Robust variance estimators for generalized regression estimators in cluster samples
Section 4. Conclusion

Table of contents

Leverage adjustments to standard variance estimators have been shown to reduce bias and improve confidence interval coverage based on general regression estimators in single-stage samples. This paper extends those results to two-stage samples by presenting new adjustments based on hat matrices. Our theory provides the justification for the adjustments and illustrates that some of the proposed estimators are related to the delete-a-cluster jackknife that is a common procedure in survey estimation.

To test the theory, we conducted a series of simulation studies on three populations designed to assess performance in a variety of situations. In a school population a large sampling fraction of first-stage units was used. In a second population, based on American Community Survey data, the effects of small sample sizes were tested. In a third simulated population, we examined large sample performance. Both simple random sampling and probability proportional to size sampling of clusters were used.

The relationships of the variance estimators were similar across all sample designs. The with-replacement variance estimator, $υ_{w r},$ which is the default choice in survey software packages, the jackknife linearization estimator, $υ_{J L},$ and the design-based variance estimator, $υ_{g},$ that assumes Poisson sampling at each stage as a computational convenience, are often negatively biased leading to confidence intervals that cover at less than the desired rate. Some of the jackknife-related estimators – $υ_{Jack},$ $υ_{J 1},$ and $υ_{J 2}$ –which explicitly or implicitly include hat-matrix adjustments, are prone to producing large, outlying values when the first-stage sample is small. This is especially true when the first-stage is selected by srs but is less so in pps sampling when an efficient measure of size is used.

The variance estimators proposed here, particularly $υ_{D},$ provide alternatives to estimating the variance of GREG estimators in complex samples. At the expense of somewhat inflating the variability of the variance estimator, the hat-matrix adjusted sandwich estimators, denoted here by $v_{D},$ $v_{J 1},$ and $v_{J 2},$ give confidence interval coverage that is closer to the nominal value in small to moderate samples. Depending on the sample design and population characteristics, hat-matrix adjusted estimators can produce less biased variance estimates and better inferences when compared to the standard methods.

Acknowledgements

The authors thank the associate editor and two referees whose comments substantially improved the presentation.

Appendix

Theoretical results

A.1 Assumptions

The assumptions used to obtain asymptotic results are listed below. The number of population and sample clusters approach infinity; however, the number of population clusters increases at a faster rate than the number of sample clusters. Certain population quantities are assumed to be bounded.

A.1.1

m / M \to 0

m \to \infty

and

M \to \infty .

A.1.2

All

N_{i}

and

n_{i}

are bounded.

A.1.3

π_{i k} = O (m / M)

for all

i k .

A.1.4

All elements of

X,

Ψ,

and

Q

are bounded.

A.1.5

The sample design is such that

\frac{\sqrt{m}}{M} ({\hat{t}}_{x π} - t_{U x}) \overset{d}{\to} N (0, V),

where

V

is a

p \times p

positive definite matrix, i.e.,

({\hat{t}}_{x π} - t_{U x}) = O_{p} (M / \sqrt{m}) .

Since $Π = O (\frac{m}{M})$ elementwise and $A = X_{s}^{⊤} Q^{- 1} Π^{- 1} X_{s}$ can be written as the sum of $n$ terms and $n_{i}$ is bounded while $m \to \infty,$ $A = O (M) .$ By definition $g_{i}^{⊤} = 1_{n_{i}} + {(t_{U x} - {\hat{t}}_{x π})}^{⊤} A^{- 1} X_{i}^{⊤} Q_{i} .$ The second term in $g_{i}$ is $O_{p} (m^{- 1 / 2});$ consequently $g_{i}$ converges to a vector of 1’s. Using $A = O (M)$ along with assumptions A.1.3 and A.1.4, $H_{i j}$ is $O (m^{- 1})$ elementwise.

