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.
4.4 - Allocate Sample
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Do curso por University of Michigan
Sampling People, Networks and Records
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James M LepkowskiResearch Professor
Survey Research Center, Institute for Social Research
Welcome, once again, to our continuing series of lectures about stratified
sampling, as we're talking about sampling people, records, and networks.
Our fourth lecture, in this series, we're going to talk about sample allocation.
Actually, we're going to talk about sample allocation in this lecture and
the next one.
We're going to do one particular allocation, actually the one we've
already looked at, proportion allocation, to understand some of its properties.
And then we will alter it in the next lecture.
With respect to proportion allocation,
we're just going to talk about allocations in general, then, and
the proportion of allocation, before we wrap this particular lecture up.
Now the stratified sampling approach has, as we've seen, some advantages.
I mentioned one about credibility.
It just makes more sense.
It's more believable to people to say that when I drew my sample,
I made sure that I had the right distribution.
Across the frame, the population, that it got replicated in the sample, as well.
That's more acceptable, more credible.
But we've also seen how we can get gains in precision,
depending on the allocation, though, and
that's part of what we're doing in these two lectures on allocations.
There's also advantages here.
We've talked about guaranteed representation of important domains.
And there are some things about flexibility that we'll see
here in allocations and administrative convenience that we can take advantage of.
But these, in order to understand these potential benefits of stratified sampling,
depend on understanding something about the allocation, as well.
We've been looking at a problem in which we have a basic sample size,
in our illustration with 400 faculty, we were drawing a sample of 80, and
we chose a particular allocation, but many allocations are possible.
Many different sampling fractions of sebetra are possible across those stratum.
So for example, for our six stratum, as shown in the last line on this display,
the allocations could have been take one from stratum 1,
one from stratum 2, one from stratum 3, one from stratum 4,
one from stratum 5, and 75, the balance, from stratum 6.
Or we could have taken two from stratum 1, and one from each of the stratum 2,
3, 4, and 5, and then the remaining 74 from stratum 6.
And so on.
There's lots and lots of possible allocations there.
It turns out that some of these allocations can be beneficial, and
some can be harmful to our overall estimates.
Some can be beneficial when we want to do domain estimation.
Some can be beneficial when we want to combine a cross estimate
of these estimates.
And the allocation that we did was related to the drawing in the lower left.
This is sort of a population-sized
distribution of the states in the United States.
California is a large share of the US population,
10% among the 50 states, and Florida and New York are fairly large,
as is Texas, in terms of their relative population sizes.
And others are very tiny, they become very small on that map.
As that population size varies, which it will,
and it's outside of our control, it's something that's going to be
there with respect to the auxiliary data that we have available.
We can take advantage of that in order to get gains and precisions.
One way to do that,
that we've already seen, was to use the allocation that we already did.
The one that was the proportionate distribution.
This particular distribution, we had a reason for using.
And that was because we actually got a gain in precision.
Taking the same percent or fraction of the elements in each of the six stratum, and
we have a nice property to it too.
That distribution, 8, 5, 4, 15, 10, 38, across the six stratum,
also is achieved by taking the same sampling fraction in each of the stratum.
Well that seems straightforward.
It has nice properties with respect to the sample design.
You're making the sample designers very happy.
But does that have any other payoff for us?
Well actually we saw that this kind of thing has a benefit that goes beyond it.
It gives us a balance then between the sample and the stratum.
That is when we draw the sample in such a way that we have the same sampling
fraction, lower case f sub h,
lower case m sub h over M sub h, is the same as lower case f,
the overall rate, lower case n over N, that's the technical detail of this.
When that kind of thing happens, there's something else that happens,
that goes with this.
It generates a distribution in the sample that is across the stratum,
the same as the distribution in the population for the frame.
For example, for our stratum 1, where in the population there were 40 of the 400,
or a W sub h of 0.1.
10% of the population resides, or is in that particular stratum.
When we sampled at the same rate in stratum 1,
as we did overall, 80 from 400 is overall.
0.2, 20%, when we applied that rate of 0.2,
we got 10% of our sample, eight from that particular stratum.
Now the sample distribution looks like the population distribution.
So by being epsom, equal chance,
we happen to also be proportionate, and this is where that term comes from.
Now I know this is confusing, I want to come back to the use of proportionate in
cluster sampling in just a second.
But here what we're talking about is the proportions we see in the population
are replicated in terms of the proportions we see in the sample across the stratum,
proportionately allocated.
Now, we used this term before with
cluster sampling when we had unequal sized clusters.
In there, we were sampling the clusters,
with probabilities that were proportionate to their size.
Same idea, but applied in a different way.
So, this term proportionate comes up in several different ways in
the sampling context.
And it's a broad label of things.
For our purposes, when you hear proportionally allocated
stratified sampling, it's the kind we've just been doing.
The same sampling rates in each of the stratum,
the sample distribution looks like the population distribution.
So when we make the sample look like the population, the W sub h,
N sub h over N is the same as the sample fraction, n sub h over n.
And that is the same thing, than just to repeat,
as having the sampling fraction be the same in all pf the stratum,
the overall sampling faction is replicated across each of the stratum.
When we have either one of these, but if I have one,
I have the other, that's where I get gains in precision.
This kind of design will give us design effects less than one for
our outcome variables.
Now, I'm going to make a very strong statement here,
which is actually not true in all cases.
But I want you to remember it.
So for teaching purposes, if we do proportionate and
allocations, and we do multipurpose design, remember now proportion and
allocation could use one or more auxiliary variables to form groups, and
then allocate the sample proportionally across them by drawing an absent sample.
If we do that, when doing multipurpose sampling,
we're going to get design effects less than one for all the variables.
Now it's too strong of a statement.
It's actually not true, but it's a good way to think about it.
We're going to get some gains of precision for almost all the variables in our study,
and that's a very powerful tool.
We won't find other allocations that do that for us in the same way.
We'll see allocations that can beat stratified random sampling,
proportionately allocated, for one variable, but
not necessarily for all of the variables, or nearly all of them.
Well, let's turn to talking about some of those other allocations,
now that we understand a base allocation.
Remember, I did this for stratified random sampling.
Stratified random sampling was there in the beginning because it was a basis
against which we could compare the others.
Proportionate allocation is very useful in its own right, but
it's also a basis for comparing other allocations, too.
Let's turn to that in lecture five for unit four next.
Thank you.
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