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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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% 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 9: Variance Estimation in Complex Surveys} % Emphasize chapter title

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\section{Chapter 9}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
{\bf Topics}
\bi
\item Linearization (Taylor Series) Methods
\item Random Group Methods
\item Resampling and Replication Methods
\item Generalized variance functions (GVF)
\ei

\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
{\bf Linearization for estimating variances}

Most of the variance formulas in Chapters 2 through 6 were for
estimators of means and totals. Those formulas can be used to find
variances for any linear combination of estimated means and totals.
$$V\left(\sum_{i=1}^k a_j \hat t_j\right) = \sum_{j=1}^k a_j^2 V(\hat t_j) + 2
\sum_{j=1}^{k-1} \sum_{l=j+1}^k a_j a_l \mbox{Cov}(\hat t_j, \hat
t_l)$$
\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
Suppose we are interested in a population quantity $\theta$ that is
a function of population means or totals
\bi
\item 
In general, let $\theta$ be a parameter of interest, 
$$\theta= h(t_1, t_2, \cdots, t_k),\quad
\hat \theta =  h(\hat t_1, \hat t_2, \cdots, \hat t_k)$$
\item 
Taylor's theorem allows us to linearize a smooth nonlinear function
$h(t_1, t_2,\cdots, t_k)$ of the population totals\\
By first-order Taylor expansion\\
$\hat \theta - \theta \approx \sum_{j=1}^ k \ds\frac {\partial
h}{\partial t_j} (\hat t_j - t_j)$
\ei
\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
Example: Ratio estimator $\theta = B = \ds\frac{t_y}{t_x}, \quad \hat \theta = \hat
B = \ds\frac{\hat t_y}{\hat t_x}$\\
$$h(x,y) = \ds\frac y x, \quad \ds\frac{\partial h}{\partial x} = \ds\frac{-y}{x^2}, \quad
\ds\frac{\partial h}{\partial y} = \ds\frac 1 x$$
$$h(t_x, t_y) = \ds\frac{t_y}{t_x}, \quad \ds\frac{\partial h}{\partial t_x} = \ds\frac{-t_y}{t_x^2}, \quad
\ds\frac{\partial h}{\partial t_y} = \ds\frac 1 {t_x}$$
\begin{eqnarray*}
\hat B - B &=& h(\hat t_x, \hat t_y) - h(t_x, t_y)\\
&\approx& \ds\frac{\partial h}{\partial x}|_{(t_x,t_y)}
\cdot (\hat t_x - t_x) + \ds\frac{\partial h}{\partial
y}|_{(t_x,t_y)} \cdot (\hat t_y - t_y)\\
&=& -\ds\frac{t_y}{t_x^2} (\hat t_x - t_x) + \ds\frac 1{t_x}
(\hat t_y - t_y)\\
&=& -\ds\frac 1 {t_x} [B(\hat t_x - t_x) - (\hat t_y -t_y)]
\end{eqnarray*}

\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\begin{eqnarray*}
 \hat B -B &\approx& -\ds\frac 1 {t_x} [B(\hat t_x - t_x) - (\hat t_y
 -t_y)]\\
 & =& \ds\frac 1 {t_x} [\hat t_y - t_y -B(\hat t_x - t_x)]\\
 & = & \ds\frac 1 {t_x} [ \hat t_y -B \hat t_x - (t_y - B t_x)]
% &\approx& \ds\frac 1 {t_x}[\hat
%t_z - t_z]
\end{eqnarray*}
Let $z_i = y_i - B x_i$,
\begin{eqnarray*}
 \hat t_z &=& \sum_{i \in S} w_i z_i = \sum_{i \in S} w_i (y_i -
B x_i)\\
& = &\hat t_y -B \hat t_x
\end{eqnarray*}
$$\hat B -B \approx \ds\frac 1 {t_x}[\hat t_z - t_z]$$

\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
$V_L (\hat B)
\approx V\left(\ds\frac{\hat t_z}{t_x}\right) = \ds\frac 1 {t_x^2}
V(\hat t_z)$\\
 $z_i = y_i - B x_i$,  $e_i = y_i - \hat B x_i$\\
Linearization variance
\begin{eqnarray*}
\hat V_L(\hat B) &\approx & \hat V(\hat t_e)/\hat
t_x^2 \\
&= &\ds\frac{N^2 \left(1-\ds\frac n N\right) \ds\frac{s_e^2}{n}}{N^2
\bar
x^2}\\
&=& \left(1- \ds\frac n N\right) \ds\frac{s_e^2}{n\bar x^2}
\end{eqnarray*}
Recall Chapter 4, SRS
$$\hat B = \bar y/\bar x = \hat t_y/\hat t_x$$
$$\hat V(\hat B) = \left(1- \ds\frac n N\right) \ds\frac{s_e^2}{n\bar x^2}$$
\end{frame}

