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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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    \vspace{-1cm} % Add space between the title page and chapter information
    \centering
    \textbf{\Large Chapter 1: Introduction} % Emphasize chapter title

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\section{Chapter 1}
\begin{frame}
{\Large Three Elements of Statistical Study}
\bi
\item Collecting Data: observational data, experimental data, survey data
\item Describing and Presenting Data: graphical and numerical
description, point estimation and interval estimation
\item Drawing Conclusions from Data
\ei
\end{frame}

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\begin{frame}
    {\Large Statistical Inference}
    \begin{itemize}
    \item We have limited visibility into the population, but we are interested in understanding the typical characteristics of the population.
    \begin{itemize}
    \item Population parameter: A numerical value that represents a characteristic of the entire population, such as the population mean $\bar{y}_U$ or population total $t$.
    \end{itemize}
    \item The only information we currently possess is based on the sample.
    \begin{itemize}
    \item Statistics derived from samples are used to estimate population parameters.
    \end{itemize}
    \end{itemize}
    \end{frame}
    

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\begin{frame}
\textbf{Goal}: Our objective is to utilize the information obtained from the sample to draw inferences about the population and its parameters.
     \begin{figure}[!htp]
       \centering
       \includegraphics[width=0.70\textwidth,
       height=0.5\textheight]{./plots/view.png}
\caption{Population, sample and statistical inference}
       \label{view}
     \end{figure}
    \end{frame}

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\begin{frame}
    {\bf Survey Sampling}
    We aim to utilize sample information to make inferences about a finite population.
    \begin{itemize}
    \item The remaining statistics, $Y_1, Y_2, \cdots, Y_{n}$, are random variables following a certain distribution, such as the normal distribution $N(u,\sigma^2)$. The observed values of these random variables are $y_1, y_2, \cdots, y_n$.
    \item In general probability sampling (design-based analysis), $Y_1, Y_2, \cdots, Y_{N}$ represent the population. We select a sample of $n$ units, denoted as $y_1, y_2, \cdots,y_n$, based on a predetermined design that assigns a probability of selection to each possible subset of the population with size $n$. Neither $Y_1, Y_2, \cdots, Y_{N}$ nor $y_1, y_2, \cdots, y_n$ are random variables. The random variables are $Z_i$'s, defined as:
    \end{itemize}
    $$Z_i = \left\{
    \begin{array}{cc}
    1 & \text{if unit}\quad i\in S \\
    0 & \text{otherwise}
    \end{array}\right.$$
    \end{frame}
    
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\begin{frame}
    \textbf{Some definitions:}
    
    \begin{itemize}
    \item Observation Units: Objects on which measurements are taken, sometimes referred to as elements.
    \item Target Population: The complete collection of observations that we aim to study.\\
    --- Defining the target population is an important and often challenging aspect of the study.\\
    --- For instance, in a political poll, should the target population encompass all eligible adults to vote, all registered voters, or all individuals who participated in the last election? The choice of target population significantly influences the resulting statistics.
    \item Sampled Population: The population from which the sample was drawn.\\
    \textbf{Note}: In an ideal survey, the sampled population would be identical to the target population, although this ideal scenario is rarely achieved precisely.
    \end{itemize}
    
    \end{frame}

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\begin{frame}
    \begin{itemize}
    \item Sample: A subset of a population.
    \item Sampling unit: The unit that is actually sampled.\\
    Example: When studying individuals and lacking a comprehensive list of all individuals in the target population, households may serve as the sampling units, with individuals living in those households as the observation units.
    \item Sampling frame: The list of sampling units.\\
    Example: In telephone surveys, the sampling frame could be a list of all residential telephone numbers in the city. In personal interviews, the sampling frame might consist of a list of all street addresses.
    \item Census: When data is collected for every unit of the population, it is referred to as a census.
    \end{itemize}
    \end{frame}