A.2 Model variance of GREG

Let $y_{s i}$ be the vector of all sample elements in cluster $i$ and $y_{i}$ be the vector of all elements in cluster $i .$ The variance of the GREG, with respect to the working model (2.1) is:

$\begin{array}{l} {var}_{ξ} ({\hat{t}}_{y}^{g r} - t_{y}) & = {var}_{ξ} (\sum_{i \in s} g_{i}^{⊤} Π_{i}^{- 1} y_{s i} - \sum_{i \in U} 1_{N_{i}}^{⊤} y_{i}) \\ = \sum_{i \in s} g_{i}^{⊤} Π_{i}^{- 1} Ψ_{s i} Π_{i}^{- 1} g_{i} - 2 {cov}_{ξ} (\sum_{i \in s} g_{i}^{⊤} Π_{i}^{- 1} y_{s i}, \sum_{i \in U} 1_{N_{i}}^{⊤} y_{i}) + 1_{N}^{⊤} Ψ 1_{N} . \end{array}$

Since $\sum_{i \in U} 1_{i}^{⊤} y_{i} = \sum_{i \in s} 1_{i}^{⊤} y_{i} + \sum_{i \in (U - s)} 1_{i}^{⊤} y_{i}$ and elements in different clusters are uncorrelated, we have,

$\begin{array}{l} {var}_{ξ} ({\hat{t}}_{y}^{g r} - t_{y}) & = \sum_{i \in s} g_{i}^{⊤} Π_{i}^{- 1} Ψ_{s i} Π_{i}^{- 1} g_{i} - 2 \sum_{i \in s} [g_{i}^{⊤} Π_{i}^{- 1} {cov}_{ξ} (y_{s i} , y_{i}) 1_{N_{i}}] + 1_{N}^{⊤} Ψ 1_{N} \\ = L_{1} - 2 L_{2} + L_{3} . \end{array}$

Since $A^{- 1} = O (M^{- 1})$ and $g_{i}$ and $Ψ_{s i}$ are bounded, we have $L_{1} = O (M^{2} / m) .$ Because $Ψ_{s i}$ is bounded, ${cov}_{ξ} (y_{s i} , y_{i}) = O (1)$ and $L_{2} = O (M) .$ $L_{3}$ is the sum of $N$ terms. Since the $N_{i}$ are bounded, $L_{3} = O (M) .$ Thus, $L_{1}$ is the dominant term of the prediction variance.

A.3 Proof that ${var}_{ξ} (e_{i}) \approx Ψ_{s i}$

In this section in order to simplify the notation, we omit the subscript $s$ on $y_{s i},$ ${\hat{y}}_{s i},$ and $Ψ_{s i} .$ The residual can be written in terms of a hat matrix as follows.

$\begin{array}{l} e_{i} & = y_{i} - {\hat{y}}_{i} \\ = (I_{n_{i}} - H_{i i}) y_{i} - \sum_{j \neq i; i , j \in s} H_{i j} y_{j} \end{array}$

where $I_{n_{i}}$ is the $n_{i} \times n_{i}$ identity matrix. The model variance of $e_{i}$ is then

$\begin{array}{l} {var}_{ξ} (e_{i}) & = {var}_{ξ} [(I_{n_{i}} - H_{i i}) y_{i} - \sum_{j \neq i} H_{i j} y_{j}] \\ = (I_{n_{i}} - H_{i i}) {var}_{ξ} (y_{i}) {(I_{n_{i}} - H_{i i})}^{⊤} + \sum_{j \neq i} H_{i j} {var}_{ξ} (y_{j}) H_{i j}^{⊤} \\ = (I_{n_{i}} - H_{i i}) Ψ_{i} {(I_{n_{i}} - H_{i i})}^{⊤} + \sum_{j \neq i} H_{i j} Ψ_{j} H_{i j}^{⊤} . (A .1) \end{array}$

As noted above, $H_{i i} = O (m^{- 1}) .$ Thus, ${var}_{ξ} (e_{i}) = Ψ_{i} + O (m^{- 1}) .$

To justify $υ_{D},$ note that the second term of (A.1) can be written as

$\sum_{j \neq i} H_{i j} Ψ_{j} H_{i j}^{⊤} = \sum_{j \in s} H_{i j} Ψ_{j} H_{i j}^{⊤} - H_{i i} Ψ_{i} H_{i i}^{⊤} .$

