Good data collection is built on good samples. But the samples can be chosen in many ways. Samples can be haphazard or convenient selections of persons, or records, or networks, or other units, but one questions the quality of such samples, especially what these selection methods mean for drawing good conclusions about a population after data collection and analysis is done. Samples can be more carefully selected based on a researcher’s judgment, but one then questions whether that judgment can be biased by personal factors. Samples can also be draw in statistically rigorous and careful ways, using random selection and control methods to provide sound representation and cost control. It is these last kinds of samples that will be discussed in this course. We will examine simple random sampling that can be used for sampling persons or records, cluster sampling that can be used to sample groups of persons or records or networks, stratification which can be applied to simple random and cluster samples, systematic selection, and stratified multistage samples. The course concludes with a brief overview of how to estimate and summarize the uncertainty of randomized sampling.
6.2 - Stratified Multistage Sampling
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Sampling People, Networks and Records
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James M LepkowskiResearch Professor
Survey Research Center, Institute for Social Research
Up until now, when talking about sampling, whether it's people or records or
networks, we've been dealing with one sampling technique at a time.
But here, as we move into the second lecture in Unit 6 on some extensions and
applications, what we're going to do is talk about how to combine our
techniques in manners that allow us to address more and more complex frames.
We're going to talk about stratified multistage sampling that combines
our ideas about randomization, as well as stratification, with a whole
lot of clustering in our sample selection, to address the problem we've referenced
before with respect to cluster sampling, what do I do when I don't have a list?
What do I do to try and reduce the sampling operation to make it
cost-feasible for me to get this work done when I don't have that list?
And we're going to do some cluster sampling, but
we're also going to stratify the clusters and
we're going to make sure that those clusters are selected at random.
But we're going to do this at several levels.
We're not just going to have one level of clustering, but in our particular
illustration, two or three levels of clustering in our selection.
We'll talk about, then, the first level of cluster selection,
what we'll call primary sampling units, or PSUs, as our first topic.
And how we deal with those, what they might look like.
And I'm going to take a particular example from the United States
just to illustrate this.
This is not to say that sample selection needs to look like this in other settings.
This is just to illustrate the ideas and indicate that these are not
ideas that are completely foreign, difficult for us to apply.
They deal with real materials, as we've been doing all along.
We'll talk about what happens after we select those units, when we still are not
at a place where we have a list of our elements that we're interested in.
We'll talk about second-stage units.
And in our particular case,
we'll even move to a third-stage unit called housing units.
So we're going to work our way all the way down for
drawing a sample in the United States of housing units.
The housing units, for us,
may be the object, they may end up being what we want to study.
But more likely, what they will represent to us is a collection of people
who usually reside together in something called a household, and
it's the people that we're after.
So there may be an additional step of selection within
a sample of housing units in order to get to our final elements, persons.
At the very end I want to say just a little bit about estimation.
That's beyond the scope of this course, but
we'll mention just a few things about it before we wrap up on this lecture.
So the first thing that one does is think about the sampling units that one's going
to have available for a particular application.
Now, many times the applications are such that other people have already done
similar work, and you can build on their work or borrow from it.
In our particular case, we're going to be looking at drawing a sample of
housing units from across the United States.
And we're going to start by using units that are readily available to us.
In this particular case, nothing to do with housing units specifically,
but to land areas.
We're going to identify the primary sampling units, our starting point,
something that has a large enough number of those units.
Not just a few of them that we're going to take an even smaller number, but
a large number of them that we're going to make a selection of them.
And then we're going to divide the selected primary sampling units up into
secondary units.
And then we're going to select those, but
only within the selected primary sampling units.
And within those secondary units, etc., we're going to select either
third-level units or housing units, however we're going to get there.
And I'm going to take you through just a few illustrations of some of the material
that might be used for something like this.
Now, our primary sampling units for
our particular application in the United States are subdivisions called counties.
There was a time in which not all the states had counties.
