12  Two-Phase Sampling

There are numerous examples showing how available auxiliary information can be used [in the sampling strategy] to achieve greater precision in the estimates. However, if the auxiliary information is not available but is known to be collectable cheaply and on a large scale, it is worthwhile to obtain that information in a first phase and then select a sample for the characteristic of interest. Raj (1968)

12.1 Introduction

Proposed by Neyman (1938), two-phase sampling is the appropriate design when there is not full knowledge of the structural behavior of the population of interest. This is reflected in a poor sampling frame that does not include auxiliary information variables of any kind, discrete or continuous, and therefore it is not possible to propose the use of an optimal sampling strategy, such as advanced size-proportional or stratified designs and regression or calibration estimators, for estimating the population parameters of interest.

In (Särndal and Sweensson 1987), a general framework develops the theory of two-phase sampling theoretically and in a way induced by the principles of the Horvitz-Thompson estimator. The two-phase sampling design, also known as biphasic or double sampling, is used when there is little or no knowledge about the behavior of the characteristic of interest across the individuals that make up the population. For example, the combined ratio estimator requires that all population elements can be stratified and that the population total of the auxiliary information characteristic, \(t_x=\sum_{U}x_k\), be known. In many practical cases, however, this type of auxiliary information is not available, such as the membership of population members in specific strata or the population total of the auxiliary information characteristics. In these cases, where the sampling frame contains little or deficient information for proposing an efficient sampling design, the statistician can turn to the following two options (Särndal et al. 1992):

1. Use a simple sampling design, such as simple random sampling without replacement or random cluster sampling, and combine it with the Horvitz-Thompson estimator to gain precision as the sample size increases.
2. Obtain information about the population to construct a new sampling frame. If the regression estimator is used, the desired precision can be achieved with a moderate sample size.

Note that assigning a large sample size or constructing a new sampling frame implies the use of economic and logistical resources that the study may not be able to bear. Thus, a third option is to use a two-phase sampling design:

1. [a)] In the first phase, a sample of size \(n_a\), moderate but not small, is selected. It will be denoted by \(S_a\). This first sample is selected through a design \(p_a(\cdot)\). For each element in \(S_a\), information must be obtained on one or more auxiliary variables¹. This sample is determined by the random variables

\[ I_k= \begin{cases} 1, &\text{if element $k$ is in the first-phase sample}\\ 0, &\text{if element $k$ is not in the first-phase sample} \end{cases} \]

Therefore, the inclusion probability of an element in the first sample \(S_a\) of the first phase is given by the following expression: \[ \pi_{ak}=Pr(I_k=1)=\sum_{s_a \ni k}p_a(s_a) \]

and the second-order inclusion probability in \(S_a\) is given by \[ \pi_{akl}=Pr(I_kI_l=1)=\sum_{S_a \ni \text{ $k$ and $l$}} p_a(s_a) \]

1. [b)] In the second phase, with the help of the information obtained in the first phase, a subsample \(S\) of size \(n\) is selected from \(S_a\) through a sampling design \(p(\cdot\mid{s_a})\). The characteristic of interest is then observed for the elements selected in the subsample. This sample is determined by the random variables \[ D_k= \begin{cases} 1, &\text{if element $k$ is in the second-phase sample}\\ 0 &\text{if element $k$ is not in the second-phase sample} \end{cases} \]

The probability that an element is in this subsample depends on what happened in the first phase. The inclusion probability of the elements in the second-phase sample is given by the following expression: \[ \pi_{k \mid s_a}=Pr(D_k=1 \mid \mathbf{I})=\sum_{s \ni k}p(s|s_a) \]

where \(\mathbf{I}=(I_1,\ldots,I_N)'\) denotes the inclusion vector of the first sample. On the other hand, the second-order inclusion probability in \(S\) is given by \[ \pi_{kl \mid s_a}=Pr(D_kD_l=1\mid \mathbf{I})=\sum_{S \ni \text{ $k$ and $l$}} p(s|s_a) \]

For example, Lohr (2000) states that, in a business survey, a first-phase sample could be drawn from tax returns and the income reported by each business selected in that first phase could be recorded. This sample may be large because obtaining the auxiliary information is assumed not to be costly. In a second phase, one could consider selecting a subsample with probability proportional to the income measured in the first phase, or using the income information to stratify the businesses in the first-phase sample and then contacting a subset of businesses in each stratum to obtain the desired information on characteristics of interest such as total expenses or declared taxes.

The author emphasizes that the sampling design that provides the sampling support covering both the first and second phases is not given by \(p_a(s_a)\) nor by \(p(s|s_a)\), but rather, using the theorem of total probability (Mood et al. 1974), is given by the following expression: \[ p(s)=\sum_{s_a \supset s}p_a(s_a)p(s|s_a) \]

Therefore, the inclusion probability of any element in the final sample \(S\) is \[ \begin{aligned} \pi_{k}=Pr(I_kD_k=1)&=\sum_{s \ni k}\sum_{s_a \supset s}p_a(s_a)p(s|s_a) \notag \\ &=\sum_{s_a \ni k}\sum_{\substack{S_a \subset S\\s\ni k}}p_a(s_a)p(s|s_a) \notag \\ &=\sum_{s_a \ni k}p_a(s_a)\sum_{\substack{S_a \subset S\\s\ni k}}p(s|s_a) \notag \\ &=\sum_{s_a \ni k}p_a(s_a)\pi_{k \mid s_a} \end{aligned} \]

Therefore, under this type of two-phase sampling scheme, it is not possible to use the principles of the Horvitz-Thompson estimator, in terms of inference for the population total, because although it is possible to know the probabilities induced by \(p_a(s_a)\) for each sample \(S_a\), it is not always possible to know the values of the second-phase inclusion probabilities \(\pi_{k \mid s_a}\) for each sample \(S_a\), since they depend on the realization of the first sample.

12.2 The \(\pi^*\) Estimator

Note that another possible estimator of the population total of the characteristic of interest is \(\sum_{S_a}y_k/\pi_{ak}\). This is another useless estimator because it could be calculated only if \(y_k\) and \(\pi_{ak}\) were known for every \(k\in{s_a}\). But \(y_k\) is known only in the subsample for \(k\in{s}\). Therefore, conditional on \(s_a\), the first-phase sample, the quantity \(\sum_{s_a}y_k/\pi_{ak}\) is unbiasedly estimated by the conditional Horvitz-Thompson estimator through \[ \hat{t}_{y,\pi^*}=\sum_{s}\frac{y_k}{\pi_{k}^*}=\sum_{s}\frac{y_k}{\pi_{ak}\pi_{k\mid{s_a}}} \]

and is defined as the \(\pi^*\) estimator (Särndal and Sweensson 1987).

