Linear or Log-linear Model

Should I use the linear or log-linear model?


The linear or log-linear model can be chosen depending on how linear the observed rates or the logarithm of the observed rates are over time. In order to check the goodness of fit of the chosen model, a user can test for normality of the residuals obtained under the linear or the log-linear fit. Select a model whose residual analysis indicates a better fit, regarding the model assumptions of normality, linearity, equal variance, and independence. One reason for using a log transformation for cancer rates is that they arise from a Poisson distribution which is skewed especially when the cancer is rare or the rates come from a small population. The log transformation is a standard way to make this skewed distribution approximately a normal distribution. Rates for common cancers or which come from a large population can be approximated as arising from a normal distribution without a transformation.

One motivation for using the log-linear model for cancer rates regardless if they are rare or not is the ease of interpretation. Under a log-linear model the rates change at a constant percent per year (i.e. a fixed annual percent change - APC), while for a linear model the rates change at a constant fixed amount per year. When comparing trends across age groups or across cancer sites where the rates are very different, the advantage of a log-linear model is that the APC is a metric which makes sense to compare across widely different scales. For example, a rare cancer and a common cancer may change at the same annual percent per year, but it is highly unlikely that they would change at the same fixed amount per year (e.g. if the rates were declining, the rare cancer rate would quickly become negative!).




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