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% Title, subtitle, and other information
\title{Stat472/572 Sampling: Theory and Practice}

\author[Yan Lu]{Instructor: Yan Lu}
\institute[UNM]{University of New Mexico}
\date{}

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% Title slide
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    \vspace{-1cm} % Add space between the title page and chapter information
    \centering
    \textbf{\Large Chapter 8: Nonresponse} % Emphasize chapter title

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\end{frame}
\section{Chapter 8}

\begin{frame}
{\bf Nonresponse}
\bi
\item Unit nonresponse: the entire observation unit is missing
\item Item nonresponse: some measurements are present for the
observation unit but at least one item is missing
\ei
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
Example: select 10,000 households for sample,

1000 ineligible for  the survey\\
700 no contact\\
200 unresolved\\
1400 refusals\\
\begin{eqnarray*}
\mbox{Nonresponse rate}& =& \ds\frac{\mbox{nonrespondents +
unresolved}}{N -\mbox{ineligible}}\\
&=& \ds\frac{1400+700+200}{9000}\\
&=&.25
\end{eqnarray*}
\end{frame}
%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
{\bf Four approaches to dealing with nonresponse:}
\bi
\item[1.] Prevent it. Design the survey so that nonresponse is low. This
is by far the best method.
\item[2.] Take a representative subsample
of the nonrespondents; use that subsample to make inferences about
the other nonrespondents.\\
----Call backs and follow ups\\
Take a subsample of nonrespondents for interviewing\\
----American community survey\\
subsample about 1/3 of households that did not respond to mail
survey
\ei
\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\bi
\item[3.] Use a model to predict values
for the nonrespondents. \\
----Weighting class adjustment methods implicitly use a model to
adjust for unit nonresponse.\\
---- Imputation often adjusts for item nonresponse,\\
----Parametric models may be used for either type of nonresponse.
\item[4.] Ignore the nonresponse (not recommended, but unfortunately common
in practice).
\ei
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
Main problem caused by nonresponse: {\bf potential bias}
\begin{table}
\begin{tabular}{ccccc}
Stratum& Size & Total &Mean & Variance\\
\hline
Respondents& $N_R$& $t_R$& $\bar y_{RU}$ & $S_R^2$ \\
Nonrespondents& $N_M$ &$t_M$ & $\bar y_{MU}$ & $S_M^2$\\
\hline
Entire Popn& $N$ & $t$ & $\bar y_U$& $S^2$\\
\end{tabular}
\end{table}
\bi
\item A probability sample from the population will likely contain
some respondents and some nonrespondents.
\item If the population
mean in the nonrespondent stratum differs from that in the
respondent stratum, estimating the population mean using only the
respondents will produce bias.
\ei
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\bi
\item Let $\bar y_R$ be an approximately unbiased
estimator of the mean in the respondent stratum, using only the
respondents.
$$\bar y_U = \ds\frac{N_R}{N} \bar y_{RU} + \ds\frac {N_M}{N} \bar
y_{MU}$$ The bias is approximately
$$E(\bar y_R) - \bar y_U \approx \ds\frac{N_M}{N}(\bar y_{RU} - \bar
y_{MU})$$
\item The bias is small if\\
---- either (1) the mean for the nonrespondents is close to the mean
for the respondents, or\\
---- (2) $N_M/N$ is small, there is little nonresponse.\\
--- But we can never be assured of (1), as we generally have no data
for the nonrespondents. Minimizing the nonresponse rate is the only
sure way to control nonresponse bias.
\ei
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
Designing Surveys to Reduce Nonsampling Errors
\begin{figure}[!htp]
       \centering
%\vspace{-2.2in}
       \includegraphics[width=0.90\textwidth,
       height=1.5\textheight]{./plots/p316.pdf}
       \label{nonresponse}
\caption{Factors affecting nonresponse}
     \end{figure}

\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
{\bf Two phase sampling (double sampling)}
\bi
\item Population
divided into two strata, respondents and initial nonrespondents
(persons who did not respond to the first call)
\item Sample size $n$
\item $n_R$ respondents
\item $n_M$ nonrespondents (missing)
\item Subsample $100v$\% of the $n_M$ nonrespondents in the sample\\
----the subsampling fraction $v$ does not depend on the data
collected\\
----suppose that through some superhuman effort all of the targeted nonrespondents
are reached
\ei
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}