TipResult

In two-phase sampling, the population total \(t_y\) is estimated unbiasedly by the \(\pi^*\) estimator. In addition, the variance of the estimator and the unbiased estimate of the variance are given by \[ Var_{Bif}(\hat{t}_{y,\pi^*})=\sum\sum_U\Delta_{akl}\frac{y_k}{\pi_{ak}}\frac{y_l}{\pi_{al}} +E_{p_a}\left(\sum\sum_{S_a}\Delta_{kl|S_a}\frac{y_k}{\pi_{k}^*}\frac{y_l}{\pi_{l}^*}\right) \] \[ \widehat{Var}_{Bif}(\hat{t}_{y,\pi^*})=\sum\sum_S\frac{\Delta_{akl}}{\pi_{kl}^*}\frac{y_k}{\pi_{ak}}\frac{y_l}{\pi_{al}} +\sum\sum_{S}\frac{\Delta_{kl|S_a}}{\pi_{kl|S_a}}\frac{y_k}{\pi_{k}^*}\frac{y_l}{\pi_{l}^*} \]

respectively, with \(\pi^*_k=\pi_{ak}\pi_{k|S_a}\), \(\pi^*_{kl}=\pi_{akl}\pi_{kl|S_a}\), \(\Delta_{akl}=\pi_{akl}-\pi_{ak}\pi_{al}\), and \(\Delta_{kl|S_a}=\pi_{kl|S_a}-\pi_{k|S_a}\pi_{l|S_a}\), where each summand in (12.2.3) is unbiased for its counterpart in (12.2.2).

Proof. Using successive conditioning from Result 7.1.3, for the probability structure of the sampling design \(p_a\), we have \[ \begin{aligned} E_{Bif}(\hat{t}_{y,\pi^*}) &=E_{p_a}\left(E_p\left(\hat{t}_{y,\pi^*}\left|\right. \mathbf{I}\right)\right)\\ &=E_{p_a}\left(E_p\left(\sum_{s}\frac{y_k}{\pi_{k}^*}\left|\right. \mathbf{I}\right)\right)\\ &=E_{p_a}\left(\sum_{S_a}E_p(D_k\mid \mathbf{I})\frac{y_k}{\pi_{ak}\pi_{k\mid{s_a}}}\right)\\ &=\sum_{U}E_{p_a}(I_k)\frac{y_k}{\pi_{ak}}=\sum_{U}y_k=t_y \end{aligned} \]

To prove the variance results, a similar argument is used because \[ \begin{aligned} Var_{Bif}(\hat{t}_{y,\pi^*})=Var_{p_a}(E_p(\hat{t}_{y,\pi^*}|\mathbf{I}))+E_{p_a}(Var_p(\hat{t}_{y,\pi^*}|\mathbf{I})) \end{aligned} \]

For the first summand, using the principles of the Horvitz-Thompson estimator, we have \[ \begin{aligned} Var_{p_a}(E_p(\hat{t}_{y,\pi^*}|\mathbf{I})) &=Var_{p_a}\left(E_p\left(\sum_{s}\frac{y_k}{\pi_{k}^*}\left|\right. \mathbf{I}\right)\right)\\ &=Var_{p_a}\left(\sum_{S_a}\frac{y_k}{\pi_{ak}}\right)\\ &=\sum\sum_U\Delta_{akl}\frac{y_k}{\pi_{ak}}\frac{y_l}{\pi_{al}} \end{aligned} \]

For the second summand, proceeding similarly and setting \(y_{ak}=y_k/\pi_{ak}\), we have \[ \begin{aligned} E_{p_a}(Var_p(\hat{t}_{y,\pi^*}|\mathbf{I})) &=E_{p_a}\left(Var_p\left(\sum_{s}\frac{y_k}{\pi_{k}^*}\left|\right. \mathbf{I}\right)\right)\\ &=E_{p_a}\left(Var_p\left(\sum_{s}\frac{y_{ak}}{\pi_{k|S_a}}\left|\right. \mathbf{I}\right)\right)\\ &=E_{p_a}\left(\sum\sum_{S_a}\Delta_{kl|S_a}\frac{y_{ak}}{\pi_{k|S_a}}\frac{y_{al}}{\pi_{l|S_a}}\right)\\ &=E_{p_a}\left(\sum\sum_{S_a}\Delta_{kl|S_a}\frac{y_k}{\pi_{k}^*}\frac{y_l}{\pi_{l}^*}\right) \end{aligned} \]

On the other hand, noting that \(E(D_kD_l|\mathbf{I})=\pi_{kl|S_a}\) and \(E(I_kI_l)=\pi_{akl}\) establishes the unbiasedness of the variance estimate.

NoteExample

Continuing with our example population \(U\) of size \(N=5\), suppose that in the first phase a sample of \(n_a=2\) elements is selected according to a simple random sampling design. In the second phase, a subsample of \(n=1\) is selected according to a simple random sampling design².

For the first phase, using Example 2.1.1, the \(\binom{N}{n_a}\) possible samples, together with their respective selection probabilities, are

        X1      X2     p_a
1     Yves      Ken     0.1
2     Yves     Erik     0.1
3     Yves   Sharon     0.1
4     Yves   Leslie     0.1
5      Ken     Erik     0.1
6      Ken   Sharon     0.1
7      Ken   Leslie     0.1
8     Erik   Sharon     0.1
9     Erik   Leslie     0.1
10  Sharon   Leslie     0.1

The inclusion probability in the first-phase sample, for each of the 5 elements of \(U\), is \[ \pi_{ak}=\frac{n_a}{N}=\frac{2}{5} \]

For the second phase, there are \(\binom{n}{n_a}\) possible subsamples for each first-phase sample. The second-phase sampling design and the overall sampling design are defined as follows:

        X1        X2     p_a     S      p( |s_a)     p(s)

1     Yves      Ken     0.1     Yves        0.5      0.05
                                Ken         0.5      0.05

2     Yves     Erik     0.1     Yves        0.5      0.05
                                Erik        0.5      0.05

3     Yves   Sharon     0.1     Yves        0.5      0.05
                                Sharon      0.5      0.05

4     Yves   Leslie     0.1     Yves        0.5      0.05
                                Leslie      0.5      0.05

5      Ken     Erik     0.1     Ken         0.5      0.05
                                Erik        0.5      0.05

6      Ken   Sharon     0.1     Ken         0.5      0.05
                                Sharon      0.5      0.05

7      Ken   Leslie     0.1     Ken         0.5      0.05
                                Leslie      0.5      0.05

8     Erik   Sharon     0.1     Erik        0.5      0.05
                                Sharon      0.5      0.05

9     Erik   Leslie     0.1     Erik        0.5      0.05
                                Leslie      0.5      0.05

10  Sharon   Leslie     0.1     Sharon      0.5      0.05
                                Leslie      0.5      0.05

Note that, using the theorem of total probability, the final sampling design, which accounts for the probability dynamics of the first and second phases, is defined as follows: \[ p(s)= \begin{cases} 0.2, &\text{if $s=\{\text{Yves}\}$},\\ 0.2, &\text{if $s=\{\text{Ken}\}$},\\ 0.2, &\text{if $s=\{\text{Erik}\}$},\\ 0.2, &\text{if $s=\{\text{Sharon}\}$},\\ 0.2, &\text{if $s=\{\text{Leslie}\}$}. \end{cases} \]

