In our last part of lecture five we were dealing with a problem concerning

unequal sized clusters and the mess it was making of our sample design.

We were either dealing with more complicated estimators, or

because of variation in sample size, or a nice clean sample design,

but the need to wait.

The need to create weights to compensate to avoid bias in our samples,

in our estimation.

And as I said, we're going to come back and talk about weighting later.

But here what we're going to do is continue in our discussion of dealing with

the real world and sampling unequal size clusters in lecture five by continuing on

our discussion by looking at an alternative to these other schemes for

dealing with unequal size clusters in our sampling.

In dealing with them through a sample selection mechanism in two stages,

that we're going to call probability proportionate to size, or PPS.

So what we require is a method that has

some of the properties of the two things that we were just dealing with.

Number one, equal chance, one in 60, for all the employees.

That was an attractive feature for us, and so, that Epsem,

equal probability of selection method was attractive, and we want to retain that.

At the same time, we would like to retain the features of the alternative

method where we had the same number of elements in each of the subsamples.

So let's consider two clusters from two hospitals from our

population of 12 in which we're going to take 15 employees in each.

And we've just set up kind of an algebra problem here, a selection equation.

Our overall f is going to come in two parts.

It's gotta be one in 60 as our overall rate.

As going to be done in two parts.

A probability of selection of the cluster P of alpha,

alpha for the cluster, and then within sampling,

where we're taking 50 employees from those that are actually there.

It depends on which cluster are selected as to what that denominator is, so

I've just put in capital B sub alpha.

It could be 50 from 60.

It could be 50 from 180.

Could be 50 from 1,860.

But that's what we've just said by saying that it has to be equal chance.

That's the 1 in 60, and in two stages on the second stage, I want the same

number of employees, the same number of elements from each of the clusters.

That's the beginning.

Now what should that piece of alpha be?

So we'll do a little manipulation.

We can do this to figure out what it would be.

But it turns out that that first stage sampling fraction, if

we look at that last expression there, the probability of selection of the hospital,

the alpha hospital, is going to be proportionate to the size of the hospital.

B subalpha over 3,000, that's what will get that equation to work.

But wait a minute?

How am I going to sample hospitals with varying probabilities?

I know how to sample employees with varying probabilities across the hospital.

That's easy, I just take the same number and then take it from how are many

are they're, but how am I going to do something like this?

Well the fractions work out this way.

This kind of thing says that what we're going to do is take two hospitals

with probabilities proportionate to size, and then take the same number of employees

in each, that's what all of these equations refer to.

Across two stages we're going to get balance in the overall sampling fraction

for all the employees,

they all have the same chance of being selected by doing it in two steps,

where the first step, we over sample big hospitals and under sample small.

But, in the second step, we over sampled employees in small hospitals and

under sample employees in large hospitals, and

the produce of the two always comes out to be our one in 60.

Boy, I need an easy way to do this because that sounds complicated to me.

It's hard to keep track of, this is an bookkeeping problem.

We know have to keep track of probabilities of selection for

the hospitals and for the employees within hospitals.

But it can be managed as long as we're careful and

go through this slowly with our available information.

So let's go back to our list of hospitals, here they are now,

here's all twelve of them, and do a PPS sample.

And I've added a new column here, the cumulative size of the hospital.

So, notice for hospital one, the actual size is 420, the cumulative size is 420.

For hospital two, the actual size is 180,

the cumulative size is 600, which is 420 plus 180.

For the third hospital, it's size, it's actual size is 120.

It's cumulative is 720.

600 plus 120. We keep adding in the sizes successively,

till we get down to the 12th hospital where its actual is 240, and

its cumulative size is 6,000, which is the total number in our population.

So here's the process, what we're going to do is generate a random number now,

not from one to 12 but from one to 6,000.

Let's suppose that we select two random numbers from one to 6,000.

The first random number is 702.

Four-digit random number from 1 to 6,000, 07, 02, 1744 is the second one.

We're going to go through and find the first hospital on the cumulative

sum whose cumulative sum is greater than or equal to the first random number, and

then we're going to do the same thing for the second number.

Find the hospital whose cumulative sum first exceeds,

as we're working our way down the list, to that number.

That chooses hospital three and seven.

So you can see now I've lined up 702 with hospital three.

Because 702 falls in the range,

if you will, from 601 one more than hospital two's, but

it's included in hospital three's up to 720, and the 1744 falls in the range,

let's see, from hospital five to six, from 1561 to 1920.

In the first case, there are 120 numbers from 600 to 720.

601, 602, 603 and so on through 720.

120 numbers we chose one of them.

But that means that I could have chose any one of the numbers

from 601 to 720 in that hospital.

The chance of that hospital with one random selection being selected

is 120 from the 6,000.

Of, that's what we wanted,

is proportionate to it's size using the cumulative numbers.

Similarly for hospital six, there are 360 numbers that go from 1561,

the one that's above hospital five, from 1561 to 1720.

360 from 6000, that's its chances of being selected, and

we happen to get one of them, 1744.

So now, we've made two draws, each with probabilities proportioned to size,

which is what we wanted in the first stage.

Now we can go to the second stage and take 50 at random from 120, and

50 at random from the 360 in hospitals three and six, respectively.

That has a danger, though.

We could end up selecting the same hospital more than once.

We could have gotten the same random number twice or

random numbers from the same hospital.

So to avoid that problem, we're going to do a form of systematic sampling.

Something that we'll come back to, so just learn the mechanics right now.

Let's start with that first random number 702.

Now let's pretend now that this was selected not from one to 6,000,

but from one to 3,000, the first half of the list.

We're going to get a selection from the first half of the list for

our first one, and then we're going to find the selected hospital as before.

But then we're going to add 3,000.

Go from the first half of the list to the second, to 3702,

which gets us a different hospital, in that case, hospital 10.

This is what's called systematic probability proportionate to size.

We're dividing the list into two groups,

because we have two samples that we're doing.

The first half of the list from which we draw one at random, and

the second half of the list from which we count the size of the group,

the list size divided by two to get the second one.

And that gives us our two selections then of hospitals three and 10,

and we will not select the same hospital more than once.

We only get one shot at each.

One from the first half, one from the second half.

So in that particular case then, we now have our sample method in terms of

not only being able to sample hospitals that are unequal in size, with equal

chance of the elements within them, and get the same size subsamples from each.

But do this in a way that avoids getting the selection of a hospital

more than once, systematic PPS selection.

All right. That's a little complicated but

I wanted you to understand this because this is what happens in the real world.

It's a more complicated version of what we've been doing.

We're going to do one more topic here on subsampling.

I want to come back to that subsample size B.

I know this is a little out of sequence but

it's kind of hard to put it in sequence because these are practical problems and

they don't necessarily fit together in some neat logical order.

But we're going to talk about how to figure out how many elements to

take for cluster.

Where do we decide to have 50 per hospital,

besides saying two hospitals for a sample of a 100, why two?

Why not three or five or something else?

Okay, that'll be our next topic on subsampling.

Please join us then for our last lecture in unit three at cluster sampling.

Thank you.

3.5 - Dealing With the Real World - Unequal Sized Clusters - Part 2

Sampling People, Networks and Records

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