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

{\bf Advantages:}
 \bi
\item If the partial derivatives are known,
linearization almost always gives a variance estimate for a
statistic and can be applied in general sampling designs
\item Theory and softwares are well developed for linearization
\ei
{\bf Disadvantages:}
 \bi
\item Calculations can be
messy, and the method is difficult to apply for complex functions
involving weights
\item Not all statistics can be
expressed as a smooth function of the population totals, such as
median and other quantiles
\item  Need large sample size for the accuracy of the linearization
approximation \ei 
\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
{\bf Random Group Methods}
\bi
\item[1.] {\bf Replicating the Survey Design}
 \item the basic survey design is replicated independently
for $R$ times.\\
---- Independently means that each of the $R$ sets of random
variables used to select the sample is independent of the other sets
after each sample is drawn, the sampled units are replaced in the
population so they are available for later samples.
\item the R
replicate samples produce $R$ independent estimators of the quantity
of interest; the variability among those estimates can be used to
estimate the variance of $\hat \theta$.
\ei
\end{frame}

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

\begin{frame}

Let $\theta$ represent the parameter of interest.\\
Define:
\begin{itemize}
    \item $\hat{\theta}_r$: the estimator of $\theta$ from the $r$th replicate.
    \item $\tilde{\theta} = \frac{1}{R} \sum_{r=1}^R \hat{\theta}_r$: the average of the replicate estimators.
    \item $\hat{V}_1(\tilde{\theta}) = \frac{1}{R} \cdot \frac{1}{R-1} \sum_{r=1}^R (\hat{\theta}_r - \tilde{\theta})^2$: an estimator of the variance of $\tilde{\theta}$.
\end{itemize}
If $\hat{\theta}_r$ is an unbiased estimator of $\theta$, then $\tilde{\theta}$ is also an unbiased estimator of $\theta$, and $\hat{V}_1(\tilde{\theta})$ is an unbiased estimator of $V(\tilde{\theta})$.

\end{frame}


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

\begin{frame}

{\bf Example 9.3.}
\bi
\item Objective: Estimate the ratio of out-of-state tuition to in-state tuition for public colleges and universities in the United States, using data from \texttt{college.csv} (see Example 3.12 for details). 
\item Method: Implement the random group method by selecting five simple random samples (SRSs) of size 10 each.\\
---- The SRSs are drawn without replacement, but the same college may appear in more than one sample.\\
---- Data for this example are in \texttt{collegerg.csv}.
\ei
\end{frame}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\begin{table}
\caption{Summary Statistics for Five SRSs of Colleges, Used in
Example 9.3. $\hat \theta_i$= average of nonresident tuitions for
sample $i$/average of resident tuitions for sample $i$}
\begin{tabular}{cccc}
\hline Replicate & Average & Average&\\
sample $r$ & ${\mbox{In-state tuition }} \bar x_r$& ${\mbox{Out-of-state tuition }} \bar y_r$&$\hat \theta_r$ \\
\hline
1& 8913.3& 21614.7& 2.4250\\
2& 9542.0& 21497.5& 2.2529\\
3& 10210.6&21323.4 & 2.0884\\
4& 9004.7&18469.0 & 2.0510\\
5& 9467.1 & 22844.0 & 2.4130\\
\hline
\end{tabular}
\end{table}
\end{frame}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\bi
 \item The sample average of the five independent estimates
is
$$\tilde
\theta= \sum_{i=1}^5 \hat \theta_r/5=2.246.$$
\item The variability across the estimates i,e., the sample
standard deviation of the five estimates is 0.175, so the standard
error of $\tilde \theta$ is
$$\sqrt{\hat V_1({\tilde\theta})} = 0.175/\sqrt{5}=0.0784.$$
\item A 95\%
CI for the ratio is $$2.246 \pm 2.78*(0.0784) = [2.03, 2.46],$$
where where 2.78 is the $t$ critical value with 4 degrees of freedom
(df).
\ei
\end{frame}

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

{\small
\bi
\item[2.]{\bf Dividing the Sample into Random Groups}
\item In practice, the complete sample is selected according to the survey design.
\item The complete sample is then divided into $R$ groups, so that each
group forms a miniature version of the survey, mirroring the sample
design. \\
----If the sample is an SRS of size $n$, the groups are formed by
randomly apportioning the $n$ observations into $R$ groups, each of
size $n/R$.\\
----In a cluster sample, the psus are randomly divided among the $R$
groups. The psu takes all its observation units with it to the
random group, so each random group is still a cluster sample. \\
----In a stratified multistage sample, a random group contains a
sample of psus from each stratum. Note that if $k$ psus are sampled
in the smallest stratum, at most $k$ random groups can be formed.\\
----The groups are then treated as though they are independent
replicates of the basic survey design.
\ei
}
\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\bi
\item
These pseudo-random groups are not quite independent replicates
because an observation unit can only appear in one of the groups; if
the population size is large relative to the sample size, however,
the groups can be treated as though they are independent replicates.
\ei
\end{frame}

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

{\bf Definitions and Variance Estimation}

\bi
\item Let $\theta$ be the parameter of interest.
\item $\hat \theta_r$: Estimator of $\theta$ calculated from the $r$th random group.
\item $\tilde \theta = \ds\frac{1}{R} \sum_{r=1}^R \hat \theta_r$: Average estimate across the $R$ random groups.
\item $\hat \theta$: Estimate of $\theta$ calculated from the complete sample.\\
--- Typically, $\hat \theta$ is a more stable estimator than $\tilde \theta$.
\item Variance estimators:
\[
\hat V_1(\tilde \theta) = \ds\frac{1}{R} \cdot \ds\frac{1}{R-1} \sum_{r=1}^R (\hat \theta_r - \tilde \theta)^2
\]
\[
\hat V_2(\tilde \theta) = \ds\frac{1}{R} \cdot \ds\frac{1}{R-1} \sum_{r=1}^R (\hat \theta_r - \hat \theta)^2
\]
--- $\hat V_2(\tilde \theta)$ is slightly larger but is often preferred.
\ei

