \beta + \epsilon \]

\(y\) : vector of non-referent log \(RR\)

\(X\) : design matrix contains the assigned doses

• Model without intercept, so that \(RR = 1|x = 0\)

• \(Cov(\epsilon)\) can be approximated depending on the study design ("cc" case-control, "ir" incidence-rate, "ci" cumulative-incidence)

\[ b_i \sim N \left( \bar b , v_{b_i} + \tau^2 \right) \]

\[ \hat {\bar b} = \frac{\sum_{i = 1}^K b_i w_i}{\sum_{i = 1}^K w_i} \] \(w_i = (v_{b_i} + \tau^2)^{-1}\)

• \(\tau^2\) is the between-studies heterogeneity

• Test for overall association: \(H_0: \bar b = 0\)

• Test (Cochran \(Q\)) and quantify the impact (\(I^2\)) of statistical heterogeneity

• "Every unit increase is associated with …" not appropriate

• Predict RRs for a set of exposure levels \(x^*\) (as compared to \(x_{ref}\))

\[\widehat {RR} = e^{(X^* - X_{ref}) \hat {\bar \beta}}\]

\[\text{95% CI: } e^{(X^* - X_{ref}) \hat {\bar \beta} \pm z_{1-\alpha/2} \left[ (X^* - X_{ref})^\top Var(\hat {\bar \beta}) (X^* - X_{ref}) \right]}\]

• Predicted RRs can be both presented in either graphical or tabular form

• To further explain heterogeneity and/or explore differences in dose-response curves

• Prespecified study-level characteristics

• Limited number of data points: interpret with caution

• Stratified (or subgroups) analysis: divide the observations in groups and evaluate differences