They all had subdivisions, but
sometimes they were called different kinds of things.
And actually, to this day, some of them still have those labels.
For example, in Louisiana, the counties are referred to as parishes.
But they're basic units, and this is a map of the United States showing the detailed
boundaries of all of these counties.
There are over 3,000 of them in the United States, and Alaska and
Hawaii are inserted here, and they, too, have these kinds of subdivisions.
They vary enormously in geographic size, but
we're going to think about these counties now as collections of housing units.
Not a collection of housing units that we can go to and list all of them.
These counties can have hundreds of thousands,
millions of housing units in them.
And some can have fewer than 10,000.
So there's a lot of variation in size.
But this is a good place for starting to sample.
We've defined the unit that we're going to use as our primary sampling,
our first-stage selection.
Knowing that there will be additional steps in the selection,
stages in the selection, clustering, to help us get to a point where we can have
a list of housing units from which we can draw our sample.
So, we've identified those first-stage units, and what we're going to do with
them is stratify them, just as we've done before when we dealt with elements.
We're now going to stratify the primary sampling units.
Stratification was a general-purpose tool and we used it to assure
representation and possibly provide us with gains in precision,
as you recall when we did proportionally allocated stratified samples of elements.
When we do that application of stratification to these PSUs,
in this case counties, we're going to follow principles that are similar to
those that we used in element stratification.
But here, we're going to use cluster characteristics.
We're going to stratify the clusters, not the elements.
Although implicitly, the housing units are getting stratified,
because they're contained within these PSUs and
we're putting the PSUs into groups where they're relatively homogeneous.
The counties within a group are similar to one another,
with respect to characteristics that we're interested in.
With respect to the nature of the counties, is it an urban or
a rural location?
Does it have high or low rates of unemployment?
Does it have a high fraction of occupied housing units or a low one?
All sorts of things that we can identify for these primary sampling units.
Our groups will be mutually exclusive and exhaustive, just as we did before.
The stratifying variables, the boundaries and the other kinds of things
follow the same kinds of element sampling stratification principles we talked about
before, but it's applied just to those units, as I've been saying.
But once we've got that creation of those groupings,
those strata, then we're going to allocate a sample of clusters across the strata.
We'll figure out how many of these things to use
using the kind of approach we talked about before with respect to cluster sampling,
where we identified the optimum subsample size, and thereby, the optimum
number of primary sampling units that we can select, the number of clusters.
We could do the allocation across our strata proportionately.
We could do it to such an extent that we have only two selections per stratum.
We could do it so that there's a different number than two but
an equal number in each of the strata, or other kinds of allocations.
We've mentioned Neyman or minimum variance kinds of allocations.
Very seldom used with this kind of thing, because there is not much
gain available beyond the kinds of things we can do proportionately.
Within each of the groups,
within each of the strata, we're going to select our samples.
And then, we are going to, after data collection, Compute estimates for
each of the strata, each of the groups separately.
Not only the statistic we're interested in, let's just say a mean or
a proportion, but also the sampling variance of that mean or
that proportion within each group, and then we're going to combine them.
We're going to use those combining rules we did for stratified sampling.
But now applying it to the strata for the primary sampling units.
Again, the stratification is there to control the distribution of the sample
to make sure that certain groups are not left out due to randomization and
chance result in our sample selection.
And we have the potential to decrease sampling rates too
depending on how we apply our allocation rules.
Well, how could we do that kind of stratification?
Well, with our PSU's, with our Primary Sampling Units being counties.
Here's another map of the United States.
At a slightly different scale and Alaska is now on the upper left and
Hawaii in the lower left and even Puerto Rico is shown here.
But this is a map showing the counties now that
are metropolitan of two levels, very large and all other.
And they're highlighted in green and then dark green and
even a lighter green shading.
And then the other counties that are in white are those that are non-metropolitan.
So this is a visual way of representing a stratification,
by taking those counties and dividing them into, let's say, four groups.
