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\begin{center}
\large\bf
36-303: Sampling, Surveys and Society\\
Exam 2\\
Solutions
\end{center}

\begin{enumerate}

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\em

\item\pts{24} {\em Multiple Choice (4 parts).  For each part, circle the
  roman numeral of the one best answer.}

\begin{enumerate}

\item\pts{6} Which of the following is {\em not} a usual part of
  post-survey processing?
\begin{enumerate}\small
% \item Data entry
\item[\fbox{ii.}] Sample size calculation
% \item Imputation
% \item Checking post-strata and building weights if needed
% \item All of the above {\em are} usually part of post-survey processing!
\end{enumerate}


\item\pts{6} Suppose we divide a sampling frame into groups, which we
  may treat as either strata for stratified sampling, or clusters for
  cluster sampling.  If we make the groups so that {\em observations
    {\bf within groups} are more {\bf similar} to each other}, and
  {\em observations {\bf between groups} are more {\bf different} from
    each other}, then, all other things being equal, we expect
\begin{enumerate}\small
% \item The variance of the stratified sample mean $\bar y_{st}$ will go
%   {\bf up} and the variance of the cluster sample mean $\bar y_{cl}$
%   will go {\bf down}.
\item[\fbox{ii.}] The variance of the stratified sample mean $\bar y_{st}$ will go
  {\bf down} and the variance of the cluster sample mean $\bar y_{cl}$
  will go {\bf up}.
% \item Both variances will go {\bf up}.
% \item Both variances will go {\bf down}.
\end{enumerate}

\item\pts{6} Which of the following is {\em not} one of the recommended
  things to work on, to reduce the tendency of survey subjects to not
  respond?
\begin{enumerate}\em
% \item  Followup.
\item[\fbox{ii.}] Choice of stratified or cluster sampling.
% \item Amount of effort it takes respondents to undeerstand/respond to
%   questions.
% \item Assurance of confidentiality, especially for sensitive
%   questions.
\end{enumerate}

\item\pts{6} Weights can be calculated and applied to individual
  observations for a variety of reasons.  Circle the reason below that
  is {\em not} appropriate.
\begin{enumerate}\em
% \item Weights may be calculated in designing a stratified sample
%   designs.
% \item Weights may be calculated in designing certain kinds of surveys
%   in which not every respondent has an equal chance of being selected.
% \item Weights may be calculated after the survey to compensate for
%   some kinds of informative (non-ignorable) missingness.
% \item Weights may be calculated after the survey to adjsut sample
%   proportions in various post-strata to equal the population
%   proportions.
\item[\fbox{iv.}] All of the above {\em are} appropriate reasons to compute weights! 
\end{enumerate}



\end{enumerate}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\item\pts{26} {\em Cluster sampling}.
% From Groves, pp 142--143, \#10, but modified since (a) it's a bit
% long, and (b) it's not clear how to deal with the ``assumption of
% simple random sampling of households'' in part (a) of his question.
A survey is conducted to find out the proportion of cell phone users
in a certain city.  From a population of 2500 residential blocks, 10
are sampled at random without replacement, and each person residing on
that block is asked whether they use a cell phone.  We will assume
there are exactly 40 people living in each block\footnote{This is a
  reasonable approximation as long as all the blocks have close to 40
  residents.}.  This yields the following table of data:
\begin{center}
\begin{tabular}{cccc}
\hline
        & Total \# of & \# of Cell & Proportion of Cell\\
Block $h$   & People $M$ & Phone Users $c_h$ & Phone Users $p_h$ \\
\hline
 1 & 40 & 10 & 0.25 \\ 
 2 & 40 &  8 & 0.20 \\
 3 & 40 & 16 & 0.40 \\
 4 & 40 & 15 & 0.38 \\
 5 & 40 & 24 & 0.60 \\
 6 & 40 & 17 & 0.42 \\
 7 & 40 & 12 & 0.30 \\
 8 & 40 & 13 & 0.32 \\
 9 & 40 & 16 & 0.40 \\
10 & 40 & 13 & 0.32 \\
\hline
Total & 400 & 144 & 
\end{tabular}
\end{center}

\begin{enumerate}

\item\pts{6} This is an example of one-stage clustered sampling.
  Circle one word in each pair of choices in the following sentences:
\begin{itemize} 
\item The primary sampling
  units (psu's) are the {\bf (\fbox{blocks}, residents)}.
\item The secondary
  sampling units (ssu's) are the {\bf (blocks, \fbox{residents})}.
\end{itemize}

\item\pts{4} Ignoring the clustering and treating this as an SRS of 400
  residents, estimate the proportion of cell phone users and its
  standard error.