{\bf The two phase sampling estimators of the population mean and total
are}
$$\hat{\bar y} = \ds\frac {n_R}{n} {\bar y}_R +\ds\frac {n_M}{n}
{\bar y}_M$$ and
$$\hat t = N \hat {\bar y} = \ds\frac N n \sum\limits_{i\in S_R}y_i
+ \ds\frac N n\cdot \ds\frac 1 v \sum\limits_{i\in S_M}y_i$$

$$E(\hat t) = t$$
\begin{eqnarray*}
\hat V(\hat{\bar y})& =& \ds\frac{n_R-1}{n-1}\cdot
\ds\frac{s_R^2}{n} +
\ds\frac{n_M-1}{n-1}\cdot \ds\frac{s_M^2}{vn}\\
& +& \ds\frac 1{n-1} \left[\ds\frac{n_R}n (\bar y_R -\hat{\bar y})^2
+\ds\frac{n_M}n (\bar y_M -\hat {\bar y})^2\right]
\end{eqnarray*}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\begin{frame}

{\bf Notes on Nonresponse and Two-Phase Sampling}

\bi
\item It is rare to achieve a 100\% response rate in the subsample of nonrespondents.
\item A common approach is to compare the subsample of nonrespondents to the original respondents to assess potential bias.
\item If everyone in the subsample responds, two-phase sampling can:
    \bi
    \item Remove the bias introduced by the original nonresponse.
    \item Properly account for nonresponse in the estimation of variance.
    \ei

\ei

\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
{\bf Mechanisms for Nonresponse}

{\small
Define:
$$Z_i = \left\{
\begin{array}{ll}
1 & \text{if unit } i \text{ is selected in the sample (included in } S\text{)} \\
0 & \text{otherwise}
\end{array}\right.
$$
$$
R_i = \left\{
\begin{array}{ll}
1 & \text{if unit } i \text{ responds} \\
0 & \text{otherwise}
\end{array}\right.
$$

Probabilities:
\bi
\item $P(Z_i = 1) = \pi_i$, the inclusion probability.
\item $P(R_i = 1) = \phi_i$, the response propensity (propensity score).
\ei

Components:
\bi
\item $y_i$: Response of interest.
\item $\bx_i$: Vector of auxiliary information known about unit $i$ in the sample.
\item If $Z_i$ (selection) and $R_i$ (response) are independent, then:
$$P(\text{unit } i \text{ is selected and responds}) = \pi_i \phi_i.$$

\ei
}
\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\bi
\item Estimate the population total $t_y = \sum_{i=1}^N y_i$ using an adjusted Horvitz–Thompson estimator:
$$\hat{t}_{\phi} = \sum_{i=1}^N Z_i R_i \ds\frac{y_i}{\pi_i \phi_i}.$$
\ei
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{frame}
{\bf Three Types of Missing Data (Little and Rubin, 1987)}

\bi
\item[1.] \textbf{Missing Completely at Random (MCAR)}: 
The probability of responding, $\phi_i$, does not depend on $\bx_i$, $y_i$, or the survey design.
\item Examples:
    \bi
    \item Laboratory error: Someone drops a test tube containing a participant’s blood sample.
    \item Survey example: A survey of student satisfaction at a university:
        \bi
        \item \textbf{X-variables}: Known for all students (e.g., race, gender, age, courses, GPA, etc.).
        \item \textbf{Y-variable}: Students' satisfaction with the university.
        \item \textbf{MCAR Condition}: Nonresponse is completely unrelated to any X-variable or the Y-variable.
        \ei
    \ei
\ei

\vspace{1em}
\textbf{Key Characteristic}: Data are missing for reasons completely unrelated to the observed or unobserved variables.
\end{frame}


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

{\bf MCAR: Key Implications}

\bi
\item Nonresponse is random:
    Nonrespondents are randomly distributed across the sampled units.
\item Respondents are representative:
    The respondents provide an unbiased representation of the entire sample.
\item Nonresponse can be ignored:
    Standard analyses can be performed without additional adjustments for nonresponse.
\item \textbf{Limitations:} 
    \bi
    \item True MCAR is rare in practice. 
    \item If MCAR does not hold, analyses assuming MCAR may lead to bias.
    \ei
\ei

\vspace{1em}
\textbf{Practical Takeaway}: Use MCAR assumptions only when there is strong evidence that missingness is unrelated to both observed and unobserved variables.
\end{frame}

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

\begin{frame}
\bi
\item[2.] \textbf{Missing at Random (MAR):}  
The probability of responding, $\phi_i$, depends on the observed variables $\bx_i$, but not on the unobserved variable $y_i$.  

