Poisson regression and independence I'm using Poisson regression to test the relationship between root number and canal number in human teeth. My question is about the independence of variables, particularly –
a) does including multiple teeth from the same individual as independent data points in a single analysis, and;
b) the possibility that morphology across different teeth of the same individual might have shared genetic or environmental influences,
violate the assumptions of the Poisson regression?
My guess is that I need to conduct some sort of test of independence (i.e., Chi-square). However, my sample is 945 individuals, with 10 teeth from each individual. Therefore, this would be a difficult test to carry out.
Any advice is welcome.
 A: I was able to solve this issue by using General Estimating Equations with an AR1 covariance matrix. A key assumption underlying Poissong General Linear Models (PGLM) is the independence of observations (Hoffmann, 2004). Thus, the inclusion of multiple teeth from the same individuals may violate assumptions of independence for the PGLM used in my study. To account for this, I fit our PGLM with Generalized Estimating Equations (GEE). GEE estimates population-averaged parameters and their standard errors based on a number of assumptions: (1) The responses variables are correlated or clustered; (2) There is a linear relationship between the covariates and a transformation of the response; and (3) within-subject covariance has a correlation structure (Zeger and Liang, 1986; Diggle et al., 2002). In order to determine  correlation structure and how root and canal number correlated within and between teeth I conducted a Pearson correlation analysis of canal and root number. I selected and Auto Regressive Order 1 (AR1) correlation structure for the GEE covariance matrix. While GEE estimates of model parameters are valid regardless of the specified correlation structure, the AR1 correlation structure is appropriate because it (a) has no distributional assumptions (Zuur et al., 2009); (b) can accurately model covariance for cross-sectional individual and clustered studies (Müller et al., 2009; Muoka et al., 2021); (c) accurately model within-subject correlation decreasing across time and/or space (Agresti, 2002); and (d) assumes observations within and individual are non-independent (Zeger and Liang, 1986). Thus, AR1 is appropriate at the individual and population levels, and for the temporospatial distances within and between individuals and groups within the indivisuals comprising my sample. GEE was caried out using ‘geepack: Generalized Estimating Equation Package’ version 1.3.2 (Halekoh et al., 2006).
