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% Title, subtitle, and other information
\title{Stat472/572 Sampling: Theory and Practice}
%\subtitle{Slides are intended for a course based on the book: \textit{Sampling: Design and Analysis, Third Edition} by Sharon L. Lohr}
\author[Yan Lu]{Instructor: Yan Lu}
\institute[UNM]{University of New Mexico}
\date{}

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% Begin the document
\begin{document}

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% Title slide
\begin{frame}[plain] % Use plain to remove header/footer on the title slide
    \titlepage
    \vspace{-1cm} % Add space between the title page and chapter information
    \centering
    \textbf{\Large Chapter 3: Stratified Sampling} % Emphasize chapter title

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\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Chapter 3}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Example: SRS vs. Stratified Sampling}

Consider a population with 1000 males and 100 females.
\bi

\item If we take a Simple Random Sample (SRS) of size 55, it is possible to end up with a sample containing no females.  
    \begin{itemize}
        \item Such a sample might not be representative of the population, especially if males and females respond differently to the item of interest.
    \end{itemize}
\item Stratified Sampling:
    \begin{itemize}
        \item Divide the population into strata based on gender (e.g., male and female).  
        \item Take a proportional sample from each stratum, e.g., 50 males from the male pool and 5 females from the female pool.  
        \item Prevents the possibility of selecting a sample with no or very few females.  
        \item Improves the precision of estimators by ensuring better representation of both strata.
    \end{itemize}
\ei
\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Stratified Sampling}
\bi
\item Divide the population into $H$ distinct subpopulations, called \textbf{strata}, such that:
    \begin{itemize}
        \item The strata are mutually exclusive (no overlap).
        \item The strata collectively cover the entire population.
    \end{itemize}
\item Each sampling unit belongs to exactly one stratum.
\item Draw an independent probability sample from each stratum.
\item Combine the information from all strata to calculate overall population estimates.
\ei
\end{frame}



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\begin{figure}\label{fig:stratification}
\begin{center}
\begin{tikzpicture}[scale=0.8]

% Population rectangle
\draw[thick] (0,0) rectangle (6,6);
\node[above] at (3,6) {\textbf{Population}};

% Divide population into strata
\draw[thick] (0,2) -- (6,2);
\draw[thick] (0,4) -- (6,4);

% Stratum labels
\node at (-1,5) {\textbf{Stratum 1}};
\node at (-1,3) {\textbf{Stratum 2}};
\node at (-1,1) {\textbf{Stratum 3}};

% Circles for Stratum 1 (Blue)
\foreach \x in {0.7, 1.4, 2.1, 2.8, 3.5, 4.2, 4.9, 5.6}
    \draw[blue!50] (\x,5.5) circle (0.2);

% Circles for Stratum 2 (Red)
\foreach \x in {1, 2, 3, 4, 5}
    \draw[red!50] (\x,3.5) circle (0.2);

% Circles for Stratum 3 (Green)
\foreach \x in {1.2, 2.4, 3.6}
    \draw[green!50] (\x,1.5) circle (0.2);

% Arrows and selected samples
\draw[->,thick] (4.9,5.5) -- (8,7); % From Stratum 1 (first sample)
\draw[->,thick] (1.4,5.5) -- (7.7,7); % From Stratum 1 (second sample)
\draw[->,thick] (3.1,3.6) -- (8.5,7.2); % Adjusted arrow for red sample
\draw[->,thick] (2.4,1.5) -- (8,6.5); % From Stratum 3

% Highlight selected circles in strata
\fill[blue] (4.9,5.5) circle (0.2); % First blue sample
\fill[blue] (1.4,5.5) circle (0.2); % Second blue sample
\fill[red] (3.1,3.6) circle (0.2);  % Red sample
\fill[green] (2.4,1.5) circle (0.2); % Green sample

% Sample circle
\draw[thick] (8,7) circle (1.5);
\node[above] at (8,8.5) {\textbf{Sample}};
\fill[blue] (7.6,7.6) circle (0.2);  % Selected blue sample 1
\fill[blue] (7.3,7.3) circle (0.2);  % Selected blue sample 2
\fill[red] (8.2,7.4) circle (0.2);   % Selected red sample
\fill[green] (8,6.8) circle (0.2);   % Selected green sample