\end{frame}


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

{\bf Example 9.4:} The 1987 Survey of Youths in Custody was divided into seven random groups.
\bi
\item The survey
design had 16 strata. Strata 6-16 each consisted of one facility
(=psu), and these facilities were sampled with probability one. In
strata 1-5, facilities were selected with probability proportional
to number of residents in the 1985 Children in Custody census.
\ei
\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\begin{table}
\caption{Survey of Youth in Custody Stratum Information}
\begin{tabular}{ccccc}
\hline &CIC size & Number of & Number of & Number of eligible\\
& number of & psu's in & residents & psu's in \\
Stratum & residents & frame & in CIC & sample\\
\hline 1 & 1-59 & 99& 2881 &11\\
2 & 60-119 & 39 & 3525 & 7\\
3 & 120-179 & 30 & 4355&7\\
4& 180-239& 13 & 2594& 7\\
5 & 240-359& 14 & 4129 &7\\
 \hline
\end{tabular}
\end{table}
\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\bi
\item Seven random groups were formed because strata 2 through 5 each
have seven psus.
\item For each self-representing facility in strata 6-16, random
group numbers were assigned as follows: The first resident selected
from the facility was assigned a number between 1 and 7. Let's say
the first resident was assigned number 6. Then the second resident
in that facility would be assigned number 7, the third resident 1,
the fourth resident 2, and so on.
\ei
\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\bi
 \item In strata 1-5, all residents in a facility (psu) were
assigned to the same random group. Thus for the seven facilities
sampled in stratum 2, all residents in facility 33 were assigned
random group number 1, all residents in facility 9 were assigned
random group number 2, and so on.
\item After all
random group assignments were made, each
random group had the same basic design as the original sample.\\
----
Random group 1, for example, forms a stratified sample in which a
(roughly) random sample of residents is taken from the self
representing facilities in strata 6-16, and an unequal-probability
sample of facilities is taken from each of strata 1-5.
\ei
\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
 \begin{table}
 \caption{Estimates of mean age of residents
for each random group, $\hat \theta_r = \sum w_i y_i/\sum w_i$,
where $w_i$ is the final weight for resident $i$, and the summations
are over observations in random group $r$.}
\begin{tabular}{cc}
Random group& Estimate of mean age $\hat \theta_r$\\
\hline
1&16.55\\
2& 16.66\\
 3 &16.83\\
  4 &16.06\\
 5 &16.32\\
6 &17.03\\
7 &17.27\\
\hline
\end{tabular}
\end{table}

\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\bi
\item $\tilde \theta = \ds\frac 1 7 \sum_{r=1}^7 \hat \theta_r=
16.67$, $\hat \theta= 16.64$ (using entire data)
\item $\hat V_1(\tilde \theta) = \ds\frac 1 7 \cdot \ds\frac 1
{7-1}
\sum_{r=1}^7(\hat \theta_r -\tilde \theta)^2 = \ds\frac {0.1704}{7} = 0.024$\\

\item $\hat V_2(\tilde \theta) = \ds\frac 1 7 \cdot \ds\frac 1 {7-1}
\sum_{r=1}^7(\hat \theta_r -\hat \theta)^2 = \ds\frac{0.1716}{7} =
0.025$
\item using $\hat \theta$, a 95\% CI for mean age is
$$16.64 \pm 2.45 \sqrt{0.025} = [16.3, 17.0],$$
where 2.45 is the $t$ critical value with $6$ df.
\ei
\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
Advantages of random group methods:
\bi
\item Easy to calculate the variance estimate.
\item The method is well-suited to multiparameter or nonparametric
problems. It can be used to estimate variances for percentiles and
nonsmooth functions as well as variances of smooth functions of the
population totals.
\item Random group methods are easily used after
weighting adjustments for nonresponse and undercoverage.
\ei
\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
Disadvantages of random group methods:
\bi
\item The number of random groups is often small. This
gives imprecise estimates of the variances.\\
----The survey design may limit the number of
random groups that can be constructed. If two psus are selected in
each stratum, then only two random groups can be formed.\\
----Generally one would like at least ten random groups to obtain a
more stable estimate of the variance and to avoid inflating the CI
by using a critical value from a $t$ distribution with few df.
\item If $\hat \theta$ is a
nonlinear statistic, $\tilde \theta$ can have large bias if the
number of observations in each group is small.
\item Setting up the random groups can be difficult in complicated
designs, as each random group must have the same design structure as
the complete survey.
\ei
\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
{\bf Resampling and Replication Methods}
\bi
\item Random group
methods are easy to compute and explain but are unstable if a
complex sample can only be split into a small number of groups.
\item Resampling
methods treat the sample as if it were itself a population; we take
different samples from this new ``population" and use the subsamples
to estimate the variance.
\item All of the methods in
this section calculate variance estimates for a sample in which psus
are sampled with replacement. If psus are sampled without
replacement, these methods may still be used, but are expected to
overestimate the variance and result in conservative CIs.
\ei
\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\bi
\item[1.] {\bf Balanced Repeated Replication (BRR)}

Restrict to surveys that are stratified to the point that only two
psus are selected from each stratum. This design gives the highest
degree of stratification possible while still allowing calculation
of variance estimates in each stratum.

