I have some difficulties in thoroughly interpreting all the outcomes of the Poisson Regression Model.
I have a Poisson regression model with Observed deaths as dependent variable, A categorical variable with 5 levels (ClusterAIR) and Education_level as a numerical variable. The expected deaths are used as an offset in the Poisson regression model, therefore the rate is observed/expected deaths in the model.
The dataset is divided into 5 clusters and of course I would expect that for the entire dataset (the sum of all 5 clusters) the observed/expected deaths approximate 1, as the expected deaths are calculated based on the entire dataset (corrected for age-structure), and the observed dataset contain the deaths observed in each sector of the dataset.
When running the entirely categorical poisson regression, the result is what I would expect. Intercept of 0.0756 --> exp(0,0756) = 1.08 : The rate observed/expected deaths in Cluster A (reference Cluster) is 1.08 and for the other clusters, the rate is (1.08 * multiplicative factor) sometimes above 1.00 and sometimes below 1.00. For example, Cluster C has an estimate of -0.18, which implies a multiplicative factor of 0.835 -> 1.08 * 0.835 = 0.90
However, when adding the education level as a numerical variable to the model, the rate observed/expected deaths is ABOVE 1 FOR ALL CLUSTERS (2nd image), despite for the entirety of the dataset the number of expected deaths equals the number of observed deaths. How is this possible ?: How is it possible that all parts of the dataset have a higher than expected mortality (after adding the education level as a covariate) while for the entirety of the dataset the observed deaths equal the expected deaths? How does the education level as numerical variable exactly interact with the determination of the intercept ? How can I interpret this finding?
For example, the intercept is 0.25 -> exp(0.25)=1.28, the lowest mortality is found in Cluster D with an estimate of -0.10 -> exp (-0.10)= 0.90 (multiplicative factor). The part of the dataset where the observed/expected mortality is the lowest, Cluster D, has still (1.28 *0.9) an observed/expected mortality of > 1.0 while for the entirety of the dataset this number has 1.0, and this only occurs after adding co-variates, when the categorical variable is the only variable, this phenomenom does not appear to happen, and the results are as expected (some clusters have an observed/expected mortality of > 1.0 and others of <1.0 as it is 1.0 for the entirety of the dataset).