In this recipe, we will be more interested in the regression part of it, instead of the ANOVA part. In the previous ANOVA chapter, we only used random effects for the intercepts, and this is usually not the price only way that random effects are introduced. Imagine that we model the sales in terms of price for certain customers, where we have several observations for each one of them. The ordinary least squares (OLS) standard approach would be to ignore this heterogeneity and pool all the observations together.

Naturally, this would introduce a problem, because the residuals would then be correlated (observations belonging to the same individual will produce similar residuals). The correct approach would be to introduce a random effect per individual, but there is a subtle point here: we are not expecting the response to differ in terms of an intercept, but in terms of the coefficient that relates prices to sales. More formally, we would expect that the coefficient relating prices to sales, would be equal to a fixed coefficient (an average/standard effect) plus an individual effect. Of course, if we look at the aggregate picture, the random effect does not really change our price-sales relationship, we will be more interested in the regression part of it, instead of the ANOVA part..