\item Half-samples:\\
randomly select one of the observations in each stratum for group 1
and assign the other to group 2. The groups in this situation are
half-samples.
\ei
\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
Altogether $2^H$ ($H$ stratum) possible half-samples could be
formed. McCarthy (1966, 1969) suggest using a balanced sample of the
$2^H$ possible half-samples to estimate the variance.\\
---- Balanced repeated replication uses the variability among $R$
replicate half-samples that are selected in a balanced way to
estimate the variance of $\hat \theta$.
\bi
\item Defining balance,

Half-sample $r$ can be defined by a vector $\bal_r =(\alpha_{r1},
\cdots, \alpha_{rH})$, let
\ei
\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
$$y_h(\bal_r) = \left\{
\begin{array}{cc}
y_{h1} & \mbox{if } \alpha_{rh}=1 \\
y_{h2} & \mbox{if } \alpha_{rh}=-1 \\
\end{array}\right.$$
The set of $R$ replicate half-samples is balanced if
$$\sum_{r=1}^R \alpha_{rh} \alpha_{rl} = 0, \mbox{ for all } l\neq
h.$$

$$\hat V_{\mbox{BRR}}(\hat \theta) = \ds\frac 1 R \sum_{r=1}^R [\hat
\theta(\bal_r) - \hat \theta]^2$$
\end{frame}

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

\begin{table}
\caption{Illustration of balanced repeated replication}
\begin{tabular}{cccccccc}
&\multicolumn{7}{c}{Stratum(h)}\\
 &1&2&3&4&5&6&7\\
 \hline
$\bal_1$& -1&-1 &-1&1 &1 &1 &-1 \\
$\bal_2$&1 & -1& -1&-1 &-1 &1 & 1\\
$\bal_3$& -1& 1& -1& -1&1 &-1 &1 \\
$\bal_4$& 1& 1&-1 &1 &-1 &-1 &-1 \\
$\bal_5$& -1&-1 &1 &1 &-1 &-1 &1 \\
$\bal_6$&1 &-1 &1 &-1 &1 &-1 &-1 \\
$\bal_7$& -1& 1& 1& -1& -1& 1&-1 \\
$\bal_8$& 1& 1&1 & 1 &1 &1 &1 \\
\hline
 \hline
 \end{tabular}
 \end{table}

\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\bi
\item[2.] {\bf The Jackknife (delete-1 jackknife)}\\
$\hat \theta_{(j)}$: the estimator without observation $j$\\
$\hat \theta$: estimator with full data set
$$ \mbox{SRS}: \hat V_{JK}
(\hat \theta) = \ds\frac{n-1} n \sum\limits_{j=1}^n (\hat
\theta_{(j)} - \hat \theta)^2$$
\ei
\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
Example: $\hat \theta = \bar y$
\begin{eqnarray*}
\hat \theta_{(j)} = \bar y_{(j)} & =& \ds\frac 1 {n-1} \sum_{i \neq
j} y_i = \ds\frac 1 {n-1} \left(\sum_{i=1}^n y_i - y_j\right)\\
&=& \bar y \cdot \ds\frac n{n-1} - \ds\frac {y_j}{n-1} \\
&=& \bar y - \ds\frac 1 {n-1} (y_j -\bar y)
\end{eqnarray*}
where $\bar y_{(j)} = \ds\frac{\mbox{sum of everyone except }
y_j}{n-1}$
\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\begin{eqnarray*}
\sum\limits_{j=1}^n (\bar y_{(j)} - \bar y)^2 & =&
\sum\limits_{j=1}^n \left(\bar y - \ds\frac 1 {n-1}(y_j - \bar y)
-\bar
y\right )^2\\
&=& \ds\frac 1 {(n-1)^2} \sum\limits_{j=1}^n (y_j - \bar y)^2\\
& =& \ds\frac 1 {n-1} s_y^2
\end{eqnarray*}
thus $$\hat V_{JK} (\bar y) = \ds\frac {n-1} n \sum\limits_{j=1}^n
(\bar y_{(j)} - \bar y)^2 = \ds\frac{s_y^2}{n},$$ which is equal to
the with-replacement estimator of the variance of $\bar y$.
\end{frame}


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

Example 9.7.  Let’s use the jackknife method to estimate the ratio of out-of-state tuition $(y)$ to in-state tuition $(x)$ described in Example 9.3. 

\[
\hat \theta = \frac{\bar y}{\bar x}, \quad \hat \theta_{(j)} = \hat B_{(j)} = \frac{\bar y_{(j)}}{\bar x_{(j)}}
\]

The jackknife variance estimator is given by:
\[
\hat V_{\hat B}(\hat \theta) = \frac{n-1}{n} \sum_{j \in S} \big(\hat B_{(j)} - \hat B\big)^2
\]

For each jackknife group (Table 9.6 page 374), omit one observation. 