\end{tikzpicture}
\end{center}
\caption{Illustration of Stratified Sampling}
\end{figure}
\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Example 3.2}
\bi
\item In Example 2.6, a random sample led to overrepresentation of some areas and no representation of others.
\item Larger counties (common in the western US) tend to have larger $y$ values, contributing to high variability.
\item Stratified sampling provides balance in the sample based on the stratifying variable.
\item Four census regions (Northeast, North Central, South, West) are used as strata. About 10\% of counties in each stratum are sampled.
\ei
\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Visualization of the Four Regions}
\vspace{-0.15in}
\begin{figure}[!htp]
    \centering
    \includegraphics[width=0.90\textwidth,height=0.75\textheight]{./plots/region.jpg}
    \caption{Boxplots of farm acreage by region: 
    - The West has higher median and variance. 
    - All regions exhibit positively skewed distributions.}
    \label{boxplot}
\end{figure}
\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Sampling Design}
\begin{table}
\centering
\begin{tabular}{ccc}
\hline
Stratum & \# of Counties & \# in Sample\\
\hline
Northeast & 220 & 21\\
North Central & 1054 & 103\\
South & 1382 & 135\\
West & 422 & 41\\
\hline
Total & 3078 & 300\\
\end{tabular}
\caption{Sample Allocation Across Strata}
\end{table}
\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Summary Statistics by Stratum}
{\small
\begin{table}
\centering
\begin{tabular}{ccccc}
Region & Stratum Size & Sample Size & Average & Variance\\
\hline
Northeast & 220 & 21 & 97,629.8 & 7,647,472,708\\
North Central & 1045 & 103 & 300,504.2 & 29,618,183,543\\
South & 1382 & 135 & 211,315.0 & 53,587,487,856\\
West & 422 & 41 & 662,295.5 & 396,185,950,266\\
\hline
\end{tabular}
\caption{Stratum-Level Summary Statistics}
\end{table}
\bi
\item Example for Northeast Region:
\[
\hat{t}_1 = 220 \cdot 97,629.8 = 21,478,558.2
\]
\[
\hat{V}(\hat{t}_1) = 220^2 \cdot \left(1 - \frac{21}{220}\right) \cdot \frac{7,647,472,708}{21} = 1.59 \times 10^{13}
\]
\ei
}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Population Estimates by Stratum}
\begin{table}
\centering
\begin{tabular}{ccc}
Region & Estimated Total & Estimated Variance\\
\hline
Northeast & $21,478,558.2$ & $1.59 \times 10^{13}$\\
North Central & $316,731,379.4$ & $2.88 \times 10^{14}$\\
South & $292,037,390.8$ & $6.84 \times 10^{14}$\\
West & $279,488,706.1$ & $1.55 \times 10^{15}$\\
\hline
Total & $909,736,034.4$ & $2.54 \times 10^{15}$\\
\end{tabular}
\caption{Stratum-Level and Overall Estimates}
\end{table}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Comparison: SRS vs. Stratified Sampling}
\begin{table}
\centering
\begin{tabular}{cccc}
 & Sample Size & $\hat{t}$ & SE \\
\hline
SRS & 300 & 916,927,110 & 58,169,381 \\
Stratified & 300 & 909,736,034 & 50,417,248\\
\hline
\end{tabular}
\caption{Comparison of Total Estimates and Variances}
\end{table}
\bi
\item Stratified sampling reduces variability in the population estimate. 
\item Variance ratio:
\[
\frac{\text{Stratified Variance}}{\text{SRS Variance}} = \frac{2.54 \times 10^{15}}{3.38 \times 10^{15}} = 0.75
\]
\item Equivalent precision achieved with fewer observations:
\[
300 \cdot 0.75 = 225
\]
\ei
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Comments on Stratified Sampling}
\bi
\item Reduces variability by avoiding poorly representative samples.
\item Enables precise estimates for subgroups of interest.
\item Often more cost-effective and practical.
\item Reduces overall variance for population estimates.
\ei
\end{frame}
 


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

\begin{frame}
\frametitle{Theory of Stratified Sampling}
\begin{table}[h!]
\centering
\begin{tabular}{cccccc}
\hline
Stratum & 1 & 2 & $\cdots$ & $H$ & Total \\
\hline
Population Size & $N_1$ & $N_2$ & $\cdots$ & $N_H$ & $\sum_{h=1}^H N_h = N$ \\
Sample Size & $n_1$ & $n_2$ & $\cdots$ & $n_H$ & $\sum_{h=1}^H n_h = n$ \\
Population Total & $t_1$ & $t_2$ & $\cdots$ & $t_H$ & \\
\hline
\end{tabular}
\end{table}

\vspace{0.2in}

\bi
\item Take an SRS of size $n_h$ from stratum $h$.
\item Population total and estimator:
\[
t_{str} = t_1 + t_2 + \cdots + t_H, \quad \hat t_{str} = \hat t_1 + \hat t_2 + \cdots + \hat t_H = \sum\limits_{h=1}^H N_h \bar{y}_h
\]

\item Population mean (a weighted average of the stratum means) and estimator
$$\bar y_U = \ds\frac {t_{str}} {N},  \quad \hat{\bar{y}}_U = \bar{y}_{str} = \ds\frac{\hat{t}_{str}}{N} = \sum\limits_{h=1}^H \ds\frac{N_h}{N} \bar{y}_h$$

\ei
\end{frame}


\begin{frame}
\frametitle{Population quantities and estimators}

Let $y_{hj}$ denote the value of the $j$th unit in stratum $h$.
{\small
\begin{table}[h!]
\centering
\begin{tabular}{|l|l|}
\hline
\textbf{Population Quantities} & \textbf{Estimators} \\ 
\hline
Stratum mean $\bar{y}_{hU} = \ds\frac{\sum\limits_{j=1}^{N_h} y_{hj}}{N_h}$ & $\bar{y}_h = \ds\frac{1}{n_h} \sum\limits_{j \in S_h} y_{hj}$ \\ \hline
Stratum total $t_h = \sum\limits_{j=1}^{N_h} y_{hj}$ & $\hat t_h = \ds\frac{N_h}{n_h} \sum\limits_{j\in S_h} y_{hj} = N_h \bar{y}_h$ \\ \hline
Population total $t = \sum\limits_{h=1}^H t_h$ & $\hat t_{str} = \sum\limits_{h=1}^H \hat t_h = \sum\limits_{h=1}^H N_h \bar{y}_h$ \\ \hline
Population mean $\bar{y}_U = \ds\frac{t}{N} = \ds\frac{\sum\limits_{h=1}^H \sum\limits_{j=1}^{N_h} y_{hj}}{N}$ & $\bar{y}_{str} = \ds\frac{\hat{t}_{str}}{N} = \sum\limits_{h=1}^H \ds\frac{N_h}{N} \bar{y}_h$ \\ \hline
Stratum variance $S_h^2 = \ds\frac{\sum\limits_{j=1}^{N_h} (y_{hj} - \bar{y}_{hU})^2}{N_h - 1}$ & $s_h^2 = \ds\frac{\sum\limits_{j \in S_h} (y_{hj} - \bar{y}_h)^2}{n_h - 1}$ \\ \hline
\end{tabular}
\end{table}
}
\end{frame}

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

\begin{frame}

\bi
\item Variance estimator of the total:
\[
\hat V(\hat t_{str}) = \hat V(\hat t_1) + \hat V(\hat t_2) + \cdots + \hat V(\hat t_H)
\]
\[
\hat V(\hat t_{str}) = \sum_{h=1}^H \left(1-\frac{n_h}{N_h}\right) \frac{N_h^2 s_h^2}{n_h}
\]
\item Variance estimator of the mean: 
$$\hat V(\bar y_{str}) = \ds\frac 1 {N^2} ( \hat V(\hat t_1) + \hat V(\hat t_2) + \cdots + \hat V(\hat t_H))$$
$$\hat V(\bar y_{str})= \sum\limits_{h=1}^H \left(1-\ds\frac
{n_h}{N_h}\right)\left(\ds\frac{N_h}{N}\right)^2 \ds\frac
{s_h^2}{n_h}$$ 

