Date: 04/24/2002 at 09:32:20 From: Dave Edwards Subject: Probability Four chiropractors examine a patient. Each can allocate the patient to a category, 1, 2, or 3. I have calculated by percentage the number of times the chiropractors agree for any combination of 2, 3, and all 4. I need to compare this finding to that of chance probability of agreement. How do I do this? I have made grid diagrams of all possible outcomes, but there must be a smarter way. Hope you can help. Many thanks. Dave.
Date: 04/24/2002 at 10:21:23 From: Doctor Mitteldorf Subject: Re: Probability The statistical test you're looking for is called chi-square. You can start reading about it here: Calculating and Interpreting Expected and Chi-Square Tables http://mathforum.org/library/drmath/view/52756.html Here's a response that I wrote to another question on this subject: You've just been hired as an expert in a gambling fraud case. The plaintiff has accused the Slippery Fingers Gambling Casino of using loaded dice. You are handed a six-sided die, and your assignment is to report to the court whether it's loaded or fair. You roll it and get a 2. Obviously that doesn't tell you anything. You roll it again and get a 1. Still, you can't say anything. But suppose you roll the die lots of times - then you should be able to tell. Say you roll it 600 times. Your expected results are: 1- 100 2- 100 3- 100 4- 100 5- 100 6- 100 If you're off by a little, you're not going to say it was a loaded die. But if you're off by a lot, you'll tell the court, "Guilty as charged!" For example, your intuition tells you that if you get Expected Actual 1- 100 105 2- 100 97 3- 100 89 4- 100 108 5- 100 106 6- 100 95 that seems about normal, but if your results are Expected Actual 1- 100 142 2- 100 94 3- 100 90 4- 100 96 5- 100 91 6- 100 87 then for sure you're going to say there were too many 1's. But how do you make this scientific and objective? Can you attach a number to how unlikely you think it is that the die was really straight? That's what chi square is all about. To calculate a chi square statistic, first construct two more columns: The third column is the difference (actual-expected), and to get the fourth column you multiply the third column by itself and divide by the first column: 4th column = chi square = (actual-expected)^2 / (expected) Expected Actual Difference Chi square 1- 100 142 42 17.64 2- 100 94 -6 .36 3- 100 90 -10 1.00 4- 100 96 -4 .16 5- 100 91 -9 .81 6- 100 87 -13 1.69 -------- 21.66 Add up the chi squares and you get 21.66. That's your measure of how far off from the expected your results are. Now comes the part that's a little mysterious. Look in a table for what you expect chi square to come out to. You have six numbers, but they had to add up to 600 no matter what. In other words, if you only had 5 numbers, the 6th would be determined. So you really have only 5 independent numbers in your system, or 5 "degrees of freedom." You look in a table of chi square for 5 degrees of freedom, and find your number 21.66 corresponds to a probability of 0.0006. So you report back to the judge: "Your Honor, an honest statistician will never report anything with certainty. But I can tell you from the experiment I did that there are only 6 chances in 10,000 that that die was true." For an on-line chi square calculator, go to SurfStat.australia: http://www.anu.edu.au/nceph/surfstat/surfstat-home/tables.html - Doctor Mitteldorf, The Math Forum http://mathforum.org/dr.math/
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