I'm analyzing a three-way mixed linear model using lmer:
Y ~ Factor1 * Factor2 * Factor3 + (1|sensor)
However, different sensors have different gains, uniformly scaling the response of each sensor across conditions. Therefore, a multiplicative random effect seems more appropriate than the additive one I currently use. Can something like this be implemented in R?
Y ~ [Factor1 * Factor2 * Factor3] $\cdot$ (1|sensor)
(The dot stands for multiplication)
My dependent measure can be negative, so using log(Y) doesn't seem like a good solution.
I'd like to try a simplified, generalized formulation of this problem, using scalar notation. We'll start with a simple (fixed effects) linear model:
In this model, given $X$ and $y$, the $\beta$ coefficients can be easily estimated by ordinary least squares. Now, we'll complicate things by assuming that different subsets of observations are sampled by different sensors and each sensor has some random additive contribution $u$. This contribution is normally distributed with 0 expectancy and an unknown variance across sensors.
This model can be estimated as a mixed effects model:
However, I'd like to consider the case of a multiplicative random effect:
Note that for each sensor, there's a single scalar gain. I wish to estimate this model by R. To my understanding, the standard random slope approach won't do it, since
Y~X1+X2+x3+(X1+X2+X3|sensor) would estimate the model
, estimating three different random gains for each sensor instead of one.
Y~0+(X1+X2+X3|sensor) won't help either.