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Sample stratification involves two steps:
(a) divide the population of sampling units into population sub-groups, called
strata
(b) select a separate sample per strata
If the same sampling fraction is used in each stratum this is termed ‘ proportionate
stratified sample’; if the sample fraction is not the same in each
stratum this is termed ‘disproportionate sampling’. More commonly
the latter would be described as ‘over-sampling of one or more sub-groups’.
Proportionate stratified sampling almost always leads to an increase in survey
precision (relative to a design with no stratification), although the increase
will often be modest, depending upon the nature of the stratifiers. Disproportionate
sampling sometimes increases precision and sometimes reduces precision. Surveys
using disproportionate sampling have to utilise survey weights more
about weighting if they are to give unbiased cross-strata estimates.
As well as using stratification in the sample design, survey
statisticians sometimes use ‘
 post-stratification
’ once the data
has been collected.
'Post-stratification'
is a weighting method that adjusts for any differences between the survey
data and the population in terms of a few key population variables (often
age and sex). The aim is to reduce any bias in the survey due to sampling
error and/or non-response effects. More detail is given below.
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In a proportionate stratified sample, the population of
sampling units are divided into sub-groups, or strata, and the sample is selected
separately in each stratum. For the sampling to be proportionate, the sampling
fraction (or interval) must be identical in each stratum.
Suppose a sample of 100 students is to be selected from
a school with 2000
students, so that the sampling fraction to be used is 1 in 20. If, before
drawing the sample, the school roll is divided by age and sex, and a separate
sample is drawn per age and sex stratum, then if the sampling fraction of
1 in 20 is used in each stratum the sample would be a proportionate stratified
sample.
This
is illustrated in the Table below where we can see that in real examples the
sampling fraction will vary a little between different strata. This is because
we can’t get a fraction of a pupil.
|
age/sex group
|
school roll
|
1 20th
|
number selected
|
actual sampling fraction
|
equivalent to 1 in
|
| under 6 M |
169 |
8.45 |
8 |
0.0473 |
21.1 |
| under 6 F |
147 |
7.35 |
7 |
0.0476 |
21.0 |
| 7-9 M |
194 |
9.7 |
10 |
0.0515 |
19.4 |
| 7-9 F |
213 |
10.65 |
11 |
0.0516 |
19.4 |
| 10 + M |
177 |
8.85 |
9 |
0.0508 |
19.7 |
| 10 + F |
198 |
9.9 |
10 |
0.0510 |
19.8 |
| TOTAL |
1098 |
54.9 |
55 |
0.0501 |
20.0 |
Table 1.1 Sampling Fractions
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Proportionate allocation is used for two reasons:
(i) to reduce
standard error
for survey estimates;
(ii) to ensure that sample sizes for strata are of their expected size.
For example, almost all large-scale GB surveys that use the
PAF
as a sampling frame use samples stratified by region,
and within region, by a measure of relative area deprivation. The first stratifier
(region) is used to ensure that the selected sample is correctly proportioned
by region. (A national sample that, just by chance, happened to under or over-represent
some of the regions would be considered by many as 'unrepresentative'.
The second stratifier (area deprivation) is used to ensure that the selected
sample is correctly proportioned by area type. In practice, many survey statisticians
would argue that of the two, only the second stratifier is strictly necessary,
and that the regional stratifier is largely cosmetic for the purpose of getting
national estimates. This is because area deprivation is strongly correlated
with many of the outcome measures social surveys collect. So ensuring that
the sample has the correct area deprivation profile means there will be less
sampling variance in the estimates and standard errors are almost bound to
be smaller than would be the case with an unstratified sample. Put another
way, if the area deprivation profile of the sample is controlled, the risk
of selecting an unrepresentative sample by chance is reduced.
Region, in contrast, tends to be only weakly associated with social survey
outcome measures, so stratification by region does not reduce sampling variance
by very much. In other words, even if the regional profile of the sample is
controlled, the risk of selecting an unrepresentative - sample by chance does
not significantly reduce. The reason for stratification by region is usually
not to improve precision of national estimates, but to get better estimates
for regional analyses. Click here for a discussion
of why this is so.
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Relative to taking a completely unstratified sample, taking
a proportionate sample is either a good thing, in that it reduced standard
errors, or a neutral thing, in that standard errors don’t change. Proportionate
stratification can never increase standard errors. The reasoning is as follows:
- total sampling variance can be decomposed into two components: within-strata
variation and between-strata variation (the split between the two depending
on how the strata are defined);
- with proportionate stratification the between-strata variance becomes zero.
So, proportionate stratification is most efficient when the stratifiers that
are used split the total variance in a way that maximises the between-strata
variance.
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In proportionate stratification a distinction is made between
‘explicit’ and ‘implicit’ stratification.
‘Explicit stratification’ is where the population of sampling
units is explicitly divided into strata and a separate sample selected per
stratum. ‘Implicit stratum’ is where the population of sampling
units is sorted by some characteristic(s) and then the sample is selected
from the sorted list using a fixed sampling interval and a random start.
For example, a population of adults might be sorted by sex, and then, within
sex by date of birth. Suppose every nth person is then selected from the population
by taking a random start between 1 and n and then every nth person after that,
working down the list. This sample would then be described as a proportionate
stratified sample with explicit stratification by sex and implicit stratification
by date of birth. Note that for explicit stratification only categorical stratifying
variables can be used (or continuous variables that have been grouped into
categories).
