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*1.5 million people have lupus in America out of a population of 308 million. 90 percent are women. My wife is white, as is 63 percent of the country. Minorities are three times more likely to have lupus than whites.

My wife has been diagnosed with lupus, but the diagnosis may not be correct.

*20 percent of lupus patients have a parent or sibling with lupus. My wife does not.

  • 70 percent of people with lupus have positive double-stranded anti-DNA test. My wife does not.

*70 percent of lupus cases are systemic, which is what she is diagnosed with.

What is the probability she has lupus?

So, to be simple... What are the chances given the first set of stats, the 1.5 million out of 308 million and further reducing stats, that she would have lupus (one out of 9 million, etc)?

second question, can the probablity and odds be calculated from the percentages. There was a one in five chance a sibling or parent has it, plus a 70 percent chance she would have the double dna test, but didn't. As we combine these percentages and figures, can it be deduced?

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migrated from math.stackexchange.com Dec 13 '12 at 0:18

This question came from our site for people studying math at any level and professionals in related fields.

14  
It is never lupus. –  Mariano Suárez-Alvarez Dec 11 '12 at 6:09
6  
I do not think this question, as written, is fit for this site. You should ask an epidemiologist. If my vote were not binding, I'd vote to close. –  Mariano Suárez-Alvarez Dec 11 '12 at 6:10
6  
In terms of mathematics this is the perfect place for such a question, but I don't know about this...I'd certainly want to base my decision in such a circumstance on more than a stackexchange answer. In terms a constructive mathematical comment, I'd say we need to know how accurate each of your doctors here are at diagnosis, because this is really a conditional probability...but I agree with Mariano, please ask a real expert for medical advice and not math.stackexchange. –  anonymous Dec 11 '12 at 6:11
4  
If this is a real question, hire a mathematical consultant. Advice is generally worth what you pay for it. –  Gerry Myerson Dec 11 '12 at 6:14
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If so, then please remove all irrelevant personal information. –  Mariano Suárez-Alvarez Dec 11 '12 at 6:17

3 Answers 3

An answer that explained why the question cannot be answered in its present form was given but then deleted after you commented on it; I'll try to explain in more detail why there is not enough information to state the desired probability. I want to emphasize that I'm not saying this because I disapprove of the question; as I wrote in a comment, I think it's not for us to decide whether you should be asking this question; it's just that there is simply not enough information, and if you really do want to find the desired probability on your own, you'd need to obtain the missing information.

First, we know nothing about the tests that were performed. You can see that the answer must depend on the reliability of the tests by considering the extreme cases: If the test is utterly unreliable and its results bear almost no relation to the actual presence of the disease, then the probability is tiny, namely roughly the same as before the test. If the test is perfectly reliable and never fails, the probability is $1$. That's a huge difference, and the only way to know where between those extremes the actual probability lies is from information about the reliability of the test, which we don't have.

Second, all the correlating properties that you list (ethnicity, sex, relatives, ...) may or may not be correlated among each other. That is, lupus might tend to be congenital in men but not in women or vice versa. Without knowing these correlations, one could only give bounds on the probability by making opposite extreme assumptions on the correlations. These bounds might be slightly more useful than the range "between tiny and $1$" due to the reliability issue, but to get a single probability you'd have to know these correlations or at least make reasonable assumptions about them.

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1  
Well reasoned, joriki! –  Kyle. Dec 13 '12 at 1:06
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+1 false positive and negative rates are what's missing. But I think the last paragraph is irrelevant. The point with those properties (not the ones in the bullet points) is that you can break it down to a more accurate percentage for your conditional calculation. i.e. you can figure out what proportion of white women have lupus in the US. –  naught101 Dec 13 '12 at 1:39

My take, as an Epidemiologist: The question really isn't answerable as given, for several reasons:

  • Without very subject specific knowledge, there's no way of knowing if there's effect measure modification between some of those estimates. For example, if you're more likely to have lupus as a woman and more likely to have it as a minority, what happens if you are a female minority member? Are the two independent? Do they interact additively? Multiplicatively?
  • There's a key factor missing: Why is he asking? He's probably not sitting at his desk going "I wonder if she has lupus..." Next week, we're likely not going to get "What is the probability my wife has dengue fever?" There's a reason he thinks this is true, which distorts all those statistics again, as those are population figures, not "Population where a family member suspects you have lupus" figures.
  • The closest thing I could peg as a thing where you could produce a specific value is the "Negative Predictive Value" of the diagnostic test, but a quick googling suggests there are several brands of that particular type of test available, and without more information, you can't really answer it.

What should this poor guy be told? To consult with his wife and her doctor. Trying to apply population level statistics to an individual is exactly not what epidemiological evidence is meant to do.

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Thank you. ${}{}{}$ –  Will Jagy Dec 13 '12 at 0:14
    
EpiGrad, if you re-post this as an answer to the migrated question, I imagine I can just delete this overall question. Yes, I can see that I do have a delete button below my question. If I don't see that, I will just wait on what you and whuber figure out. –  Will Jagy Dec 13 '12 at 0:32
    
Hmmm. Can they migrate this answer including the votes? It's a good answer. –  Will Jagy Dec 13 '12 at 0:55
    
Re: the second point. that was included in the original question, but was removed because it was irrelevant on maths.SE. Check it out at math.stackexchange.com/revisions/256079/4 –  naught101 Dec 13 '12 at 2:27

There is no way anyone can give the answer to the question you ask based on the information you provided. Most of the information you do provide might, if it was thoughly completed (you certainly need the correlations between the individual factors you provide) give the a priori prababilty that a white woman with no sibling etc. would have lupus. That probability could also be computed simply by counting the number of lupus cases among people in that exact category compared to the total number of people in that category (but counting lupus cases is not that simple: your question boils down to asking whether your wife should be included in such a count).

However the key ingredient in the question is that your wife has been diagonsed (maybe incorrectly) with lupus. There is no way in which one can derive the effect this has on the probability. Certainly if the diagonosis is worth anything, it greatly increases the probability above the a priori probability. But to know by how much this affects the probability, one would need to know all the factors the diagonsis is based on (and you supply none), together with a very detailed analysis of how those factors correlate with this disease or other ones. So people are right that you should ask an epidimiologist, who has some chance of knowing the relevant information, unlike the poeple on this site. And even for an epidimiologist the question is very hard to give a reliable answer to.

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