The sum over the full cluster sample is

$\sum_{j \in s} H_{i j} Ψ_{j} H_{i j}^{⊤} = X_{i} A^{- 1} (\sum_{j \in s} X_{j}^{⊤} Q_{j} Π_{j}^{- 1} Ψ_{j} Π_{j}^{- 1} Q_{j} X_{j}) A^{- 1} X_{i}^{⊤} .$

In the special case of $Q_{j} = Ψ_{j}^{- 1}$ and $Π_{i} = c I_{n_{i}}$ for some constant $c \in (0, 1)$ (i.e., the sample is self-weighting), we have

$\sum_{j \in s} H_{i j} Ψ_{j} H_{i j}^{⊤} = c^{- 2} X_{i} A^{- 1} (\sum_{j \in s} X_{j}^{⊤} Ψ_{j}^{- 1} X_{j}) A^{- 1} X_{i}^{⊤},$

along with $H_{i i} = c X_{i} A^{- 1} X_{i}^{⊤} Ψ_{i}^{- 1}$ and $A = c^{- 1} X Ψ^{- 1} X .$ Using these simplifications, $\sum_{j \in s} H_{i j} Ψ_{j} H_{i j}^{⊤} = H_{i i} Ψ_{i} .$ Substituting this result in (A.1) and simplifying gives

$\begin{array}{l} {var}_{ξ} (e_{i}) & = (I_{n_{i}} - H_{i i}) Ψ_{i} {(I_{n_{i}} - H_{i i})}^{⊤} + \sum_{j \neq i} H_{i j} Ψ_{j} H_{i j}^{⊤} \\ = (I_{n_{i}} - H_{i i}) Ψ_{i} . (A .2) \end{array}$

This is the basis for the adjustment of $υ_{R}$ to obtain $υ_{D} .$

A.4 Proof that ${\hat{B}}_{(i)} = \hat{B} - R_{i}$ for cluster samples

In this section, we omit the subscript $s$ on $X_{s},$ $y_{s},$ $X_{s i},$ $y_{s i},$ $X_{s (i)},$ and $y_{s (i)}$ to simplify the notation. The subscript $(i)$ denotes removal of the $i^{th}$ cluster from the full sample matrix or vector. For example, ${\hat{B}}_{(i)}$ is an estimate of $B$ based on all sample clusters except cluster $i$ and is

${\hat{B}}_{(i)} = {(X_{(i)}^{⊤} W_{(i)} X_{(i)})}^{- 1} X_{(i)}^{⊤} W_{(i)} y_{(i)}$

where $W_{(i)} = Q_{(i)} Π_{(i)}^{- 1} .$ Using Lemma 9.5.1 in Valliant et al. (2000), we have

${\hat{B}}_{(i)} = (A^{- 1} + A^{- 1} X_{i}^{⊤} W_{i} {(I_{n_{i}} - H_{i i})}^{- 1} X_{i} A^{- 1}) X_{(i)}^{⊤} W_{(i)} y_{(i)} .$

Since $X_{(i)}^{⊤} W_{(i)} y_{(i)} = X^{⊤} W y - X_{i}^{⊤} W_{i} y_{i}$ and $\hat{B} = A^{- 1} X^{⊤} W y,$ we have

$\begin{array}{l} {\hat{B}}_{(i)} & = A^{- 1} (X^{⊤} W y - X_{i}^{⊤} W_{i} y_{i}) \\ + A^{- 1} X_{i}^{⊤} W_{i} {(I_{n_{i}} - H_{i i})}^{- 1} X_{i} A^{- 1} (X^{⊤} W y - X_{i} W_{i} y_{i}) \\ = \hat{B} - A^{- 1} X_{i}^{⊤} W_{i} {(I_{n_{i}} - H_{i i})}^{- 1} (I_{n_{i}} - H_{i i}) y_{i} + A^{- 1} X_{i}^{⊤} W_{i} {(I_{n_{i}} - H_{i i})}^{- 1} {\hat{y}}_{i} \\ - A^{- 1} X_{i}^{⊤} W_{i} {(I_{n_{i}} - H_{i i})}^{- 1} H_{i i} y_{i} \\ = \hat{B} - A^{- 1} X_{i}^{⊤} W_{i} {(I_{n_{i}} - H_{i i})}^{- 1} e_{i} . \end{array}$

