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\large\bf
36-303: Sampling Surveys and Society\\
Clustered Sampling\\
\rm Brian Junker\\
132E Baker Hall\\
brian@stat.cmu.edu\\
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\rhead{Cluster Sampling}
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\tableofcontents

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\section{Why Use Clustering?}

After stratification the most natural extension to simple random
sampling involves the use of clusters of the population of interest.
In stratified sampling, we divide the population into distinct
subpopulations called {\em strata}, and within each stratum we select
a separate sample.  In cluster sampling, we divide the population up
into clusters, and we select a sample of clusters and include all of
the elements from these clusters in the sample.  Figure~\ref{f:lohr},
reproduced from Lohr (1999), indicates some similarities and
differences between these approaches.

There are two primary reasons for clustering:

\begin{enumerate}
\item {\em A reliable list of elements of the population may be
  unavailable and it may be unreasonably expensive to try to compile
  such a list.}  We can, however, make a list of clusters and thus it
  is sensible to use them as the sampling units.  For example:
\begin{itemize}
\item This is often the case when we sample human populations and the
  clusters are households.  This is because it is relatively easy to
  prepare and maintain a list of household locations, whereas it is
  virtually impossible to maintain a list of individuals in
  identifiable locations.
\item You could also imagine doing this on-campus.  C-Book is a flawed
  frame for CMU undergraduates, but the Hub has an exhaustive list of
  classes and their locations, so you could take an SRS of classes
  from the Hub's list, rather than an SRS of students: the clusters
  are the classes.
\end{itemize}
\item {\em Even if a reliable list of population elements is
  available, it may be difficult, expensive or disruptive to take an
  SRS of individuals.}  On the other hand, and SRS of clusters may be
  easier.  For example:
\begin{itemize}
\item The travel costs associated in going from one housing unit to
  another for a random sample of individuals may be substantial.
  Further, when the cluster consists of a household, one individual
  (e.g. ``head of household'') can provide information on all the
  other members.
\item In the National Assessment of Educational Progress, the survey
  form is an achievement test in math, science, or some other
  subject. While it would be possible to pull out individual students
  from a class and send them to a special room for testing, it is much
  easier and less disruptive to sample classrooms (so the classrooms
  are the clusters) and give the test to all eligible students in the
  class.
\end{itemize}
\end{enumerate}     

\begin{figure}
\begin{center}
% \fbox{\hbox to 5in {\vbox to 6in {~}}} 
%\psfig{file=lohr.eps,width=5in}
\includegraphics[width=6in]{lohr.eps}
%\includegraphics[width=6in]{lohr.pdf}
\end{center}
\caption{Similarities and Differences Between Cluster Sampling and
  Stratified Sampling.  From Lohr (1999).}\label{f:lohr}
\end{figure}


A key reference is Lohr's (1999) textbook [recommended text for the
  class], Chapter 5.

\section{Some Basic Elements of Cluster Sampling} 

\subsection{Clustering vs. Stratification}
        
You might remember that stratification usually increases the precision
of our sample (reduces standard errors), relative to an SRS.  {\em
  Clustering has the opposite effect}: it tends to decrease the
precision of the sample (increase the standard errors).  This is
because individuals in the same cluster tend to be more alike than
different, and this causes their responses to be positively
correlated.
%  
One does not have to choose either clustering or stratification, and
in large scale surveys the two methods are often combined, % .  
% In coming
% lectures, we will examine some actual surveys that combine clustering
% and stratification 
to achieve the advantages in cost from clustering
while preserving some of the precision that stratification confers.


It is simpler to introduce the ideas when all the clusters have the
same size (all families have exactly 4 members, all classes have
exactly 30 students, etc.).  The same ideas also work in the more
realistic situation of clusters of differing sizes, and we will look
briefly at that situation as well.


\subsection{Example 1: Using Family Clusters to Estimate the
  Proportion Eligible for Medicare}

Suppose we have a population of size $2N$ composed of $N$ families of
size 2, a husband and wife.  We say that the families are clusters of
size $M=2$.  Further suppose that the husband and wife in any given
pair are exactly the same age. We are interested in the proportion of
the population eligible for Medicare corresponding to those over the
age of 65, and we take a sample of $n$ families.  Since both members of
each family have the same age we in effect have redundant information
and instead of ending up with an overall sample of size $2n$
individuals, our effective sample size is only $n$.
      

