4.5 - Other Allocations - Using auxiliary data to be more efficient

We've seen in our last lectures how allocation can play a role in a number

of things with respect to our stratified samples.

And in this lecture on Sampling People, Records, and Networks in our unit four.

We're going to be looking at in stratified sampling some other allocations and

seeing what their properties are.

In doing this, we're going to talk about three things.

We're going to talk about an equal sample size allocation.

About domain estimation,

which often goes hand in hand with equal sample size allocation.

And then the effect on precision of some other allocations.

Let's start though by looking at another allocation besides the one we looked at

in our last lecture which was the proportionate allocation.

But one that alters that to gives us about the same number of sample

elements in each of the strata.

Sort of an approximately equal to distribution in what we did before.

So let's look at our six strata formed in our population of 400 faculty.

Grouping them by sex, and then by rank, assistant, associate, and full professor.

And an alternative allocation the gives us roughly the same number.

The problem is that in a sample of size 80, 80 is not divisible equally by six.

And so what we're going to do in this allocation is approximately equal to.

If we could we would have exactly the same number.

But I guess I blew it by not having a sample size that was

an even multiple of six.

But here we can see what the allocation looks like.

In that last column we've got 13 or 14 in each of the groups.

And the sum of those is still 80.

And that's the key part of what we're attempting to do here.

We are in terms of the individual groups now having small sample sizes in

each of the groups.

We probably would not want to form separate estimates in each of

these groups.

But there's an interesting property, we do have almost the same number of assistant,

associate, and full professors.

Equal numbers there.

Compared to the proportionate allocation where we had 23, 15, and 38,

now we've got 27, 27, 26 across those three groups.

All right, well, that's beneficial because now we've got

somewhat larger sample sizes.

And if we wanted to compute separate estimates for each of those rank levels,

we've got about the same number.

And our precision should be about the same if our standard deviations,

our variances for our income are about the same.

Now they won't be.

But nonetheless this is a good allocation for

producing separate estimates for each of those rank groups.

It's also going to give us 40 females and 40 males.

Little better sample sizes for males and females, half of our sample female,

half of our sample male.

Now that's not necessarily because the population distribution is that way.

We have actually much smaller than half.

Only about 20% of the population happens to be female, but

we're going to force it to be 50% in our sample.

We're departing a long way away from the proportionate allocation.

Recall from the proportionate allocation, well actually you didn't see this.

But if we go back and add these up, there are 17 females and

63 males in that allocation.

About 20% are female.

And so in that proportionate allocation we wouldn't have enough females to compute

a separate estimate.

If what we wanted to do was produce an estimate for females and

males that were equal quality, equally precise.

So by going with this disproportionate allocation, right there,

the proportionate allocation, there's only one.

But it gives us a distribution that is useful for

combining things across this strata.

But this disproportionate allocation,

unequal allocation across the strata in terms of sample sizes.

It has better properties for other purposes.

For example, as I've mentioned, we get better estimates for small sized groups.

We've got only 17 females in the proportionate allocation, now we have 40.

We're going to get more precision for the females, worse precision for the males,

but still the same precision as for the females.

That, by the way, gives us, also, good sample sizes,

a good sample size allocation for comparing the different sized groups.

As long as the standard deviations,

those element variances within each of the groups, are about the same.

The equal sample size allocation is the best one for when we do comparisons.

Getting a small an estimate of the variance of the difference.

Right, that's what a comparison is.

What's the mean salary for females, what's the mean salary for

males, what's the difference between them?

To get the smallest variance of the difference in equal allocation will,

provided we have equal standard deviations among the groups,

give us the best smallest variance outcome for a given allocation.

However when we do this if I also still want to combine the data I have a problem.

If I want to combine the data across assistant, associate and

full professors, I have a problem.

I have a weighting problem.

I'm going to have to resort to the capital Wh.

Or as we will see when we talk about weighting at the very end,

our last lesson here, a different way of expressing those weights.

I'm going to have to weight the elements in a different way.

Because I have an over representation now of assistant professors, and

associate professors.

And an under representation of full professors.

I have an over-representation of females and

an under-representation of males in that equal allocation.

And I'm going to need to correct that when putting things back together.

Again, coming back to Canada.

Remember what I said about how in Canada with the ten provinces, and

in their labor force survey.

They take the provinces and

allocate to them the same sample size in each of the groups.

Including those small maritime provinces out on the right-hand coast,

on the Atlantic Coast.

Now by doing that, they're ignoring the population distribution,

in terms of the allocation of the sample.

Here it is, 38% of the population of the country,

a third of the population lives in one province.

That's Ontario, that's the one that's closest to us here in Michigan.

But there's another quarter of the population in Quebec.

And so on, there are several provinces that have more than their share of 10%.

