# How do I manually predict Tweedie GLM

I have a simple Tweedie model in GLM, or $$Y_i=\beta_0+\beta_1x_1$$. As usual, $$Y$$ is the output and $$x_i$$ is its dependent variable. Suppose that $$p>1$$, my link function would be $$\frac{\mu^{1-p}}{1-p}$$.

I thought this would be a simple exercise of $$\frac{\mu^{1-p}}{1-p}=\beta_0+\beta_1x_1$$. However, I can't solve for $$\mu$$ since I can't take natural log of a negative value. I'm wondering if anyone ran into this situation before.

• At which point are you taking the logarithm of a negative value? – Sextus Empiricus Jul 8 at 20:29
• In my model, all $\beta$s are positives. My $p$ is 1.5 since $p>1$. After I moved $1-p$ to the right side, I realized I cannot take $ln$ of both side. – tkhu Jul 8 at 21:00
• Why are all $\beta$s positive? If the left side is negative due to that $1-p$ term then just make the right side negative as well. If you don't like that, then get rid of the $1-p$ term. – Sextus Empiricus Jul 9 at 1:19
• My $\beta$s were positive because that's what the rxGLM summary results gave me. Turns out the package also got rid of $1-p$ on the left too, so I think $1-p$ is already baked into the estimated values. – tkhu Jul 9 at 2:56

I was able to answer my own question. I was using rxGLM in R with Tweedie distribution. The code indicates that "Family-link: Tweedie-mu^-0.5 ". Thus, the link function is actually $$\mu^{-0.5}$$.
It is then simply $$\mu^{-0.5}=β_0+β_1x_1$$. Flipping side gives $$\mu^{0.5}=\frac{1}{β_0+β_1x_1}$$. Finally the answer is $$\mu=\frac{1}{(β_0+β_1x_1)^2}$$. I checked this by hand and it works!