Now suppose that husbands and wives don't have identical ages, but on
average older husbands have older wives and younger husbands have
younger wives.  This positive association or correlation between the
age of the husband and the age of the wife in a pair again reduces the
``effective sample size'' associated with our cluster sample of $n$
families.  We get an estimate that is more accurate than a simple
random sample of $n$ individuals from the population, but still less
than a simple random sample of size $2n$.
        

This example illustrates the basic impact of clustering that we will
tend to observe in the sampling of human populations.  In general we
consider a population of $NM$ elements subdivided into $N$ clusters of
size $M$.  We take a sample of $n$ of these clusters and incorporate
into our sample information on all $M$ elements in each of the
selected clusters.  Thus we record information on $nM$ units.  If the
information from individuals within a cluster is positively related,
then there is less variation among individuals within a cluster than
for the same number of individuals drawn from different clusters.
Thus we expect that our cluster sample will be less accurate than a
simple random sample of the same size, $nM$.
       

After introducing some terminology and notation we will turn to the
actual formulas for the variance of an estimate for a proportion or
for a mean from a cluster sample, which involve measures of variation
within and between clusters.

\section{Terminology and Notation}

In cluster sampling the notation is a little bit messier than in SRS
or even stratified sampling, because of the need to keep track of
which observations come from which clusters, and the need to keep
track of posive correlations between elements from the same cluster.

\begin{itemize}

\item In a clustered sample, the clusters are sometimes called {\em
  primary sampling units} or {\em psu's}.  The individuals within a
  cluster are called {\em secondary sampling units} or {\em ssu's}.  

\item In {\em one-stage cluster sampling}, we first take an SRS of
  psu's (clusters).   Then \underline{\em all} of the ssu's
  (individuals) within each cluster are included in the sample.   This
  is the situation we will focus on in these notes.

\item In {\em two-stage cluster sampling}, we first take an SRS of
  psu's.  Then within each psu, we take an SRS of ssu's.\\

  For example in a survey of a school district we might take an SRS of
  schools (psu's, or clusters) and then take another SRS of students
  (ssu's, or individuals) because it is too expensive to go to every
  student in every sampled school.

\end{itemize}
I will try to stick with ``clusters'' and ``individuals'' or
``clusters'' and ``units'', but keep in mind that psu $=$ cluster, and
ssu $=$ an individual or unit within a cluster.

In SRS, we talked of a population of $N$ units.  Now $N$ will refer to
the clusters or psu's.  Within each cluster there are ssu's.  The
basic data we observe on each observation is 
\begin{eqnarray*}
	y_{ij} & = & \mbox{ measure for $j^{th}$ element of $i^{th}$ cluster}\\
	       & = & \mbox{ measure for $j^{th}$ ssu in the  $i^{th}$ psu}
\end{eqnarray*}
So, $y_{ij}$ might be 0 or 1 depending on which candidate the $j^{th}$
member of household $i$ supports, or $y_{ij}$ might be the GPA of the
$j^{th}$ member of classroom $i$, etc.

\begin{itemize}
\item Some population quantities for psu's are:
\begin{eqnarray*}
  N   & = & \mbox{ number of psu's (clusters) in the population}\\
  M_i & = & \mbox{ number of ssu's (individuals) in cluster $i$} \\
  K   & = & \sum_{i=1}^N M_i 
      ~ = ~ \mbox{ total number of ssu's in the population} \\
%  t_i & = & \sum_{j=1}^{M_i} y_{ij} 
%      ~ = ~ \mbox{ total in psu $i$} \\
%  t   & = & \sum_{i=1}^N \sum_{j=1}^{M_i} y_{ij} 
%      ~ = ~ \mbox{ total in entire population} \\
%  S^2_t & = & \frac{1}{N-1}\sum_{i=1}^N \left(t_i - t/N\right)^2
%      ~ = ~ \mbox{ population variance of the psu totals}
\end{eqnarray*}

\item Some population quantities for ssu's are:
\begin{eqnarray*}
  \bar y_{pop} & = & \frac{1}{K}\sum_{i=1}^N \sum_{j=1}^{M_i}
  y_{ij} ~ = ~ t/K
      ~ = ~ \mbox{ the population mean}\\
  \bar y_{i,pop} & = & \frac{1}{M_i} \sum_{j=1}^{M_i}
  y_{ij} ~ = ~ \mbox{ population mean in the $i^{th}$ psu}\\
  S^2 & = & \frac{1}{K-1}\sum_{i=1}^N \sum_{j=1}^{M_i} (y - \bar
  y_{pop})^2 ~ = ~ \mbox{ population variance of ssu's}\\
  S^2_i & = & \frac{1}{M_i-1} \sum_{j=1}^{M_i} (y - \bar
  y_{i,pop})^2 ~ = ~ \mbox{population variance of ssu's within a single psu}
\end{eqnarray*}