\ei
\end{frame}


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




%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
{\bf Properties of the estimators:}
\bi
\item $E[\hat t_{str}] = t$
\item $E[\bar y_{str}] = \bar y_U$
\item $\hat V(\hat t_{str})$ is an unbiased estimator of $V(\hat t_{str})$
\item $\hat V(\bar y_{str})$ is an unbiased estimator of $V(\bar y_{str})$
\ei
\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\begin{eqnarray*}
E[\hat t_{str}] &=& E\left[\sum_{h=1}^H N_h \bar y_h\right] \\
&=& \sum_{h=1}^H N_h E(\bar y_h) \\
&=& \sum_{h=1}^H N_h \bar y_{hU} = \sum_{h=1}^H t_h = t
\end{eqnarray*}

\begin{eqnarray*}
E[\bar y_{str}] &=& E\left[\ds\frac{\hat t_{str}}{N}\right] \\
&=& \ds\frac{t}{N} \\
&=& \bar y_U
\end{eqnarray*}
\end{frame}


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

\begin{frame}
\frametitle{Confidence Intervals for Stratified Samples}
\bi
\item Approximation applies under the following conditions:
    \begin{itemize}
        \item (1) The sample sizes within each stratum are large, \textbf{or}
        \item (2) The sampling design includes a large number of strata.
    \end{itemize}

\item According to the central limit theorem (Krewski and Rao, 1981), an approximate $100(1-\alpha)\%$ confidence interval for the population mean $\bar{y}_U$ is:
$$\bar{y}_{str} \pm z_{\alpha/2} SE(\bar{y}_{str})$$

\item Some survey software packages use the percentile of a $t$-distribution with $n - H$ degrees of freedom instead of the percentile of the normal distribution.
\ei
\end{frame}


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

\begin{frame}
\frametitle{Stratified Sampling for Proportions}
\bi
\item A special case of the mean occurs when:
$$y_i = 
\begin{cases} 
1 & \text{if the unit has the characteristic,} \\
0 & \text{otherwise.}
\end{cases}$$
\item In this case, we estimate mean as proportions.\\
----Within each stratum:
$$\bar{y}_h = \hat{p}_h, \quad s_h^2 = \frac{n_h}{n_h-1} \hat{p}_h (1-\hat{p}_h)$$
\ei
\end{frame}

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

\begin{frame}
\frametitle{Formulas for Stratified Sampling of Proportions}
\bi


\item Stratified proportion:
$$\hat{p}_{str} = \sum_{h=1}^H \ds\frac{N_h}{N} \hat{p}_h$$

\item Variance of stratified proportion:
$$\hat{V}(\hat{p}_{str}) = \sum_{h=1}^H \left(1- \ds\frac{n_h}{N_h}\right) \left(\ds\frac{N_h}{N}\right)^2 \ds\frac{\hat{p}_h (1-\hat{p}_h)}{n_h-1}$$

\item Stratified total and its variance:
$$\hat{t}_{str} = \sum_{h=1}^H N_h \hat{p}_h$$
$$\hat{V}(\hat{t}_{str}) = N^2 \hat{V}(\hat{p}_{str})$$
\ei
\end{frame}


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

\begin{frame}
\frametitle{Example 3.4: ACLS Stratified Random Sampling}
\bi
\item The American Council of Learned Societies (ACLS) conducted a stratified random sample of societies across seven disciplines.
\item The study aimed to analyze publication patterns, computer use, library use, and female membership in these disciplines.
\item The data is summarized in the following table:
\ei
\end{frame}

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

\begin{frame}
\frametitle{ACLS Sampling Data}
{\small
\begin{tabular}{lcccc}
\hline
Discipline & Membership  & \# mailed & valid returns & female\\
&$N_h$&&$n_h$& members(\%)\\
\hline
Literature        & 9,100  & 915  & 636 & 38\\
Classics          & 1,950  & 633  & 451 & 27\\
Philosophy        & 5,500  & 658  & 481 & 18\\
History           & 10,850 & 855  & 611 & 19\\
Linguistics       & 2,100  & 667  & 493 & 36\\
Political Science & 5,500  & 833  & 575 & 13\\
Sociology         & 9,000  & 824  & 588 & 26\\
\hline
Totals            & 44,000 & 5,385 & 3,835 & \\
\hline
\end{tabular}
}
\bi
\item Goal: Estimate the percentage and total number of female members across the societies.
\item Assumptions: Nonresponse is ignored, and there are no duplicate memberships.
\ei
\end{frame}

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

\begin{frame}
\frametitle{Estimating Female Membership}
\bi
\item Proportion of female members:
\begin{eqnarray*}
\hat{p}_{str} &=& \sum_{h=1}^7 \frac{N_h}{N} \hat{p}_h \\
&=& \frac{9,100}{44,000} \times 0.38 + \cdots + \frac{9,000}{44,000} \times 0.26 \\
&=& 0.2465
\end{eqnarray*}

\item Standard error of $\hat{p}_{str}$:
\begin{eqnarray*}
SE(\hat{p}_{str}) &=& \sqrt{\sum_{h=1}^7 \left(1 - \frac{n_h}{N_h}\right)\left(\frac{N_h}{N}\right)^2 \frac{\hat{p}_h (1 - \hat{p}_h)}{n_h - 1}} \\
&=& 0.0071
\end{eqnarray*}

\item Estimated total number of female members:
\begin{eqnarray*}
\hat{t}_{str} &=& 44,000 \times 0.2465 = 10,847 \\
SE(\hat{t}_{str}) &=& 44,000 \times 0.0071 = 312
\end{eqnarray*}
\ei
\end{frame}



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

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Using Weights in Stratified Sampling}