The inclusion probability of an element of \(S_a\) in the subsample of the last phase, conditional on the realization of a particular sample, is given by \[ \pi_{k|S_a}=\frac{n_a}{n}=\frac{1}{2} \]

Then the conditional inclusion probability of an element of \(U\) given by \(\pi_k^*\) is \[ \pi_k^*=\pi_{ak}\pi_{k|S_a}=\frac{n_a}{N}\frac{n_a}{n}=\frac{n}{N}=\frac{1}{5} \]

which, for this particular case, coincides with the element’s inclusion probability, properly speaking, given in (12.1.6). However, almost always \(\pi_k^* \neq \pi_k\), as shown by the following configuration induced by a sampling design with unequal selection probabilities.

        X1        X2     p_a     S      p( |S_a)       p(s)

1     Yves      Ken     0.25     Yves        0.9      0.225
                                 Ken         0.1      0.025

2     Yves     Erik     0.15     Yves        0.8      0.120
                                 Erik        0.2      0.030

3     Yves   Sharon     0.15     Yves        0.7      0.105
                                 Sharon      0.3      0.045

4     Yves   Leslie     0.10     Yves        0.6      0.060
                                 Leslie      0.4      0.040

5      Ken     Erik     0.10     Ken         0.5      0.050
                                 Erik        0.5      0.050

6      Ken   Sharon     0.05     Ken         0.4      0.020
                                 Sharon      0.6      0.030

7      Ken   Leslie     0.05     Ken         0.3      0.015
                                 Leslie      0.7      0.035

8     Erik   Sharon     0.05     Erik        0.2      0.010
                                 Sharon      0.8      0.040

9     Erik   Leslie     0.05     Erik        0.1      0.005
                                 Leslie      0.9      0.045

10  Sharon   Leslie     0.05     Sharon      0.5      0.025
                                 Leslie      0.5      0.025

Note that, for this configuration, and once again using the theorem of total probability, the final sampling design is defined as follows: \[ p(s)= \begin{cases} 0.510, &\text{if $s=\{\text{Yves}\}$},\\ 0.110, &\text{if $s=\{\text{Ken}\}$},\\ 0.140, &\text{if $s=\{\text{Sharon}\}$},\\ 0.095, &\text{if $s=\{\text{Erik}\}$},\\ 0.145, &\text{if $s=\{\text{Leslie}\}$}. \end{cases} \]

In this case, for the first phase, the inclusion probability in the first-phase sample for each of the 5 elements of \(U\) is

\[ \pi_{ak}= \begin{cases} 0.65, &\text{if $k=\text{Yves}$},\\ 0.45, &\text{if $k=\text{Ken}$},\\ 0.35, &\text{if $k=\text{Erik}$},\\ 0.30, &\text{if $k=\text{Sharon}$},\\ 0.25, &\text{if $k=\text{Leslie}$}. \end{cases} \]

The inclusion probability of an element of \(S_a\) in the second-phase subsample, conditional on the realization of a particular sample, is given by the following 10 cases, as many cases as first-phase samples:

• If \(S_a=S_1\), then \[ \pi_{k|S_a}= \begin{cases} 0.90, &\text{if $k=\text{Yves}$},\\ 0.10, &\text{if $k=\text{Ken}$}. \end{cases} \]
• If \(S_a=S_2\), then \[ \pi_{k|S_a}= \begin{cases} 0.80, &\text{if $k=\text{Yves}$},\\ 0.20, &\text{if $k=\text{Erik}$}. \end{cases} \]
• And so on, until
• If \(S_a=S_{10}\), then \[ \pi_{k|S_a}= \begin{cases} 0.50, &\text{if $k=\text{Sharon}$},\\ 0.50, &\text{if $k=\text{Leslie}$}. \end{cases} \]

Therefore, there will also be 10 cases for calculating the quantity \(\pi_k^*\), as follows:

• If \(S_a=S_1\), then \[ \pi_{k}^*= \begin{cases} 0.65 \times 0.90 = 0.585, &\text{if $k=\text{Yves}$},\\ 0.45 \times 0.10 = 0.045, &\text{if $k=\text{Ken}$}. \end{cases} \]
• If \(S_a=S_2\), then \[ \pi_{k}^*= \begin{cases} 0.65 \times 0.80 = 0.520, &\text{if $k=\text{Yves}$},\\ 0.35 \times 0.20 = 0.007, &\text{if $k=\text{Erik}$}. \end{cases} \]
• And so on, until
• If \(S_a=S_{10}\), then \[ \pi_{k}^*= \begin{cases} 0.30 \times 0.50 = 0.150, &\text{if $k=\text{Sharon}$},\\ 0.25 \times 0.50 = 0.125, &\text{if $k=\text{Leslie}$}. \end{cases} \]

The above shows that \(\pi_k^* \neq \pi_k\), because the inclusion probability is given by \[ \pi_{k}= \begin{cases} 0.510, &\text{if $k=\text{Yves}$},\\ 0.110, &\text{if $k=\text{Ken}$},\\ 0.140, &\text{if $k=\text{Erik}$},\\ 0.095, &\text{if $k=\text{Sharon}$},\\ 0.145, &\text{if $k=\text{Leslie}$}. \end{cases} \]

Note that in practice, with fairly large populations, it is not possible to calculate \(\pi_k\). As an exercise, using the data from Example 2.1.3, the unbiasedness of the \(\pi_k^*\) estimator should be verified both in the first configuration and in this last one.

