I've performed $m$ fits of datasets $Y=\theta X+b$ coming from different experiments. As a result, I have $m$ estimates $\theta_1,\theta_2,\dots \theta_m$ where $\theta_{k}=(\theta_{k,1},\dots\theta_{k,n})$ is a vector of slopes. For each slope $\theta_{k,i}$, I have a confidence interval $[\underline{\theta}_{k,i},\bar{\theta}_{k,i}]$.

I want to test a hypothesis, which these slopes (i.e. $\theta_{\cdot,j}$ with data $\theta_{1,j},\dots\theta_{m,j}$) come from Gamma distribution. The easiest way is to use Kolmogorov-Smirnov or Chi-square tests for mean values. Nevertheless, I don't want to lose information, which contains in confidence intervals $[\underline{\theta}_{k,i},\bar{\theta}_{k,i}]$.

Do you have any idea how to deal with such a problem?

  • $\begingroup$ What is the origin of the multiple fits? Did you use different subsets of the training sets? $\endgroup$ – Karel Macek Jan 18 '17 at 14:27
  • $\begingroup$ It is different participants in a study. I study performance in task Y versus performance in task X during several days. $\endgroup$ – zlon Jan 18 '17 at 14:30
  • $\begingroup$ I see, you can run the Kolmogorov-Smirnov with estimated parameters (en.wikipedia.org/wiki/… or stats.stackexchange.com/questions/20648/… ) I would say that it would be the most appropriate way. $\endgroup$ – Karel Macek Jan 18 '17 at 14:33
  • 1
    $\begingroup$ I know it. But my problem is not to estimate parameters of possible distribution. My problem is to test not only means, but means +-CI. For example i have slope1+-CI1, slope2+-CI2,slopeN+-CIN. So, i know how to test that slope(1-N) come from Gamma distribution. The question is how to test that slope(1-N)+-CI(1-N) come from a Gamma distribution. In other words if my CI are really big (+/- several means), than good test for means has no sense. $\endgroup$ – zlon Jan 18 '17 at 14:48
  • $\begingroup$ What other distribution can the slopes have if not Gamma? $\endgroup$ – Karel Macek Jan 18 '17 at 15:05

In fact, you have $m$ Student distributions, one for each $\theta_{k,i}$ out of your data and you want to know if the mixture of these distributions is equivalent to a Gamma distribution.

Essentially, they are of different shapes. However, you try the following for each slope $i$

  • Generate $N$ of samples from each of them. You will have $m\cdot N$ results.
  • Fit these data to Gamma and obtain parameter estimates.
  • Validate by Kolmogorov-Smirnov for fitted values if the distributions match.
  • $\begingroup$ Thank you! It is not analytical solution, but it sounds perfect! $\endgroup$ – zlon Jan 18 '17 at 15:28
  • $\begingroup$ Upvote done :). I start to implement this idea and I have last 2 questions to accept the answer:1) Why is it Student distribution; 2) how many degrees of freedom it has i my case? $\endgroup$ – zlon Jan 18 '17 at 21:48

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.