\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
For instance,  for the first replicate group, $\bar x_{(1)}$ is the average of all $x$ values in the sample except $x_1$:
\begin{eqnarray*}
\bar x_{(1)} &= & \frac{1}{9} \sum_{i=2}^{10} x_i \\
&=& \frac{1}{9} (7140+9808+\cdots +11976+8935+8316)\\
& =& 8802
\end{eqnarray*}
Please refer to Table 9.6 on page 374 for Jackknife calculation. 
We can calculate, 
$$\hat B = 2.425,  \sum (\hat B_{(j)} - \hat B)^2 = 0.0595,$$ and
\[
\hat V_{JK} (\hat B) = (0.9)(0.0595) = 0.05358
\]
\end{frame}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}[fragile]

{\bf Example 9.7 using R}
\begin{verbatim}
data(collegerg) 
# replicate group 1
collegerg1<-collegerg[collegerg$repgroup==1,]
collegerg1[,23:24]
> collegerg1[,24:25]
   tuitionfee_in tuitionfee_out
 1          9912          23640
 2          7140          14810
 3          9808          26648
 4          8987          35170
 5          7930           8674
 6          7200          17550
 7          8929          21692
 8         11976          22488
 9          8935          27199
10          8316          18276

\end{verbatim}
\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}[fragile]

{\bf Linearization estimates (default)}
\begin{verbatim}
collegerg1$sampwt<-rep(500/10,10)
dcollegerg1<-svydesign(id=~1, weights=~sampwt,
data=collegerg1)
> svymean(~tuitionfee_in+tuitionfee_out, dcollegerg1)
                  mean      SE
tuitionfee_in   8913.3  454.46
tuitionfee_out 21614.7 2325.15

\end{verbatim}
\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}[fragile]

Calculte estimate and SEs of ratio using  linearization
\begin{verbatim}
> ratio.lin<-svyratio(~tuitionfee_out,~tuitionfee_in,
dcollegerg1)
> ratio.lin
Ratio estimator: svyratio.survey.design2(~tuitionfee_out, 
~tuitionfee_in, dcollegerg1)
Ratios=
               tuitionfee_in
tuitionfee_out      2.424994
SEs=
               tuitionfee_in
tuitionfee_out     0.2311776
> confint(ratio.lin,df=degf(dcollegerg1))
                                2.5 %   97.5 %
tuitionfee_out/tuitionfee_in 1.902034 2.947954
\end{verbatim}

\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}[fragile]

{\bf Jackknife estimates}
\begin{verbatim}
> ## define jackknife replicate weights design object
> dcollegerg1jk <- as.svrepdesign(dcollegerg1, type="JK1")
> dcollegerg1jk
Call: as.svrepdesign(dcollegerg1, type = "JK1")
Unstratified cluster jacknife (JK1) with 10 replicates.
# JK estimates are same as linearization estimates 
# since we are estimating mean from an SRS and 
# we didn't include fpc in design object
> svymean(~tuitionfee_in + tuitionfee_out, dcollegerg1jk)
                  mean      SE
tuitionfee_in   8913.3  454.46
tuitionfee_out 21614.7 2325.15
\end{verbatim}
\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}[fragile]
\begin{verbatim}
# jackknife SE for ratio
svyratio(~tuitionfee_out, ~tuitionfee_in, 
design = dcollegerg1jk)
> svyratio(~tuitionfee_out, ~tuitionfee_in, 
design = dcollegerg1jk)
Ratio estimator: svyratio.svyrep.design(~tuitionfee_out, 
~tuitionfee_in, design = dcollegerg1jk)
Ratios=
               tuitionfee_in
tuitionfee_out      2.424994
SEs=
          [,1]
[1,] 0.2314828
\end{verbatim}
$\hat B= 2.425,  SE = 0.2314828$ and 
$\hat V_{JK} (\hat B) = .05358$.
\end{frame}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
 {\bf Using weights for Jackknife}
 
$$\bar y = \ds\frac{\sum_{i\in S} w_i y_i}{\sum_{i \in S} w_i}, \quad \bar y_{(i)} =
\ds\frac{\sum_{k \in S} w_{k(i)} y_k}{\sum_{k \in S} w_{k(i)}}$$
SRS: $w_i = N/n$.
$$w_{k(i)} =
\left\{
\begin{array}{cc}
\ds\frac n{n-1} w_k & k \neq i\\
0 & k = i
\end{array}\right.
$$
\end{frame}


\begin{frame}

{\bf Extension to a complex survey data:}
\bi
\item
Cluster samples: delete one psu instead of deleting one unit
\item Stratified multistage
cluster sample: the jackknife is applied separately in each stratum
at the first stage of sampling, with one psu deleted at a time
\item[a.] $H$ strata, $n_h$ psus are chosen for the
sample from stratum $h$. Assume these psus are chosen with
replacement
%once we have jackknife weights, we can apply them to any variable.
%Jackknife works for many cases but not work for median and
%percentile.
\item[b.] $n$: number of psus
\ei
\end{frame}

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

\bi
\item[c.] $\hat \theta_{(j)}$: estimate of $\theta$ that would be
obtained by deleting psu $j$ 
$$w_{i(hj)} =
\left\{
\begin{array}{cc}
w_i & \mbox{if observation unit i is not in stratum h}\\
0 & \mbox{if observation unit i is in psu j of stratum h}\\
\ds\frac {n_h}{n_h-1} w_i & \mbox{if observation unit i is in
stratum h}\\
&\mbox{but not in psu j}
\end{array}\right.
$$

$$\hat V_{JK}(\hat \theta) = \sum\limits_{h=1}^H
\ds\frac{n_h-1}{n_h} \sum\limits_{j=1}^{n_h} (\hat \theta_{(hj)}-\hat \theta)^2$$
\ei