12.3 Stratification in Two-Phase Sampling

Hidiroglou and Rao (2003) state that Neyman (1938)’s first proposal was stratification in two-phase sampling, where in the first phase a random sample \(S_a\) of size \(n_a\) is selected. The next step is to observe an auxiliary information variable \(x_k\) for each element \(k\in S_a\) and, based on the behavior of this characteristic, stratify the sample \(S_a\); that is, every element \(k\in S_a\) is classified into one and only one stratum \(h\) with \(h=1\ldots, H\), such that \[ S_a=\bigcup_{h=1}^H S_{ah} \ \ \ \ \ \ \ \ n_a=\sum_{h=1}^H n_{ah} \]

where \(S_{ah}\) corresponds to the \(h\)-th stratum of size \(n_{ah}\), which is commonly considered random. In the second phase, a sample \(S_{h}\) of fixed size \(n_h\) is selected for each stratum \(h=1,\ldots,H\), such that \[ S=\bigcup_{h=1}^H S_{h} \ \ \ \ \ \ \ \ n=\sum_{h=1}^H n_{h} \]

where \(S\) corresponds to the second-phase subsample of size \(n\). Note that the first-phase sample \(S_a\) is selected through an arbitrary design \(p_a(s_a)\), while the second-phase subsample \(S_h\) within each stratum \(h=1,\ldots,H\) is also selected through an arbitrary design in each stratum³, denoted by \(p_h(S_h|S_a)\).

TipResult

Under this framework, the population total \(t_y\) is estimated unbiasedly by \[ \hat{t}_{y,\pi^*}\sum_{h=1}^H\sum_{S_h}\frac{Y_k}{\pi_k^*} \] In addition, the variance of the estimator and the unbiased estimate of the variance are given by \[ \begin{aligned} Var_{Bif}(\hat{t}_{y,\pi^*})&=\sum\sum_U\Delta_{akl}\frac{y_k}{\pi_{ak}}\frac{y_l}{\pi_{al}}\notag\\ &+E_{p_a}\left(\sum_{h=1}^H\sum\sum_{S_{ah}}\Delta_{kl|S_a}\frac{y_k}{\pi_{k}^*}\frac{y_l}{\pi_{l}^*}\right) \end{aligned} \] \[ \widehat{Var}_{Bif}(\hat{t}_{y,\pi^*})=\sum\sum_S\frac{\Delta_{akl}}{\pi_{kl}^*}\frac{y_k}{\pi_{ak}}\frac{y_l}{\pi_{al}} +\sum_{h=1}^H\sum\sum_{S_h}\frac{\Delta_{kl|S_a}}{\pi_{kl|S_a}}\frac{y_k}{\pi_{k}^*}\frac{y_l}{\pi_{l}^*} \]

respectively, where each summand in (12.3.3) is unbiased for its counterpart in (12.3.2).

Suppose that, in the first phase, a simple random sample \(S_a\) of size \(n_a\) is drawn from a population of size \(N\). Therefore, \[ \pi_{ak}=\frac{n_a}{N}\ \ \ \ \ \ \ \ \ \pi_{akl}=\frac{n_a(n_a-1)}{N(N-1)} \]

Then, with the information collected in the first phase, it is possible to separate the units into \(H\) different strata; the stratum to which an element belongs is known only after the first-phase sample is selected. Next, for each stratum, a sample of size \(n_h\) is selected through a simple random sampling design, assuming that the strata have size \(n_{ah}\) with \(h=1,2,...,H\). Thus, for the second phase, the inclusion probability of an element is given by \[ \pi_{k\mid s_a}=\frac{n_h}{n_{ah}}\ \ \ \ \ \ \ \text{for $k\in S_{ah}$ with $h=1,\ldots,H$} \]

and the second-order inclusion probability is \[ \pi_{kl\mid s_a}= \begin{cases} \frac{n_h}{n_{ah}} &\text{if $k=l \in S_{ah}$}\\ \\ \frac{n_h(n_h-1)}{n_{ah}(n_{ah}-1)} &\text{if $k\neq l$, $k,l \in S_{ah}$}\\ \\ \frac{n_h}{n_{ah}}\frac{n_{h'}}{n_{ah'}} &\text{if $k\in S_{ah}$, $l\in S_{ah'}$} \end{cases} \]

From the above, the estimator of the population total is \[ \hat{t}_{y,\pi^*}= \sum_{S}\frac{y_k}{\pi_{k}^*}= \frac{N}{n_a}\sum_{S_h}\frac{n_{ah}}{n_h}y_k \]

To calculate the variance, successive conditioning is used as follows: \[ \begin{aligned} Var_{Bif}(\hat{t}_{y,\pi^*}) &=Var_{SRS}(E_{STSI}(\hat{t}_{y,\pi^*}\mid \mathbf{I}))+E_{SRS}(Var_{STSI}(\hat{t}_{y,\pi^*}\mid \mathbf{I}))\\ &=Var_{SRS}\left(\frac{N}{n_a}\sum_{S_a}y_k\right)\\ &=+E_{SRS}\left(Var_{STSI}\left(\frac{N}{n_a}\sum_{S_h}\frac{n_{ah}}{n_h}y_k\mid \mathbf{I}\right)\right)\\ &=\underbrace{\frac{N^2}{n_a}\left(1-\frac{n_a}{N}\right)S^2_{y_U}}_{V_1} +\underbrace{\frac{N^2}{n_a^2}E_{SRS}\left(\sum_{h=1}^H\frac{n_{ah}^2} {n_h}\left(1-\frac{n_h}{n_{ah}}\right)S^2_{y_{ah}}\right)}_{V_2} \end{aligned} \]

where the first term refers to the variance of the sample in the first phase, while the second term refers to the additional variance due to subsampling in the second phase. Note that \(S^2_{y_{ah}}\) is the variance of the characteristic of interest in the \(h\)-th stratum of the first-phase sample. It is important to emphasize that, in the second term, the operator \(E_{SRS}\) is specified over each and every possible stratified sample in the second phase.

Rao (1973) proposed the estimation of these variance components, which are estimated unbiasedly by the following expressions: \[ \begin{aligned} \hat{V}_1= \frac{N^2}{n_a}\left(1-\frac{n_a}{N}\right) \sum_{h=1}^H\frac{n_{ah}}{n_a}\left\{(1-Q_h)S^2_{y_{S_h}}+\frac{n_a}{n_a-1}(\bar{y}_{S_h}-\bar{y}_S)\right\} \end{aligned} \] \[ \begin{aligned} \hat{V}_2=\frac{N^2}{n_a^2}\left(\sum_{h=1}^H\frac{n_{ah}^2}{n_h}\left(1-\frac{n_h}{n_{ah}}\right)S^2_{y_{ah}}\right) \end{aligned} \] respectively, where \(Q_h=\dfrac{(n_a-n_{ah}}{n_h(n_a-1)}\). The proof of this result can be consulted in Hidiroglou and Rao (2003).

12.4 Selection Proportional to Size

The previous sections have shown how auxiliary information can be used to gain precision and efficiency in estimating the total of a characteristic of interest. In some cases, this information can be used at the design stage and, in others, at the estimation stage. When the goal is to use it at the design stage, a sampling design proportional to some auxiliary information characteristic \(x\) can be used. This option is presented here.

