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\large\bf Some Useful Formulas From the Statistics of Survey Sampling, I
% \\
% \normalsize Do not bring this to the exam.  A copy will be provided for you.
\end{center}

\noindent
{\bf Equally-Likely Outcomes \& Counting}

\begin{itemize}

\item If $K$ outcomes $O_1, \ldots, O_K$ are equally likely, then the
  probability of any one of them is $1/K$. 

\item Consider taking a sample of $n$ objects from a population of $N$
  objects. 
\begin{itemize}
\item Sampling with replacement, there are $N^n$ possible samples of
  size $n$; the probability of any one of them is $1/N^n$.
\item Sampling without replacement, there are 
$
	{N \choose n} = \frac{N!}{n!(N-n)!}
$ 
possible samples of size $n$ [where
$N!=N\cdot(N-1)\cdot(N-2)\cdots3\cdot2\cdot1$], so the probability of
any one of them is $1\left/{N\choose n}\right.$.
\end{itemize}

\end{itemize}

\noindent
{\bf Discrete Random Variables}

Let $X$ and $Y$ be random variables with sample spaces
$\{x_1,\ldots,x_K\}$ and $\{y_1,\ldots,y_K\}$ and distributions
\begin{eqnarray*}
  P[X=x_i,Y=y_j] ~ = ~ p_{ij} ~~,~~~
  P[X=x_i]      ~  = ~ p_{i\cdot} ~ = ~ \sum_{j=1}^K p_{ij} ~~,~~~
  P[Y=y_j]       ~ = ~ p_{\cdot j} ~ = ~ \sum_{i=1}^K p_{ij}
\end{eqnarray*}
Then, for example
\begin{eqnarray*}
  E[X] ~ = ~ \sum_{i=1}^K x_i p_{i\cdot} ~~,~~~
  Var(X) ~ = ~ \sum_{i=1}^K (x_i  - E[X])^2p_{i\cdot} &,~&
  Cov(X,Y) ~ = ~ \sum_{i=1}^K (x_i  - E[X])(y_i-E[Y])p_{ij}\\
  P[X=x_i|Y=y_j] ~ = ~ p_{ij}/p_{\cdot j}~~,~~~
  E[X|Y=y_j] ~ = ~ \sum_{i=1}^K x_i P[X=x_i|Y=y_j] &,~&
  E[aX + bY + c] ~ = ~ aE[X] + bE[Y] + c
\end{eqnarray*}

\noindent
{\bf Random Sampling From a Finite Population}

Consider a population of size $N$ and a sample of size $n$.  Let $y_i$
be the (fixed) values of some variable of interest in the population
(such as a person's age, or whether they would vote for Obama).  Let
\[
	Z_i = \left\{\begin{array}{l}
        1, \mbox{~if $i$ is in the sample}\\
        0, \mbox{~else}
        \end{array}\right.
\]
be the random sample inclusion indicators, and let $Y_i$ be the random
observations in the sample.  Then the sample average can be written
% \def\cond{\parbox{2.9em}{\scriptsize\em $i$ is in the sample}}
\[
    \bar Y ~ = ~ \frac1n\sum_{i=1}^n Y_i 
    ~ = ~ \frac1n\sum_{i=1}^N Z_i y_i
\]
The $Z_i$'s are Bernoulli random variables with
\begin{eqnarray*}
  E[Z_i] ~ = ~ \frac{n}{N} ~~,~~~ Var(Z_i) =
  \frac{n}{N}\left(1-\frac{n}{N}\right) 
  ~~,~~~ Cov(Z_i,Z_j) = - \frac{1}{N-1} \frac{n}{N}\left(1-\frac{n}{N}\right) 
\end{eqnarray*}

\noindent
{\bf Confidence Intervals and Sample Size}

\begin{itemize}

\item[(a)] A CLT-based $100(1-\alpha)\%$ confidence interval for the
  population mean is $(\bar Y - z_{\alpha/2}SE~,~~\bar Y +
  z_{\alpha/2}SE)$.
\item[(b)] For sampling with replacement from an infinite population, $SE =
  SD/\sqrt{n}$.
\item[(c)] For sampling without replacement from a finite population, the
  SE has to be multiplied by the finite population correction (FPC).
\item[(d)] For a given margin of error (ME, half the width of the CI) and
  confidence level $1-\alpha$, we can find the sample size by solving
\[
      z_{\alpha/2}SE < ME
\]
for $n$.  The same approach works for both SRS with replacement (using
the SE in (b)) and SRS without replacement (using the SE in (c)).

\end{itemize}
% {\em You should memorize the formula for the FPC, and the two formulas
%   for sample size calculations (one for sampling with replacement, one
%   for sampling without replacement).}

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\begin{center}
\large\bf Some Useful Formulas From the Statistics of Survey Sampling, II
% \\
% \normalsize Do not bring this to the exam.  A copy will be provided for you.
\end{center}