And then there are four of them that have that.

And the other six provinces all have much smaller shares.

Including one, Prince Edward Island,

that has a very small share of the total population.

Yet the sample allocation is like a pie.

It's ten equal size slices of this thing.

And that looks very different.

And in order to take that pie that's equally divided into ten slices.

And get results from each of those slices and then put it together to look like this

distribution, we're going to have to do something.

We're going to have to take the values for

people who are in Ontario who are underrepresented in that sample.

3,000 households is 10% not 38% of the total.

We're going to have to increase their contribution by a factor of 3 to 4 in

order to get back to the population distribution.

And that kind of reallocation is necessary if we want to achieve two purposes.

Number one, get the best allocations for comparisons.

Or in the Canadian context what they're actually doing is providing

equally precise estimates for every one of the strata.

Because they have a confederation that is very important to them.

That all the provinces have the same contribution or benefit,

if you will, the size of the sample across these national studies.

So they have a political reason for doing this, as well as a statistical reason.

But when they've got all done computing and

providing estimates for Ontario and Quebec and Prince Edward Island.

They also produce a national estimate.

They put them back together, and they do that by a weighting.

All right, so, now we've got two allocations, right?

We've got the proportionate, which we understand a little bit better and

understand some of its benefits.

It's good for combining things across groups.

And we have the equal sample size allocation.

In which, it's better for providing domain estimates,

separate estimates for each of the provinces.

Or for comparing the provinces, in terms of the outcome, in that case,

unemployment rate.

But, in order to put things back together,

we have to weight them, so two different purposes.

We're going to have to sometimes make a compromise in the allocation.

We couldn't do this for a country like the United States that has 50 states.

It's much too many for us to do this kind of thing,

and the sample sizes in each would be fairly small anyway.

So there are different things that are done in those cases that's kind of a blend

of proportionate and equal.

There are other allocations that we can use.

There's one that's called an optimum allocation, or

a minimum variance allocation.

Remember we ran into minimum variance before when we talked about cluster sample

sizes.

The number of elements we're going to take per cluster.

And we talked about a way to choose that sub sample size that would

minimize the variance of the mean.

Well, there's a similar kind of thing that can be done here.

Among all those possible allocations, the proportionate, the equal sample size,

and all the other ones that are in between or beyond those two.

Which of those will give us the smallest variance?

Now, this is beyond the purposes of this course.

It requires, in terms of that allocation to look at the size of the strata and

the variability within the strata.

And sometimes even the cost within the strata.

Sometimes the cost may vary.

Maybe it's much cheaper to do the sample selection in one province than another.

All of those things factoring together will lead to this minimum variance

allocation.

But it's really as I say beyond the scope of this course to deal with it.

Just know that it's out there and

know that it arises primarily in the study of a single variable.

Multipurpose allocation, this is not good for it,

because the minimum variance allocation is variable specific.

There's a minimum variance allocation for each variable, and

it may differ across the variables.

In that case, what are we going to use when we have to do a multipurpose study?

Sometimes we may do the minimum variance allocation for

one variable because it's so much more important than all the others.

We'll still measure other variables.

But we know that we're not going to get as precise a set of estimates for

the other variables as we do for that one.

But that kind of thing comes up more often.

When dealing with variables that have skewed distributions, such as income, or

expenditures, or wealth.

There's where you can get real gains in precision.

But the minimum variance allocation doesn't arise often.

In sociological and psychological studies, public health, medicine, other sciences.

It really works best in economics and finance,

those kinds of areas where you've got some extremes in your distributions.

Let's consider just one more pass through this, just to fix these ideas in our mind.

Let's consider a simpler example to see how allocation

can affect sampling variance, one more time before we wrap up.

And go on to our last lecture.

In particular, suppose that we had this distribution.

I'm going to go back to Qatar.

I know I've introduced Qatar before.

But it has a very interesting set of properties.

In its population distribution,

there is a very large share of the population that are expatriate.

They are not native to the population.

There are white and blue collar expatriates there who are working.

And let's just suppose,

this is a little bit of a departure from the actual distribution.

Suppose there are a million people in Qatar.

Actually there are more than 2 million there.

But a million people and 20% of them are native Qataris, the other 80%,

800,000 of them are white or blue collar expatriates.

And we happen to see distributions we know from past data for

a characteristic we're measuring, say income.

That we've got very different variances between the strata.

You'll notice it in row two there.

The S squared, the overall S squared, and then the S1 squared, S2 squared for

each of the two strata are very different.

And the means are very different.

Here's a good case for doing stratified sampling.

We know the means are different.

A proportionate allocation ought to get us gains in precision.

So will other allocations, potentially.

So why don't we do the following exercise?