If it is known that the structural behavior of the auxiliary information characteristic is proportional to the behavior of the characteristic of interest, then it would be desirable to select the sample with probability proportional to \(x\). However, this information \(x\) is not available at the population level, although it is known to be cheap to obtain at least in a large sample. Therefore, it is collected in an initial sample \(s_a\) of size \(n_a\) induced by a simple random sampling design from a population of size \(N\). After this auxiliary information becomes available, a subsample \(s\) of size \(m\) is selected with replacement proportional to the auxiliary information variable \(x\).

TipResult

Under this framework, where the initial sample \(s_a\) of size \(n_a\) is selected by simple random sampling and the subsample \(s\) of size \(m\) is selected proportional to \(x\), the unbiased estimator of the population total, its variance, and its estimated variance are given by \[ \hat{t}_y=\frac{N}{n_a}\hat{t}_{ay} =\frac{N}{n_a}\frac{1}{m}\sum_{k\in S}\frac{y_k}{p_{ak}} =\frac{N}{n_a}\frac{t_{ax}}{m}\sum_{k\in S}\frac{y_k}{x_k} \] \[ \begin{aligned} Var_{Bif}(\hat{t}_y)&=\frac{N^2}{n_a}\left(1-\frac{n_a}{N}\right)S^2_{y_U}\notag\\ &+\frac{N(n_a-1)}{(N-1)n_a}\frac{1}{m}\sum_U\frac{1}{p_{k}}\left(\frac{y_k}{p_{ak}}-t_y\right)^2 \end{aligned} \] { \[ \begin{aligned} \widehat{Var}_{Bif}(\hat{t}_y) &=\frac{N^2}{n}\frac{t_{ax}^2}{m(m-1)}\left[\sum_{k\in S}\frac{y_k^2}{x_k^2}-\frac{1}{m}\left(\sum_{k\in S}\frac{y_k}{x_k}\right)^2\right]\\ &+\frac{N(N-n_a)}{mn_a(n_a-1)}\left(t_{ax}\sum_{k\in S}\frac{y_k^2}{x_k}+\frac{t_{ax}^2}{n_a(m-1)} \left[\sum_{k\in S}\frac{y_k^2}{x_k^2}-\frac{1}{m}\left(\sum_{k\in S}\frac{y_k}{x_k}\right)^2\right]\right)\notag \end{aligned} \] } respectively, with \(\hat{t}_{ay}=\frac{1}{m}\sum_{s}\frac{y_k}{p_{ak}}\), \(p_{ak}=\frac{x_k}{t_{ax}}\), and \(t_{ax}=\sum_{S_a}x_k\).

Proof. Using the property of successive conditioning once again, we have \[ \begin{aligned} E(\hat{t}_y)&=E_{SRS}\left(\frac{N}{n}E_{PPS}\left(\sum_{s}\frac{y_k}{p_{ak}}|\mathbf{I}\right)\right)\\ &=E_{SRS}\left(\frac{N}{n}\sum_{s_a}y_k\right)=t_y \end{aligned} \]

And for the first term of the variance, we have \[ \begin{aligned} Var_{SRS}(E_{PPS}(\hat{t}_y))=Var_{SRS}\left(\frac{N}{n_a}\sum_{s_a}y_k\right) =\frac{N^2}{n_a}\left(1-\frac{n_a}{N}\right)S^2_{y_U} \end{aligned} \]

For the second term, using Result 2.2.14 and Result 4.2.6, note that \[ \begin{aligned} Var_{PPS}(\hat{t}_y|\mathbf{I})&=\frac{N^2}{n_a^2}Var_{PPS}\left(\frac{1}{m}\sum_{s}\frac{y_k}{p_k}|\mathbf{I}\right)\\ &=\frac{N^2}{n_a^2}\frac{1}{m}\sum_{k\in S_a}p_{ak}\left(\frac{y_k}{p_{ak}}-t_{ay}\right)^2 =\frac{N^2}{n_a^2}\frac{1}{m}\sum_{S_a}\sum_{k<l}p_kp_l\left(\frac{y_k}{p_k}-\frac{y_l}{p_l}\right)^2 \end{aligned} \]

Therefore, we have \[ \begin{aligned} E_{SRS}(Var_{PPS}(\hat{t}_y)) &=E_{SRS}\left(\frac{N^2}{n_a^2}\frac{1}{m}\sum_{S_a}\sum_{k<l}p_kp_l\left(\frac{y_k}{p_k}-\frac{y_l}{p_l}\right)^2\right)\\ &=E_{SRS}\left(\frac{N^2}{n_a^2}\frac{1}{m}\sum_{U}\sum_{k<l}p_kp_l\left(\frac{y_k}{p_k}-\frac{y_l}{p_l}\right)^2I_kI_l\right)\\ &=\frac{N^2}{n_a^2}\frac{1}{m}\sum_{U}\sum_{k<l}p_kp_l\left(\frac{y_k}{p_k}-\frac{y_l}{p_l}\right)^2E_{SRS}(I_kI_l)\\ &=\frac{N^2n_a(n_a-1)}{n_a^2N(N-1)}\frac{1}{m}\sum_{U}\sum_{k<l}p_kp_l\left(\frac{y_k}{p_k}-\frac{y_l}{p_l}\right)^2\\ &=\frac{N(n_a-1)}{(N-1)n_a}\frac{1}{m}\sum_U\frac{1}{p_{k}}\left(\frac{y_k}{p_{ak}}-t_y\right)^2 \end{aligned} \]

The above uses the alternative form of the variance of the \(PPS\) sampling design. The proof of the unbiased estimate of the estimator’s variance can be consulted in Raj (1968).

12.5 Other Applications

This two-phase sampling design has many applications in practice, and the topics discussed so far are only a brief introduction to the complex and vast world of sample surveys, with all their shortcomings and limitations. However, this chapter has shown that it is possible to address these limitations from a theoretical point of view and to find a practical solution to these problems. A brief summary of other applications of two-phase sampling follows.

12.5.1 Improving the Estimator

This chapter focused on the search for an optimal sampling design and on improving the way samples are selected in the second stage. However, it is possible to consider a very simple sampling design in both stages and, with the help of auxiliary information collected in the first-phase sample, improve the estimator by using the general regression estimator framework or calibration estimators. Of course, depending on the quality of the information obtained, it is possible to improve both the sampling design and the estimator.