\item Some sample quantities of interest are: 
\begin{eqnarray*}
  {\cal S} & = & \mbox{ set of $i$'s (psu's; clusters) sampled}\\ 
  {\cal S}_i & = & \mbox{ set of $j$'s (ssu's; individuals) sampled in
    $i^{th}$ psu} 
    ~ = ~ \mbox{ set of all ssu's in the sample}\\
  n & = & \mbox{ number of psu's in sample (size of ${\cal S}$)} \\
  m_i & = & \mbox{ number of ssu's in the sample from the $i^{th}$
    psu (size of ${\cal S}_i$)}\\
  \bar y_i & = & \frac{1}{m_i}\sum_{j\in {\cal S}_i} y_{ij}  
  ~ = ~ \mbox{ sample mean for the $i^{th}$ ssu}\\
%   \hat t_i & = & \sum_{j\in {\cal S}_i} \frac{M_i}{m_i} y_{ij}
%   ~ = ~ \mbox{ estimated total for $i^{th}$ psu}\\
%   \hat t_{unb} & = & \sum_{i\in {\cal S}} \frac{N}{n} \hat t_i
%   ~ = ~ \mbox{ unbiased estimator of population total}\\
%   s^2_t & = & \frac{1}{n-1} \sum_{i\in{\cal S}} \left(
%   \hat t_i - \hat t_{unb}/N\right)^2 
%   ~ = ~ \mbox{ estimated variance of psu totals}\\
%   s^2_i & = & \frac{1}{m_i-1} \sum_{j\in{\cal S}_i} \left(
%   y_{ij} - \bar y_i\right)^2
%   ~ = ~ \mbox{ sample variance within the $i^{th}$ psu}
\end{eqnarray*}

\end{itemize}

\section{Basic Ideas for Estimating Means}

From the $N$ clusters in the population we take an SRS without
replacement of $n$ of them.  Let ${\cal S}$ be the set of clusters $i$
sampled, so that ${\cal S}$ has $n$ elements.
\begin{itemize}
\item
The size of each cluster is $M_i$.  For simplicity we assume {\em equal
cluster sizes}: $M_i\equiv M$ $\forall~i$.  
\item
We also will assume that every individual in the cluster is in our
sample; this is {\em one-stage cluster sampling}.
\end{itemize}
For example, we could consider all two-person households ($M=2$ for
all clusters) in a survey to estimate mean income.  $y_{ij}$ is the
income of the $j^{th}$ person in the $i^{th}$ household. 

For each cluster $i$ in ${\cal S}$, we can calculate the 
cluster mean
\[
  \bar y_i  =  \frac{1}{M}\sum_{j=1}^M y_{ij}
\]
In our example, $\bar y_i = \frac12(y_{i1} + y_{i2})$, the average of
the two person's incomes in the $i^{th}$ household.

Since we have an SRS of clusters, we can apply our formulas for SRS's
to estimate population quantities, but now we think of an SRS of {\em
  clusters}, where the measurement on each {\em cluster} is $\bar y_i$:
\[
  \bar y_{cl} = \frac{1}{n} \sum_{i\in {\cal S}} \bar y_i
\]
The standard error (SE) needed for constructing confidence intervals
is the {\em square root of}
\begin{eqnarray*}
   \Var(\bar y_{cl}) & = & (1-f) S^2_{cl,pop}/n \\
  & = & (1-f) \frac1n \left[
     \frac{1}{N-1} \sum_{i=1}^N (\bar y_i - \bar y_{pop})^2\right]\\
  & \approx & (1-f) \frac1n \left[\frac1{n-1} 
     \sum_{i\in{\cal S}} (\bar y_i - \bar y_{cl})^2\right]\\
  & = &  \left(1-\frac{n}{N}\right) \frac1n \left[\frac1{n-1} 
     \sum_{i\in{\cal S}} (\bar y_i - \bar y_{cl})^2\right]\\
  & = & \left(1-\frac{n}{N}\right) \frac1n s^2_{\bar y_i}
\end{eqnarray*}