That is, ${\hat{B}}_{(i)} = \hat{B} - R_{i} .$

A.5 Jackknife variance estimator of clustered GREG in terms of leverages

We now simplify the delete-a-cluster Jackknife variance estimator of the clustered GREG. As in Sections A.3 and A.4, we omit the subscript $s$ on various terms to simplify the notation. The estimated total after removing the $i^{th}$ cluster is defined as

$\begin{array}{l} {\hat{t}}_{y (i)}^{g r} & = \frac{m}{m - 1} {\hat{t}}_{y (i)}^{π} + [t_{U x} - \frac{m}{m - 1} {\hat{t}}_{x (i)}^{π}] {\hat{B}}_{(i)} \\ = \frac{m 1_{n}^{⊤} Π^{- 1} y}{m - 1} - \frac{m 1_{n_{i}}^{⊤} Π_{i}^{- 1} y_{i}}{m - 1} + [1_{N}^{⊤} X_{U} - \frac{m 1_{n}^{⊤} Π^{- 1} X}{m - 1} + \frac{m 1_{n_{i}}^{⊤} Π_{i}^{- 1} X_{i}}{m - 1}] (\hat{B} - R_{i}) \\ = \frac{m 1_{n}^{⊤} Π^{- 1} y}{m - 1} - \frac{m 1_{n_{i}}^{⊤} Π_{i}^{- 1} y_{i}}{m - 1} \\ + \frac{m}{m - 1} (1_{N}^{⊤} X_{U} - 1_{n}^{⊤} Π^{- 1} X) (\hat{B} - R_{i}) - \frac{1}{m - 1} (1_{N}^{⊤} X_{U} - m 1_{n_{i}}^{⊤} Π_{i}^{- 1} X_{i}) (\hat{B} - R_{i}) \\ = \frac{m}{m - 1} {\hat{t}}_{y}^{g r} - \frac{m 1_{n_{i}}^{⊤} Π_{i}^{- 1} y_{i}}{m - 1} - \frac{m}{m - 1} (1_{N}^{⊤} X_{U} - 1_{n}^{⊤} Π^{- 1} X) R_{i} - \frac{1}{m - 1} K_{i} . \end{array}$

Adding and subtracting $\frac{m}{m - 1} 1_{n_{i}}^{⊤} Π_{i}^{- 1} {(I_{n_{i}} - H_{i i})}^{- 1} e_{i}$ and doing a substantial amount of simplification leads to

${\hat{t}}_{y (i)}^{g r} = \frac{m}{m - 1} {\hat{t}}_{y}^{g r} - \frac{m}{m - 1} g_{i}^{⊤} Π_{i}^{- 1} {(I_{n_{i}} - H_{i i})}^{- 1} e_{i} + \frac{m}{m - 1} G_{i} - \frac{1}{m - 1} K_{i} .$

Taking the difference between the delete-one estimates and the average of those estimates gives

$\begin{array}{l} {\hat{t}}_{y (i)}^{g r} - {\hat{t}}_{y (\cdot)}^{g r} & = - \frac{m}{m - 1} (D_{i} - \bar{D}) + \frac{m}{m - 1} (G_{i} - \bar{G}) - \frac{1}{m - 1} (K_{i} - \bar{K}) \\ = - \frac{m}{m - 1} (D_{i} - \bar{D}) + \frac{m}{m - 1} [(G_{i} - \bar{G}) - \frac{1}{m} (K_{i} - \bar{K})] . \end{array}$

Letting $F_{i} = (G_{i} - \bar{G}) - m^{- 1} (K_{i} - \bar{K})$ leads to the formula for $υ_{Jack}$ in equation (2.12). Next, since $H_{i i} = O (m^{- 1})$ and ${\hat{y}}_{i} = X_{i} \hat{B},$

$\begin{array}{l} F_{i} & = (G_{i} - \bar{G}) - \frac{1}{m} (K_{i} - \bar{K}) \\ \approx [- 1_{n_{i}}^{⊤} Π_{i}^{- 1} {\hat{y}}_{i} + \frac{1}{m} \sum_{i \in s} 1_{n_{i}}^{⊤} Π_{i}^{- 1} {\hat{y}}_{i}] - \frac{1}{m} [- m 1_{n_{i}}^{⊤} Π_{i}^{- 1} X_{i} \hat{B} + \sum_{i \in s} 1_{n_{i}}^{⊤} Π_{i}^{- 1} X_{i} \hat{B}] \\ = 0 . \end{array}$

Thus, $F_{i} = o (1),$ and $υ_{Jack}$ in (2.6) and (2.12) is asymptotically equivalent to $υ_{J 1}$ in (2.13).