{\em

Under SRS with replacement,
\begin{eqnarray*}
\hat p_{SRS} & = & 144/400 ~ = ~ \fbox{0.36} \\
\mbox{Var}_{SRS}(\hat p) & = & (1-n/N)\hat p(1-\hat p)/n \\
   & = & (1 - 40*10/(40*2500))*(0.36)*(1-0.36)/400 = 0.000573696\\
\mbox{SE}_{SRS}(\hat p) & =&  \sqrt{0.000573696} ~ = ~ \fbox{0.02395195}
\end{eqnarray*}
}

\item\pts{6} Now re-estimate the proportion of cell phone users and
  its standard error, using appropriate cluster sampling methods.
  {\em Hint: to reduce calculation, use the fact that in the table
    above, $s^2_{p_h} = 0.01214333$.} {\bf NOTE: on the exam, I wrote
    0.1214333 by mistake, but corrected it in class.}

{\em
Under single stage clustering with equal cluster sizes, the sample is
self-weighting, so we know without any calculation that \fbox{$\hat
  p_{CL} = \hat p_{SRS} = 0.36$} again.  Of course you can obtain that
by averaging the numbers in the $p_h$ column in the data table as
well (with rounding error in the table, you might end up with 0.359 if
you do the average of the $p_h$'s by hand).

For the variance, we treat clusters as sampling units and cluster
means as the unit-level data.  So
\begin{eqnarray*}
  \mbox{Var}_{CL}(\hat p) & = & (1-n/N) s^2_{p_h}/n \\
  & = & (1 - 10/2500)*(0.01214333)/10 ~ = ~ 0.001209476\\
  \mbox{SE}(\hat p) & =& \sqrt{0.001209476} ~ = ~ \fbox{0.03477752}
\end{eqnarray*}
}

\item\pts{6} Calculate the design effect DEFF for this design.

{\em
\[
	\mbox{DEFF} ~ = ~ \frac{\mbox{Var}_{CL}(\hat
          p)}{\mbox{Var}_{SRS}(\hat p)}
        ~ = ~ \frac{0.001209476}{0.000573696} ~ = ~ \fbox{2.108218}
\]
}

\item\pts{4} Calculate $\rho$, the correlation between responses from
  residents on the same block.

{\em
\[
	\mbox{DEFF} = 1 + (M-1)\rho
\]
}%
so%
{\em
\[
	\rho = (\mbox{DEFF} - 1)/(M-1) = (2.108218 - 1)/(10-1) =
        \fbox{0.1231353} 
\]
}
\end{enumerate}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% groves, p. 213, \#4, with some elaboration.
\item\pts{24} {\em Response rates and missing data.}  You are
  completing a telephone survey of an SRS of 1000 members of a much
  larger professional organization, regarding their level of
  involvement in support of the organization.  Currently the response
  rate is 80\%, with 52.5\% of those responding saying they attend
  every monthly meeting of the local chapter of the organization, and
  47.5\% saying they do not.

\begin{enumerate}

\item\pts{8} Does it seem likely that the 200 (20\% of 1000) who did
  not respond to your survey are missing completely at random (MCAR,
  ignorable missingness) or missing not at random (MNAR, non-ignorable
  missingness)?  Choose MCAR or MNAR and {\em briefly} explain your
  reasoning.

{\em
It seems likely that these 200 respondents are \fbox{MNAR}.  The
survey is about {\em participation}, and {\em nonresponse is a form of
  nonparticipation}.  Therefore I expect a lower proportion of the 200
nonrespondents to say that they attend every monthly meeting, than the
800 who did respond/participate in the survey. Since not responding is
related to the response we would have seen, the missing data is MNAR.
}

\item\pts{8} Which of the following is more likely to be correct
  (circle \ul{one}):
\begin{itemize}\em
% \item An unbiased estimate of the population proportion that attends
%   every meeting is 52.5\%
\item[\fbox{ii.}] An unbiased estimate of the population proportion that attends
  every meeting would probably be less than 52.5\%.
% \item An unbiased estimate of the population proportion that attends
%   every meeting woudl probably be more than 52.5\%
\end{itemize}


