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

\clearpage

\section{Key Ideas}

\subsection{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}


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.

We will begin exploring the basic ideas of clustering by looking at
examples of the influence of clustering on estimating proportions
($p_{pop}$'s).  Then we will look at how clustering impacts
estimationg numerical quantities ($\bar y_{pop}$'s).  

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 the former
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.

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

\subsection{Some Basic Elements of Cluster Sampling} 
        

\subsubsection*{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.

\subsection{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}

\subsection{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*}
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.

\subsubsection*{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 2.648 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.}  $s^2_{y_{ij}} = 2.648$, so that
\[
   \Var(\bar y_{srs}) = \left(1 - \frac{Mn}{MN}\right) \frac{1}{Mn}
   s^2_{y_{ij}}
   = (1- 20/400)\frac{1}{20}(2.648) = 0.126
\]
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{Cluster Sampling for Attributes}

We now consider the estimation of the proportion $p$ of the population
of size $NM$ possessing an attribute.  Let $p_i$ be the proportion of
elements in the $i$th cluster possessing the attribute.  Then the
cluster sample estimate of $p$, $\bar{p}_c$, is just the average
of the values of $p_i$ for the sampled clusters,
\begin{eqnarray}
  \bar{p}_c=\frac{1}{n}\sum^n_{i=1}p_i
\end{eqnarray}
and, since the clusters are of equal size, this estimate is simply the
overall sample proportion of individuals with the attribute of
interest.  Thus our cluster sample selection procedure assigns each
individual in the sample the same chance of selection and the sample
is {\em self-weighting.}
      

We get the variance of $\bar{p}_c$ by treating the values of
$p_i$ as a sample of $n$ measurements and looking at their variation
when used to estimate the overall population proportion $p$, i.e.,
\begin{eqnarray}
  Var(\bar{p}_c) =
  \frac{(N-n)}{(N-1)}\frac{\Bigg(\frac{\sum^N_{i=1}(p_i-p)^2}{N}\Bigg)}{n}.
\end{eqnarray}

We typically want to compare the accuracy of our estimate,
$\bar{p}_c$, with that of a simple random sample of the same
overall sample size, $nM$, i.e.,
\begin{eqnarray}
Var(\bar{p})=\frac{(NM-nM)}{(NM-1)}\frac{p(1-p)}{nM}.
\end{eqnarray}         

One way to compare the variances involves the ``correlation'' between
elements in the same cluster, $\rho$.  This quantity $\rho$ is called
the {\em intracluster correlation coefficient.}  It turns out that\\
  
%box around
\hspace*{1in}\fbox{\parbox[c][.5in]{4in}{
\begin{eqnarray*}
\frac{Var(\bar{p}_c)}{Var(\bar{p})}=1+(M-1)\rho.
\end{eqnarray*}}}\hfill
\parbox{1in}{\begin{eqnarray}\end{eqnarray}}\\       


\noindent Since variances cannot be negative, the quantity
$1+(M-1)\rho$ can't be negative and the minimum possible value for
$\rho$ is $\frac{-1}{(M-1)}$, which tends to 0 as the cluster size $M$
gets large.  Unlike a regular correlation coefficient which takes
values between 1 and -1, the intracluster correlation coefficient runs
between 1 and $\frac{-1}{(M-1)}$ .  If $\rho>0$, the cluster provides
less precision than a random sample of $M$ individuals, whereas when
$\rho<0$, something which occasionally happens, the use of clusters is
more precise.
      
\subsubsection*{Example 3: Limited English Proficiency Students}  

The U. S. Department of Education is interested in determining the
number of elementary school children in public schools with limited
proficiency in English (LEP).  Suppose there are $N=20$ schools in a
given district and that each school has $M=100$ students.
Investigators take a sample of $n=5$ schools and gather information on
the proportion of LEP students in each school. The total sample size
is number of schools sampled times the number of children in each
school, i.e., 500.  Because of housing patterns we expect children in
a given school to be more alike with respect to their proficiency in
English than those in different schools. This is because immigrants to
the district from a given country often live close to one another for
economic or other reasons.  Suppose we can determine that the
intra-school correlation coefficient for the attribute LEP is 0.0383.
Then the ratio of the variance of the cluster sample of 5 clusters to
the variance we would have had if we had taken a simple random sample
of students from a across the school district of size 500 is

\[
  1+(M-1)\rho=1+99 \times0.0383=4.79        
\]


\noindent i.e., the variance of the cluster sample is almost 5 times
greater than that of a simple random sample of equivalent size.  Put
another way, we could have taken a simple random sample of 100
children from across the school district and achieved an estimate of
the proportion of LEP students with equivalent accuracy.


