Confidence Intervals using a Binomial Distribution

[#1]

I need to calculate 95% confidence intervals for the reporting fraction of adverse events at different time intervals. I'd like to use the "Exact Binomial Confidence Interval".

The link below shows how to do it using Excel; is there a comparable way to do this in JSL?

http://www.itl.nist.gov/div898/handbook/prc/section2/prc241.htm

Thanks!

[#2]

It looks like that is using the Clopper-Pearson Exact CI. Here is what I have used for calculating the upper confidence interval for alpha. Unfortunately, the theory on how it was derived has faded from my memory.

Clopper Pearson Exact Upper Bound
(1 + (:Total - :Fail) / ((:Fail + 1) * F Quantile(
1 - :alpha,
2 * (:Fail + 1),
2 * (:Total - :Fail)
))) ^ (-1)

[#3]

There are a couple of ways to imitate the Excel example directly using the JMP Binomial Distribution function. One way is to create a column with that formula and other columns for the independent variables and use the Profiler to achieve the target right hand side by varying the proportion p. The other way is to set it up as a nonlinear fit using p again to achieve the target right hand side. They both work.

[#4 · Accepted solution]

Here is an example of the nonlinear approach. The script below creates a data table with same data as in the Excel example. Run the table script to calculate the confidence limits.

(I first tried a neat solution that included the nonlinear platform (invisibly) in a column formula to enable automatic calculation. It worked, but not always as JMP sometimes got stuck inside an apparently infinite loop.)

[#5]

Thank you all for your answers. MS your solution worked when I pulled the New Script section out and ran it separately (in JMP 8.0.2 and 9.0.0).

[#6]

MS is there a way I can generalize your solution for a table full of
values?  For example:




style="font-family: Courier New,Courier,monospace; font-weight: bold; text-decoration: underline;">Preferred
Term    
Count     Total  
Fraction

style="font-family: Courier New,Courier,monospace;">

style="font-family: Courier New,Courier,monospace;">Headache          
18       
513     3.5%

style="font-family: Courier New,Courier,monospace;">

style="font-family: Courier New,Courier,monospace;">Nausea            
24       
513     4.7%

style="font-family: Courier New,Courier,monospace;">

style="font-family: Courier New,Courier,monospace;">Dizziness          14       
513     2.7%

style="font-family: Courier New,Courier,monospace;">

style="font-family: Courier New,Courier,monospace;">.

etc.



It appears from your solution that I would need to create a separate
one-row table for each row
to calculate the confidence limits.



Regards,

Peter

[#7]

Yes, I tried that, it worked but it was a bit shaky. I did not save the script but the principle was something like this:

I used script that excluded all rows then a for loop where each iteration unexcludes the current row, run the nonlinear script (incl. saving the parameter value), and finally re-excludes the current row. However, sometimes I got erroneous results probably because the row exclusion/unexclusion completion was running ahead of completion the numerical nonlinear fitting. One could try to find the sweet spot using Wait()-commands or use some other way to force completion.

Another possibility would be to run the non-linear with a by-argument using column with unique ID for each row. and then loop through the different nonlinear objects to finish and get estimates:
Something like

[#8]

I figured out a way to do it that might be faster than using the
nonlinear function.  Takes about 12-18 tries in the loop
before finding an acceptable value.  Adjust
the epsilon parameter to get higher or lower precision.
 Takes less than a second on a table of 2400 values!



This was written in JMP 9 btw.


style="font-size: 10pt; font-family: "Courier New"; color: green;">/*



style="font-size: 10pt; font-family: "Courier New"; color: green;">Binomial
Confidence Intervals



style="font-size: 10pt; font-family: "Courier New"; color: green;">
style="font-size: 10pt; font-family: "Courier New"; color: green;">Search
for upper and lower
confidence intervals: 
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style="font-size: 10pt; font-family: "Courier New"; color: green;">.05
and .95

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a table with columns called:



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style="font-size: 10pt; font-family: "Courier New"; color: green;">Case
Count   
number of cases for this
Preferred Term



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Cases  
total number of cases



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style="font-size: 10pt; font-family: "Courier New"; color: green;">Lower
CI Limit