\bi
\item **Sampling Weights:** The number of units in the population represented by each sample member $(h, j)$, where:
\begin{itemize}
    \item $h$: Stratum index, 
    \item  $j$: Element index within the stratum
\end{itemize}
\item Estimator for Total:
\begin{eqnarray*}
\hat{t}_{str} &=& \sum_{h=1}^H N_h \bar{y}_h \\
&=& \sum_{h=1}^H \sum_{j \in S_h} \frac{N_h}{n_h} y_{hj} \\
&=& \sum_{h=1}^H \sum_{j \in S_h} w_{hj} y_{hj},
\end{eqnarray*}
where $w_{hj} = \frac{N_h}{n_h}$.
\ei


\end{frame}

\begin{frame}
\bi
\item Estimator for Mean:
\begin{eqnarray*}
\bar{y}_{str} &=& \frac{\sum_{h=1}^H \sum_{j \in S_h} w_{hj} y_{hj}}{\sum_{h=1}^H \sum_{j \in S_h} w_{hj}}.
\end{eqnarray*}
\ei
\end{frame}

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


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Example: Sampling Weights}
\bi
\item Population Details:
A population of 2000 units consists of:
\begin{itemize}
    \item 1600 males (Stratum 1)  
    \item 400 females (Stratum 2)
\end{itemize}
\item Sample Details:
The sample contains 400 units:
\begin{itemize}
    \item 200 units from each stratum.
\end{itemize}

\item Calculations:
\begin{eqnarray*}
\pi_{1j} &=& \frac{200}{1600} = \frac{1}{8}, \quad w_{1j} = \frac{1}{\pi_{1j}} = 8, \\
\pi_{2j} &=& \frac{200}{400} = \frac{1}{2}, \quad w_{2j} = \frac{1}{\pi_{2j}} = 2.
\end{eqnarray*}

\item Interpretation:
\begin{itemize}
    \item Each man in the sample represents 8 men in the population.
    \item Each woman in the sample represents 2 women in the population.
\end{itemize}
\ei
\end{frame}



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Extension of Weights Notation to Stratified Random Sampling}
\begin{center}
\begin{tabular}{cccccc}
\hline
 Strata & 1 & 2 & $\cdots$ & $H$ & Total \\
\hline
 Population size & $N_1$ & $N_2$ & $\cdots$ & $N_H$ & $\sum_{h=1}^H N_h = N$ \\
 Sample size & $n_1$ & $n_2$ & $\cdots$ & $n_H$ & $\sum_{h=1}^H n_h = n$ \\
 Population total & $t_1$ & $t_2$ & $\cdots$ & $t_H$ & \\
\hline
\end{tabular}
\end{center}
\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Key Formulas for Stratified Random Sampling}
{\footnotesize
\bi
\item $\pi_{hj} = \frac{n_h}{N_h}$
\item $w_{hj} = \frac{N_h}{n_h}$
\item Estimator for Population Total:
$\hat t_{str} = \sum\limits_{h=1}^H \hat t_h
=\sum\limits_{h=1}^H N_h \bar y_h = \sum\limits_{h=1}^H
\sum\limits_{j\in S_h} \ds\frac {N_h}{n_h} y_{hj} =
\sum\limits_{h=1}^H \sum\limits_{j\in S_h} w_{hj} y_{hj}$

\item Variance of $\hat t_{str}$:
$$V(\hat t_{str}) = \sum\limits_{h=1}^H N_h^2 \left(1 - \frac{n_h}{N_h}\right) \frac{S_h^2}{n_h}$$
\item Estimator for Population Mean:
$$\bar y_{str} = \frac{\hat t_{str}}{N} = \sum\limits_{h=1}^H \frac{N_h}{N} \bar y_h = \frac{\sum\limits_{h=1}^H \sum\limits_{j\in S_h} w_{hj} y_{hj}}{\sum\limits_{h=1}^H \sum\limits_{j\in S_h} w_{hj}}$$
\item Variance of $\bar y_{str}$:
$$V(\bar y_{str}) = \frac{V(\hat t_{str})}{N^2} = \sum\limits_{h=1}^H \frac{N_h^2}{N^2} \left(1 - \frac{n_h}{N_h}\right) \frac{S_h^2}{n_h}$$
\ei
}
\end{frame}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Comments on Weights in Stratified Sampling}
\bi
\item Let $\pi_{hj}$ denote the probability of selecting unit $j$ from stratum $h$:  
$$\pi_{hj} = \frac{n_h}{N_h}, \quad w_{hj} = \frac{1}{\pi_{hj}} = \frac{N_h}{n_h}$$
\item The sum of weights equals the population size:  
\begin{eqnarray*}
\sum_{h=1}^H \sum_{j \in S_h} w_{hj} &=& \sum_{h=1}^H \sum_{j \in S_h} \frac{N_h}{n_h} \\
&=& \sum_{h=1}^H N_h = N
\end{eqnarray*}
The weights ensure that the sample represents the entire population.
\ei
\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
{\Large Proportional Allocation:}

 The number of sampled units in each stratum
is proportional to the size of the stratum
$$\ds\frac {n_h}{N_h} = \ds\frac n N, \quad n_h = n \cdot \ds\frac {N_h} N$$
$$\pi_{hj} = \ds\frac {n_h}{N_h} = \ds\frac n N \quad \mbox{and} \quad w_{hj} =
\ds\frac 1 {\pi_{hj}} = \ds\frac {N} {n}$$ Sample is self-weighting
\begin{eqnarray*}
\bar y_{str} &=& \sum_{h=1}^H \ds\frac {N_h} N \bar y_h=
\sum_{h=1}^H \ds\frac {N_h} N \ds\frac{\sum_{j\in
S_h} y_{hj}}{n_h}\\
&=& \sum_{h=1}^H \ds\frac 1 n \sum_{j\in S_h} y_{hj}
= \ds\frac 1 n \sum_{h=1}^H \sum_{j\in S_h} y_{hj}\\
&=& \bar y
\end{eqnarray*}
\end{frame}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Example}
\bi
\item 
Population Details:
A population of 2000 units consists of:
\begin{itemize}
    \item 1600 males (Stratum 1)  
    \item 400 females (Stratum 2)
\end{itemize}
\item Different Sample Details from previous example:
A sample containing 400 units is drawn using proportional allocation:
\begin{itemize}
    \item 320 males from Stratum 1  
    \item 80 females from Stratum 2
\end{itemize}

\item Calculations:
\begin{eqnarray*}
\pi_{1j} &=& \frac{320}{1600} = \frac{1}{5}, \quad w_{1j} = \frac{1}{\pi_{1j}} = 5, \\
\pi_{2j} &=& \frac{80}{400} = \frac{1}{5}, \quad w_{2j} = \frac{1}{\pi_{2j}} = 5.
\end{eqnarray*}

\item Key Feature:
The number of sampled units in each stratum is proportional to the size of the stratum, 20\%.  Sample is self-weighting with a weight of 5.