Implicit stratification, in contrast, which only involves sorting a population
rather than grouping. It can be used for continuous variables as well as those with a large number of classes some of which can be rare in certain strata.
Large-scale surveys often use a combination of explicit and implicit stratification.
The sampling frame will firstly be grouped into a number of explicit strata,
and within each of these the sampling frame will be sorted by a continuous
variable or one with many classes. See exemplar 2 for an example of this.
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Software packages that calculate standard errors for complex surveys usually only
allow
for explicit stratification. The way around this for a survey that uses
implicit stratification
is to:
(a) Keep the sample of
PSUs
in the same order as it was selected in.
(b) Put achieved cases into pairs, working down the list (i.e. the first
two achieved cases working down the list are the first pair, the third and
fourth achieved are the second pair, etc.).
(c) If there are an uneven number of achieved cases then put the last three
achieved cases together to give a triplet.
(d) Treat each pair/triplet as if they were selected from the same explicit
stratum. So there will be half as many explicit strata as there are achieved
cases.
This ‘trick’ needs some care when calculating standard errors
for sub-groups, since the approach only works if there are two achieved
cases per ‘pair’. For a sub-group this can easily drop to one.
One option would be to re-pair the sample for each sub-group, but this is
too onerous in practice.Survey packages vary in
how they can handle this which is an example of the
lonely PSU
problem.
Surveys within this type of implicit stratiifcation may be better analysed
by replication methods, but rather few packages
can handle this.
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For large-scale government sponsored surveys it is common
practice to
spread fieldwork over a period, often of a year.
In these cases the sample for a whole year is selected at one point in time
(usually using a combination of implicit and explicit stratification) and
then the primary sampling units are systematically allocated to the 12 months
of the year. The allocation is done in such a way that, within each month,
the original stratification is maintained.
With this design the decision on how to deal with the stratification in estimating
standard errors is not so straightforward. If the pairing follows the sample
stratification then, in all pairs, the two primary sampling units will be
from different months of the survey. This means that the ‘within-stratum
between-psu’ estimated component of variance will incorporate both a
genuine between-psu element plus a between-month element (which will often
be a seasonal effect). This latter component tends to over-estimate the standard
errors for estimates. To avoid this one approach is to treat the sample for
each month as an independent sample and treat the sample within each month
as a stratified sample. This in effect means that the original sample is resorted,
firstly by month, and then within month, by the original order. The pairs
are then constructed from this new list.
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In a disproportionate stratified sample, the population
of sampling units are divided into sub-groups, or strata, and a sample selected
separately per stratum. Crucially, the sampling fraction is not the same within
all strata: some strata are over-sampled relative to others.
Suppose a sample of 50 white students and 50 non-white
students is to be
selected from a school with 2000 students, of whom 100 are non-white. To achieve
this the school roll would need to be divided into two strata: white and non-white,
and separate samples selected per strata. The sampling fraction to be applied
in the white stratum would be 1 in 38; the sampling fraction to be applied
in the non-white stratum would be 1 in 2.
Disporportionate
sampling is more common in surveys of organisations. Larger organisations will
tend to be more variable, for example in their labour force numbers. More precise
estimates could then be obtained from samples with an over representation of
larger organisations.
Disproportionate stratification is used for two purposes:
A. To give larger than proportionate sample sizes in one or more sub-groups
so that separate analyses by sub-group will be possible; and, far more rarely
B. To increase the precision of key survey estimates.
Disproportionate stratification will only reduce standard errors (relative
to a proportionate stratified sample) if the population standard deviation
for the variable of interest is higher than average within the over-sampled
strata. (In practice, standard errors will be minimised if the sampling fraction
used per stratum is proportional to the population standard deviation within
the stratum). For surveys that want to estimate the porportion of certain
types of respondent this means reducing the sampling fraction when the types
of interest is uncommon. The fact that most surveys collect data on a wide
range of variables means that disproportionate stratified sampling to reduce
standard errors is very rarely used in household surveys – since the
optimal sample design for one variable is unlikely to be optimal for others.
Furthermore, the population standard deviations are often not known at the
design stage. It is more common in other types of survyes, such as those of
farms or businesses where some groups are much more variable than others.
To obtain unbiased estimates for a disproportionate stratified
sample, the survey estimates have to be weighted.
This is achieved within most software packages by defining a weight variable
that gives a weight per case. The cases are then ‘weighted by’ this weight variable in the analysis.
The calculation of the weight is fairly straightforward: it is simply the
inverse of the sampling fraction used in the stratum that the case belongs
to. So, in a stratum where the sampling fraction is 1 in 10 all cases would
get a weight of 10; and in a stratum where the sampling fraction is 1 in 22
all cases would get a weight of 22. In practice the weights applied to a particular
survey may be more complex than this if, for instance, within strata not all
cases are selected with equal probability, of if non-response weights have
been included.
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As you might expect, within strata that have higher sampling fractions estimates are more precise than if a smaller ample had been drawn as part of a proportionate sampling scheme. Similarly the precision will be worse within strata with lower sampling fractions.
The effect of disproportionate stratification on overall estimates for the population will depend on a variety of factors. It depends on the relative sampling fractions and on how the variability of the survey responses differs in the different strata. This is discussed in detail in the section on weighting.
When you are reading the weighting section remember that an over sampled stratum needs to be given a lower weight in calculating an overall total, mean or percentage to get an unbiased estimate. Similarly strata with low sampling fractions need to be given large weights.
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peas project 2004/2005/2006.
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