I am a bit confused about how the exGaussian model works in brms v2.7.0.
In the vignette " Parameterization of Response Distributions in brms" you say the following about the parameters:
“β is the scale (inverse rate) of the exponential component, ξ is the mean of the Gaussian component, σ is the standard deviation of the Gaussian component, and erfc is the complementary error function. We parameterize μ=ξ+β so that the main predictor term equals the mean of the distribution.”
From this and the post here I understand that the point estimates in the model reflect the sum of the mean of the Gaussian component and the scale of the exponential component. Thus, the estimates are estimates of μ and not of ξ? -> Is this an accurate interpretation?
If the estimates represent ξ+β, then I’m a bit confused on how to interpret the way the formula is written in the case of the exGaussian model.
If I write the formula: “RT ~ predictor”
–> does this mean that I state that the mean of the Gaussian component (ξ) depends on the predictor or am I stating that the sum of the Gaussian mean and scale of the exponential component (μ) depends on the predictor? Since I get 1 point estimate of the scale of the exponential component (β) in my summary, I assume that the formula means that only the Gaussian mean (ξ) depends on the predictor?
If only the mean of the Gaussian component (ξ) is modeled as a function of the predictor when the formula is written as “RT ~ predictor” and not as “bf(RT~predictor, beta~predictor)”, why can’t the estimates of the mean of the Gaussian component (ξ) be derived from subtracting the predicted value of β from the point estimate (μ)?
I’m sorry if this is a stupid question, but it’s all very new to me…