 # Modelling mean of gaussian component in brms version 2.3.0

Hi there,
up until version 2.1.0, the main parameter modelled with family = exgaussian was the mean of the Gaussian component. Since version 2.3.0, this has changed to model the mean of the distribution. Is there a way of modelling the mean on the Gaussian component in version 2.3.0 or newer?

Many thanks!

• Operating System: Mac
• brms Version: 2.3.0
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You can implement it via the `custom_family` feature. See `vignette("brms_customfamilies")`.
And I recommend you update brms to version 2.4.0 from CRAN.

Dear Paul,

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…

Thanks!
Hanne

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Yes, in recent version of brms, the main formula predicts mu and not xi anymore. This used to be different but I changed it for consistency with most other families where we predict the mean via the main formula. You may of course derive the predictions for xi by computing mu - beta.

Thank you for the quick response!

Given this “new” parameterization, am I correct in my understanding that the model structure for μ and β imply a structure for ξ?

That is, if I have `RT ~ a, beta ~ 1`, this implies a structure such that `mu ~ a` (i.e. we express that the mean can vary by `a` but that the exponential component explicitly does not, implying that the gaussian mean component must be permitted to vary by `a`).

Similarly, if we have `RT ~ 1, beta ~ a` then by necessity ξ will be perfectly anti-correlated with β? This latter strikes me as troublesome for sampling. Arguably this is only a problem in the case where the user allows β to vary by a given predictor but does not allow the overall mean to also vary by that predictor, but I can imagine scenarios where users building up increasingly complex models might not catch on to this implied structure. Possibly a warning is warranted?

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You are right with your conclusion that mu and beta imply a strucutre for xi as is the case when we predict xi and beta which then imply a structure for mu.

I am not sure I can follow why a warning should be warranted though. I would need to see some (simulated or real) examples that show this pathodological behavior.