\item\pts{8} To get the sample size nearer to the target of $n=1000$,
  you could either 
\begin{itemize}
\item Ask the organization to pay for you to call a new SRS of size
  250, hoping that $(0.80)(250)=200$ people choose to respond {\em
    (cost: \$200 because it just involves routine single calls to each
    new phone number)}; or
\item Ask the organization to pay for you to followup with the 200 in
  your original sample who didn't respond yet, to try to get their
  responses {\em (cost: \$800 because it involves repeated call-backs
    until each of the 200 non-respondents either responds or
    refuses)}.
\end{itemize}
Which do you choose, and why {\em (briefly)}?

{\em 
Although it is more expensive, \fbox{I prefer following up the 200
  non-respondents}.  We want an unbiased estimate of participation,
and the people we get in the 2nd sample are again likely to be the
participators, rather than the nonparticipators/nonresponders.  Since
the 200 are likely MNAR it is important to convert them to responders,
rather than looking for other easy-responders in a new sample of 250.
We will never find out how nonrespondents would have responded, if we
only collect data from those who find it easy to respond, we will just
get more data for a biased estimate that way.
}

\end{enumerate}



\item\pts{26} {\em Imputation methods.}

\begin{enumerate}

% gates, p 368 \#4
\item One method of imputation for missing responses to
  individual survey items is {\em mean imputation}.
\begin{enumerate}
\item\pts{5} Explain briefly how mean imputation works.

{\em
For each missing value, we fill in the average of all of the respondents
who did provide a response to that question.
}

\item\pts{4} Under what assumption (MCAR, MAR, MNAR) is mean imputation
  OK?  (Choose one, no explanation needed.)\\

{\em
MCAR.
}

\item\pts{4} Identify a possible problem with mean imputation.

{\em
The mean value of everyone else may not even be an admissible
response.  For example if the question is how many times a week do you
go to the gym, admissible answers are whole numbers.  However the
average over all the responders might well not be a whole number.
}

\end{enumerate}

\item Another method of imputation for missing responses is {\em
  hot-deck imputation}. 
\begin{enumerate}
\item\pts{5} Explain briefly how hot-deck imputation works.

{\em
In hot deck imputation we find someone who (a) did respond to the
question that is missing for this respondent; and (b) looks like this
respondent in every other way (has similar demographics, similar
responses to other questions, etc.).  We then copy the response from
the person who had the response to the person who didn't.
}

\item\pts{4} Under what assumption (MCAR, MAR, MNAR) is hot-deck imputation
  OK?  (Choose one, no explanation needed.)\\

{\em
MAR (MCAR too, but MAR is a better answer).
}

\item\pts{4} Identify a possible problem with hot-deck imputation.

{\em
Here are two possible problems:
\begin{itemize}
\item There might not be a good ``matching'' person among the other
  respondents in the survey.
\item The less good the match, the less likely it is that that the two
  people would have responded the same way.  Even if the match is very
  good, there's no guarantee that the two people would have responded
  the same way.
\end{itemize}
}

\end{enumerate}

\end{enumerate}


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\comment{

\clearpage

% Examples from lecture 09...

\item\pts{??} Another 36-303 group wants to do a survey of PAT bus
  riders' attitudes toward a proposed schedule of bus fare increases.
  In their survey proposal they specify the {\em target population} to
  be all adult residents of Pittsburgh's East End (Oakland, Shadyside,
  Squirrell Hill \& Point Breeze).  They plan to collect data by
  approaching people waiting at six specific bus stops near where
  these students live, between 8am and 10am for one whole work-week
  (Mon-Fri), for face-to-face interviews.
\begin{enumerate}

\item\pts{??} Did this group specify a {\em sampling frame} for their
  survey?  If so, say what it is.  If not, specify a sampling frame
  that the group could work with.

\vspace*{\fill}

\item\pts{??} Write down two very likely sources of {\em coverage
  error} for this survey as currently proposed.

\vspace*{\fill}

\item\pts{??} Make one big suggestion that this group could implement
  to improve their sampling plan.

\vspace*{\fill}

\end{enumerate}

}

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\end{enumerate}

\end{document}