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CI Limit



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Upper confidence interval
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style="font-size: 10pt; font-family: "Courier New"; color: navy;"><=
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max_iterations
style="font-size: 10pt; font-family: "Courier New"; color: navy;">,
style="font-size: 10pt; font-family: "Courier New"; color: black;">
i
style="font-size: 10pt; font-family: "Courier New"; color: navy;">++,
style="font-size: 10pt; font-family: "Courier New"; color: black;">



style="font-size: 10pt; font-family: "Courier New"; color: black;">
style="">           
test_b0
style="font-size: 10pt; font-family: "Courier New"; color: navy;">=
style="font-size: 10pt; font-family: "Courier New"; color: black;">
(lower_limit

style="font-size: 10pt; font-family: "Courier New"; color: navy;">+
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upper_limit)

style="font-size: 10pt; font-family: "Courier New"; color: navy;">/
style="font-size: 10pt; font-family: "Courier New"; color: black;">

style="font-size: 10pt; font-family: "Courier New"; color: teal;">2
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style="font-size: 10pt; font-family: "Courier New"; color: black;">



style="font-size: 10pt; font-family: "Courier New"; color: black;">
style="">           
bd
style="font-size: 10pt; font-family: "Courier New"; color: navy;">=
style="font-size: 10pt; font-family: "Courier New"; color: black;">

style="font-size: 10pt; font-family: "Courier New"; color: rgb(0, 0, 221);">binomial
distribution
style="font-size: 10pt; font-family: "Courier New"; color: black;">(
style="font-size: 10pt; font-family: "Courier New"; color: black;">test_b0
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style="font-size: 10pt; font-family: "Courier New"; color: black;">

style="font-size: 10pt; font-family: "Courier New"; color: navy;">:
style="font-size: 10pt; font-family: "Courier New"; color: black;">Total
Cases
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style="font-size: 10pt; font-family: "Courier New"; color: black;">

style="font-size: 10pt; font-family: "Courier New"; color: navy;">:
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Count)
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style="font-size: 10pt; font-family: "Courier New"; color: black;">



style="font-size: 10pt; font-family: "Courier New"; color: black;">
style="">           

style="font-size: 10pt; font-family: "Courier New"; color: rgb(0, 0, 221);">if
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((
style="font-size: 10pt; font-family: "Courier New"; color: rgb(0, 0, 221);">abs
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target)

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epsilon)
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style="font-size: 10pt; font-family: "Courier New"; color: black;">
style="">               

style="font-size: 10pt; font-family: "Courier New"; color: rgb(0, 0, 221);">break
style="font-size: 10pt; font-family: "Courier New"; color: black;">()
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style="font-size: 10pt; font-family: "Courier New"; color: rgb(0, 0, 221);">if
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target
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style="font-size: 10pt; font-family: "Courier New"; color: black;">
style="">           

style="font-size: 10pt; font-family: "Courier New"; color: green;">//
then
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style="font-size: 10pt; font-family: "Courier New"; color: black;">
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upper_limit
style="font-size: 10pt; font-family: "Courier New"; color: navy;">=
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style="font-size: 10pt; font-family: "Courier New"; color: black;">
style="">           

style="font-size: 10pt; font-family: "Courier New"; color: green;">//
else
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style="">               

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style="font-size: 10pt; font-family: "Courier New"; color: black;">
style="">               
lower_limit
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)
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test_b0
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style="font-size: 10pt; font-family: "Courier New"; color: black;">
style="">    )
style="">   

style="font-size: 10pt; font-family: "Courier New"; color: green;">//
formula
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style="font-size: 10pt; font-family: "Courier New"; color: black;">)
style="font-size: 10pt; font-family: "Courier New"; color: navy;">;
style="font-size: 10pt; font-family: "Courier New"; color: black;">
style="">   
style="font-size: 10pt; font-family: "Courier New"; color: green;">//
end new column
style="font-size: 10pt; font-family: "Courier New"; color: black;">



style="font-size: 10pt; font-family: "Courier New"; color: black;">

Thanks again for everyone's suggestions and help!

[#9]

The Chadd's formula and it's lower bound counterpart should give the same upper and lower bounds as the method in the nist manual. I'll look up the other bound if you are interested. That would require no coding at all except to create the column forumulas.

It may be worth mentioning that the "exact" calculation is not necessarily considered "best" in the sense that the resulting interval may be unnecessarily wide. Here are a couple of papers in PDF format that are readily available and which you may find interesting.

http://projecteuclid.org/DPubS/Repository/1.0/Disseminate?view=body&id=pdf_1&handle=euclid.ss/1009213286


http://www.stat.iastate.edu/preprint/articles/2002-05.pdf

[#10]

+1 on that. I'm an Adjusted Wald fan, personally.