Proportions for the Binomial Distribution ©2005 Dr. B. C. Paul

We Did an Example in which we somehow knew all the values for p How do we really know the proportion of successes and failures Done with statistical tests like all the other values

Example We saw the gassifier system was the IGCC Achilles Heal last time around Coolstick Candles has a metal candle they believe can replace ceramic candles and increase available up time  Coolstick claims they can get 96% availability  You go to actual operations using the Coolstick Candles and look at availability statistics Taking random checks of availability you enter 1 if the candles were available and 0 if not.

First Enter Data in SPSS

Next Test the Proportion Highlight and Click on Analyze to Pull down the menu Go to Non Parametric Tests and Highlight to bring out the pop-out Menu Highlight and click on Binomial

Set Up the Test Select Candleswork for our variable We will need to change the test Proportion since the program wants To compare every proportion to 0.5

Set the Proportion and Look at Options I reset the proportion to 0.95 (the company’s claimed Availability) I clicked on options to bring up The option menu I will check-off descriptive statistics And hit continue.

After Clicking OK I get my output Our 60 observations Showed 90% Availability not 95%

Looking at Our Test Table The test tells us that our chances of pulling a sample of 60 with a proportion of 90% when the actual value of p is 95% is just under 8% (Since 5% is standard in statistics we probably will not reject that 0.95 is possible)

Paying Attention to the Footnotes The program does a one tailed test on the odds of the observed value being in Which ever direction chosen from the actual value. It also mentions a Z Approximation.

The Z Approximation If you have a p value of 0.5 and a decent size sample (above 50 or more)  And you plot the number of cases with 0 positives, 1 positive etc.  The distribution comes out looking like discrete value points under a normal distribution curve This suggests that the normal distribution can be used for getting odds on binomial events  Program uses that fact.

Problems with the Z approximation It works well when p=0.5  Which is the reason SPSS tried to default to 0.5 Odds are actually skew compared to normal if p is different from 0.5  P=0.95 is an extreme case  We would need probably over 1000 samples to get a good approximation with the normal distribution SPSS has exact options for people who know their Z approximations are at risk of being screwed up

Using An Exact Option I clicked on exact to bring up the next menu I can approach it two ways Monte Carlo Exact The default is asymptotic (ie meaning as Your data set gets large enough you will tend To converge toward a normal distribution)

Explaining the Options Monte Carlo means making a large number of random trials and then counting results Exact means exhaustively computing every possibility  Better have a fast machine with a lot of memory

For a Sample of 60 I could get exhaustive calculations fairly easily Note that in this case the Z approximation gave us amazingly good results We got just under 8% as an exact value

Z Approximation Allows one to Build Confidence Intervals Have not found an option to do this in SPSS Doing it manually We already know for our Data set r= 54 (running) n= 60 (tries) p’=0.9

Setting Up a 95% Confidence Interval Plugging and Chugging – p is between and

Interesting Conclusions 0.9 seems suspiciously below 0.95 but I cannot rule out that it might be 0.95 Flip side is that regular ceramic candles do 0.87 availability which is also in the confidence interval  Thus I not only cannot say these candles are better than ordinary ceramics – I cannot say they are not worse

May be times I will want to compare two binomial data sets Example  Gasifiers tend to eat refractory (causes down time)  Suppose I try a new refractory and then compare availability before and after the change I pick the same 24 days for months before and after the change and compare.

Enter my Before and After Availability Data in SPSS

Set Up to Analyze My Data Pull down the Analyze Menu Select Nonparametric Highlight two related samples

Set Up My Test Highlight and Click on Before Note it appeared in the variable 1 box

Continue With My Set-Up Highlight the After Note that both selections are now highlighted And variable two is listed as after

More of the Set-Up I clicked the arrow to move the Group over into the test variables list I reset my test to a McNemar test (which is the type used for this kind Of problem)

Clicking on Ok We Get Results We have two days of the month That had run the previous month That did not run this month And three days that ran this month That had not run the previous Month. The significance is nothing (We cannot say the refractory change Has helped based on this data)

Basic Portfolio of tests on proportions SPSS can also test proportions on two sample sets to see if they are the same These are the methods of measuring proportions with binomial data