Rather than trying to come up with a minimum variance allocation by some

formula, which exists.

Suppose we just start with our proportionate allocation.

We're going to do a sample of 1,200.

1,200 from the 1 million that are there.

And we're going to do it proportionately.

20% of our sample of 1,200 should come from stratum 1.

20% of 1,200 is 240.

And the remainder from stratum 2.

What will the variance of the mean be?

Well, there's our formula that we have for the variance of the mean.

Where we take the square of the fraction of the population in the group.

Times 1 minus the sampling fraction divided

by the sample size nh times the sh squared.

And we have those elements and at least approximately in this case because

the sampling fractions are fairly small.

1200 from a million is so small.

We can just round it.

We're not going to worry about it in this particular case.

It just complicates the calculation.

But here let's just take the essence of it.

Wh squared, sh squared divided by nh for each of the two strata.

W1 squared S1 squared over n1, W2 squared S2 squared over n2 for stratum two.

And put in the numbers and do the calculation, and

we see we get a variance of the mean, in this particular case, of about 1300, 1333.

Now that's the proportionate allocation.

We expect to get gains in precision, how large?

Well, it depends on our computation, what happens for

simple random samples of the same size.

Here's the simple random sampling variance for the same data.

Now again, I've put in the finite population correction here in the middle.

But we're going to ignore it, we're going to round it to 1 essentially and

our variance now comes out to be 1500.

That is when we look at the design effect.

The variance of the given design relative to the variance of

the simple random sample of the same size.

This design effect turns out to be 0.89.

We got an 11% gain in precision by

the simple expedient of proportionate allocation.

But there are other allocations out there.

It may be that we think, whoa, what would happen if I under-allocate to the one

group and over-allocate to the other?

I'm going to do smaller sample to the native Qataris and

more to the expatriates.

What would happen to our variances?

And then we'll go the other way.

We'll look and see what happens if we start shifting sample not

away from the stratum one but into stratum one from stratum two.

So allocation one goes to the direction of putting more sample into

the expatriate part of the population.

Example two is an allocation that we've already looked at, the proportionate.

And 3, 4, 5, and 6 are putting more and more into that native Qatari group,

including the equal sample size allocation in allocation 4.

All right, so for the first one, what do we actually get?

Well, we get a variance of 2,133.

And we get a design effect of 1.45, we are losing precision.

Now this is interesting, with stratified sampling

we don't necessarily only get gains in precision, we can lose.

So some of these are not good for us.

Depending on what's going on with respect to the distribution of

the population sizes.

And the standard deviations or element variances across the strata.

Now, we already know that we got gains in precision for the proportion.

So there we can see, we're not only doing better than the first allocation.

But we're actually doing better than simple random sampling.

We're going in the right direction.

Let's take more and more of the sample away from the expatriates and

put it into the Qatari population.

Can we get even better variances?

So for allocation three, we see that we get a gain again of 1200.

Our sampling variance again is down to 1200 and our design effect is 0.8.

Gee we're doing even better than the proportionate allocation.

That one, 400 and 800 in allocation 3, is actually the optimum.

It's the minimum variance allocation.

If we were to put in all possible combinations,

that's the one that gives us the smallest variance.

That's what we were referring to earlier in this lecture.

But then if we keep pushing it, now we keep pushing it,

now we have equal number in design for equal numbers in the two strata.

We see that our variances all of sudden start bumping back up.

This is the combined.

It's a good allocation for comparing the two groups, but

our design effect is back up to 0.89.

It's still better than simple random sampling.

If we push it even further, the opposite of our proportionate allocation

in design five, we see that we actually get the worst case.

We get a doubling of variance.

It looks like a cluster sample.

But the allocation drives then this design effect, and that's worth keeping in mind.

These allocations, there are many of them.

Proportionate gives us gains in precision uniformly across our variables.

The equal allocation will give us good properties for

comparisons, sometimes gains in precision, sometimes not.

There is one that is best, the optimum allocation.

In this case, we've seen what it is.

And so we see that there's a variety of things that we can do.

When we have a decision to make about how to allocate our sample across

these strata.

And it will depend on our purposes.

With these allocations that we've been looking at,

there is a weighting that's being done, a weighting at the stratum level.

We're going to combine the stratum estimates to compute our overall result.

That stratum level weighting is fine for us to think about conceptually.

But it's not what happens when we use statistical software to analyze these

stratified random samples.

The statistical software that we use actually weights at the element level.

So what we're going to do in out last lecture is look at how that stratum level

weighting can be converted into an element level weighting.

So that we can analyze these stratified samples using available software,

which doesn't do what we've been talking about up till now.

So let's look at that in our last lecture in unit four

on stratified sampling, thank you.

4.5 - Other Allocations

Sampling People, Networks and Records

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