\noindent
{\bf Stratified Sampling}

\noindent
Consider $H$ strata with population counts $N = \sum_{h=1}^H N_h$ and
sample counts $n=\sum_{h=1}^H n_h$.  Let $f_h = n_h/N_h$; $W_h =
N_h/N$; and $\bar y_h = \frac{1}{n_h}\sum_{i=1}^{n_h} y_{ih}$ in each
stratum, and let $s_h^2=\frac{1}{n_h-1}\sum_i(y_{ih}-\bar y_h)^2$ be the
sample variance in each stratum.  Then
\begin{eqnarray*}
\overline y_{st} ~ = ~ \sum_{h=1}^H W_h \overline y_h ~~,~~~~
\Var(\overline y_{st}) 
	~ \approx ~  \sum_{h=1}^H W_h^2 (1-f_h) \frac{s_h^2}{n_h} ~~,~~~~
DEFF ~ = ~ \frac{\Var(\overline y_{st})}{\Var(\overline y_{srs})}
	=\frac{\sum_{h=1}^H W_h^2 (1-f_h) \frac{s_h^2}{n_h}}
	{(1-f)\frac{s^2}{n}}
\end{eqnarray*}



\noindent
{\bf Cluster Sampling}

\noindent
Consider a population of $N$ clusters.  We take an SRS ${\cal S}$ of
$n$ clusters, and all units within each sampled cluster (one-stage
clustering).  Assume clusters all have same size $M$.  Let $\bar y_i
= \frac{1}{M}\sum_{j=1}^M y_{ij}$ in each cluster.  Then
\begin{eqnarray*}
  \bar y_{cl} ~ = ~ \frac{1}{n} \sum_{i\in {\cal S}} \bar y_i
~~,~~~~
   \Var(\bar y_{cl}) 
  ~ \approx ~  \left(1-\frac{n}{N}\right) \frac1n s^2_{\bar y_i}
  ~ = ~ \left(1-\frac{n}{N}\right) \frac1n  \left[\frac1{n-1} 
     \sum_{i\in{\cal S}} (\bar y_i - \bar y_{cl})^2\right]
\end{eqnarray*}
and 
\[
 DEFF = \frac{\Var(\bar y_{cl})}{\Var(\bar y_{srs})}
    =  \frac{ M s^2_{\bar y_i}}{s^2_{y_{ij}}}
    \approx 1 + (M-1)\rho 
\]
where $s^2_{\bar y_i}$ is the sample varance of the cluster means,
$s^2_{y_{ij}}$ is the sample variance of the individual observations,
and $\rho$ is the intraclass (intracluster) correlation, or ICC.

\noindent 
{\bf Post-Stratification Weights and Means}

\noindent
As part of survey data collection it is a good idea to get general
demographic information (e.g. in our surveys: sex, age, class, major,
hometown, etc.).  After data collection we compare the proportions in
each of these categories in our sample with the same proportions in
the population. If they agree, great.  If not, calculate
\[
    w_i = (N_h/N)/(n_h/n) \mbox{~for each $i$ in post-stratum $h$}~~,~~~
    \mbox{\em and ~~} \overline y_w = \frac{\sum_i w_i y_i}{\sum_i w_i} 
\]


\noindent
{\bf Post-Stratification Variance Calculations}

\noindent
\ul{\em Taylor series}:
\[
\Var_{TS}(\overline y_w) \approx 
	\frac{1}{\left(\sum_i w_i\right)^2}
	\left[
	\Var(\sum_i w_i y_i) - 2\overline y_w \Cov(\sum_i w_i y_i, \sum_i w_i)
	+ (\overline y_w)^2\Var(\sum_i w_i)
	\right]
\]
where $\bar y_w$ is as above, $\bar w=\frac1n\sum_i w_i$,~~
$\bar{wy}=\frac1n\sum_i w_i y_i$,~~\\
\begin{center}
$\Var(\sum_{i=1}^n w_i) 
	 \approx  n  \cdot \frac{1}{n-1}\sum_{i=1}^n (w_i - \bar w)^2$,~~
$\Var(\sum_{i=1}^n  y_i w_i) 
	~ \approx ~ n \cdot \frac{1}{n-1} \sum_{i=1}^n (w_i y_i - \bar{wy})^2$,~~\\[1em]
% \begin{eqnarray*}
%     \Var(\sum_{i=1}^n w_i) 
%	~ \approx ~ n  \cdot \frac{1}{n-1}\sum_{i=1}^n (w_i - \bar w)^2 
%~~,~~~
%	\Var(\sum_{i=1}^n  y_i w_i) 
%	~ \approx ~ n \cdot \frac{1}{n-1} \sum_{i=1}^n (w_i y_i - \bar{wy})^2
%\\
$	\Cov(\sum_{i=1}^n y_i w_i, \sum_{i=1}^n w_i) 
	 \approx 
	n \cdot \frac{1}{n-1} \sum_{i=1}^n (w_i y_i - \bar{wy}) (w_i - \bar w)
$
%\end{eqnarray*}
\end{center}
\noindent
\ul{\em Jackknife}:
\begin{itemize}
\item Replicate $n$ times (by removing one obs. each time and
  recalculating weights):
\[
	\overline y_w^{(r)} = \frac{\sum_{i=1}^n w_i^{(r)}
          y_i^{(r)}}{\sum_{i=1}^n w_i^{(r)}} 
\]
\item Calculate
\begin{eqnarray*}
	\bar y_{JK} ~ = ~ \frac1n\sum_{r=1}^n \bar y_w^{(r)}
~~,~~
	Var_{JK}(\bar y_w) 
	~ \approx ~ \frac{n-1}{n} \sum_{r=1}^n (\bar y_w^{(r)} - \bar y_{jk})^2
\end{eqnarray*}
\end{itemize}

}

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