Learn fundamental concepts in data analysis and statistical inference, focusing on one and two independent samples.

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From the course by Johns Hopkins University

Mathematical Biostatistics Boot Camp 2

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Learn fundamental concepts in data analysis and statistical inference, focusing on one and two independent samples.

From the lesson

Two Binomials

In this module we'll be covering some methods for looking at two binomials. This includes the odds ratio, relative risk and risk difference. We'll discussing mostly confidence intervals in this module and will develop the delta method, the tool used to create these confidence intervals. After you've watched the videos and tried the homework, take a crack at the quiz!

- Brian Caffo, PhDProfessor, Biostatistics

Bloomberg School of Public Health

Okay, hi troops my name is Brian Caffo and this is Mathematical Biostatistics Boot

Camp Lecture five, where we are going to

be talking about relative risks and odds ratios.

So the goal of today is to talk about some

relative measures, like the odds ratio and the relative risk.

And there's many reasons why you might want to consider a odds ratio,

or a relative measure, rather than an absolute measure, like a risk difference.

So, as an example if you're considering comparing

a an event that's somewhat rare. So say for

example, if you're interested in whether or not

some environmental effect causes a fairly rare disease,

where you're comparing a, a small proportion of

people who contract the disease, among the unexposed group.

And a small proportion of people who

contract the disease among the exposed group.

and so it might be interesting to

know that even though the absolute difference in

rates is very small, you might the relative difference might be very large

so frequently we're interested in relative differences than absolute differences.

And two particular in the case of proportions that

are interesting are a, dividing the two proportions that's

the relative risk, and b dividing the associated odds

which is called the odds ratio for obvious reasons.

Okay, so let's put some context on this where we use a

data set similar to or exactly the same as we've used before.

So, consider a randomized trial where 40 subjects were randomized, 20 each,

to two drugs with the same

active ingredient, but, say, different expedients.

so one's a capsule, and one's a tablet, or something like that, let's say.

So, consider counting the number of subjects

having side effects for each drug.

So drug A had 11 people with side effects, nine with none, 20 total.

Drug B had five people with side effects, 15 with non and 20 total

which left 16 total people with side effects, 14 with none and 40 total.

And the interest is whether drug A has a statistically higher percentages of

side effects than drug B accounting for what would be expected by chance.

So lets look into that.

So there's several ways to approach this problem.

In fact the two by tables are surprisingly complex out of

four numbers but we're going to approach it from the following way.

We're going to assume that the count of the number

of people with side effects from group one, is binomial.

With N1 equal to 20, and p1 a proportion that we'd like to know.

And Y, the proportion of, the count of

people with side effects from drug B, will also

be binomial. with number of subject n2, which is 20, in

this case, and p2 being the proportion of people with side effects from that group.

The obvious estimate of p1 is p1 hat, which is x over n1,

and the obvious estimate of p2 hat is y over n2, of course.

So we're going to use the following notation below in addition to the X and

Y notation. So Xis the side of the count for drug a, y

is the count for drug b and so on but we'll also, given

that it's a little two by two matrix, we can index the elements by I and j.

So n11 is the upper left hand cell.

N21 is the, lower left hand cell. And so on.

And, and then if we need a margin.

A point that's fixed by the,

I'm sorry, point that aggregates over, either rows or columns.

Will mark that with a plus, so n plus one means that we, sum

of the two ah,ah first indices

and what plus means we sum over the two lateral indices.

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