\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}[fragile]

{\bf Example 9.8 coots data using R}\\
Delete one psu at a time
\begin{verbatim}
> data(coots)
> coots$relwt<-coots$csize/2
> dcoots<-svydesign(id=~clutch,weights=~relwt,data=coots)
> dcootsjk <- as.svrepdesign(dcoots, type="JK1")
> dcootsjk
Call: as.svrepdesign(dcoots, type = "JK1")
Unstratified cluster jacknife (JK1) with 184 replicates.
> svymean(~volume,dcootsjk)
         mean    SE
volume 2.4908 0.061
> confint(svymean(~volume,dcootsjk),df=degf(dcootsjk))
          2.5 %   97.5 %
volume 2.370354 2.611203
\end{verbatim}
\end{frame}



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
 Advantages:
\bi
\item The jackknife is an all-purpose method. The same
procedure is used to estimate the variance for every statistic for
which jackknife can be used.
\item  The jackknife provides a
consistent estimator of the variance when $\theta$ is a smooth
function of population totals (Krewski and Rao, 1981).
\item Replication methods such as the jackknife can be used to account
for some of the effects of imputation on the variance estimates (Rao
and Shao, 1992).
\ei
\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
Disadvantages:
\bi
 \item For some sampling designs, the
jackknife may require a large amount of computation.
\item The
jackknife performs poorly for estimating the variances of some
statistics that are not smooth functions of population totals. For
example, the jackknife does not give a consistent estimator of the
variance of quantiles and median.
\ei
\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\bi
\item[3.] {\bf Bootstrap}\\
Treat sample as population, then draw ``resamples" with
replacement from the original sample\\
Take $R$ bootstrap resamples, obtaining $\hat \theta_1^*, \cdots,
\hat \theta_R^*$.
\ei
Example 9.9, estimate the variance of the median height $\theta$ in
the population, using data from {\it htsrs} (refer to example 7.4).
\bi
\item The
median height in population {\it htpop} is $\theta = 168$
\item  Sample median from {\it htsrs}, $\hat \theta = 169$
\item Resample 2000 times, $R  = 2000$.
\ei
\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\begin{tabular}{c|ccccccc}
Median of resample & 165 & 166 & 166.5 & 167 & 167.5 & $\cdots$ &
172\\
\hline frequency & 1 & 5 & 2 & 40 & 15 & $\cdots$ & 4
\end{tabular}
Sample mean: $\ds\frac{\hat \theta_1^* + \cdots + \hat \theta_{2000}^*}{2000} =169.3$\\
Sample variance:
$$\hat V_B(\hat \theta) = \ds\frac 1 {R-1} \sum\limits_{r=1}^R \left[
\hat \theta^*_{r} - \ds\frac 1 R \sum\limits_{r=1}^R \hat
\theta^*_r\right]^2 = 0.9148$$
An approximate 95\% CI may be constructed using the bootstrap variance as $$169 \pm 1.96*\sqrt{0.9148} =
[167.4, 171.2]$$ or using 2.5\% and 97.5\% quantiles as 
$$[q_{2.5\%}, q_{97.5\%}] = [167.5,171]$$
\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}[fragile]

{\bf Direct code to find bootstap variance and CI for median of height}
\begin{verbatim}

# bootstrap by direct coding
# number of iteration
R <- 10000
# init location for bootstrap theta
thetahat <- rep(NA, R)
# draw R bootstrap resamples 
for (i in 1:R) {
  # 
  resam <- sample(htsrs$height, 199, replace = TRUE)
thetahat[i] <- median(resam)
}
\end{verbatim}
\end{frame}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}[fragile]

{\bf Variance and CI estimate by normal approximation using bootstrap variance}
\begin{verbatim}

> # variance and CI estimate by normal approximation
> sebs<-sqrt(var(thetahat))
> sebs
[1] 0.9685265
> m<-median(htsrs$height)
> m
[1] 169
> CI.bs1 <- c(m-1.96*sebs,m+1.96*sebs)
> CI.bs1
[1] 167.1017 170.8983
\end{verbatim}
An approximate 95\% CI of height median may be constructed using the bootstrap variance as $$169 \pm 1.96*0.9685 =
[167.1017, 170.8983]$$ 

\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}[fragile]

{\bf Sort the bootstrap estimates to obtain bootstrap CI}
\begin{verbatim}

> # 0.025th and 0.975th quantile gives equal-tail 
> # bootstrap CI
> thetahat.sorted <- sort(thetahat)
> CI.bs2 <- c(thetahat.sorted[round(0.025*R)], 
thetahat.sorted[round(0.975*R+1)])
> CI.bs2
[1] 167 171
\end{verbatim}
The 0.025th and 0.975th quantile of the $R$ sorted bootstrap estimates gives equal-tail bootstrap CI
for median as $[167, 171]$.