\item \textbf{Key Characteristics:}  
\bi
\item Nonresponse is related to observed data but unrelated to the missing values themselves, conditional on $\bx_i$.
\item Unlike MCAR, adjustments should be made to account for nonresponse using the observed data $\bx_i$.
\ei

\item \textbf{Example:}  
\bi
\item In the National Crime Victimization Survey (NCVS), suppose:  
\bi
    \item The probability of responding to the survey depends on \textit{observed factors} such as race, sex, and age (all known quantities).  
    \item However, within each race/sex/age group, the probability of response does not depend on whether the individual experienced victimization (unobserved).  
\ei
\ei

\ei
\end{frame}


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

\begin{frame}

{\bf MCAR and MAR: Ignorable Nonresponse}
\bi
\item Ignorable nonresponse means that the nonresponse mechanism can be explained and accounted for using a suitable model.
\item Once the model accounts for the nonresponse, its effects can be ``ignored" in the analysis.
\item {\bf Important:} ``Ignorable" does not mean that the nonresponse can be completely disregarded; adjustments or modeling are still necessary to address potential bias.
\ei
\end{frame}

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

\bi
\item[3.] {\bf Nonignorable Nonresponse (Not Missing at Random - NMAR)}: nonresponse is considered {\bf nonignorable} when the probability of responding ($\phi_i$) is directly related to the outcome variable ($y_i$), even after accounting for known variables ($\bx_i$).
\item Example: In the NCVS (National Crime Victimization Survey), it is suspected that individuals who have been victims of crime are less likely to respond to the survey than nonvictims, regardless of shared characteristics such as race, age, or sex.
\item {\bf Key Challenge:} Nonresponse bias cannot be corrected using observed data alone; it requires additional assumptions or external information about the relationship between nonresponse and $y_i$.
\ei
\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
{\bf Methods to Address Nonresponse}

In survey sampling methodology, two primary methods are commonly used to address nonresponse:  
\bi
\item {\bf Weighting:}  
    \begin{itemize}
        \item Adjusts the weights of responding units to compensate for nonrespondents.  
        \item Example: If younger participants are less likely to respond, their responses are given higher weights to represent their population segment adequately.
    \end{itemize}
\item {\bf Imputation:}  
    \begin{itemize}
        \item Fills in missing values using observed data or predictive models.  
        \item Example: Estimate missing income values based on variables like age, education, and occupation.
    \end{itemize}
\ei
{\bf Goal:} Reduce bias introduced by nonresponse and improve the reliability of survey estimates.  
\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
{\bf Weighting class method}
\bi
\item Divide sample into classes based on variables known for everyone
\item Assume MCAR within those classes i.e. assume that nonrespondents are
similar to respondents within each weighting class
\ei
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
Example:
\begin{tabular}{ccc}
& Male & Female\\
\hline
R & 10,000 & 12,000\\
NR & 8000 & 5000\\
\hline Sample & 18,000 & 17,000
\end{tabular}
\bi
\item Started one with $w_i =100$ for everyone
\item New weight $w_i^*$ for each male respondent
$$w_i^* = w_i \times \ds\frac{18,000}{10,000}= 180$$
each male respondent represents 180 men in population
\ei
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\bi
\item Females
\[
w_i^* = w_i \times \ds\frac{17,000}{12,000} = 141.7
\]
---- Each female respondent now represents 142 females in the population. \\
---- Assumption: The distribution of $y_i$ (the variable of interest) is similar for both responding and nonresponding women.

