Linear regression with temporally delayed covariates (distributed-lag linear regression) is a standard approach to lag exposure assessment, but it is limited to a single biomarker of interest and cannot provide insights on the relationships holding among the pathogen exposures, thus precluding the assessment of causal effects in a general context. In this paper, to overcome these limitations, distributed-lag linear regression is applied to Markovian structural causal models. Dynamic causal effects are defined as a function of regression coefficients at different time lags. The proposed methodology is illustrated using a simple lag exposure assessment problem.
Almon S. (1965): The Distributed Lag between Capital Appropriations and Net Expenditures. Econometrica 33: 178–196.
Andrews W.K., Fair R.C. (1992): Estimation of Polynomial Distributed Lags and Leads with End Point Constraints. Journal of Econometrics 53: 123–139.
Gasparrini A., Leone M. (2014): Attributable Risk from Distributed Lag Models. BMC Medical Research Methodology 14(1): 14–55.
Gasparrini A., Scheipl F., Armstrong B., Kenward M.G. (2017): A Penalized Framework for Distributed Lag Non-Linear Models. Biometrics 73(3): 938–948.
Granger C.W.J., Newbold P. (1974): Spurious Regressions in Econometrics. Journal of Econometrics 2(2): 111–120.
Haavelmo T. (1943): The Statistical Implications of a System of Simultaneous Equations. Econome,rica 1(1): 1–12.
Koopmans T.C., Rubin H., Leipnik R.B. (1950): Measuring the Equation Systems of Dynamic Economics. In T. C. Koopmans (ed.), Statistical Inference in Dynamic Economic Models, pages 53–237. John Wiley & Sons, New York, US-NY.
Martins L.C., Pereira L.A.A., Lin C.A., Santos U.P., Prioli G., do Carmo Luiz O., Saldiva P.H.N., Ferreira Braga A.L. (2006): The effects of air pollution on cardiovascular diseases: Lag structures. Revista de Saúde Puública 40(4). doi: 10.1590/S0034-89102006000500018.
Pearl J. (2000): Causality: Models, Reasoning, and Inference. Cambridge University Press, Cambridge, UK.
Schmidt P. (1974): A Modification of the Almon Distributed Lag. Journal of the American Statistical Association 69: 679-681.
Schwartz J. (2000): The Distributed Lag between Air Pollution and Daily Deaths. Epidemiology 11(3): 320–326.
Solow R.M. (1960): On a Family of Lag Distributions. Econometrica 28: 393–406.
Welty L.J., Peng R.D., Zeger S.L., Dominici F. (2009): Bayesian Distributed Lag Models: Estimating Effects of Particulate Matter Air Pollution on Daily Mortality. Biometrics 65(1): 282–291.
Wright S. (1934): The Method of Path Coefficients. Annals of Mathematical Statistics 5(3):161–215.
Zanobetti A., Wand M.P., Schwartz J., Ryan L.M. (2000): Generalized Additive Distributed Lag Models: Quantifying Mortality Displacement. Biostatistics 1(3): 279–292.