I have wondered about the extent to which that interquartile range in NHS costs informs us about the uncertainty of a cost.

It describes variation of costs among NHS trusts. Therefore it seems a good measure of uncertainty if you are interested in in the cost in one NHS trust that is randomly selected from all NHS trusts. Is that what you are interested in?

Or are you interested in the uncertainty around the national average of a cost? I would think that does not actually depend on variation among trusts. If the national average was estimated from a random sample of trusts, then we could calculate the random error in the estimate. However, the national averages are, as far as I know, pretty much based on the average of all trusts instead of a random sample from them, therefore there is no random error in the national average.

I think that the uncertainty around a national average cost depends on other things then random error, e.g. uncertainty about future developments, uncertainty about future technology, political influences on costs, etc.

I realize that using the interquartile range is an easy way of complying with the request to include uncertainty analysis, but I think that doing an uncertainty analysis right requires a lot more than estimating random error in estimates.

 Thanks Robert for your  comprehensive email. I am interested in the uncertainty around the mean .As you previously mentioned, it is hard to estimate the standard error which is required to estimate the alpha and beta parameters of gamma distribution. Thus, I am quite not sure about how I can estimate these parameters based on the current information available  

Hi Rimal, it seems that my previous post here was not clear enough, sorry: I think that the published average NHS cost is not an estimated average based on a sample with an associated random error but it is the actual average which has -no- random error.

Rob

Hi Robert. Thanks for your reply. Ok. I think I get your point now 

Dear Rimal,

We also work with NHS data and therefore we have faced the same problem.When you know the mean and any two quantiles (like in NHs reference costs where 0.25 and 0.75 quartiles are reported), it is possible to estimate the parameters of the distribution (in this case, for costs, Gamma) by solving two equations. This can be done in multiple ways, for example like this:

www.johndcook.com/quantiles_parameters.pdf

In that pdf there is a link to a software application that can be used to estimate the parameters of different distributions given two quantiles. I think this is the kind of tools that you are looking for.

https://biostatistics.mdanderson.org/SoftwareDownload/SingleSoftware.aspx?Software_Id=6

In the meantime, I have also found out how to do this directly in R. There is an example here:

http://stackoverflow.com/questions/14266354/how-can-i-estimate-the-shape-and-scale-of-a-gamma-dist-with-a-particular-mean-a

I hope this helps!

Isaac

Dear Issac. Many thanks for your reply, I have already generated the alpha and Beta using R commands. However, I did not know the logic behind this calculation, i.e. the method or equation that was used to generate these parameters. Do you know the actual equation?

Dear Rimal,

For the theory behind it you can see this:

www.johndcook.com/quantiles_parameters.pdf

If you mean the logic behind the R code in the example provided, R is solving numerically the equation below as a function of shape, given the mean and the 95% quantile:

pgamma(300, shape, scale=scale) - 0.95

I hope this helps!

Dear Isaac:

Many thanks for the PDF. I looked through it but it seems the calculation very complicated. Do you know what is the calculations that have been used? i.e. like algebra calculation rather using function term.

F(x; α, β) = F(x/β; α, 1).

Thus for any α > 0,

β =

x1/ F −1(p1; α, 1)

is the unique β satisfying

F(x1; α, β) = p1.

 I am not sure how this calculation can be solved into a simple calculation 

Dear Rimal,

F denotes the cumulative distribution function (CDF) of a probability distribution. In case of the Gamma distribution, the parametric form of the CDF is shown for example here:

http://www.itl.nist.gov/div898/handbook/eda/section3/eda366b.htm

I'm afraid that substitution into your equation above is the simplest you can get.

Best wishes,

Isaac

Use AgriMetSoft distribution calculators. It has several distributions consists of Gamma.

https://agrimetsoft.com/distributions-calculator/