As Estevao and Särndal (2001) states, a distinctive feature of two-phase sampling is that auxiliary information can be found at several levels:

• Complete population level: the value of each auxiliary information characteristic is known for every individual belonging to the population.
• Incomplete population level: only the totals of the auxiliary information characteristics are known, not their individual values.
• First-phase level \(S_a\): the value of each auxiliary information characteristic is known for every individual belonging to the first-phase sample \(S_a\).
• Second-phase level \(S\): the value of each auxiliary information characteristic is known for every individual belonging to the second-phase subsample \(S\).

Some information resides at the population level, while other information resides at the sample level in the first phase of sampling. Even with access to both, the researcher decides at their discretion whether to use both, one of them, or even neither, to obtain efficient estimates. The variance of the estimator, whether regression or calibration, will then depend on the level at which the auxiliary information chosen for use is found. It is important to identify which type of auxiliary information is relevant for the study, since complete auxiliary information is not always available. Even if it is available, it must be decided whether it will be used, because

1. In some situations, efficiency can decrease dramatically if some auxiliary information characteristic is ignored in the calibration process. It is even possible to obtain a calibration estimator whose variance is smaller than that of an estimator constructed using complete auxiliary information.
2. Complete auxiliary information is not always available, so the objective of improving estimation must be achieved with the information at hand. It is important to understand how this type of limitation affects the variance of the estimator.

Estevao and Särndal (2001) have shown that there are exactly ten different cases containing different auxiliary information configurations for calibration estimators, and they account for their variance depending on the case. The treatment by Särndal and Sweensson (1987) for the general regression estimator is exhaustive and is a very good reference source for two-phase sampling strategies that, at the estimation stage, consider a superpopulation model to assist the efficiency of the estimator. This reading can be complemented with Chapter 9 of Särndal et al. (1992).

12.5.2 A Model for Nonresponse

People who do not respond often differ crucially from those who do. Thus, the following classification is possible: a) unit nonresponse, where the entire observation unit is missing, which usually occurs because the interviewer could not contact the household, the selected person is ill, or the person refuses to participate. At this stage, the interviewer must determine some demographic characteristics of the household for subsequent imputation; and b) item nonresponse, where some records of the observation unit are missing although others have in fact been answered. The following are some perspectives for dealing with nonresponse:

• Prevention: design the survey so that nonresponse is small. This is the best method for dealing with it.
• Subsample: select a representative subsample of the units that did not respond and make inferences.
• Models: use a model to predict the values of the units that did not respond; that is, replace the records of the missing unit with predicted records resulting from the model.
• Ignoring: it is very common practice to ignore survey nonresponse and make inferences with the data collected from responding units.

Nonresponse has large effects⁴ on the quality of the estimates. For example, if the sample size were increased to address nonresponse, one might end up with a larger number of people from the same class of respondents, increasing homogeneity. Note that bias can increase because resources that could have helped remedy nonresponse were wasted. On the other hand, if the effect of nonresponse is omitted in a victimization survey, the total number of victims is underestimated. Now, two strata are formed in the population, “respondents” and “nonrespondents”, and bias is reduced if the mean is similar in the two strata, an option that is impossible to know because the “nonrespondents” simply do not respond, or if there is little nonresponse.

Lohr (2000) suggests that some factors affecting the increase in nonresponse may be:

1. Content: surveys related to drug use or finances. The response rate can be bounded if the questions are ordered appropriately.
2. Survey timing: some periods produce higher nonresponse rates than others.
3. Interviewers: apply standard quality-improvement methods to increase the accuracy and response rate of the interviewers involved in the study.
4. Collection method: telephone and mail surveys have lower response rates than personal interviews⁵.
5. Questionnaire design: wording of the questions.
6. Burden: excessively long surveys that make the respondent unwilling.
7. Survey presentation: this is the first contact between the respondent and the interviewer.
8. Incentives: financial incentives or “gifts” increase the response rate. Anti-incentives are also useful, for example suspending a driver’s license when a person refuses to answer.

Brewer (2002) states that nonresponse and two-phase sampling are related as follows: the simplest way to deal with nonresponse is to treat the sample of respondents as if they constituted the target sample, or equivalently as if the population of actual respondents and nonrespondents were governed by the same probability structure. In this way, the target sample is treated as the first-phase sample and the set of actual respondents is treated as the second-phase subsample.

Särndal and Lundström (2004) mentions that this approach begins with the assumption that the response distribution is known, although in practice it is not. This implies that the first- and second-order response probabilities are given by \[ Pr(k\in r|S)=\theta_k \ \ \ \ \ \ Pr(k,l\in r|S)=\theta_{kl} \]

which are assumed known, where \(r\) denotes the group of actual respondents and \(S\) the total sample made up of respondents and nonrespondents. In this way, it is possible to calculate the combined weights, noting the similarity with the construction of the quantity \(\pi_k^*\), \((1/\pi_k)\times(1/\theta_k)\) and calculate the following unbiased two-phase estimator: \[ \hat{t}_y=\sum_{k\in r}\frac{y_k}{\pi_k\theta_k} \]

Because the response probabilities \(\theta_k\) are unknown, the previous estimator is impossible to calculate. Therefore, to make it operational, an estimate of them must be found. Suppose that auxiliary information characteristics are available that allow an estimator, or also a predictor, of this probability to be obtained, denoted by \(\hat{\theta}_k\). Therefore, a two-phase estimator that accounts for nonresponse has been obtained by replacing \(\theta_k\) with \(\hat{\theta}_k\) and is given by \[ \hat{t}_y=\sum_{k\in r}\frac{y_k}{\pi_k\hat{\theta}_k} \]

There are different ways to find estimators \(\hat{\theta}_k\); some of them are discussed in Chapter 9 of Särndal et al. (1992).

12.5.3 Sampling on Occasions

In many research studies, samples are selected repeatedly over time from the same population, and the same characteristic of interest is measured on each occasion. In this way, its structural behavior can be measured over time. Sampling on two occasions considers a finite population and, on the first occasion, a sample \(S_a\) is selected through a sampling design \(p_a(\cdot)\) and the characteristic of interest \(y\) is measured. On the second occasion, two independent samples are selected: an overlapping sample, \(S_t\), from the previous sample \(S_a\), and a nonoverlapping sample, \(S_{nt}\), taken from the complement of the first sample \(S_a^c\). Chapter 9 of Särndal et al. (1992) addresses the theory for treating this sampling configuration.

12.6 Frame and Lucy

The population of industrial-sector firms is used below to illustrate the development of two-phase sampling and how it can substantially improve the sampling strategy. This section considers three configurations that clearly show difficult but common scenarios in practice, where surveys and sampling frames suffer from imperfections and it is necessary to sharpen the statistical tools available for dealing with these problems.