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\begin{frame}
    Example: Telephone survey of likely voters\\
    Target population: All likely voters\\
    Sampling frame: A list of telephone numbers\\
    --- (1) Not all households have telephones\\
    --- (2) Some individuals with phones are not registered to vote and are therefore ineligible for the survey\\
    --- (3) Certain eligible individuals with phones may be unreachable, refuse to respond, or incapable of responding\\
    Sampling unit: A phone number\\
    Observation unit: An individual associated with the phone number
    \end{frame}


        
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\begin{frame}
    \begin{figure}[!htp]
        \centering
        \includegraphics[width=0.9\textwidth]{./plots/fig0101.pdf}
        \caption{Telephone survey of likely voters}
    \end{figure}
    \vspace{-0.3cm} % Adjust spacing as needed
    \begin{center}
        \textit{\scriptsize Source: Figure 1.1 of Sampling: Design and Analysis, 3rd edition, by Sharon L. Lohr}
    \end{center}
\end{frame}



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\begin{frame}
{\bf Why Sampling?}
\begin{itemize}
    \item Cost: A census is expensive.
    \item Time: Conducting a census is very time-consuming.
    \item Impractical: In some applications, a census can be impractical.\\
    Example: The government requires automakers who want to sell cars in the U.S. to demonstrate that their cars can survive certain crash tests. Obviously, the company can't be expected to crash every car to see if it survives! So the company crashes only a sample of cars.
    \end{itemize}
    \end{frame}
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\begin{frame}
    {\bf Types of Samples:}
    
    1. Non-probability (non-random) samples: These samples focus on volunteers, easily available units, or those that happen to be present when the research is conducted. Non-probability samples are useful for quick and cost-effective studies, case studies, qualitative research, pilot studies, and developing hypotheses for future research. However, non-probability samples often introduce bias.
    \begin{itemize}
    \item Convenience sample: Also known as an ``accidental" sample or ``man-in-the-street" sample. The researcher selects units that are convenient, easily accessible, or nearby.
    \newpage
    \item Purposive sample: The researcher selects units with a specific purpose in mind, such as students who live in dorms on campus or females.
    \item Quota sample: The researcher constructs quotas for different types of units. For example, interviewing a fixed number of shoppers at a mall, with half being male and half being female.
    \end{itemize}
    \end{frame}
    
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\begin{frame}
    2. Probability-based (random) samples: These samples are based on probability theory. Every unit of the population of interest must be identified, and all units must have a known, non-zero chance of being selected into the sample.
    \begin{itemize}
    \item Simple random sample (SRS): Randomly select a sample of size $n$ from a population of size $N$. 
    \begin{itemize}
    \item[a)] The sampling unit and observation unit are the same.
    \item[b)] Each subset of size $n$ has an equal probability of being selected as the sample.
    \item[c)] Each unit has an equal chance of being selected in the sample.
    \end{itemize}
    --- Random number generators\\
    --- Lottery method
    \end{itemize}
    \end{frame}
    
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\begin{frame}
    \begin{itemize}
    \item Systematic random sampling: Randomly select the first item or subject from the population and then select every $n$th subject from the list. 
    \begin{itemize}
    \item The results are representative of the population unless certain characteristics of the population are repeated for every $n$th individual, which is highly unlikely.
    \item Systematic sampling is useful for selecting large samples, such as $100$ or more. It is less cumbersome than a simple random sample using either a table of random numbers or the lottery method.
    \item If the selection interval matches some pattern in the list (e.g., the list is male, female, male, female, $\cdots$), and you select the No.1, No.3, No.5, $\cdots$ observations to form a systematic sample, you will introduce systematic bias into your sample.
    \end{itemize}
    \end{itemize}
    \end{frame}
    

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\begin{frame}

\bi
\item Stratified random sampling: Divide population into $H$ strata,
take an SRS of size $n_h$ from stratum $h$, $h= 1, \cdots, H$,
select the sample independently.\\
Example: You want to find out the attitudes of students on your
campus about immigration.\\
----- 27,000 students: 22,000 West; 3,000 East; 1000 Midwest; 600 South; 400
Foreign.\\
------Select a simple random sample of 1500 students, you might not
get any from the Midwest, South, or Foreign. \\
------Divide the students into these five groups (Stratum), and then select the same percentage
of students from each group using a simple random sampling method.
This is proportional stratified random sampling.\\
------Divide students into the five groups and then select the same number of
students from each group using a simple random sampling method. This
is disproportionate stratified random sampling.
\ei

\end{frame}
    
    
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\begin{frame}
\bi
\item Cluster sampling: A cluster is a naturally-occurring grouping
of the members of the population. For example, city residents are
also residents of neighborhoods, blocks, and housing structures.


