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IntraOcular Trauma Test (IOTT)

 

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IntraOcular Trauma Test *

*


(Plot the data.  If the result hits you between the eyes, then it's significant.)


Using an appropriate probability grid, plot the cdf (cumulative distribution function) data. The data will appear as a straight line on the correct grid. Fortunately, there are two grids that will cover most of the distributions we need, and making additional ones isn't too complicated.

To make a Normal (or Lognormal) grid notice that the y-axis is in terms of number of standard deviations, although it's not labeled that way. So the middle of the graph is at y=0 and that corresponds to cdf, F(x) = 0.5 = 50%. One standard deviation unit up (or down) is F(x) = 0.8413 (or 0.1587). Two units up (or down) is 0.9772 (or 0.0228). Three units up (down) is 9987 (0.0013). And so on. These values, and intermediate values chosen for graphing purposes, are tabulated everywhere and can be found using MS EXCEL also. If the x-axis is to represent a normally distributed x, then it's Cartesian. If lognormal is what you want, then the x-axis is logarithmic.

To make a grid for the exponential distribution, we can take advantage of knowing that the exponential distribution is a special case of the Weibull, when the slope parameter, beta, equals one. The Weibull grid is even easier to make than the Normal grids because F(x) has a closed form (unlike the Normal), viz.

 

A little arithmetic shows that

This is a linear equation, Y=M*X+B, where X=ln(x) and Y=ln(-ln(1-F(x))), with slope of M = b, and intercept = -b ln(h). (Remember that b and h are constants for a given fit so b ln(h) is also a constant.)

The grid then is simply Y=ln(-ln(1-F(x))), and X=ln(x). (Logarithmic x-axis, and log(logarithmic) y-axis.) Notice that the (0, 0) point occurs at x=1 (so that ln(x)=0, and y= ~0.632, since ln(-ln(1-0.632))~0. (The actual value is at y=1-exp(-1) ~0.6321205588)

While mean rank and median rank plotting positions are often used, the following rank position has the useful property that the MLE line goes through the points if there is nothing wrong with the data.  The rank for the ith uncensored observation is

yi = (i-0.5)/n

where i = 1, 2, 3, ... n, and n is the sample size.

That's it. You can program EXCEL to do the plots if you don't have software to do it already.

 

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