Judging the quality of fitting an EventData object to a distribution using LogRankTest

Question

I have $389$ data points defined in an EventData object and want to judge the quality of fit of this data to a fitted $3$-parameter Weibull distribution. This particular data set contains no censoring but in general there will be censoring in my future data sets. My approach is to find the three parameters using EstimatedDistribution and then to represent the found distribution by generating a number of random variates from it so that they can be compared to the original EventData object. I’m using LogRankTest to produce the $p$-value to judge the fit. My fundamental question is, is there a more elegant way of doing this in Mathematica? My data and code are below.

Here are the raw data:

data1=Table[40.8,{10}];
data2=Table[42.075,{23}];
data3=Table[43.35,{48}];
data4=Table[44.625,{80}];
data5=Table[45.9,{63}];
data6=Table[47.175,{65}];
data7=Table[48.45,{47}];
data8=Table[49.725,{33}];
data9=Table[51,{14}];
data10=Table[53.55,{6}];
data=Flatten[{data1,data2,data3,data4,data5,data6,data7,data8,data9,data10}];

Here is where I define the EventData object and determine the fitted distribution, distribution.

censoring=Table[0,{Length[data]}];
eventdata=EventData[data,censoring]
dist=EstimatedDistribution[eventdata,WeibullDistribution[a,b,c]]

EventData[ ]

WeibullDistribution[2.62045,7.14193,39.7656]

And here is where I non-parametrically determine the $p$-values via LogRankTest.

n=200000;
i=0;
Do[
 {i=i+1,
 If[i>4,Break[]],
 samplevalues=RandomVariate[dist,{n}],
 logrankpvalue=LogRankTest[{eventdata,samplevalues}] ,
 Print["i=",i,"    Log Rank p Value=",logrankpvalue]},
 {5}]

i=1    Log Rank p Value=0.999826

i=2    Log Rank p Value=0.973828

i=3    Log Rank p Value=0.948566

i=4    Log Rank p Value=0.976803

Note that with $n$ set at a relatively large value ($200\,000$), $p$-values of around $0.97$ are produced indicating that the hazard rates of the two data sets are in pretty good agreement. I think this tells me that EstimatedDistribution did a good job in the fitting. Conversely, in general, small values of $n$ (e.g., $10$) produce a much poorer agreement as expected and as evidenced by smaller $p$-values. I generate four $p$-values because I’ve observed that depending upon the particular random values chosen that it is possible to produce a large $p$-value with small values of $n$.

My questions are

1. Is there a more elegant way of determining the degree of similarity between the EventData object and the found distribution? Perhaps comparing to the theoretical distribution, dist directly instead of sampling from dist?

2. If this approach (or any improved suggested approach) is robust for the no-censoring case is it also robust if the EventData object contains censored values?

Answer 1

One function you might want to look at is DistributionFitTest. Applying this to your data and the estimated Weibull distribution:

DistributionFitTest[data, 
    WeibullDistribution[2.62045, 7.14193, 39.7656],{"TestDataTable", All}]

The values in this table show three different tests for the quality of the fit of the distribution to the data. I'm no expert in statistics, but the fact that these distribution tests seem to show one thing and the LogRankTest seems to show something else might make me nervous. Look closely at what the LogRankTest is doing:

This is not really the same thing as determining the degree of similarity between the data and the estimated distribution, as the OP suggests.