\end{frame}


\begin{frame}[fragile]

{\bf The following is bootstrapping mean of height}
\begin{verbatim}
> data(htsrs)
> nrow(htsrs)
[1] 200
> head(htsrs)
     rn height gender
1   257    159 F     
2  1016    174 M     
3  1264    186 M     
4   817    158 F     
5   374    178 F     
6  1063    177 M  
> wt<-rep(10,nrow(htsrs))
> dhtsrs<-svydesign(id=~1, weights=~wt,data=htsrs)
> dhtsrs
Independent Sampling design (with replacement)
svydesign(id = ~1, weights = ~wt, data = htsrs)
\end{verbatim}
\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}[fragile]
\begin{verbatim}
> set.seed(9231)
> dhtsrsboot <- as.svrepdesign(dhtsrs, type="subbootstrap",
replicates=1000)
# linearization
> svymean(~height,dhtsrs)
         mean     SE
height 168.94 0.7831
> svymean(~height,dhtsrsboot)
         mean     SE
height 168.94 0.7978
> degf(dhtsrsboot) # 199 = n - 1
[1] 199
> confint(svymean(~height,dhtsrsboot),df=degf(dhtsrsboot))
          2.5 %   97.5 %
height 167.3667 170.5133
\end{verbatim}
An approximate 95\% CI may be constructed using the bootstrap variance as $168.94 \pm 1.971957*0.7978 =
[167.4, 171.5]$
\end{frame}



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
 Distribution of the
$\hat \theta_r^*$s should be like distribution of $\hat \theta$ from
the original sample.
$$\hat F^*(x) - \hat F(x) \rightarrow 0$$
Estimate quantiles of distribution of $\hat \theta - \theta$ using
histrogram of bootstrap values.
%$$1 -\alpha = p\{\hat \theta - q_{1 - \alpha/2} \le \theta \le \hat
%\theta - q_{\alpha/2}\}$$
%
%$q_{1 - \alpha/2}$: estimate from bootstrap of $\hat \theta^* - \hat
%\theta$
$$\hat V_B(\hat \theta) = \ds\frac 1 {R-1} \sum\limits_{r=1}^R \left[
\hat \theta^*_r - \ds\frac 1 R \sum\limits_{r=1}^R \hat
\theta^*_r\right]^2$$ or
$$\hat V_B (\hat \theta) = \ds\frac 1 {R-1} \sum\limits_{r=1}^R
(\hat \theta^*_r - \hat \theta)^2$$

%bootstrap is consistent
%estimator of variance for smooth functions of means for iid samples.
\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
The rescaling bootstrap of Rao and Wu (1988) for a stratified
multistage sample\\
--- $n_h$: number of psus sampled from stratum $h$\\
--- $R$: number of bootstrap replicates to be created.
Typically, $R= 500$ or $1,000$\\
--- bootstrapping is applied within each stratum
\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\bi
\item[a.] For bootstrap replicate $r (r =1,\cdots, R)$, select an SRS of size $n_h -1$ psus with replacement
from the $n_h$ psus in stratum $h$. Do this independently for
each stratum.\\
Let $m_{hj}(r)$ be the number of times psu $j$ of stratum $h$ is
selected in replicate $r$.
\item[b.] Create the replicate weight vector for replicate $r$ as
$$w_i(r) = w_i \times \ds\frac {n_h}{n_h-1} m_{hj} (r)$$ for
observation $i$ in psu $j$ of stratum $h$.\\
 The result is $R$ vectors of replicate weights.
\ei
\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\bi
\item[c.]
Use the vectors of replicate weights to estimate $V(\hat \theta)$.
Let $\hat \theta_r^*$ be the estimator of $\theta$, calculated the
same way as $\hat \theta$ but using weights $w_i(r)$ instead of the
original weights $w_i$. Then,
$$\hat V_B (\hat \theta) = \ds\frac 1 {R-1} \sum\limits_{r=1}^R(\hat
\theta_r^* -\hat \theta)^2$$
\ei
\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}[fragile]

Example 9.10. We use the bootstrap to estimate variances from the data in file
{\it htstrat.csv}, discussed in Example 7.6. The bootstrap weights are constructed by taking 
1000 stratified random samples with replacement from the data set; 
\bi
\item select 159 women ($n_{women} =160$)
\item select 39 men ($n_{men} = 40$) 
\ei
with replacement for each resample.
\begin{verbatim}