Note that the cluster sample size is $n$ but the individual sample
size is $M\cdot n=2n$.  We might compare our SE here with the SE we
would calculate if this were an SRS without replacement {\em of
  individuals}.  So let
\begin{eqnarray*}
  \bar y_{srs} & = & \frac{1}{Mn} \sum_{i,j} y_{ij}
  ~ = ~ \frac{1}{Mn} \sum_{j\in {\cal S}} \sum_{i=1}^M
    y_{ij}
\end{eqnarray*}
(Note that
\begin{eqnarray*}
\bar y_{srs}
    & = & \frac1n \sum_{j\in {\cal S}} \frac1M \sum_{i=1}^M
    y_{ij}\\
    & = & \frac{1}{n} \sum_{i\in {\cal S}} \bar y_i \\
    & = & \bar y_{cl}
\end{eqnarray*}
so that cluster sampling with equal cluster sizes is {\em
  self-weighting}: the complex estimator $\bar y_{cl}$ equals the simpler
estimator $\bar y_{srs}$.)

The variance under SRS would be
\begin{eqnarray*}
  \Var(\bar y_{srs}) & = & (1-f) S^2_{pop}/(Mn) \\
  & = & (1-f) \frac1{Mn}\left[\frac{1}{MN-1} \sum_{i=1}^N
  \sum_{j=1}^M (y_{ij} - \bar y_{pop})^2\right]\\
  & \approx & (1-f) \frac1{Mn}\left[\frac{1}{Mn-1} \sum_{i\in {\cal S}}
  \sum_{j=1}^M (y_{ij} - \bar y_{srs})^2\right]\\
  & = & \left(1 - \frac{Mn}{MN}\right) \frac{1}{Mn} s^2_{y_{ij}}
\end{eqnarray*}

\subsection{Design Effect}

As with stratified sampling we can calculate a {\em design effect}
\[
    DEFF = \frac{\Var(\bar y_{cl})}{\Var(\bar y_{srs})}
    = \frac{\left(1-\frac{n}{N}\right) \frac1n \left[\frac1{n-1} 
     \sum_{i\in{\cal S}} (\bar y_i - \bar y_{cl})^2\right]}{
    \left(1-\frac{Mn}{MN}\right) 
    \frac1{Mn}\left[\frac{1}{Mn-1} \sum_{i\in {\cal S}}
  \sum_{j=1}^M (y_{ij} - \bar y_{srs})^2\right]}
    = \frac{ M \frac1{n-1} 
     \sum_{i\in{\cal S}} (\bar y_i - \bar y_{cl})^2}{
    \frac{1}{Mn-1}  \sum_{i\in {\cal S}}
  \sum_{j=1}^M (y_{ij} - \bar y_{srs})^2}
    = \frac{ M s^2_{\bar y_i}}{s^2_{y_{ij}}}
\]
to see what the effect on precision of clustering is.  In stratified
sampling, we also calculated a design effect $DEFF$ (it has a
different formula).
\begin{itemize}
\item In stratified sampling we usually get $DEFF<1$ if we design the
  strata carefully.  
\item In clustered sampling, we usually get $DEFF>1$. 
\end{itemize}
We will see more about the design effect below.

\subsection{Example 2: Estimating Average GPA (Lohr, 1999)}

A student wants to estimate the average GPA in his dormitory.  There
are $N=100$ suites that hold $M=4$ students each.  There are three
random sampling schemes he could use:
\begin{itemize}
\item {\em SRS without replacement:} From a list (frame) of all 400
  students in the dorm, take an SRS without replacement of, say, 20
  students.
\item {\em Stratified sample:} From each of the 100 suites he could
  take an SRS without replacement of, say, 2 students (for a total
  sample size of $2\times100=200$).  Here, {\em the suites are strata.}
\item {\em Clustered sample:} He could take an SRS of, say, $n=5$
  suites from the list of suites, and then take all four students in
  each suite.  Again the sample size is $Mn= 4\times5=20$.  Here, {\em
    the suites are clusters.}
\end{itemize}
In this case, the least effort is probably the clustered sample, so
that is what the student does.  The results are contained in
Table~\ref{t:gpa} (page \pageref{t:gpa}).