12.6.1 First Configuration: Stratification

In this first scenario, the sampling frame is considered deficient and only includes the location and identification of firms in the industrial sector. Under this framework, it is assumed that absolutely nothing is known about the structural behavior of the population through the variables of interest: Income, Expenses, and Taxes declared during the previous year.

Suppose that the researcher knows that the industrial sector is divided into three levels: Large, Medium, and Small, and that the behavior of the characteristics of interest is substantially different in each of these population subgroups. If the sampling frame were good enough to determine the classification of each firm into one of these three strata, a stratified sampling design could be used to improve estimation. However, suppose that such information is not available at the population level. Some private entities sell this information at a reasonable price. The bad news is that, because of conflicts of interest, they do not provide the complete list, but rather a subset of 1000 of the 2396 firms in the industrial sector. The good news is that the researcher can choose the one thousand firms as desired.

Under this configuration, a two-phase sampling design can be used as follows: in the first phase, select a sample of size \(n_a=1000\) and obtain the level information for each firm included in this first sample. For this, the S.SI function from the TeachingSampling package is used to obtain the first sample, which will be called phase1_data.

data(BigLucy)
N <- dim(BigLucy)[1]
n <- 4000
sam <- S.SI(N, n)
phase1_data <- BigLucy[sam, ]
attach(phase1_data)
head(phase1_data)

             ID        Ubication Level    Zone Income Employees Taxes SPAM ISO
11 AB0000000011 C0109686K0192211 Small County1    374        34     6  yes  no
14 AB0000000014 C0189067K0112830 Small County1    330        23     4  yes  no
32 AB0000000032 C0036536K0265361 Small County1    380        18     6  yes  no
48 AB0000000048 C0054436K0247461 Small County1    422       101     8  yes  no
83 AB0000000083 C0206936K0094961 Small County1    260        84     2  yes  no
84 AB0000000084 C0224613K0077284 Small County1    481        65    10  yes  no
   Years  Segments
11    50 County1 2
14    35 County1 2
32    48 County1 4
48    23 County1 5
83    33 County1 9
84    17 County1 9

The sample taken in the first phase has size 1000 and is divided across the three strata. On the other hand, in the second phase, using the information on stratum membership, a second stratified sample of size \(n=2000\) is selected. For this, the S.STSI function from the TeachingSampling package is configured.

na1 <- summary(Level)[[1]]
na2 <- summary(Level)[[2]]
na3 <- summary(Level)[[3]]
n.a <- c(na1, na2, na3)
n.a

[1]  141 1248 2611

n1 <- 120
n2 <- 880
n3 <- 1000
n <- c(n1, n2, n3)

sam <- S.STSI(Level, n.a, n)
phase2_data <- phase1_data[sam, ]
head(phase2_data)

              ID        Ubication Level    Zone Income Employees Taxes SPAM ISO
32  AB0000000032 C0036536K0265361 Small County1    380        18   6.0  yes  no
119 AB0000000119 C0113018K0188879 Small County1     84        81   0.5  yes  no
131 AB0000000131 C0117430K0184467 Small County1    222        34   2.0  yes  no
253 AB0000000253 C0051568K0250329 Small County1    232        47   2.0   no  no
328 AB0000000328 C0173183K0128714 Small County1    193        81   1.0   no  no
349 AB0000000349 C0098285K0203612 Small County1    314        54   4.0   no  no
    Years   Segments
32     48  County1 4
119    26 County1 12
131    34 County1 14
253    23 County1 26
328    13 County1 33
349    32 County1 35

attach(phase2_data)

The subsample taken in the second phase has size 400 and is divided across the three strata. Once the information has been obtained, the quantities of interest are estimated. For this, the E.STSI function from the TeachingSampling package is used; it returns the estimates expanded to the first-phase sample. To expand them to the population, it is enough to multiply them by the inverse of the inclusion probability of the first sample. The results are shown below.

target_variables <- data.frame(Income, Employees, Taxes)
(N / sum(n.a)) * E.STSI(Level, n.a, n, target_variables)[1, , ]

               N   Income Employees   Taxes
Big         3007  3668981    394276  212447
Medium     26612 17528437   2161437  583627
Small      55677 15905238   2914967  219646
Population 85296 37102657   5470681 1015720

Note that this strategy is recommended when efficient estimates by population subgroups are desired.

12.6.2 Second Configuration: Selection Proportional to Size

In this subsection, suppose that the same conditions as in the previous scenario hold. However, the interest is no longer centered on efficient estimation of the totals of the characteristic of interest within some population subgroups, but rather on efficient estimation of the population total of the characteristics of interest. Thus, the goal is to implement a simple random sampling design in a first stage in order to incorporate auxiliary information in the second stage. As before, the S.SI function from the TeachingSampling package is used to select this first sample.

data(BigLucy)
N <- dim(BigLucy)[1]
na <- 4000
sam <- S.SI(N, na)
phase1_data <- BigLucy[sam, ]
attach(phase1_data)

Once the sample has been selected, the researcher is forced to collect auxiliary information that can improve the sampling strategy. In this case, the researcher knows that the Income characteristic is directly related to the characteristics of interest, Number of Employees and Taxes. In addition, this information is easy to obtain because, as in the previous configuration, an entity provides it, although only for 1000 firms under confidentiality clauses. Thus, the researcher collects Income data for the 1000 firms included in the first-phase sample and decides to improve the sampling strategy by incorporating this auxiliary information into the sampling design. In this sense, the researcher decides to use a sampling design proportional to firm Income. The S.PPS function from the TeachingSampling package is used to select the subsample. The subsample has size \(m=400\) and is selected with replacement.

n <- 2000
res <- S.PPS(n, Income)
sam <- res[, 1]
pk.s <- res[, 2]
sum(pk.s)

[1] 0.69

phase2_data <- phase1_data[sam, ]
attach(phase2_data)
target_variables <- data.frame(Income, Employees, Taxes)

For estimating the population total of the characteristics of interest, the E.PPS function from the TeachingSampling package is used; it provides the estimate expanded in the Phase 1 sample. To expand the results to the population, once again, it is enough to multiply these results by the inverse of the first-phase inclusion probability, given by 2396/1000.