\item {\bf General Formula for Adjusted Weights:}
\[
w_i^* = w_i \times \ds\frac{\text{Sum of weights for the selected sample in class}}{\text{Sum of weights for respondents in class}}
\]
---- Adjust weights within each response class (e.g., gender, age group) to account for nonresponse.
\ei

\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
Weighting-class adjustment
\bi
\item assumption: respondents and nonrespondents in the same
weighting-adjustment class are similar
\item weights of respondents
in the weighting-adjustment class are increased so that the
respondents represent the nonrespondents share of the population as
well as their own
\ei
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
Estimate response propensity score for class $c$ by
$$\hat \phi_c = \ds\frac{\mbox{sum of weights for respondents in
class} \quad c}{\mbox{sum of weights for selected sample in
class}\quad c}$$
$$w_i^* = \sum\limits_{c} \ds\frac{w_i x_{ci}}{\hat \phi_c},$$ where
$x_{ci} =1$ if unit $i$ is in class $c$

$$\hat t_{wc} = \sum\limits_{i\in S} w_i^* y_i$$
$$\hat {\bar y}_{wc} = \ds\frac{\hat t_{wc}}{\sum\limits_{i\in S}
w_i^*}$$
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
In an SRS,
$$\hat\phi_{c} =\ds\frac{n_{cR}}{n_c}$$

\begin{eqnarray*}
\hat t_{wc} &= &\sum\limits_{i\in S} \sum\limits_{c} \ds\frac N n
\ds\frac{n_c}{n_{cR}} x_{ci} y_i\\
&=& N\sum\limits_{c}\ds\frac{n_c}n \bar y_{cR}
\end{eqnarray*}
$$\hat{\bar y}_{wc} = \sum\limits_{c} \ds\frac{n_c}n \bar y_{cR}$$
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
{\bf Poststratification}
\bi
\item Similar to weighting class adjustment, except that population
counts are used to adjust the weights
\item SRS sample is collected
\item Units are grouped into $H$ different post strata based on
demographic variables such as race and sex
\item $N_H$ units in poststratum, $n_h$ units were selected for the sample and $n_{hR}$
respond
\ei
\end{frame}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
Assumptions:
\bi
\item within each stratum each unit selected to be in the sample has the same probability of being a
respondent
\item the response or nonresponse of a unit is independent of the responses or nonresponses of all other units
\item nonrespondents in a poststratum are like the respondents\\
Data are MCAR within each poststratum
\ei
\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
The poststratified estimator for $\bar y_U$ is
\begin{equation}\label{eq:est1}
\bar y_{post} = \sum\limits_{h=1}^H \ds\frac{N_h} N \bar y_{hR}
\end{equation}
The weighting-class estimator for $\bar y_U$, if the weighting
classes are the poststrata, is
\begin{equation}\label{eq:est2}
\bar y_{wc} = \sum\limits_{h=1}^H \ds\frac{n_h} n \bar y_{hR}
\end{equation}
\bi
\item Estimate (\ref{eq:est1}) and estimate (\ref{eq:est2}) are similar in form
\item Difference is that in poststratification the $N_h$ are known,
whereas in weighting-class adjustments the $N_h$ are unknown and
estimated by $N* n_h/n$
\ei
\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
For the poststratified estimator, often the conditional
variance given $n_{hR}$ is used. For an SRS,
$$V(\bar y_{post}|n_{hR},h=1,\cdots,H) = \sum\limits_{h=1}^H
\left(\ds\frac{N_h}{N}\right)^2
\left(1-\ds\frac{n_{hR}}{N_h}\right)\left(\ds\frac{S_h^2}{n_{hR}}\right)$$
\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\textbf{Poststratification Using Weights}\\
Let
$$x_{hi} = \left\{
\begin{array}{cc}
1 & \mbox{if unit } i \mbox{ is a respondent in poststratum } h \\
0 & \mbox{otherwise}
\end{array}\right.$$

\bi
\item The weight for each respondent adjusts for unequal selection probabilities and nonresponse within the poststratum.
\item The sum $\sum_{i \in S_h} w_i \cdot x_{hi} = \sum\limits_{i\in S_h} w_i$ estimates the population count $N_h$ for that subgroup of respondents.
\ei
\end{frame}


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

Poststratification uses a \textbf{ratio estimator within each subgroup} to adjust the sample estimates according to the known true population count for each subgroup.
$$w_i^* = \sum\limits_{h=1}^H w_i x_{hi}
\ds\frac{N_h}{\sum\limits_{j\in R}w_j x_{hj}} = w_i
\ds\frac{N_h}{\sum\limits_{h=1}^H \sum\limits_{j\in R} w_j x_{hj}}$$