\ei
\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
{\bf Variances of mean and total from proportional stratified random sampling:}

$$V_{prop} (\bar y_{str}) = \left(1-\ds\frac n N\right)\ds\frac 1 n
\sum\limits_h \ds\frac {N_h} N S_h^2$$
$$V_{prop} (\hat t_{str}) = \left(1- \ds\frac n N\right) \ds\frac N n
\sum\limits_h N_h S_h^2$$
\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
{\bf ANOVA Table}

\begin{tabular}{ccc}
SS & df & Sum of Squares\\
\hline\\
Between strata SSB& $H-1$ & $\sum\limits_{h=1}^H
\sum\limits_{j=1}^{N_h} (\bar y_{hU} -\bar
y_U)^2$\\
&& $=\sum\limits_{h=1}^H N_h (\bar y_{hU} -\bar y_U)^2$\\
\hline
Within strata SSW & $N-H$ & $\sum\limits_{h=1}^H
\sum\limits_{j=1}^{N_h} (y_{hj} -\bar y_{hU})^2$\\
&&$=\sum\limits_{h=1}^H (N_h -1) S_h^2$\\
\hline
Total corrected SSTO& $N-1$ &$\sum\limits_{h=1}^H
\sum\limits_{j=1}^{N_h} (y_{hj} -\bar y_U)^2$\\
 & & $=(N-1)S^2$\\
\hline
\end{tabular}

SSTO = SSB +SSW
\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
{\Large Comparison between SRS and proportional allocation}

{\normalsize \begin{eqnarray*} V_{prop}(\hat t_{str}) &=&
V\left(\sum\limits_{h=1}^H N_h \bar y_h\right) = \sum\limits_{h=1}^H
N_h^2 \left(1- \ds\frac {n_h}{N_h}\right)
\ds\frac{ S_h^2}{n_h}\\
&=& \sum\limits_{h=1}^H \left(1- \ds\frac n N\right) \ds\frac {N}{n
N_h} N_h^2 S_h^2
= \sum\limits_{h=1}^H \left(1-\ds\frac n N\right) \ds\frac N n N_h S_h^2\\
&=& \left(1-\ds\frac n N\right) \ds\frac N n \left[SSW +
\sum\limits_{h=1}^H S_h^2\right]
\end{eqnarray*}

\begin{eqnarray*}
V(\hat t_{srs}) &=& \left(1-\ds\frac n N\right) N^2 \ds\frac {S^2}n \\
&=& \left(1- \ds\frac n N\right) \ds\frac {N^2} n \ds\frac 1 {N-1} (SSW+SSB)\\
&\approx& \left(1-\ds\frac n N\right) \ds\frac N n (SSW +SSB)
\end{eqnarray*}
}

\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Efficiency of Proportional Stratification}

\bi
\item Proportional stratification is more efficient if:
$$\sum\limits_{h=1}^H S_h^2 < SSB$$ 
where $SSB = \sum\limits_{h=1}^{H} N_h (\bar y_{hU} - \bar y_U)^2$.
\item This condition is usually satisfied since the large population sizes of the strata force:  
$$N_h (\bar y_{hU} - \bar y_U)^2 > S_h^2$$
\ei

\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
{\bf Comments:}
\bi
\item The variance of the estimator $\hat t$ from a stratified sample with proportional allocation is generally smaller than the variance of the estimator from SRS, given the same number of observations.
\item The more unequal the stratum means $\bar y_{hU}$ and the more homogeneous the units within each stratum, the greater the precision gained from proportional allocation.
\ei
\end{frame}

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

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Optimal Allocation}

{\bf Example:} Sampling American Corporations to Estimate Trade with Europe

\bi
\item The variation among large corporations is often greater than the variation among small ones:
    \begin{itemize}
        \item Large units tend to exhibit more variability than smaller units.
    \end{itemize}

\item To account for this variability:
    \begin{itemize}
        \item A higher percentage of large corporations needs to be sampled.
    \end{itemize}

\item Limitations of Proportional Allocation:
    \begin{itemize}
        \item Proportional allocation samples the same percentage within each stratum.
        \item Works well if the variances $S_h^2$ are similar across strata.
        \item Performs poorly when the variances $S_h^2$ differ significantly between strata.
    \end{itemize}

\item Optimal Allocation:
    \begin{itemize}
        \item Take more units from strata with larger variances $S_h^2$.
        \item Balances sampling effort with variability to achieve greater precision.
    \end{itemize}
\ei
\end{frame}


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

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Cost Function}

\begin{itemize}
    \item Cost function:
    \[
    c = c_0 + \sum\limits_{h=1}^H c_h n_h
    \]
    where:
    \begin{itemize}
        \item $c_0$: Overhead costs (e.g., maintaining an office)
        \item $c_h$: Cost of sampling a unit in stratum $h$
        \item $n_h$: Number of sampled units in stratum $h$
    \end{itemize}
    
    \item Objective:
    \begin{itemize}
        \item Minimize $V(\hat{t}_{str})$ for a fixed cost $c$
        \item Minimize $c$ for a fixed $V(\hat{t}_{str})$
    \end{itemize}

\end{itemize}
\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\begin{itemize}
    \item Variance of $\hat{t}_{str}$:
    \[
    V(\hat{t}_{str}) = \sum\limits_{h=1}^H N_h^2 \left(1-\ds\frac{n_h}{N_h}\right)\ds\frac{S_h^2}{n_h}
    \]
    Expanding and simplifying:
    \[
    V(\hat{t}_{str}) = \sum\limits_{h=1}^H N_h^2 \ds\frac{S_h^2}{n_h} - \sum\limits_{h=1}^H N_h S_h^2
    \]