The very large metropolitan areas, the medium sized metropolitan areas,
the smaller sized metropolitan areas, and the non-metropolitan areas,
all represented in terms of the colors on the graph.
That's a form of stratification.
Well there are other kinds of things that can be done.
Usually what we're going to be doing is not looking at a map, but
a list of the primary sampling units.
And this is a partial list of primary sampling units that are counties in one
part of the United States that has already been organized for.
So, they've already been grouped into a stratum, they've got numbers on them,
we've got their names, and for each of them we have size,
we have even accumulation of the size.
And what we're going to do here is order these,
not by alphabetic means which it may appear that these are, but geographically.
We're going to put these counties next to each other
on the list that are next to each other physically.
And so we're going to work our way through this particular location, and organize our
list, order our list geographically, and then draw a sample from it systematically.
That will give us implicit geographic stratification, even within
our larger strata, where we've explicitly grouped these counties together.
So, we will use combinations here of stratification and
clustering, along with randomized selection, but we'll even use an implicit
stratification through systematic selection with random starts.
That's the selection of the counties from a geographically ordered list.
What happens then at the second stage now?
Supposed we've selected those counties, what should we do at that point?
Well, we have some decisions to make with respect to sampling units again.
We're now going to need to look to see what can I do within this county?
Can I go right to the housing unit?
Well, most of these counties are way too large for us to do that.
We would end up listing thousands and thousands of housing units.
And in our sample drawing, only a small number, a few hundred perhaps.
Well, we've spent a lot of money on this.
We wouldn't want to do that.
So we're going to introduce another level of unit
there to reduce potentially the amount of listing that we have to do.
Here, one kind of second stage unit that comes from the in the US case.
The census operations are called census tracts.
Geographic areas that are defined to be like neighborhoods
they're designed to have roughly the same number of housing units in each.
But there's a lot of variation in size.
They're created for purposes of reporting, statistical reporting.
So that these areas actually have more detailed data presented for
them from the census information.
And here you can see in this particular county,
this is a county that we've zoomed in on.
We're showing the basic boundaries of the census tracks, but you see there's
concentration's dense areas where those census tracks come out smaller and
others where they're much larger.
Population is more dense in those areas where we have lots and
lots of census tracks.
So that's one possible unit for our selection that we would use to select
an additional location or areas within our county before we start our listing.
The difficulty with census tracts is because they're fairly large in size.
We're talking about thousands of housing units in them.
They are following boundaries that are convenient to follow.
Most of the time, these are roads that people can follow so
they're not hard to do, but they are a fairly large forms.
And so we might seek another unit that's smaller in terms of the number of housing
units within it and indeed there is such a unit called a census block.
And here's a map of census blocks within that same county, and
you see the heavy dark lines there to find tracts.
Now these are tract abbreviations, they all start with a 4,000.
But these are the tract boundaries.
And within them, you can see finer gradations which are census blocks.
All of the land area within our county is subdivided into not only tracts,
but even finer gradations, blocks.
And this happens to be around the University of Michigan
where we're recording these lectures.
And so we have a choice, should we use sets of tracts, larger units and then
try and figure out what to do with housing units, or should we go right to blocks?
Or should we do tracts and then blocks?
There are decisions to be made.
On the basis of practical considerations that are a little bit beyond our scope,
one would make a choice.
In this particular case,
we decided that because the census blocks cover all the land area and
their group within tracts, we may as well just use them directly.
It means that our sample may be a little more spread out because we've got blocks
all across that county.
But the travel considerations and other cost considerations indicated
that that's not a substantial problem for us to deal with as opposed to having
a few fairly large areas, called tracts, to deal with.
Again, with the secondary sampling units, we can create a list.
These kinds of materials are readily available in this particular case at
the block level.
Here's a listing of blocks in which we indicate the tract and block number.
All the blocks have numbers rather than names.
They have counts of housing units that we can use as measures of size.