Why then did investigators choose their sample in this way?  The
answer is cost.  Suppose that the cost of going to a school and
setting up a language test is \$1000, whereas the cost of
administering the test to the student one set up is \$10.  Taking a
random sample of 100 students would have meant that they would have
gone to at least, say, 10 schools, perhaps more.  The cost of
administering the test in 10 schools is

\[
(10 \times\$1000)+(100 \times\$10)=\$11,000.
\]
         
\noindent Instead, the investigators went to only 5 schools and thus
their cost was


\[ 
(5 \times\$1000)+(100 \times\$10)=\$6,000.
\]        


\noindent Thus, by taking a cluster sample the investigators incurred
only (6/11)th of the cost associated with a simple random sample of
equivalent precision.


When the population size is large and we can ignore the finite
population correction, and we can rewrite the ratio of the variance
for cluster sample versus that for a simple random sample directly as


\begin{eqnarray}
  \frac{Var(\bar{p}_c)}{Var(\bar{p})} \cong
  \frac{M\sum^N_{i=1}(p_i-p)^2}{Np(1-p)}
\end{eqnarray}
        

What we have in these formulas for the variances is three different
measures of variability: (i) the overall population variance, i.e.,
the variance of the observations for samples of size 1, $p(1-p)$; (ii)
the subpopulation for the $i$th cluster, i.e., variance of the
observations for samples of size 1 within the $i$th cluster,
$p_i(1-p_i)$; and (iii) the variation of the cluster proportions about
the overall population proportion, $\sum(p_i-p)^2$.
        

A very famous formula in statistics links these quantities as
follows:\\

\hspace*{1in}\fbox{\parbox[c][.7in]{4in}{
\begin{eqnarray*}
NMp(1-p)=M\sum^N_{i=1}(p_i-p)^2+M\sum^n_{i=1}p_i(1-p_i)\\
\mbox{total SS=SS between, clusters+SS within clusters}\nonumber
\end{eqnarray*}}}\hfill
\parbox{1in}{\begin{eqnarray}\end{eqnarray}}\\

        
By substituting for the sum of squares between clusters the formula
for the ratio of variances we get

\begin{eqnarray}
\frac{Var(\bar{p}_c)}{Var(\bar{p})}
& \cong &\frac{M\sum^N_{i=1}(p_i-p)^2}{Np(1-p)}\\
 & = &\frac{NMp(1-p)-M\sum^N_{i=1}p_i(1-p_i)}{Np(1-p)}\nonumber\\
& = &M\Bigg(1-\frac{\sum^N_{i=1}p_i(1-p_i)}{Np(1-p)}\Bigg)\nonumber
\end{eqnarray}\label{num7}
        
The fraction in expression (7) in brackets tends is always less than
or equal to 1.  Thus the variance of cluster sampling is never more
than M times that of an equivalently sized simple random sample. But
often the fraction is non-negligible and we get little degradation
from the clustering relative to simple random sampling.  In the other
extreme, when all of the clusters are identical and $p_i=p$ for all
clusters, the fraction is 1 and ratio of the variances is
approximately equal to 0, i.e., cluster sampling has yielded a
fantastic gain: in this instance, once you've seen one cluster you in
effect have seen them all!  Williams (1978, Chapter 11) gives a simple
example of this phenomenon.
       
\subsubsection*{Example 4: Numerical Illustrations.}  

Suppose that the proportion of individuals in a population with a
specific attribute is $p=0.5$.  If the population consists of $N=20$
clusters, where for 10 clusters $p_i=0.25$ and for the other 10
$p_i=0.75$, then


\begin{eqnarray}
\frac{Var(\bar{p}_c)}{Var(\bar{p})} & \cong &
M\Bigg( 1-\frac{\sum^N_{i=1}p_i(1-p_i)}{Np(1-p)} \Bigg)\\
& = & M\Bigg( 1-\frac{10(0.25 \times0.75)+10(0.75 \times0.25)}{20(0.5
\times0.5)} \Bigg)\nonumber\\ 
 & = & M \bigg(1-\frac{3}{4} \bigg)= \frac{M}{4}. \nonumber
\end{eqnarray}


Thus for clusters of size $M=2$ and $M=3$, there is less variability
in a cluster sample than in a simple random sample!  For $M \geq4$,
however, cluster sampling is less efficient, since the ratio is less
than 1. As the cluster size $M$ increases in this example, we have
greater and greater degradation in precision associated with cluster
sampling.
       
