3. Chi-squared tests in dual frame surveys

Yan Lu

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In this section, we consider the case of chi-squared tests in a dual frame survey. Some hypotheses of interest may include: a simple hypothesis $H_0\text{: }{q}_{ia}=p_{ia}{N}_a/N_A=q_{ia0}^A\mathrm{,}{q}_{iab}^A=p_{iab}{N}_{ab}/N_A=q_{iab0}^A\mathrm{,}$ $q_{iab}^B=p_{iab}{N}_{ab}/N_B=q_{iab0}^B\mathrm{,}{q}_{ib}=p_{ib}{N}_b/N_B=p_{ib0}$ (note that $q_{ia}$, etc., are used to simplify the notations); $H_0\text{: }{p}_{i\mathrm{,}PML}=p_{i\mathrm{0,}PML},$ in which we test whether the PMLE of proportions from the union of the two frames in (2.9) are some specific values (note that $p_i$ can be estimated by other methods); $H_0:p_{ia}=p_{iab}=p_{ib},$ testing whether the proportions are equal in the three domains; or $H_0:p_{ij}=p_{i+}{p}_{+j},$ testing independence of the row classification and column classification.

Let $\eta ={\left({p^{\prime }}_a{N}_a/N_A\mathrm{,}{p^{\prime }}_{ab}{N}_{ab}/N_A\mathrm{,}{p^{\prime }}_b{N}_b/N_B\mathrm{,}{p^{\prime }}_{ab}{N}_{ab}/N_B\right)}^{\prime }\mathrm{,}$ $p_a={\left(p_{1a}\mathrm{,}{p}_{2a}\mathrm{,}\cdots p_{ka}\right)}^{\prime },$ $p_b={\left(p_{1b}\mathrm{,}{p}_{2b}\mathrm{,}\cdots p_{kb}\right)}^{\prime }\mathrm{,}$ $p_{ab}={\left(p_{1ab}\mathrm{,}{p}_{2ab}\mathrm{,}\cdots p_{(k-1)ab}\right)}^{\prime },$ and $h_i$ ’s are continuous functions. A more general hypothesis of interest may be denoted as the following:

$$H_0\text{: }{h}_i(\eta )=0,i=1,2,\cdots r\mathrm{.}\text{ (3}\text{.1)}$$

Let ${\eta }_j$ be the $j\text{-th}$ element of $\eta$ and let $h(\eta )={\left(h_1(\eta ),h_2(\eta ),\cdots h_r(\eta )\right)}^{\prime }.$

Assume that $\partial h_i(\eta )/\partial {\eta }_j$ is continuous in a neighborhood of $\eta$ and that

$$\nabla =\frac{\partial h_i(\eta )}{\partial {\eta }_j}\text{ (3}\text{.2)}$$

has full rank. Also assume

1. A₁. There is a sequence of superpopulations $U_{A1}\subset U_{A2}\subset \dots \subset U_{At}\subset \dots$ as defined in Isaki and Fuller (1982).

2. A₂. Let ${\tilde{n}}_A$ and ${\tilde{n}}_B$ as defined in Section 2 and assume that ${\tilde{n}}_A$ and ${\tilde{n}}_B$ both increase such that ${\tilde{n}}_A/{\tilde{n}}_B\to \gamma$ for some $0<\gamma <1.$

3. A₃. Let ${\pi }_{it}^A=p\left(\text{psu }i\text{ is in sample from Frame }A\mathrm{,}\text{using population }{U}_{At}\right)$ and

   ${\pi }_{ijt}^A=p\left(\text{psus }i\text{ and }j\text{ are in sample from Frame }A\mathrm{,}\text{ using population }{U}_{At}\right)$ be the inclusion and joint inclusion probabilities for the frame- $A$ sample from population $U_{At},$ and define ${\pi }_{it}^B,$ ${\pi }_{ijt}^B$ and $U_{Bt}$ similarly for frame $B.$ Assume there are constants $c_1$ and $c_2$ such that

   $$0<c_2<{\pi }_{it}^F<c_1<1\text{ (3}\text{.3)}$$

   for all $i$ and any superpopulation in the sequence, where $F$ denotes frame $A$ or frame $B.$ Also assume there exists an ${\alpha }_t$ with ${\alpha }_t=o(1)$ such that

   $${\pi }_{it}^F{\pi }_{jt}^F-{\pi }_{ijt}^F\le {\alpha }_t{\pi }_{it}^F{\pi }_{jt}^F\mathrm{.}\text{ (3}\text{.4)}$$

4. A₄. $N_{ab}/N\to \psi$ for some $\psi$ between 0 and 1.

Theorem 1. With assumptions $A_1-A_4$ set out beforehand, we have the following conclusion: ${\tilde{n}}^{1/2}h\left(\widehat{\eta }\right)$ is asymptotically normal with mean $0$ and asymptotic variance $\nabla \Sigma \nabla \text{'},$ where $\Sigma$ is a block-diagonal matrix with blocks ${\Sigma }_A$ and ${\Sigma }_B$ and $\tilde{n}={\tilde{n}}_A+{\tilde{n}}_B.$ ${\Sigma }_A$ is the asymptotic covariance matrix of ${\tilde{n}}^{1/2}{\widehat{\eta }}_A$ with ${\widehat{\eta }}_A={\left({{\widehat{p}}^{\prime }}_a{\widehat{N}}_a/N_A\mathrm{,}{\widehat{p}}_{ab}^A{}^{\prime }{\widehat{N}}_{ab}/N_A\right)}^{\prime },$ ${\Sigma }_B$ is the asymptotic covariance matrix of ${\tilde{n}}^{1/2}{\widehat{\eta }}_B$ with ${\widehat{\eta }}_B={\left({{\widehat{p}}^{\prime }}_b{\widehat{N}}_b/N_B\mathrm{,}{\widehat{p}}_{ab}^B{}^{\prime }{\widehat{N}}_{ab}/N_B\right)}^{\prime }$ and $\widehat{\eta }={\left({{\widehat{\eta }}^{\prime }}_A\mathrm{,}{{\widehat{\eta }}^{\prime }}_B\right)}^{\prime }.$