They can be cumulated, they can be numbered, and
this kind of thing then corresponds to that geographic area in the lower left.
I'm showing an 18th century map of the city of Detroit, one of the oldest cities
in the United States, but even back then it was divided into blocks because
the land area was divided geographically for purposes of agricultural production.
And there were housing units located on different tracts of land.
But those tracts of land often were bounded by roads or
in those days dirt tracks.
But that list can, again, be used for
drawing a sample of blocks from a particular area.
Here, for example, within a multi-block area, in
the white unshaded area are a set of three blocks that were selected for our study.
They were selected together because they were fairly small in size and we decided
to link them together to give something sufficient in size for us to work with.
But nonetheless,
that selection leads us to then the next level which is the housing unit.
Now are we down to the level where we can actually list housing units and
have a small enough number in our list that we will select enough of them
to make it worth our while to have listed the housing units?
So here's a magnification of that area,
a sketch map that we're going to need to fill in where the housing units are.
Now, this particular illustration that I'm giving us here
comes from a sampling operation in which there were no listed housing
units available from any commercial source or any government source.
So we had to create them ourselves.
So we actually sent someone physically to that block to list the housing units.
And as they were doing it, they made annotations on the map that we gave them
to tell us, look, here's where I started, look at the X.
And the arrow indicates which way I went,
I went in a clockwise fashion around the block.
And by the way, there's a public library here on this corner.
There no housing units over here.
There's a park in this particular corner.
That kind of thing then will help us orient for
after we've selected housing units, getting back to those housing units,
finding them, and conducting data collection.
But we don't actually try and list things on that map, but in a separate listing.
You have a list of housing, just a list like this one,
a housing unit list of the mapped area.
Showing the address of the housing unit, and in some cases, a little more detail,
when it's not entirely clear that it would be easy to find them.
It might be an apartment building, it might be a building that
was a single family home divided now into apartments.
We may need some specification that goes with it.
But that listing there is defining what our housing units are.
And they're done geographically.
Listed geographically, so that we can make a selection that has
implicit stratification by geographically by selecting from it systematically again.
Notice, systematic selection gets used in this process, because from an ordered
list, it has valuable properties without us having to form explicit strata.
Alright, so, multistages.
In that particular case, we started with counties, we went to blocks, and
then we went to housing units.
In each case we thought about stratification either explicitly,
formal groupings, or implicitly through an ordered list and
a systematic selection from it.
However we end up doing our final sample, combining our sampling techniques.
We are still faced with the task of estimation.
In this case, estimation from a stratified multistage sample is just beyond
the scope of the course.
It's not that it's going to be unfamiliar to you in looking at it, but
you'll find that there's a subsequent course in the specialization that
addresses the issues that are there.
Such as computing estimates for complex samples, and
computing estimates of sampling variances from these complex samples.
Complex samples involving stratification, clustering, and randomization in them.
And then drawing inferences, how do we calculate from those standard errors,
confidence intervals that take into account the design,
what degrees of freedom should we use, and so on.
So, estimation would follow from this particular sample design.
But all these features are going to be built into the estimation,
just as they're built into the sample selection that we've just gone through.
Okay, so it's an extension of what we have been doing,
an extension to stratified multistage sampling.
So as you get some idea about what we mean by this, it's done in many different ways
across countries around the world, depending on the available materials.
Whether they're available from the census or from registry systems.
But nonetheless, those samples would follow very similar kinds of designs.
Let's turn in lecture 3 then,
to a series of three lectures that will deal with weighting.
As part of our extensions and applications, because weights are,
as we've seen already, problematic in estimation.
And we want to talk in a little more detail about how weights are created and
used in three instances.
One, having to do with over and undersampling, in our next lecture.
As well as non-response and non-coverage adjustments and network sampling.
But our next lecture, lecture 3, will be on weights for over and undersampling, and
I look forward to you joining us there as we develop that topic more fully.
Thank you.
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