\begin{eqnarray*}
\sum\limits_{h=1}^H \sum\limits_{i\in S} w_i^* x_{hi}& =&
\sum\limits_{h=1}^H \sum\limits_{i\in S} w_i
\ds\frac{N_h}{\sum\limits_{h=1}^H
\sum\limits_{j\in R} w_j x_{hj}} x_{hi}\\
&=& \sum\limits_{h=1}^H \sum\limits_{i\in S}w_i x_{hi}
\ds\frac{N_h}{\sum\limits_{h=1}^H \sum\limits_{j\in
R} w_j x_{hj}}\\
&=& N_h
\end{eqnarray*}
\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
$$\hat t_{post} = \sum\limits_{h=1}^H \sum\limits_{i\in S} w_i^* x_{hi} y_i$$
$$\hat{\bar y}_{post}= \sum\limits_{h=1}^H \sum\limits_{i\in S} w_i^*
y_i/N = \sum\limits_{h=1}^H \ds\frac{N_h} N \bar y_{hR}$$ 

\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
{\bf Raking adjustment}

A poststratification method that can be used when poststrata are
formed using more than one variable, but only the marginal
population totals are known

Assumption
\bi
\item within each poststratum, each unit selected to be in the
sample has the same probability of being a respondent
\item the response or nonresponse of a unit is independent of the
behavior of all other units
\item nonrespondents in a poststratum are like the respondents
\item response probabilities depend only on the row and column and
not on the particular cell
\ei
\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
Example: have population information on two variables, but don't
have cell counts information

\begin{tabular}{c|cc|c}
& M & F & \\
\hline
Assistant professor & & &80\\
Associate professor & & & 180\\
Full professor & & & 540\\
\hline & 600 & 200 & 800
\end{tabular}

Sum of weights from survey\\
\begin{tabular}{c|cc|c}
& M &F & \\
\hline
Assistant professor & 40 & 60 & 100\\
Associate professor & 80 & 60 & 140\\
Full professor & 320 &40 &360\\
\hline & 440 & 160 &600
\end{tabular}

Iteration 1 (adjust columns)\\
Multiply weight of males by $600/440$, females by $200/160$
\end{frame}



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\begin{tabular}{c|cc|c}
& M &F &\\
\hline
Assistant professor &54.5 & 75 & 129.5\\
Associate professor & 109.1 & 75 & 184.1\\
Full professor & 436.4 & 50 & 486.4 \\
\hline & 600 &200 & 800
\end{tabular}

Iteration 2 (adjust rows)\\
multiply weight of assistant professor by $80/129.5$\\
multiply weight of associate professor by $180/184.1$\\
 multiply weight of full professor by $540/486.4$\\
\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
 \begin{tabular}{c|cc|c}
  & M &F &\\
  \hline
 Assistant professor & 33.7 & 46.3 & 80\\
 Associate professor & 106.7 & 73.3 & 180\\
 Full professor & 484.5 & 55.5 & 540\\
 \hline
 & 624.9 & 175.1 & 800
 \end{tabular}
\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
 Iterate until the marginal counts are equal to the marginal counts of the population (the first table) \\
 \begin{tabular}{c|cc|c}
 & M &F &\\
 \hline
 Assistant professor & 29.6 & 50.4 & 80\\
 Associate professor & 97.3 & 82.7 & 180\\
 Full professor & 473 & 67 & 540\\
 \hline
 & 600 & 200 & 800
 \end{tabular}

Note:
\bi
 \item if some of the cell estimates are zero, may not converge
 \item overadjustment: if there is little relation between the extra dimension in raking and
 the cell means, raking may increase the variance
\ei
\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
{\bf Some Comments on Weighting Adjustments}
\bi
\item Weighting class adjustments and poststratification can help reduce nonresponse bias.\\
\item In each weighting cell or poststratum, respondents and nonrespondents are assumed to be similar.\\
---- Specifically, each individual in a weighting class is assumed to have an equal likelihood of responding to the survey or a response propensity uncorrelated with $y$.\\
\item Some practitioners treat the weighting adjustment as a complete remedy for nonresponse and proceed as though nonresponse no longer exists.\\
---- This approach should be avoided, as it overlooks potential biases.
\ei
\end{frame}


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

\bi
\item Weighting can improve many estimates but rarely eliminates all nonresponse bias.\\
\item When weighting adjustments are applied, practitioners should explicitly state the assumed response model and provide evidence to support it.\\
\item Weighting adjustments are typically used for unit nonresponse and not for item nonresponse, which requires different weights for each item.\\
\item Poststratification is a special case of calibration methods used in survey sampling.
\ei
\end{frame}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
{\bf Item Missing}

