Parameter interpretation for discrete weibull regression

Question

Please can someone provide an accessible interpretation of the parameter estimates from a discrete weibull regression model, e.g in R:

library(DWreg)
library(COMPoissonReg)

data(freight)

dw.reg(broken ~ transfers, data = freight,
       para.beta=FALSE,para.q1=FALSE,para.q2=TRUE)

produces:

Maximum Likelihood estimation
Newton-Raphson maximisation, 11 iterations
Return code 1: gradient close to zero
Log-Likelihood: -18.82916 
3  free parameters
Estimates:
            Estimate Std. error t value  Pr(> t)    
(Intercept) -26.7814     7.3943  -3.622 0.000292 ***
transfers    -2.5857     0.7579  -3.412 0.000646 ***
beta         10.8723     2.9356   3.704 0.000213 ***
---
Signif. codes:  
0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 

I am trying to understand how one would communicate the results to a layman in an analogous way that one might interpret the parameter estimates from linear (unit increase in x is assoc with beta increase in y) or poisson (mulitplicative) etc.

Answer 1

The Discrete Weibull distribution is defined by:

\begin{align} f(x) = q^{x^b}-q^{(x+1)^b} , b>0,q\in (0,1),x=0,1,2,\ldots \end{align}

Having a set of covariates, $X$, there are three ways of connecting distribution parameters to the covariates,

\begin{align} 1. &\, logit(q)=X\beta\\ 2.& \,\log(\log(q)) =X\beta\\ 3.& \,\log(b) =X\beta\\ \end{align}

In your case, the logit transformation is used without any transformation on $\beta$.

You can find more about the discrete Weibull distribution and the Bayesian implementation on: http://bura.brunel.ac.uk/bitstream/2438/14135/1/FulltextThesis.pdf#page=84

Answer 2

Disclaimer - I do not know anything about this package. I'll say some generalities which might give you a hint. I haven't looked at implementations of discrete Weibull-regression but for continuous Weibull every implementation uses its own paremetrization/ definition of the parameters.

I prefer the parametrization: $$F_W(x)=1-e^{-(\frac{x}{\alpha})^\beta}$$ and so in the discrete case $$Pr( W\leq x+1) =Pr(W_d \leq x) = F(x+1)$$ which is the one used in the wiki-article (disclaimer, I wrote it).

Typically output from these types of packages outputs parameters for location-scale parametrization-transformed variables s.t we estimate the parameters of a Gumbal-distribution: $$F_G(y)=1-\exp(-\exp(\frac{y-\mu}{\sigma}))=F_G(\log(x))=F_W(x)$$

with $\mu = \log(\alpha)$, $\sigma = \frac{1}{\beta}$.

The estimation is then done s.t $\log(\alpha)=m+kx$ is a linear combination of data. Since $\alpha = e^{m}e^{kx}\ldots$ you get some kind of multiplicative effect if this is the case. Other link functions are possible so it depends.

Depending on the true value of $\alpha$ i.e the resolution of your discrete distribution the estimated $\alpha$ and $\beta$ might not be unique and in particular, your estimators might not be consistent i.e your expected to get the right result. (I've yet found proof telling me anything else). From this it follows that I'd be very careful in interpreting it too much.

I'll give you some hints from the continuous Weibull:

• Look at the Gumbal distribution and think about the normal distribution.. draw analogies from that..
• About 63% of events will happen before $x=\alpha$ since $F_W(\alpha)=1-e^{-1}\approx0.632120559$
• Assume unit $\alpha$. If $\beta< 1$ we have a decreasing hazard: $\beta$ tells you how fast it's decreasing. If $\beta>1$ it tells you how fast it's increasing.

Check: https://www.stat.washington.edu/people/fritz/DATAFILES498B2008/ISSTECH-96-022.pdf http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.151.2376&rep=rep1&type=pdf

for the best references on Weibull