\subsubsection*{Example 3 (Continued)} 

Suppose the distribution of the number of LEP students in the 20
schools in our school district is as follows:

\vspace{.25in}
\begin{tabular}{c|c}
    {\bf Number of School} & {\bf Proportion of LEP Students}\\ \hline
2 & 0.1\\
3 & 0.2\\
3 & 0.3\\
10 & 0.4\\
2 & 0.5\\
\end{tabular}

\vspace{.25in} The overall proportion of LEP students is $p=0.326$.
Thus $Np(1-p)=20 \times0.326 \times0.674=4.39$ and from the
information in the table we calculate
                
\begin{eqnarray}
\sum^N_{i=1}p_i(1-p_i)=4.19.
\end{eqnarray}


Since $M=100$ the ratio of the variances is

\begin{eqnarray}
\frac{Var(\bar{p}_c)}{Var(\bar{p})} \cong
        M\Bigg(1-\frac{\sum^N_{i=1}p_i(1-p_i)}{Np(1-p)}
        \Bigg)=100\Bigg(1-\frac{4.19}{4.39}\Bigg)=4.56.
\end{eqnarray} 
        

Thus we get the precision of the cluster sample is equivalent to that
of a simple random sample of about 20\% the sample size.  The
difference between the ratio of 4.56 computed here and that of 4.79
computed in our earlier look at this example is due to ignoring the
finite population correction here.

\section{Cluster Sampling for Measurements}

The same ideas carry over for cluster samples of measurements.  We now
consider the estimation of the mean $\mu$ of the population of size
$NM$.  Let $y_i$ be the sum of measurements for the $i$th cluster.
Then the cluster sample estimate of $\mu$, $\bar{y}_c$, is just
the average of the all of measurements for all n sampled clusters

\begin{eqnarray}
\bar{y}_c=\frac{1}{nM}\sum^n_{i=1}y_i.
\end{eqnarray}


Our cluster sample selection procedure assigns each individual in the
sample the same chance of selection and the sample is {\em
  self-weighting}.  We get the variance of $\bar{y}_c$ by
treating the values of $y_i$ as a sample of $n$ measurements and
looking at their variation about their mean value for all $N$
clusters.  As we did with attributes, we compare the accuracy of our
estimate, $\bar{y}_c$, with that of a simple random sample
$\bar{y}$ of the same overall sample size, $nM$, and the ratio
turns out to be the same as it was for attributes:
        

\begin{eqnarray}
  \frac{Var(\bar{y}_c)}{Var(\bar{y})}=1+(M-1)\rho,
\end{eqnarray}


\noindent where $\rho$ is the intracluster correlation coefficient.
        
\subsubsection*{Example 5} 

Henry (1990, pp. 107-109) gives an example where and , in which the
estimated variances are
 
\begin{eqnarray}
  s^2(\bar{y}_c)=30.17\nonumber\\
  s^2(\bar{y})=20.23\nonumber
\end{eqnarray}


Thus there is an increase in the variance of about 50\% due to
clustering. This means that a simple random sample of equivalent
precision to the cluster sample would have required only a sample of
size

\[
       \bigg(\frac{20.23}{30.17}\bigg)15 = 10.05
\] 
or 10.  For this example, the estimated intracluster correlation
coefficient is
         
\begin{eqnarray}
  \bar{\rho} 
  = \frac{1}{M-1}\bigg(\frac{s^2(\bar{y}_c)}{s^2(\bar{y})}-1\bigg)
  = \frac{1}{4}\bigg(\frac{30.17}{20.23}-1\bigg)=0.123.
\end{eqnarray}


\section{Unequal Cluster Sizes}

The results for clustering described here work out in a similar way
when the clusters are not of equal size although the actual formulas
are somewhat more complicated.  Choosing compact clusters of roughly
equal size turns out to be the most efficient way to do cluster
sampling, but in real life clusters often come in varying sizes and
there is little we can do about this. For example, households come in
sizes that typically vary from 1 (for single person households) to as
many as 12, or even more.  The ratio of the variance for a cluster
sample to that of a simple random sample still takes approximately the
same form,

\begin{eqnarray}
  \frac{Var(\bar{y}_c)}{Var(\bar{y})}
  \cong1+(\bar{M}-1)\rho,
\end{eqnarray}
        

where $\bar{M}$ is the average cluster size.

\newpage
\vspace{.25in}
\noindent {\bf Historical Note:} The formal ideas for the use of
cluster sampling were introduced in the same the pioneering paper by
Jerzy Neyman in 1934 in which he advocated the use of optimal
allocation for stratified sampling, and it was his combination of
stratification and clustering this had such a profound influence on
subsequent developments in the field of sampling and in large-scale
survey practice.

\vspace{.25in}
\noindent {\bf Note on Proofs of Results:} Cochran (1977) provides
formal derivations for some of the formulas presented here for cluster
sampling as well as for multi-stage cluster sampling and cluster
sampling when the clusters are of unequal size. Lohr (1999) also has
an excellent presentation on the relevant formulas and their
derivation.


\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