    \item Equivalent to minimizing:
    \[
    \sum\limits_{h=1}^H N_h^2 \ds\frac{S_h^2}{n_h}
    \]
\end{itemize}
\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Deriving Optimal Allocation}

\begin{itemize}
    \item Objective function:
    \[
    f = \sum\limits_{h=1}^H N_h^2 \ds\frac{S_h^2}{n_h} + 
    \lambda \left(c_0 + \sum\limits_{h=1}^H c_h n_h - c \right)
    \]

    \item Take the partial derivative with respect to $n_h$ and set to 0:
    \[
    \ds\frac{\partial f}{\partial n_h} = \ds\frac{-N_h^2 S_h^2}{n_h^2} + \lambda c_h = 0
    \]

    \item Solving for $n_h$:
    \[
    n_h = \ds\frac{N_h S_h}{\sqrt{c_h \lambda}}
    \]

    \item Using the constraint $\sum_{h=1}^H n_h = n$, we find:
    \[
    \ds\frac{1}{\sqrt{\lambda}} = \ds\frac{n}{\sum_{l=1}^H \ds\frac{N_l S_l}{\sqrt{c_l}}}
    \]
\end{itemize}
\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\bi
\item Final result for optimal allocation:
    \[
    n_{h,opt} = n \times \left( \ds\frac{N_h S_h / \sqrt{c_h}}{\sum_{l=1}^H N_l S_l / \sqrt{c_l}} \right)
    \]

    \item Simplified proportionality:
    \[
    n_{h,opt} \propto \ds\frac{N_h S_h}{\sqrt{c_h}}
    \]
    \ei
We take a larger sample from stratum $h$ if
\bi
\item The stratum size $N_h$ is large
\item The variance within the stratum $S_h$ is large
\item The sampling within the stratum $c_h$ is inexpensive
\ei
\end{frame}

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

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Neyman Allocation}
{\small
\begin{itemize}
    \item Optimal allocation formula:
    \[
    n_{h,opt} = n \times \left( \ds\frac{N_h S_h / \sqrt{c_h}}{\sum_{l=1}^H N_l S_l / \sqrt{c_l}} \right)
    \]

    \item **Neyman Allocation**: Assumes equal costs for all strata ($c_h$'s are equal):
    
    \[
    n_{h,\mbox{Neyman}} = n \times \left( \ds\frac{N_h S_h}{\sum_{l=1}^H N_l S_l} \right)
    \]

    \item Simplify with a constant factor $a$:
    \[
    a = \ds\frac{n}{\sum_{l=1}^H N_l S_l}
    \]
    \[
    n_{h,\mbox{Neyman}} = a \times N_h S_h
    \]

    \item Key insight:
    \begin{itemize}
        \item The sample size for each stratum is proportional to the product of the population size $N_h$ and standard deviation $S_h$.
    \end{itemize}
\end{itemize}
}
\end{frame}

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




%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Variance of Neyman Allocation}

\begin{eqnarray*}
V(\hat t_{str,\mbox{Neyman}}) &=& \sum\limits_{h=1}^H
\left(1-\ds\frac {n_h}{N_h}\right)
\ds\frac{N_h^2 S_h^2}{n_h} \\
&=& \sum\limits_{h=1}^H \left(1- \ds\frac{a N_h S_h}{N_h}\right)
\ds\frac{N_h^2 S_h^2}{a N_h S_h} \\
&=& \sum\limits_{h=1}^H \left(1 - a S_h\right) \ds\frac {N_h S_h}{a} \\
&=& \sum\limits_{h=1}^H \left(1 - \ds\frac{n}{\sum_{l=1}^H N_l S_l} S_h\right)
\ds\frac{N_h S_h \sum_{l=1}^H N_l S_l}{n} \\
&=& \ds\frac{\sum\limits_{h=1}^H N_h S_h \sum\limits_{l=1}^H N_l S_l}{n} - \sum\limits_{h=1}^H N_h S_h^2
\end{eqnarray*}
\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Variance of Proportional Allocation}

\begin{eqnarray*}
V(\hat t_{str,\mbox{Prop}}) &=& \sum\limits_{h=1}^H
\left(1-\ds\frac{n_h}{N_h}\right) \ds\frac{N_h^2}{n_h} S_h^2 \\
&=& \sum\limits_{h=1}^H \left(1 - \ds\frac{n}{N}\right) \ds\frac{N}{n} N_h S_h^2 \\
&=& \sum\limits_{h=1}^H \ds\frac{N}{n} N_h S_h^2 - \sum\limits_{h=1}^H N_h S_h^2
\end{eqnarray*}

\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Comparison of Neyman and proportional allocation,  and SRS given the same sample size}

\begin{eqnarray*}
\sum\limits_{h=1}^H N_h S_h \sum\limits_{l=1}^H N_l S_l &=& \sum\limits_{h=1}^H N_h^2 S_h^2 + 2 \sum_{i=1}^H \sum_{j>i}^H N_i N_j S_i S_j \\
\sum\limits_{h=1}^H N_h S_h^2 &=& \sum\limits_{h=1}^H N_h^2 S_h^2 + \sum\limits_{i=1}^H \sum\limits_{j>i}^H N_i N_j (S_i^2 + S_j^2)
\end{eqnarray*}

\[
V(\hat t_{str, \mbox{Neyman}}) \leq V(\hat t_{str, \mbox{Prop}})
\]

\[
\text{Relative Precision: } V (\hat t_{str,\mbox{Neyman}}) \leq V(\hat t_{str, \mbox{Prop}}) \leq V_{srs}(\hat t)
\]
\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Example 3.3: Siniff and Skoog (1964) - Estimation of Nelchina Herd}

Siniff and Skoog (1964) used stratified random sampling to estimate the size of the Nelchina herd of Alaska caribou in February of 1962. 

