How to calculate the area under a Weibull curve

Hi all,

I'm practising my survival analysis skills and I'm trying to conduct a partitioned survival model by means of a Weibull distribution. I can easily manage to do so for exponential distributions, where the survival function is denoted as s(t)=EXP(-(EXP(-intercept)*t)). The area under the curve is then A(t)=(1-s(t))/EXP(-intercept). The intercept is derived from an analysis in R.

The survival curve for a Weibull distribution is S(t)=EXP(-EXP(-(intercept/EXP(log-scale)))*t^(1/EXP(log-scale))).
The intercept and log-scale are again derived from R. I do not have a formula to calculate an area under this survival curve, nor can I find one. Could anyone help me with this?

Thanks!

*I've posted the same question on the mrexel forum. I can't post the link since I don't have a sufficient amount of posts yet.

Can I assume you are referring to the probability density function for the Weibull distribution? Your expression for the PDF looks a little different from what is given by Wikipedia (https://en.wikipedia.org/wiki/Weibull_distribution ) which is a little different from what is given in Excel's help file for Excel's built WEIBULL.DIST() function (https://support.office.com/en-us/art...9-a83ef82b45db ). I can readily see how Wikipedia's and Microsoft's expressions for the PDF are the same, but I don't readily see how your expression is equivalent -- but it could easily be.

If I assume your expression is equivalent to Microsoft's expression for the PDF and that the "integral" you want is correctly represented by the cumulative distribution function (CDF), then I would expect you could use the WEIBULL.DIST() function with TRUE for the 4th "Cumulative" argument. If you cannot use the built in function, both Wikipedia and the help file give longhand expressions for the CDF, which you should be able to use.

Alright it took me some time to figure out. I am using a survival function: S(t) = exp(−αtγ)
scale = exp(log-scale)
γ = 1/Scale
α =exp(−(Intercept)γ)

This is not the same as the probability density function.

I guess i am only missing the information on the beta parameter to use the WEIBULL.DIST() function.
Any chance you could tell me how I can derive beta?

Not specifically, but it seems like it should mostly be an algebra problem. How are you planning to map s(t)=exp(-aty) onto W(t)=1-exp(-power(x/beta,alpha))?

If your s(t) were 1-exp(-aty), then it would seem to me that alpha=1 and beta=1/(ay). I only mention because the s(t) given in post 3 might have been F(t) in the Wikipedia page for the survival function https://en.wikipedia.org/wiki/Surviv...ion#Definition If that is the case, then I think that is correct, but double check my algebra. I did it quick and only half understanding your problem, so I may have missed or overlooked something.

In either case, it feels to me like an algebra problem at this point. Figure out how to algebraically translate your desired CDF into the Weibull CDF, and you will have alpha and beta.

That is correct, you can always use a Riemann sum on your function to get the integral https://en.wikipedia.org/wiki/Riemann_sum There are many examples around the internet that you should be able to find by putting something like "numerical integration in Excel" into your favorite internet search engine. You need to be clear in your own mind whether you are doing the integral for the PDF or the CDF, and what that equation is.