Finally, to justify $υ_{J 2}$ in (2.14), we write $υ_{J 1}$ in the computational form

$υ_{J 1} = \frac{m}{m - 1} [\sum_{i \in s} {(g_{i}^{⊤} U_{i} e_{i})}^{2} - \frac{1}{m} {(\sum_{i \in s} g_{i}^{⊤} U_{i} e_{i})}^{2}] (A .3)$

where $U_{i} = Π_{i}^{- 1} {(I_{n_{i}} - H_{i i})}^{- 1} .$ Note that the model variance of $D_{i}$ is

$\begin{array}{l} {var}_{ξ} (D_{i}) & = {var}_{ξ} (g_{i}^{⊤} U_{i} e_{i}) \\ = g_{i}^{⊤} U_{i}^{⊤} {var}_{ξ} (e_{i}) U_{i} g_{i} . \end{array}$

Because $U_{i} = O (M / m)$ and the sum in $\sum_{i \in s} {var}_{ξ} (D_{i})$ contains $n = m \bar{n}$ terms, the variance of $\sum_{i \in s} g_{i}^{⊤} U_{i} e_{i}$ is $O (M^{2} / m) .$ Next, scaling $υ_{J 1}$ to be appropriate for a mean, the first term in the brackets in (A.3) is $N^{- 2} \sum_{i \in s} D_{i}^{2} = O (m^{- 1}) .$ Since the second term in brackets has model expectation 0 and variance that is $O (m^{- 1}),$ it converges in probability to 0, and $υ_{J 2}$ is asymptotically equivalent to $υ_{J 1} .$

A.6 Asymptotic equivalence of variance estimators

In this appendix we sketch arguments for why several variance estimators are asymptotically equivalent. Using design-based arguments, Yung and Rao (1996, Appendix) showed that the jackknife linearization estimator, $υ_{J L}$ , for the GREG is asymptotically equivalent to the design-consistent estimator, $υ_{Jack},$ in stratified multistage designs with a large number of strata and a bounded number of sample clusters selected from each stratum. Using regularity conditions in Rao and Shao (1985), that result can be extended to cover designs in which either (i) the number of strata is large and the number of clusters per stratum is bounded or (ii) the number of strata is limited and the number of sample clusters per stratum is large, as is the case in this article.

The jackknife linearization estimator in Section 2 can be expanded as

$N^{- 2} υ_{J L} = N^{- 2} \sum_{i \in s} g_{i}^{⊤} Π_{i}^{- 1} e_{i} e_{i}^{⊤} Π_{i}^{- 1} g_{i} - N^{- 2} m {(m^{- 1} \sum_{i \in s} g_{i}^{⊤} Π_{i}^{- 1} e_{i})}^{2} . (A .4)$

The first term in (A.4) equals $v_{R} .$ Because, under some reasonable assumptions, $g_{i}$ and $e_{i}$ are bounded, and $Π_{i}^{- 1} = O (M / m)$ by assumptions A.1.2 and A.1.3, the first term in (A.4) is $O (1 / m) .$ The second term is also $O (1 / m),$ but the model expectation of ${\bar{e}}_{2} = m^{- 1} \sum_{i \in s} g_{i}^{⊤} Π_{i}^{- 1} e_{i}$ is zero as long as (2.1) holds. Since ${\bar{e}}_{2}$ is a mean, its model-variance will approach 0 as $m \to \infty .$ Thus, the second term in (A.4) will converge in probability to 0 and $υ_{J L} \approx υ_{R} .$