(N / na) * E.PPS(target_variables, pk.s)[1, ]

        N    Income Employees     Taxes 
    83370  36653418   5251528   1006500 

12.6.3 Third Configuration: Calibration Estimation

For this last scenario, suppose that the researcher selects a simple random sample for the first sampling phase in order to collect information that will improve the sampling strategy.

data(BigLucy)
N <- dim(BigLucy)[1]
na <- 4000
sam <- S.SI(N, na)
phase1_data <- BigLucy[sam, ]
attach(phase1_data)

Now suppose that the entity providing the information is willing to provide, for each firm included in the first-phase sample, not only Income information but also information on the Number of Employees. Thus, the researcher proposes selecting a subsample using a simple random sampling design and combining it with a calibration estimator using the raking method.

t.ax <- c(na, sum(Income), sum(Employees))
n <- 2000
sam <- S.SI(na, n)
phase2_data <- phase1_data[sam, ]
attach(phase2_data)

To estimate the results expanded to the first phase, the calib function from the Sampling package is used; it provides the calibrated weights for Phase 1. Similarly, these results are expanded to the population by multiplying by the inverse of the inclusion probability of the first sample.

library(sampling)
y.as <- data.frame(Income, Employees, Taxes)
x.as <- cbind(1, Income, Employees)
pi.ak <- rep(n / na, times = n)
w.ak <- calib(x.as, d = 1 / pi.ak, t.ax, method = "raking")

tc.a <- t(w.ak / pi.ak) %*% as.matrix(y.as)
(N / na) * tc.a

       Income Employees   Taxes
[1,] 36504065   5388340 1009476

12.6.3.1 Comparison of Results

Although at first glance the results may not seem very close to the true population totals, note that, in particular for the Income characteristic of interest, a large gain is obtained compared with a simple random sampling design. Note also that, in this case, the calibration estimator gives better results.

Table 12.1: Estimates obtained under different scenarios for two-phase sampling.

Method	Population total	Estimated total	Dev. %
Strata	28654	27854	-2.79
Proportional	28654	30031	4.81
Calibration	28654	27995	-2.29

12.7 Exercises

1. Suppose a longitudinal study proposes three semipanel-type surveys at different times. For the third measurement, a sampling design with a 20% rotation was used for the following possible specifications:

• Of size \(n_1\), selected only from the sample of the first measurement.
• Of size \(n_{12}\), selected from the samples of measurements one and two.
• Of size \(n_{123}\), selected from the samples of all three measurements.
• Of size \(n_{23}\), selected from the samples of measurements two and three.
• Of size \(n_{3}\), selected from the sample of the third measurement.

1. Draw a diagram illustrating sample rotation across the three measurements and the relative sizes of the five configurations above.

2. Propose a formula for estimating the population total of the characteristic of interest in the third measurement for the five configurations above.

3. Without writing any statistical formula for the variances, indicate which of these configurations induces greater efficiency in the estimates and why.

4. Suppose a two-phase sampling design. In the first phase, a simple random sample without replacement \(s_a\) of size \(n_a=150\) was selected. In this phase, information on a characteristic of interest \(x\) was collected. In the second phase, it was decided to select a sample \(s\) using a Poisson sampling design with expected sample size \(n_s=10\), through inclusion probabilities proportional to the auxiliary information characteristic. The information for the second-phase sample is as follows:

y	x
2653	33
17949	247
1060	12
1324	12
2223	18
2553	30
2216	20
13205	138
3475	35
7072	62
4623	47

1. Calculate an unbiased estimate for the population total of \(y\), taking into account that the total of the characteristic of interest in the first-phase sample is 4060. Use the following expression to calculate the corresponding estimated coefficient of variation:

\[ \begin{aligned} \widehat{Var}(\hat{t}_{y,\pi^*})&= \left(\frac{N}{n}\bar{x}_{s_a}\right)^2\frac{1}{n_a-1} \left[(n_a-f_a)\sum_s \left(\frac{y_k}{x_k}\right)^2-(1-f_a)\left(\sum_s\frac{y_k}{x_k}\right)^2\right] - \frac{N}{n}\bar{x}_{s_a}\sum_s\frac{y_k^2}{x_k} \end{aligned} \]

1. Assume that the second-phase sample from the previous exercise was obtained using PPS sampling. Calculate an unbiased estimate for the population total of \(y\) and calculate the corresponding estimated coefficient of variation.

2. Suppose a two-phase sampling design. In the first phase, a simple random sample without replacement \(s_a\) of size \(n_a=160\) was selected. In this phase, the population was stratified into four subgroups, each of size 40. In the second phase, it was decided to select a stratified random sample of \(20\) elements in each stratum, and the characteristic of interest was observed. The results obtained are shown below:

Stratum \(h\)	\(\bar{y}_{s_h}\)	\(S^2_{y_{s_h}}\)
1	17.05	19945
2	19.75	24179
3	22.40	28359
4	31.25	42829

1. Calculate an unbiased estimate for the population total of \(y\).
2. Obtain an estimate of the variance and report the corresponding estimated coefficient of variation.
3. Obtain an estimate of the variance and report the corresponding estimated coefficient of variation, assuming that the sample had been obtained from a single-phase stratified random sampling design of size \(n=80\).

Brewer, K. R. W. 2002. Combined Sampling Inference, Weighting Basu’s Elephants. London: Arnorld.

Estevao, V. M., and C-E. Särndal. 2001. “The Ten Cases of Auxiliary Information for Calibration Estimators in Two-Phase Sampling.” Journal of Official Statistics 18: 233–55.

Hidiroglou, M. A., and J. N. K. Rao. 2003. “Variance Estimation in Two-Phase Sampling.” In Proceedings of Statistics Canada Symposium, edited by Statistics Canada.

Lohr, S. 2000. Sampling: Design and Analysis. Thompson.

Mood, A. M., F. A. Graybill, and D. C. Boes. 1974. Introduction to the Theory of Statistics. 3rd ed. McGraw Hill.

Neyman, J. 1938. “Contribution to the Theory of Sampling Human Populations.” Journal of the American Statistical Association 33: 101–16.

Raj, D. 1968. Sampling Theory. McGraw Hill.

Rao, J. N. K. 1973. “On Double Sampling for Stratification and Analityc Surveys.” Biometrika 60: 125–33.

Särndal, C. E., and S. Lundström. 2004. Estimation in Surveys with Nonresponse. Wiley.

Särndal, C. E., and B. Sweensson. 1987. “A General View of Estimation for Two Phases of Selection with Aplications to Two-Phase Sampling and Nonresponse.” International Statistical Review 55: 279–94.

Särndal, C. E., B. Swensson, and J. Wretman. 1992. Model Assisted Survey Sampling. Springer, New York.

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1. Note that this process is less costly than obtaining the information directly from the population.

2. Although using a simple random sampling design in both phases is not realistic in practice, this example helps provide a better understanding of the probability structure induced by two-phase sampling.

3. The initial proposal by Neyman (1938) was to use a simple random design both for selecting the first sample in the first phase and for selecting the second-phase subsamples in each stratum.

4. If one insists on calculating and estimating totals and means without accounting for nonresponse, the technical report must include the corresponding response-rate figure.

5. Using a CATI system, computer-assisted telephone interviewing, improves data accuracy.