Note that there are considerably differences in the nine estimated slopes for NAP. Instead of the loop in the code above, you can also use the lmList command from the nlme package to produce the same results. This option also gives a nice graphical presentation of estimated intercepts and slopes (Pinheiro and Bates, 2000).
In the second step, the estimated regression coefficients are modelled as a function of exposure.
This is 'just' a one-way ANOVA. The response variable is the estimated slopes from step 1, Exposure is the (nominal) explanatory variable, t is the corresponding regression parameter, n is the intercept, and bi is random noise. The matrix notation for this is below. It looks intimidating, but this is only because exposure is a factor with levels 0 and 1. Level 0 is used as the baseline. The model in Equation (5.4) is written in matrix notation as
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