\begin{table}
\begin{center}
\let\m\multicolumn
\begin{tabular}{c|ccccc}
Person & \m{5}{c}{Suite (Cluster) ($i$)}\\
Number ($j$) & 1 & 2 & 3 & 4 & 5 \\
\hline
1 & 3.08 & 2.36 & 2.00 & 3.00 & 2.68 \\
2 & 2.60 & 3.04 & 2.56 & 2.88 & 1.92 \\
3 & 3.44 & 3.28 & 2.52 & 3.44 & 3.28 \\
4 & 3.04 & 2.68 & 1.88 & 3.64 & 3.20 \\
\hline
$\bar y_i$ & 3.04 & 2.84 & 2.24 & 3.24 & 2.77
\end{tabular}
\end{center}
\caption{GPA data from a clustered random sample of dorm suites.}\label{t:gpa}
\end{table}

For the clustered sample we have
\[
   \bar y_{cl} = \frac{1}{5}\sum_{i=1}^5 y_i = \frac15(3.04 + 2.84 +
   2.24 + 3.24 + 2.77) = 2.826
\]
and the sample variance of the $\bar y_i$'s is
\[
    s^2_{\bar y_i} = \frac{1}{5-1} \left[
      (3.04 - 2.826)^2 + \cdots + (2.77 - 2.826)^2
      \right]
    = 0.14098
\]
Therefore
\[
  \Var(\bar y_{cl}) = \left(1-\frac{n}{N}\right) \frac1n s^2_{\bar
    y_i}
  = \left(1-\frac{5}{100}\right) \frac{1}{5} (0.14098)
  = 0.0268
\]
So $SE=\sqrt{0.0268} = 0.164$, and a 95\% CI for the mean GPA in the
dorm would be
\[
    (2.826 - (1.96)\cdot(0.164), ~ 2.826 + (1.96)\cdot(0.164))
\]
which runs from about 2.51 to about 3.15.

If the data had been collected as an SRS of size $n=20$ we would have
gotten
\[
    \bar y_{srs} = 2.862 
\]
(same answer, since the cluster sample was self-weighted!), with
sample variance\footnote{Lohr (1999, p. 142) points out that the
  simple sample variance 0.2648 is an underestimate because the data
  was in fact collected as a clustered sample. She proposes a less
  biased estimator based on ANOVA sums of squares, but we will use the
  simpler estimate for our purposes.}  $s^2_{y_{ij}} = 0.2648$, so that
\[
   \Var(\bar y_{srs}) = \left(1 - \frac{Mn}{MN}\right) \frac{1}{Mn}
   s^2_{y_{ij}}
   = (1- 20/400)\frac{1}{20}(0.2648) = 0.01258
\]
We can see that the design effect in this case is
\[
   DEFF = \frac{\Var(\bar y_{cl})}{\Var(\bar y_{srs})}
   = \frac{ M s^2_{\bar y_i}}{s^2_{y_{ij}}} =
   \frac{(4)(0.14098)}{(0.2648)} 
   = 2.13
\]
So we would need a little over twice\footnote{Lohr (1999) calculates
  $DEFF=2.02$ based on her less-biased estimate of $s^2_{y_{ij}}$.} as
  many observations in a clustered sample, in this case, to get the
  same precision as an SRS.

It turns out that 
\[
    DEFF = \frac{\Var(\bar y_{cl})}{\Var(\bar y_{srs})}  
    \approx 1 + (M-1)\rho
\]
where $\rho$ (``rho'') is the {\em intracluster correlation
  (ICC)}. The ICC is the correlation between all pairs of observations
within each cluster (remember that observations within a cluster tend
to be positively correlated).

We can use this formula to see how correlated the observations are in
this example, by solving for $\rho$:
\[
   \rho \approx (DEFF - 1)/(M-1) = (2.13 - 1)/3 = 0.38
\]
That is to say, there is a correlation of abotu 0.38 between any two
people's GPA in the same dorm suite, in this survey!


\section{References} 
\begin{description}

\item Cochran, W. G. (1977). {\em Sampling Techniques.}
  3rd. Edition. Wiley: New York.

\item Neyman, J. (1934). On the two different aspects of the
  representative method: the method of stratified sampling and the
  method of purposive selection (with discussion). {\em Journal of the
    Royal Statistical Society,} {\bf 97}, 558--625.

\item Henry, G. T. (1990). {\em Practical Sampling.} Sage: Newbury
Park, CA.

\item Lohr, S. (1999). {\em Sampling: Design and Analysis.}  Duxbury:
  Pacific Grove, CA.

\item Williams, B. (1978). {\em A Sampler on Sampling.} Wiley: New
  York.

\end{description}

\end{document}

Statistics 36-203 Winter, 1999