Missing survey items can occur for various reasons:
\bi
\item The interviewer may forget to ask a question.
\item The respondent may refuse to answer or may not know the information.
\item A data entry clerk might inadvertently skip the value.
\item Sometimes, responses are marked as missing during data editing or cleaning:
\bi
\item For example, if discrepancies arise—such as a 3-year-old reported to have voted in the last election—both values (age and voting status) may be set to missing.
\ei
\ei

\end{frame}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
{\bf Imputation: Filling in Missing Items}

Imputation is a common technique to address missing survey data:
\bi
\item A replacement value is assigned to the missing item, often based on data from another respondent who is similar to the nonrespondent on other variables.
\item To maintain transparency, an additional variable should be created in the data set to indicate whether a response was measured or imputed.
\item Imputation helps:
\bi
\item Reduce nonresponse bias.
\item Create a ``clean," rectangular data set without gaps for missing values.
\ei
\ei
\end{frame}

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

\begin{frame}[fragile]

{\bf Example}
\begin{verbatim}
#data impute 
     person age gender education crime violcrime
 1      1    47 M             16     0         0
 2      2    45 F             NA     1         1
 3      3    19 M             11     0         0
 4      4    21 F             NA     1         1
 5      5    24 M             12     1         1
 6      6    41 F             NA     0         0
 7      7    36 M             20     1        NA
 8      8    50 M             12     0         0
 9      9    53 F             13     0        NA
10     10    17 M             10    NA        NA

\end{verbatim}
\end{frame}

\begin{frame}[fragile]
\begin{verbatim}
    person age gender education crime violcrime
11     11    53 F             12     0         0
12     12    21 F             12     0         0
13     13    18 F             11     1        NA
14     14    34 M             16     1         0
15     15    44 M             14     0         0
16     16    45 M             11     0         0
17     17    54 F             14     0         0
18     18    55 F             10     0         0
19     19    29 F             12    NA         0
20     20    32 F             10     0         0
\end{verbatim}
\end{frame}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{frame}
{\bf Deductive Imputation}

Deductive imputation uses logical relationships between variables to fill in missing values:
\bi
\item Values are imputed during data editing based on known information. 
\item Example 1: A respondent person 9 in the ``impute'' data is missing the response for being a victim of violent crime, but she reported not being a victim of any crime. The logical imputed value for violent crime would be 0.
\item Example 2: In a longitudinal survey, a woman reports having two children in year 1 and year 3, but the value is missing for year 2. The logical imputed value for year 2 would be two.
\item Deductive imputation is particularly useful in scenarios where strong logical or temporal relationships exist among variables.
\ei
\end{frame}

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\begin{frame}
{\bf Cell Mean Imputation}

\bi
\item Respondents are grouped into classes (cells) based on known variables.
\item The mean value of the responding units in each cell, $\bar y_{cR}$, is used to replace missing values within that cell.
\item Assumes missing items are missing completely at random (MCAR) within each cell.\\

\textbf{Example:} \\
- The mean annual income ($y$) for black males aged 19–24 is \$19,242.\\
- Impute \$19,242 for every black male aged 19–24 who has a missing income value.\\

\item \textbf{Limitations:}
- Can create artificial spikes in the data (e.g., in histograms).
- Distorts multivariate relationships by ignoring individual variability within cells.
\ei
\end{frame}

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\begin{frame}

{\bf  Example 8.9, page 336}

\begin{table}
\caption{The four cells are constructed using 
variables age and sex.}
\begin{tabular}{c|c|c}
& $\le 34$ &$\ge 35$ \\
\hline
M&Persons & Persons\\
&3,5,10,14 &1,7,8,15,16 \\
\hline
F& Persons&Persons\\
&4,12,13,19,20 & 2,6,9,11,17,18\\
\hline
\end{tabular}
\end{table}
\bi
\item Persons 2 and 6, who have missing values for years of education, would be assigned the average value of 12.25, calculated from the four women aged 35 or older who responded to this question.
\ei
\end{frame}