\bi
\item The biologists used preliminary estimates of caribou densities to divide the area of interest into six strata. \\
      \quad --- Each stratum was then divided into a grid of 4-square-mile sampling units.
\item Stratum A, for example, contained $N_1 = 400$ sampling units; $n_1 = 98$ of these were randomly selected to be in the survey.
\ei

\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Example 3.10: Caribou Survey - Neyman Allocation}

The caribou survey in Example 3.3 used Neyman allocation to determine the $n_h$. 

\bi
\item Before taking the survey, the investigators constructed strata to be relatively homogeneous in terms of population density.
\item Total sample size was set as $n = 225$.
\item Estimated count in each stratum is served as a rough estimate of the standard deviation.
\item Wanted the sampling fraction to be at least $1/3$ in smaller strata.
\item Used the Neyman allocations as a guideline for determining the final sample sizes in the last column of Table \ref{tab:optallo}.
\ei

\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}

{\small
\begin{table}[h!]
\centering
\begin{tabular}{|c|c|c|c|c|c|}
\hline
{Stratum} & $N_h$ & $s_h$ & $N_h s_h$ & $n_h = 225 \ds\frac{N_h s_h}{\sum N_l s_l}$ & {Sample Size} \\
\hline
A & 400 & 3,000 & 1,200,000 & 96.26 & 98 \\
B & 30 & 2,000 & 60,000 & 4.81 & 10 \\
C & 61 & 9,000 & 549,000 & 44.04 & 37 \\
D & 18 & 2,000 & 36,000 & 2.89 & 6 \\
E & 70 & 12,000 & 840,000 & 67.38 & 39 \\
F & 120 & 1,000 & 120,000 & 9.63 & 21 \\
\hline
\textbf{Total} & \textbf{699} & &  \textbf{2,805,000} & \textbf{225.00} & \textbf{221}  \\
\hline
\end{tabular}
\caption{Optimal allocation for Caribou Herd Example}\label{tab:optallo}
\end{table}
}
\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
{\Large Some design issues of stratified random sampling}

\bi
\item Allocating observations to strata\\
----Proportional allocation: $\ds\frac {n_h}{N_h} =\ds\frac{n}N$\\
----Neyman allocation (special case of optimal allocation when $c_h$'s are all equal)
$$n_{h,\mbox{Neyman}} = n \left(\ds\frac{N_h S_h}{\sum\limits_{l=1}^H
N_l S_l}\right)$$
\item Sample size
\item Defining strata: variables and number of strata
\ei
\end{frame}


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

\begin{frame}
\frametitle{Example 3.12: Stratified Sampling for U.S. Colleges and Universities}

The file \texttt{college.csv}, abstracted from online data published by the U.S. Department of Education (2020), contains selected variables from a population of 1,372 U.S. colleges and universities.

\bi
  \item The goal is to estimate the total instructional budget or the number of students who have full-time employment five years after graduation. These variables are thought to be related to the size of the institution.
  \item The institutions are grouped into ten strata based on size and residential status, as defined by the variable \texttt{ccsizset}.
  \item For each stratum \( h \), \( S_h \) represents the population standard deviation of the undergraduate enrollment (\texttt{ugds}).
\ei
\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\begin{figure}[!htp]
       \centering
       \includegraphics[width=1.0\textwidth,
       height=0.6\textheight]{./plots/table0308}
\caption{Proportional and Neyman allocation example for college data}
       \label{neyman}
       \begin{center}
        \textit{\scriptsize Source: Table 3.8 of Sampling: Design and Analysis, 3rd edition, by Sharon L. Lohr}
    \end{center}
     \end{figure}

\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
{\bf Determining sample size}
\begin{eqnarray*}
V(\hat t_{str}) &=& \sum\limits_{h=1}^H N_h^2\left(1-\ds\frac
{n_h}{N_h}\right)
\ds\frac{S_h^2}{n_h}\\
&\leq& \sum\limits_{h=1}^H N_h^2 \cdot \ds\frac{S_h^2}{n_h} =
\ds\frac 1 n \sum\limits_{h=1}^H \ds\frac n {n_h} N_h^2 S_h^2 = v/n
\end{eqnarray*}
\bi
\item $v$ depends on stratum size $N_h$, variances $S_h^2$, and on the relative sample sizes
$n_h/n$
\item $v$ can be thought of as the ``average" variability per observation unit in a stratified
random sample with the specified allocation

95 \% CI: $\hat t_{str} \pm z_{\alpha/2} \sqrt{v/n}$\\
$z_{\alpha/2} \sqrt{v/n} =e$, $n = z_{\alpha/2}^2 v/e^2$
\ei
\end{frame}


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

\begin{frame}
\frametitle{Defining Strata}

\textbf{1. Variables for Stratification}
\bi
  \item Variables should be highly associated with the variables of interest.
  \item For example:
  \begin{itemize}
    \item To estimate total business expenditures on advertising, we might stratify by the number of employees, size of the business, and type of product or service.
    \item For farm income, we could use farm size as a stratifying variable, as larger farms are expected to have higher incomes.
  \end{itemize}
  \item The stratifying variables must be known for all sampling units in the population.
\ei
\end{frame}



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}

\bi
  \item[2.] \textbf{Number of Strata:}
  \item The number of strata depends on several factors:
  \begin{itemize}
    \item The difficulty in constructing a sampling frame with stratifying information.
    \item The cost of stratification.
  \end{itemize}
  \item Various formulas in the literature provide guidelines for determining the number of strata.
  \item A pilot study can help in deciding the number of strata.
  \item General rule: The more information you have about the population, the more strata you should use. If little prior information about the target population is available, you should consider using simple random sampling (SRS).
\ei
\end{frame}


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

\begin{frame}
\frametitle{Stratification or SRS?}

Recall: Relative precision of stratification and SRS
\[
V (\hat t_{str,\mbox{opt}}) \leq V(\hat t_{str,\mbox{prop}}) \leq V_{srs}(\hat t)
\]
\bi
  \item[1.] Stratified sampling provides higher precision than SRS, but why conduct SRS?
  \item Stratification adds complexity to the survey, which may not justify the small gain in precision.
  \item Information is needed about which units belong to each stratum and how many units there are.
  \item[2.] When is stratified sampling efficient?
  \item SSB (between-strata variation) is large (strata means differ greatly).
  \item SSW (within-strata variation) is small (low variability within stratum).
\ei
\end{frame}


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

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Example 3.13: Homeless Counts in the United States}

In the United States, state and local agencies responsible for coordinating homeless services produce annual estimates of the number and characteristics of persons experiencing homelessness. 