In Section A.5 it was shown that $υ_{Jack}$ and $υ_{J 1}$ are asymptotically equivalent. Under A.1.1-A.1.4, $H_{i i} = O (m^{- 1}) .$ Consequently, $υ_{J 2}$ and $υ_{D}$ are approximately the same as $υ_{R}$ as $m \to \infty .$ Thus, $υ_{Jack} \approx υ_{J L}$ by extension of Yung and Rao (1996), both of which are design-consistent. Further, $υ_{J L}$ is asymptotically equivalent to $υ_{J 1},$ $υ_{J 2},$ $υ_{D},$ and $υ_{R} .$ As a result, the alternative variance estimators considered here all have both model-based and design-based justifications.

References

Kott, P.S. (1988). Model-based finite population correction for the Horvitz-Thompson estimator. Biometrika, 75(4), 797-799.

Krewski, D., and Rao, J.N.K. (1981). Inference from stratified samples: Properties of the linearization, jackknife and balanced repeated replication methods. The Annals of Statistics, 9(5), 1010-1019.

Li, J., and Valliant, R. (2009). Survey weighted hat matrix and leverages. Survey Methodology, 35, 1, 15-24. Paper available at https://www150.statcan.gc.ca/n1/en/pub/12-001-x/2009001/article/10881-eng.pdf.

Long, J.S., and Ervin, L.H. (2000). Using heteroscedasticity consistent standard errors in the linear regression model. The American Statistician, 54(3), 217-224.

MacKinnon, J.G., and White, H. (1985). Some heteroskedasticity consistent covariance matrix estimators with improved finite sample properties. Journal of Econometrics, 29(3), 305-325.

Rao, J.N.K., and Shao, J. (1985). Inference from stratified samples: Second-order analysis of three methods for nonlinear statistics. Journal of the American Statistical Association, 80(391), 620-630.

Royall, R.M., and Cumberland, W.G. (1978). Variance estimation in finite population sampling. Journal of the American Statistical Association, 73(362), 351-358.

Särndal, C.-E., Swensson, B. and Wretman, J. (1989). The weighted residual technique for estimating the variance of the general regression estimator of the finite population total. Biometrika, 76(3), 527-537.

Särndal, C.-E., Swensson, B. and Wretman, J. (1992). Model Assisted Survey Sampling. Springer Series in Statistics. New York: Springer-Verlag.

Valliant, R. (2002). Variance estimation for the general regression estimator. Survey Methodology, 28, 1, 103-114. Paper available at https://www150.statcan.gc.ca/n1/en/pub/12-001-x/2002001/article/6424-eng.pdf.

Valliant, R., Dorfman, A.H. and Royall, R.M. (2000). Finite Population Sampling and Inference: A Prediction Approach. Wiley Series in Probability and Statistics: Survey Methodology Section. New York: John Wiley & Sons, Inc.

Yung, W., and Rao, J.N.K. (1996). Jackknife linearization variance estimators under stratified multi-stage sampling. Survey Methodology, 22, 1, 23-31. Paper available at https://www150.statcan.gc.ca/n1/en/pub/12-001-x/1996001/article/14388-eng.pdf.

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Robust variance estimators for generalized regression estimators in cluster samples
Section 4. Conclusion

Acknowledgements

Appendix

Theoretical results

A.1 Assumptions

A.2 Model variance of GREG

A.3 Proof that ${var}_{ξ} (e_{i}) \approx Ψ_{s i}$

A.4 Proof that ${\hat{B}}_{(i)} = \hat{B} - R_{i}$ for cluster samples

A.5 Jackknife variance estimator of clustered GREG in terms of leverages

A.6 Asymptotic equivalence of variance estimators

References

Robust variance estimators for generalized regression estimators in cluster samples Section 4. Conclusion

Acknowledgements

Appendix

Theoretical results

A.1 Assumptions

A.2 Model variance of GREG

A.5 Jackknife variance estimator of clustered GREG in terms of leverages

A.6 Asymptotic equivalence of variance estimators

References

Editorial policy

Submission of Manuscripts

Note of appreciation

Standards of service to the public

Copyright

Robust variance estimators for generalized regression estimators in cluster samples
Section 4. Conclusion