Randomly select $n$ clusters, then observe all the elements in the
selected clusters or partial of the elements in the selected
clusters.
\ei
\end{frame}

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\begin{frame}

Example: To obtain information about the drug habits of all high
school students in New Mexico.\\
----Obtain a list of all the high schools in NM\\
----Select an SRS of high schools\\
----Within each selected high school, list all classes,
and select an SRS of classes\\
----The students in the selected classes are the observations in
your sample
\end{frame}

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\begin{frame}
{\bf Biases}
\bi
\item Selection Bias: If some part of the target population
is not in the sampled population, a bias called Selection Bias
occurs. \\
Examples:\\
---- In a survey to estimate per capita income, if
transient people are ignored.\\
---Mis-specification of the target population\\
----Failure to include all the target population in the sampling frame,
also called undercoverage\\
----Substituting a convenient member of a population for a designated
member not readily available
\ei
\end{frame}

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\begin{frame}

----Non Response: failure to obtain responses from those chosen in
the sample \\
----Allowing a sample to consist entirely of volunteers (Radio, TV, or
call-in polls) \\
Note that large samples are generally considered
good but if the sample is unrepresentative, it can be quite bad. The
design of the survey is far more important than the absolute size of
the sample.

\end{frame}

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\begin{frame}
\bi
\item Measurement Bias: Measurement bias occurs when the measuring
instrument has a tendency to record in one direction more often than
the other. Measurement biases are more common when dealing with people.\\
----People may not tell the truth \\
----Lack of understanding of questions\\
----Lack of proper account of events in memory\\
----Variations in responses due to interviewer\\
----Misreading questions, or miss recording responses\\
----Desire to impress the interviewer\\
----Ordering and wording of questions have effects on responses\\
 Many
of these problems can be avoided by proper questionnaire design 
\ei
\end{frame}

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\begin{frame}
{\bf Questionnaire Design:}
\bi
\item Decide what you want to find out; this is the most important step in writing a questionnaire
\item Pilot study: Test questions before sending out the questionnaire.
\item Keep the questions {\it Simple} and {\it Clear}: Questions should be neither too
lengthy nor too technical. They should be easily understood by non
experts
\item Questions should be specific and not general
\item Decide whether to use open or closed questions.\\
----Open Question: The respondent is not prompted with categories for
responses. It allows responses to form their own response
categories.\\
-----Closed Question: A question is closed when specific response
categories are provided. \\
----- Closed questions with well thought and researched categories
elicit more accurate responses.
\ei
\end{frame}

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\begin{frame}
\bi
\item Avoid questions that prompt or motivate the
respondent to say what investigator wants to hear
\item Use choices rather than Agree/Disagree type questions
\item Ask only one concept in one question
\item Pay attention to question-order effect. Ask general
questions first then follow with specific questions. 
\ei

\end{frame}

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\begin{frame}
{\bf Sampling and Non-Sampling Errors:}
\bi
\item Sampling Errors: Sampling errors are results of inherent variability in the sampling process.
These arise because the results vary from sample to sample. Margin
of errors reported are a result of sampling error. These can only be
reduced by increasing the sample size but not be eliminated.
\item Non-Sampling Errors: These result from selection bias, measurement
error and inaccuracies of responses. These can not be attributed to
sample-to-sample variability. Such errors can be eliminated by
proper precautions. Selection bias can be reduced by using
probability sample. Accurate responses can be achieved through
proper and careful design of survey instrument and training of
interviewers.
\ei

\end{frame}

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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.
\end{frame}

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