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I want to look at an objective measure test to get the probability that a given patient falls under a category, in this case, functional level (K-Level)

+==============+==============+===============+===============+===============+===============+
|    AMPPRO    |   K-level    |     K0-K1     |      K2       |      K3       |      K4       |
+==============+==============+===============+===============+===============+===============+
| max score 47 | Mean score   | 25            | 34.65         | 40.5          | 44.67         |
+--------------+--------------+---------------+---------------+---------------+---------------+
|              | SD           | 7.37          | 6.49          | 3.9           | 1.75          |
+--------------+--------------+---------------+---------------+---------------+---------------+
|              | Range        | 17.63 - 32.37 | 28.16 - 41.14 | 36.60 - 44.40 | 42.92 - 46.42 |
+--------------+--------------+---------------+---------------+---------------+---------------+

Especially in the higher K-Levels as the SD goes down, the distinction between classifications becomes blurred. The test results are an integer score ranging from 0-47.

If I eyeball it, I can say that a patient who scores 43 is probably about 48.5% likely a K4, 48.5% likely a K3, 2% likely a K2, and 1% likely a K0-K1. I'm not good at formal statistics, so this would just give general the shape of the bar chart I would like to generate.

X axis is the bins you put your population into based on the score. Y axis is the probability, ranging from 0-1. You get two tiny bars for the bins "K0-K1" and "K2". Two big, roughly equal sized bars for "K3" and "K4".

Is there a way to do this... more mathematically? Is it possible to get a probability distribution with just Means and Standard deviations?

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The standard solution to generating a probability distribution only from certain moments is the Maximum Entropy Distribution. Since in your case you care about the mean and std/variance of your random variable that would boil down to a Normal distribution with precisely these moments.

So what you could do here would be fit a Gaussian to each of your K-Levels and compare the PDFs for each of the K-Levels to find which one is most likely. If you have very different numbers of points in each K-Level you will have to adjust the probabilities you would assign by the relative weights for a point to come from each class in the first place.

Alternatively you could fit a Gaussian Mixture Model to your data and use that to get the posterior probabilities for the different K-Levels. But I think this might just go a little too far beyond what you want to do.

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  • $\begingroup$ I like the approach of doing a Gaussian for each K-level. Since variance is simply the square of the SD, I will use a normal Gaussian distribution for the middle two bins. The final goal is a bar chart with a normalized probability of each K-level given the numeric score. How should I deal with the lowest and highest bins? I don't want my PDF of K4 going down with a higher AMP score on the high end. And I don't want my PDF K0-K1 going down as the AMP score goes lower $\endgroup$ – Arri Ferrari Jun 14 '18 at 20:28
  • $\begingroup$ One thing I'd like to make clear though is that the table above is all the data I have. I don't have the individual results. $\endgroup$ – Arri Ferrari Jun 14 '18 at 20:29
  • $\begingroup$ Since you are going to compare the PDFs with each other, only the relative values are going to matter in the end. Therefore, even if your PDFs for the lowest and highest bins are very small, the ones from the other bins would be much smaller already and you would still pick the correct bin in the end. $\endgroup$ – qeschaton Jun 15 '18 at 4:48

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