The use of polynomial regression analysis with indicator variables for interpretation of mercury in fish data

Mercury levels in fish in reservoirs and natural lakes have been monitored on a regular basis since 1978 at the La Grande hydroelectric complex located in the James Bay region of Quebec, Canada. The main analytical tools historically used were analysis of covariance (ANCOVA), linear regression of the mercury-to-length relationship and Student-Newman-Keuls (SNK) multiple comparisons of mean mercury levels. Inadequacy of linear regression (mercury-to-length relationships are often curvilinear) and difficulties in comparing mean mercury levels when regressions differ lead us to use polynomial regression with indicator variables. For comparisons between years, polynomial regression models relate mercury levels to length (L), length squared (L-2), binary (dummy) indicator variables (B-n), each representing a sampled year, and the products of each of these explanatory variables (L x B-1, L-2 x B-1, L x B-2, etc.). Optimal transformations of the mercury levels (for normality and homogeneity) were found by the Box-Cox procedure. The models so obtained formed a partially nested series corresponding to four situations: (a) all years are well represented by a single polynomial model; (b) the year-models are of the same shape, but the means may differ; (c) the means are the same, but the year-models differ in shape; (d) both the means and shapes may differ among years. Since year-specific models came from the general one, rigorous statistical comparisons are possible between models. Polynomial regression with indicator variables allows rigorous statistical comparisons of mercury-to-length relationships among years, even when the shape of the relationships differ. It is simple to obtain accurate estimates of mercury levels at standardized length, and multiple comparisons of these estimations are simple to perform. The method can also be applied to spatial analysis (comparison of sampling stations), or to the comparison of different biological forms of the same species (dwarf and normal lake whitefish).