A two-stage Generalized Method of Moments model for feedback with time-dependent covariates

Abstract

Correlated observations in longitudinal studies are often due to repeated measures on the subjects. Additionally, correlation may be realized due to the association between responses at a particular time and the predictors at earlier times. There are also feedback effects (relation between responses in the present and the covariates at a later time), though these are not always relevant and are often ignored. All these cases of correlation must be accounted for as they can have different effects on the regression coefficients. Several authors have provided models that reflect the direct and delayed impact of covariates on the response, utilizing valid moment conditions to estimate the relevant regression coefficients. However, there are applications when one cannot ignore the effect of the responses on future covariates. A two-stage model to account for the feedback, modeling the direct as well as the delayed effects of the covariates on future responses and vice versa is presented. The use of the two-stage model is demonstrated by revisiting child morbidity and its impact on future values of body mass index using Philippines health data. Also, obesity status and its feedback effects on physical activity and depression levels using the Add Health dataset are analyzed.

Keywords

Longitudinal data valid moment conditions binary and continuous outcomes marginal models cross-sectional associations lagged associations

Get full access to this article

View all access options for this article.

References

Diggle

Heagerty

Liang

K-Y

et al.

Analysis of longitudinal data. Oxford, UK: Oxford University Press, 2002.

Irimata

Broatch

Wilson

JR.

Partitioned GMM logistic regression models for longitudinal data.

Stat Med 2019; 38: 2171–2183.

Wilson

Vazquez-Arreola

Chen

D-G.

Marginal models in analysis of correlated binary data with time dependent covariates. Switzerland: Springer Nature, 2020.

McCullagh P and Nelder JA. An outline of generalized linear models. In: Generalized Linear Models. 2nd. Ed. London: Chapman and Hall, 1989, pp. 21--54.

Zeger

Liang

Albert

PS.

Models for longitudinal data: a generalized estimating equation approach.

Biometrics 1988; 44: 1049–1060.

Pepe

Anderson

GL.

A cautionary note on inference for marginal regression models with longitudinal data and general correlated response data. Commun Stat Simul Comput 1994; 23: 939–951.

Zeger

Liang

K-Y.

Feedback models for discrete and continuous time series. Stat Sin 1991; 1: 51–64.

Keele

L a

nd Kelly

NJ.

Dynamic models for dynamic theories: the ins and outs of lagged dependent variables. Polit Anal 2006; 14: 186–205.

Achen

CH.

Why lagged dependent variables can suppress the explanatory power of other independent variables. In: Annual meeting of the political methodology section of the American political science association, Los Angeles, CA, 2000, www-personal.umich.edu/∼franzese/Achen.2000.LDVstealingExplanPower.pdf (2000, accessed 30 March 2020).

10.

Lai

Small

Marginal regression analysis of longitudinal data with time-dependent covariates: a generalized method-of-moments approach. J Royal Stat Soc Ser B Stat Methodol 2007; 69: 79–99.

11.

Lalonde

Wilson

Yin

GMM logistic regression models for longitudinal data with time-dependent covariates and extended classifications.

Stat Med 2014; 33: 4756–4769.

12.

Bhargava

Modelling the health of Filipino children. J R Stat Soc Ser Stat Soc 1994; 157: 417.

13.

Martorell

TJ.

Malnutrition, morbidity, and mortality. Popul Dev Rev 1984; 10: 49.

14.

Sayers

Heron

Smith

, et al. Joint modelling compared with two stage methods for analysing longitudinal data and prospective outcomes: a simulation study of childhood growth and BP. Stat Meth Med Res 2016; 26: 437–452.

15.

Berhane

Thomas

DC.

A two-stage model for multiple time series data of counts.

Biostatistics 2002; 3: 21–32.

16.

Irimata KM. Three Essays on correlated binary outcomes: detection and appropriate models. PhD Thesis, Arizona State University, 2018.

17.

Lindsay

Improving generalized estimating equations using quadratic inference functions. Biometrika 2000; 4: 823–836.

18.

Irimata

Wilson

JR.

Using SAS to estimate lagged coefficients with the %partitionedGMM macro. In: SAS global forum 2018 conference proceedings, Denver, CO, USA, 2018.

19.

Wehby

Yang

Depression, antidepressant use and weight gain. J Appl Econ 2012; 2: 1–38.

20.

Lanza

Grella

Chung

PJ.

Does adolescent weight status predict problematic substance use patterns?

Am J Health Behav 2014; 38: 708–716.

21.

Zhou

Lefante

Rice

et al.

Using modified approaches on marginal regression analysis of longitudinal data with time-dependent covariates.

Stat Med 2014; 33: 3354–3364.