When is comes to polling data, I assume the MOE between two values is different than the MOE of a value. This of course makes one’s head spin and is hard to explain to folks. I think some stats folks call it the “margin of error of estimate” for the first and the “margin of error of difference” for the second. Is this right? Is there a rule of thumb for how much biger the error of difference is than the error of estimate?
Of course, there is no meaningful concept of margin of error for categorical data.
That aside, I’m not completely sure what you’re asking. Can you give an example?
If I have a margin of error for estimate from my sample on who will vote for A, the MOE tells me that within some confidence interval, that is where my answer lies. What the MOE doesn’t tell me, is how does this relate to my estimate from my sample of who will vote for B. Is that right?
As I understand it, you have a series of unknown proportions p[sub]1[/sub], p[sub]2[/sub], …, p[sub]k[/sub] that add up to 1. You’re estimating them, and wondering how the confidence interval for p[sub]1[/sub] compares to the confidence interval p[sub]2[/sub], and that for p[sub]3[/sub], and so on and so forth.
They’re the same. They depend on your sample size and that’s it.
Sorry, can you give an example of the ‘error of difference’ as well? I’ll take a stab at it.
Bush 47% (+/- 4 % )
Kerry 48% (+/- 4%)
Those two +/- 4%'s would be your “error of estimate,” yes?
Furthermore you might have “Kerry is leading Bush by 1% (+/- 5.7%)”, *that margin of error being your “error of difference.”
Is this what you’re asking, the difference between those two types of margin of error?
As an aside, it’s fairly to easy to find a confidence interval for the difference of two point estimates–for when you have specific values*–simply add the variances to get the variance for the difference; but I can’t remember how this works for percentages.
*what’s the general term for when you have a count of things, for example, 50 miles per gallon; 200 calories; 98.6 degrees F–as opposed to a percentage?
Ah, I think I understand. You want to know the relation between the confidence intervals for p[sub]1[/sub], p[sub]2[/sub], and p[sub]1[/sub] - p[sub]2[/sub].
I was mistaken earlier–the confidence interval for p[sub]i[/sub] depends on the estimate of p[sub]i[/sub] as well as the sample size (usually denoted n).
The standard error for a proportion p is given by sqrt(p(1 - p)/n), and the standard error for a difference of proportions p[sub]1[/sub] and p[sub]2[/sub] is given by sqrt((p[sub]1[/sub](1 - p[sub]1[/sub]) + p[sub]2[/sub](1 - p[sub]2[/sub]))/n). So the standard error for p[sub]1[/sub] - p[sub]2[/sub] is the square root of the sum of the squares of the standard errors for p[sub]1[/sub] and p[sub]2[/sub].
Confidence intervals depend on the estimator, the standard error for that estimator, and the level of confidence you want, so if you hold that last parameter constant, you’ve got your relationship.
That sounds about right. I’ll noodle on this some more tomorrow.
Don’t neglect the research of Marge Inoverra. 
Here’s a page with an online calculator on this topic.