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\begin{frame}
{\bf Hot deck imputation : impute value from record in
the data set}
\bi
\item {\bf Sequential Hot-Deck Imputation:} impute the value in the same subgroup that was
last read by the computer.
\begin{table}
\caption{The four cells for our example are constructed using the
variables age and sex.}
\begin{tabular}{c|c|c}
& $\le 34$ &$\ge 35$ \\
\hline
M&Persons & Persons\\
&3,5,10,14 &1,7,8,15,16 \\
\hline
F& Persons&Persons\\
&4,12,13,19,20 & 2,6,9,11,17,18\\
\hline
\end{tabular}
\end{table}
\ei
In this example, person 19 is missing the response for crime
victimization. Person 13 had the last response recorded in her
subclass, so value 1 is imputed.
\end{frame}

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\begin{frame}
\bi
 \item {\bf Random Hot-Deck Imputation:} A donor is randomly chosen from the persons in the
subgroup with information on all the missing items. To preserve
multivariate relationships, usually values from the same donor are
used for all missing items of a person.

Example: In ``impute'' data set, person 10 is missing both variables
for victimization. Persons 3, 5, and 14 in his cell have responses
for both crime questions, so one of the three is chosen randomly as
the donor. In this case, person 14 is chosen, and his values are
imputed for both missing variables.
\ei
\end{frame}

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\begin{frame}
\bi
 \item {\bf Nearest neighborhood Hot-Deck Imputation:} Define a distance measure between observations,
and impute the value of a respondent who is closest to the
person with the missing item, where closeness is defined using the
distance function.
\ei

Example: If age and sex are used for the distance function, so that
the person of closest age with the same sex is selected to be the
donor, the victimization responses of person 3 will be imputed for
person 10.

\end{frame}

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\begin{frame}
{\bf Regression Imputation}
\bi
 \item Build a regression model to predict missing values, e.g., victimization responses, using other variables.
 \item Impute the predicted values from the regression model.
 \item Cell mean imputation is a special case of regression imputation.
 \item May result in spikes in the data distribution.
\ei
\end{frame}


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\begin{frame}

{\bf Example: Logistic Regression Imputation}
\bi
\item We have only 18 complete observations for the response variable crime victimization 
(not sufficient to fit a robust model), but a logistic regression with age as the explanatory variable provides the following 
model for predicted probability of victimization:
$$\log \ds\frac{\hat p}{1-\hat p} = 2.5643 - 0.0896 \times \mbox{age}$$
\item For a 17-year-old, the predicted probability of being a crime victim is 0.74. 
\item Since 0.74 exceeds the predetermined cutoff of 0.5, the value 1 is imputed for Person 10.
\ei
\end{frame}

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\begin{frame}
{\bf Multiple imputation}

In multiple imputation, each missing value is imputed m ($\ge2$)
different times. \\
----Typically, the same stochastic model is used for each
imputation.\\
----This create $m$ different ``data" sets with no missing values.\\
----Each of the $m$ data sets is analyzed as if no imputation had been
done; \\
----The different results give the analyst a measure of the additional
variance due to the imputation.\\
---- Multiple imputation with different models for nonresponse can
give an idea of the sensitivity of the results to particular
nonresponse models. (Rubin (1987, 1996, 2004))
\end{frame}

\begin{frame}
{\bf Multiple Imputation}

Multiple imputation is a statistical technique used to handle missing data by imputing each missing value $m$ ($m \geq 2$) different times, generating multiple complete datasets.
\bi
\item Typically, the same stochastic model is used for each imputation.
\item This process creates $m$ complete datasets with no missing values.
\item Each of the $m$ datasets is analyzed independently as if no imputation had been done.
\item The variation across the $m$ results provides a measure of the additional uncertainty (variance) introduced by the imputation process.
\item Using multiple imputation with different nonresponse models allows researchers to assess the sensitivity of results to 
specific assumptions about the nonresponse mechanism.
\ei

\end{frame}

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\begin{frame}

\bi
\item Multiple imputation is widely regarded as a robust approach for handling missing data, especially when data are missing at random (MAR).\\
\ei

{\footnotesize (See Rubin, 1987, 1996, 2004 for foundational work on multiple imputation.)}

\end{frame}

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\begin{frame}
{\bf Problems with All Imputation Methods}

\bi
\item Imputation creates synthetic (or "fake") data, which may not fully represent the true missing values.
\item It is essential to flag which values are imputed to distinguish them from observed data.
\item Variance estimates tend to be underestimated:
    \bi
    \item The effective sample size is artificially increased due to the inclusion of imputed values.
    \item Imputed data are often treated as though they were collected during the actual data collection process, 
    ignoring the uncertainty associated with the imputation process.
    \ei
\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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