\end{verbatim}
\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}[fragile]
\begin{verbatim}
> data(htstrat)
> nrow(htstrat)
[1] 200
> head(htstrat)
     rn height gender
1   201    166 F     
2   965    163 F     
3   490    166 F     
4   249    155 F     
5   260    154 F     
6   324    160 F     
> dhtstrat <- svydesign(id = ~1, strata = ~gender, 
fpc = c(rep(1000,160),rep(1000,40)), data = htstrat)
> dhtstrat
Stratified Independent Sampling design
svydesign(id = ~1, strata = ~gender, 
fpc = c(rep(1000, 160), rep(1000, 40)), data = htstrat)
\end{verbatim}
\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}[fragile]
\begin{verbatim}
> set.seed(982537455)
> dhtstratboot <- as.svrepdesign(dhtstrat,
 type="subbootstrap",replicates=1000)
> svymean(~height,dhtstratboot)
         mean     SE
height 169.02 0.7296
> degf(dhtstratboot)
[1] 198
> confint(svymean(~height,dhtstratboot),
df=degf(dhtstratboot))
          2.5 %   97.5 %
height 167.5769 170.4543
\end{verbatim}
The average height is estimated
as $\bar y_{str} = 169.02$ with bootstrap standard error of $0.7296$; the standard error calculated using
the stratified sampling formula (equation (3.6), page 85), ignoring the fpc, is 0.739.
\end{frame}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
Advantage:
\bi
\item The bootstrap works for smooth functions of
population means and for some nonsmooth functions such as quantiles
in general sampling designs.
\item The
bootstrap is well suited for finding CIs directly.
\ei
Disadvantages:
\bi
\item In some settings, the bootstrap may require
more computations than BRR or jackknife, since $R$ is typically a
very large number.
\item Less theoretical work has been done on properties of the
bootstrap in complex sampling designs.
\ei
\end{frame}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
{\bf Generalized variance functions (GVF)}
\bi
\item In many large government surveys such as the U.S. Current
Population Survey (CPS) or the Canadian Labour Force Survey,
hundreds or thousands of estimates are calculated and published.
\item
The agencies could calculate standard errors for each published
estimators, but that would add greatly to the labor involved in
publishing timely estimates from the surveys.
\ei
\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\bi
\item
In addition, other analysts of the public-use data files may wish to
calculate additional estimates, and the public-use files may not
provide enough information to allow calculation of standard
errors.\\
-----
Want a flexible formula to allow calculating standard errors for
most estimators wanted
\item Generalized variance functions (GVFs) are provided in a number of
surveys to calculate standard errors. They have been used for the
CPS since 1947. Villiant (1987), Wolter (2007, Chapter 7) describes
the theory underlying GVFs.
\ei
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
Given a survey statistics $\hat t_i$, such as the estimated
number of persons employed.
\bi
\item Let $\hat p_i$ be an estimated proportion,
$\hat p_i = \hat t_i /N$, where $N$ is a control total (i.e.
constant), provided by the Census Bureau,
\item Let $d_i$ be the design
effect related to variable $i$. We have
$$
V({\hat p_i}) = d_i\times \ds\frac{p_i(1-p_i)}{n}.
$$
$$V({\hat t_i}) =N^2 d_i\times \ds\frac{p_i(1-p_i)}{n} = a_i t_i^2 +
b_i t_i$$

 where $a_i = -\ds\frac {d_i} n, b_i ={d_i} \ds\frac{N}{n}$. If the deffs are similar for different
estimates, we have $a_i \approx a$ and $b_i \approx b$.
\ei
\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
The general procedure for constructing a generalized variance
function is as follows:
\bi
\item[a.] Using replication or some other method, estimate variances for $k$ population totals
of special interest, $t_1, t_2, \cdots, t_k$ . Let $v_i$ be the
relative variance for $t_i$, for $i=1, 2, \cdots, k$.
\begin{eqnarray*}
v_i &=&V(\hat t_i)/\hat t_i^2\\
&=& \ds\frac{a_i \hat t_i^2 + b_i \hat t_i}{\hat t_i^2}\\
&=& a_i + \ds\frac{b_i}{\hat t_i}
\end{eqnarray*}
\ei
\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\bi
\item[b.] Postulate a regression model relating a set of $v_i$ to
$\hat t_i$ by $$v_i = \alpha + \beta/ {\hat t_i}$$ to find the
parameter estimates $a$ and $b$ for $\alpha$ and $\beta$.

This is a linear regression model with response variable $v_i$ and
explanatory variable $1/\hat t_i$. Valliant (1987) found that this
model produces consistent estimators of the variances for the class
of superpopulation models he studied.
\ei
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\bi
\item[c.] Use regression techniques to estimate $\alpha$ and $\beta$ by $a$ and $b$.
Valliant (1987) suggests using weighted least squares to estimate
the parameters, giving higher weight to items with small $v_i$.

\item[d.] The GVF estimate of relative variance is predicted from the
regression function $a + b/\hat t_i$ or
$$\hat V({\hat t_{\mbox{new}}}) = a \hat t_{\mbox{new}}^2 +
b \hat t_{\mbox{new}}$$
\ei
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
Advantages of Generalized Variance Functions (GVFs):
\bi
\item The GVF may be used when insufficient information
is provided in the public-use data files to allow direct calculation
of standard errors. The data collector can calculate the GVF, and
often has more information for estimating variances than is released
to the public.
\item Valliant (1987) found that if design effects
for the $k$ estimated totals are similar, the GVF variances are
often more stable than the direct estimates of variance, as they
smooth out some of the fluctuations from item to item.
\item A GVF saves a great deal of time and speeds
production of annual reports. It is also useful for designing
similar surveys in the future.
\ei
\end{frame}

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

Disadvantages of Generalized Variance Functions (GVFs)
\bi
\item If the design effects (deffs) vary significantly across different responses, the simple GVF model may produce inaccurate variance estimates for some responses.

\item Caution is needed when applying GVFs to subpopulations, especially if the subpopulation exhibits an unusually high level of clustering (and thus a high design effect). In such cases, the GVF estimate of variance may be substantially underestimated because it does not account for the higher clustering.

\item GVFs may not perform well for variables not included in the original calculation of regression parameters. Such estimates may not align with the assumptions of the GVF model, leading to unreliable results.
\ei
\end{frame}

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

\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.
\end{frame}


\end{document}