 

Proof. The arguments given in Theorem $1$ in Lu and Lohr (2010) show that $\widehat{\eta }$ is consistent for $\eta$ and that $\widehat{\eta }$ obeys the central limit theorem, as ${\tilde{n}}_A$ and ${\tilde{n}}_B$ both increase such that ${\tilde{n}}_A/{\tilde{n}}_B\to \gamma .$ Thus, since the samples $S_A$ and $S_B$ are selected independently, we have

$${\tilde{n}}^{1/2}\left(\widehat{\eta }-\eta \right)\stackrel{d}{\to }N\left(\mathrm{0,}\Sigma \right)\mathrm{.}$$

$h\left(\widehat{\eta }\right)$ is consistent for $h\left(\eta \right)$ because $\widehat{\eta }$ is consistent for $\eta$. Using the delta method, ${\tilde{n}}^{1/2}h\left(\widehat{\eta }\right)$ is asymptotically normal with mean $0$ and asymptotic variance $\nabla \Sigma \nabla \text{'}.$

Based on Theorem 1, the following results follow immediately.

Result 1. (Extended Wald Test) If a consistent estimator of the variance $\Sigma$ is available, by Theorem 1, the generalized Wald statistic can be formed as follows:

$$X_W^2=\tilde{n}h{\left(\widehat{\eta }\right)}^{\prime }{\left(\widehat{\nabla }\widehat{\Sigma }{\widehat{\nabla }}^{\prime }\right)}^{-1}h\left(\widehat{\eta }\right).\text{ (3}\text{.5)}$$

This test statistic is distributed asymptotically as ${\chi }^2(r)$ under $H_0$ (refer to equation 3.1), where $r$ is the rank of $\nabla .$

As we have noted previously, the estimate of the variance may be unstable or no closed-form estimate of $\Sigma$ is available. One way we can modify the statistic in (3.5) is to first act as though the sample is a simple random sample, then modify the reference distribution used in the test to get the correct level. Equation (3.6) gives the modified statistic.

Result 2. Let

$$X_{MW}^2=\tilde{n}h{\left(\widehat{\eta }\right)}^{\prime }{\left({\widehat{\nabla }}_0{\widehat{P}}_0{{\widehat{\nabla }}^{\prime }}_0\right)}^{-1}h\left(\widehat{\eta }\right)\mathrm{,}\text{ (3}\text{.6)}$$

where ${\widehat{\nabla }}_0{\widehat{P}}_0{{\widehat{\nabla }}^{\prime }}_0$ can be any estimate of $\nabla P\nabla$ that is consistent when $H_0$ is true. Matrix $P_0$ is a block diagonal matrix with diagonal blocks: covariance matrix from frame $A$ and covariance matrix from frame $B$ when $H_0$ is true and when sampling is SRS. Suppose the matrix $\nabla$ has rank $r$ under the null hypothesis $H_0:h(\eta )=0.$ Then $X_{MW}^2\approx {\displaystyle {\sum }_1^r{\lambda }_{0i}{W}_i},$ where the ${\lambda }_i$ ’s are the eigenvalues of ${\left(\nabla P{\nabla }^{\prime }\right)}^{-1}\left(\nabla \Sigma {\nabla }^{\prime }\right),$ $W_1,\dots ,W_r$ are independent ${\chi }_1^2$ random variables and ${\lambda }_{0i}$ is the value of ${\lambda }_i$ under $H_0.$

Result 3. (Extended Rao-Scott first order correction) Suppose matrix $\nabla$ has rank $r.$ Let $X_{MW}^2$ be as defined in (3.6). Under the null hypothesis $H_0:h(\eta )=0,$ the statistic $X_{MW}^2/\widehat{\lambda }\mathrm{.}$ has expectation $r,$ where $\widehat{\lambda }.={\displaystyle \sum {\widehat{\lambda }}_i/r},$ ${\widehat{\lambda }}_i$ is a consistent estimate of ${\lambda }_i$ under $H_0.$ For example, ${\widehat{\lambda }}_i$ ’s could be the eigenvalues of ${\left({\widehat{\nabla }}_0{\widehat{P}}_0{\widehat{\nabla }}_0{}^{\prime }\right)}^{-1}\left(\widehat{\nabla }\widehat{\Sigma }{\widehat{\nabla }}^{\prime }\right).$

Result 4. (Extended Rao-Scott second order correction) Suppose matrix $\nabla$ has rank $r.$ Define

$$X_S^2=\frac{X_{MW}^2}{\widehat{\lambda }\mathrm{.}\left(1+{\widehat{a}}^2\right)}$$

where ${\widehat{a}}^2={\displaystyle {\sum }_{i=1}^{k-1}{\widehat{\lambda }}_i^2}/\left[\left(k-1\right)\widehat{\lambda }{\mathrm{.}}^2\right]-1$ is an estimate of the population value $a^2.$ Under null hypothesis, $X_S^2$ is distributed asymptotically as ${\chi }_v^2,$ a chi-square random variable with degrees of freedom $v=\left(k-1\right)/\left(1+a^2\right).$

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