\bi
\item Each agency selects one of the last ten nights in January to conduct the count of the:
\begin{enumerate}
    \item Sheltered population (persons in emergency shelters or transitional housing).
    \item Unsheltered population (persons staying in places not intended for human habitation, such as on the streets, under bridges, in cars, tents, bus or subway stations, or abandoned buildings).
\end{enumerate}
\item The survey is conducted on one night to reduce possible double-counting and minimize the effort required for teams of volunteers who canvass areas to survey the unsheltered population.
\ei
\end{frame}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Example 3.13 (Cont.): Survey Design}

\bi
\item Most agencies also use the survey to disseminate information about resources and social services available for persons experiencing homelessness.
\item Collect data about needs and demographics for future community outreach efforts.
\item The goal is to have a sampling plan that: 
\begin{enumerate}
    \item Produces an accurate estimate of the unsheltered population's size.
    \item Allows contact with as many unsheltered persons as possible.
\end{enumerate}
\item In many cities, the large land area makes it impossible for volunteers to visit every location in the city during one night.
\ei
\end{frame}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Homeless Counts in New York City}

New York City, with a land area of about 800 square kilometers, uses a carefully designed stratified random sample for its annual point-in-time count (Schneider et al., 2016; New York City Department of Homeless Services, 2019).

\bi
\item The city is divided into approximately 7,000 areas, each classified as "high-density" or "low-density":
\begin{enumerate}
    \item High-density: Areas in Manhattan or the subway system are considered high-density if expected to contain at least two persons experiencing homelessness.
    \item High-density: Areas in the Bronx, Queens, Staten Island, or Brooklyn are considered high-density if expected to contain at least one person.
    \item Low-density: Areas not classified as high-density.
\end{enumerate}
\item Strata: Six high-density strata, six low-density strata.
\ei
\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Sampling Strategy in New York City}

\bi
\item Sampling fraction: 100\% for high-density strata (all areas in high-density strata are canvassed).
\item Sampling fractions for low-density strata vary by borough and year, designed to give high-precision estimates for each borough and the subway system.
\item In 2019, about 21\% of the 7,000 areas in the city were canvassed.
\ei
\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Advantages of Stratified Sampling in NYC}

The stratified sampling design allows New York City to:
\bi
\item Achieve its objectives for the one-night count with the number of volunteers available (typically between 2,000 and 2,500).
\item Deploy most volunteers to areas thought to contain unsheltered persons, obtaining a large number of unsheltered persons in the sample, and gather more information about their characteristics. This still produces statistically valid estimates due to random sampling from each low-density stratum.
\item However, the point-in-time count still has measurement error for estimating the number of persons experiencing homelessness on the night of the count:
\begin{enumerate}
    \item Some persons remain out of sight or in locations deemed unsafe for volunteers to visit.
    \item Volunteers may mistakenly identify persons as not homeless when they are, or as homeless when they are not.
    \item Volunteers may fail to survey some individuals along their routes.
\end{enumerate}
\ei
\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
{\Large Model-based inference for stratified sampling}
\bi
\item The one-way ANOVA model with fixed effects provides an underlying structure for
stratified sampling:
\begin{equation}\label{eq:anova}
y_{hj} = \mu_h +\epsilon_{hj}
\end{equation}
where $\epsilon_{hj}$ are independent with mean $0$ and variance
$\sigma_h^2$.
\item The least squares estimator of $\mu_h$ is
$\bar y_h$, the average in stratum $h$
\ei

\end{frame}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
{\bf Estimators and Properties:}
\bi
\item  $T_h = \sum\limits_{j=1}^{N_h} y_{hj}$: the total in stratum
$h$
\item $T = \sum\limits_{h=1}^H T_h$: the overall total
\item Note that both $T_h$ and $T$ are random variables under model (\ref{eq:anova})
\item The best linear unbiased estimator for $T_h$ is
$\hat T_h = \ds\frac {N_h}{n_h} \sum\limits_{j\in S_h} y_{hj}.$
\item $E_M [\hat T_h -T_h] =0$
\item $E_M [(\hat T_h -T_h)^2] =
N_h^2\left(1-\ds\frac{n_h}{N_h}\right)\ds\frac{\sigma_h^2}{n_h}$
\ei
\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}

By the fact that observations in different strata are independent
under model (\ref{eq:anova}),
\begin{eqnarray*}
Var(\hat T)  &=& E_M \left[\left\{\sum\limits_{h=1}^H(\hat
T_h
-T_h)\right\}^2\right]\\
&=& E_M \left[ \sum\limits_{h=1}^H (\hat T_h -T_h)^2 +
\sum\limits_{h=1}^H \sum\limits_{k\neq h} (\hat T_h -T_h)(\hat T_k
-T_k)\right]\\
&=& E_M \left[\sum\limits_{h=1}^H (\hat T_h -T_h)^2\right]\\
&=& \sum\limits_{h=1}^H N_h^2
\left(1-\ds\frac{n_h}{N_h}\right)\ds\frac{\sigma_h^2}{n_h}
\end{eqnarray*}

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\textbf{Comments:}
\bi
\item The theoretical variance $\sigma_h^2$ can be estimated by $s_h^2$.
\item Adopting the model in (\ref{eq:anova}) results in the same estimators for $t$ and its standard error as found under randomization theory.
\item However, if a different model is used, different estimators will be obtained.
\ei
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\begin{frame}
\frametitle{Disclaimer}

Slides are intended for a course based on the book: \textit{Sampling: Design and Analysis, Third Edition} by Sharon L. Lohr

All examples, datasets, and pages directly scanned and included in this chapter are from the textbook.

References to papers or books cited in these slides can be found in the Bibliography section of the textbook.
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