Optimal linear estimation in two-phase sampling
Section 7. Simulation study

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We have conducted a simulation study to assess the performance of the proposed two-phase estimators of the total $t_y,$ for scalar variables $y,$ $x_1$ and $x_2,$ and compare them with the competing regression estimators considered above. Distributions of these variables were specified as follows. The distribution of $x_1$ is the lognormal with mean and variance parameters $\mathrm{<mo\; stretchy="false">(</mo>}{\mu }_{x_1}=\mathrm{4,}{\sigma }_{x_1}^2=\mathrm{4\; <mo\; stretchy="false">)</mo>\; .}$ The distribution of $x_2$ is specified by the linear model $x_2=5+x_1+\in ,$ where $\in ~N\mathrm{<mo\; stretchy="false">(</mo>0,}{\sigma }_{\in }^2),$ and the distribution of $y$ is specified by the linear model $y=10+2x_1+3x_2+\eta ,$ where $\eta ~N\mathrm{<mo\; stretchy="false">(</mo>\; 0,}{\sigma }_{\eta }^2\mathrm{<mo\; stretchy="false">)</mo>\; .}$ The value of ${\sigma }_{\in }^2$ determines the linear relationship between $x_2$ and $x_1,$ as defined by the population square correlation coefficient $r_{x_1\mathrm{,}{x}_2}^2,$ and the value of ${\sigma }_{\eta }^2$ determines the linear relationship of $y$ with $x_1$ and $x_2,$ as defined by the coefficient of determination $r^2=\mathrm{<mo\; stretchy="false">[</mo>}{r}_{y\mathrm{,}{x}_1}^2+r_{y\mathrm{,}{x}_2}^2-2r_{y\mathrm{,}{x}_1}{r}_{y\mathrm{,}{x}_2}{r}_{x_1\mathrm{,}{x}_2}\mathrm{<mo\; stretchy="false">]</mo>}/\mathrm{<mo\; stretchy="false">(</mo>\; 1}-r_{x_1\mathrm{,}{x}_2}^2\mathrm{<mo\; stretchy="false">)</mo>}.$

Three values, 0, 0.25, 0.75, were specified for $r_{x_1\mathrm{,}{x}_2}^2,$ and two values, 0.25 and 0.75, for $r^2,$ giving six combinations of values $\mathrm{<mo\; stretchy="false">(</mo>}{r}_{x_1\mathrm{,}{x}_2}^2\mathrm{,}{r}^2\mathrm{<mo\; stretchy="false">)</mo>\; .}$ For the value $r_{x_1\mathrm{,}{x}_2}^2=0,$ in particular, the bivariate lognormal distribution for $\mathrm{<mo\; stretchy="false">(</mo>}{x}_1\mathrm{,}{x}_2\mathrm{<mo\; stretchy="false">)</mo>}$ with parameters $\mathrm{<mo\; stretchy="false">(</mo>}{\mu }_{x_1}=\mathrm{4,}{\sigma }_{x_1}^2=\mathrm{4\; <mo\; stretchy="false">)</mo>},$ $\mathrm{<mo\; stretchy="false">(</mo>}{\mu }_{x_2}=\mathrm{9,}{\sigma }_{x_2}^2=\mathrm{9\; <mo\; stretchy="false">)</mo>}$ and zero correlation was used. The required values of ${\sigma }_{\in }^2$ and ${\sigma }_{\eta }^2$ are readily determined, while values for $r_{y\mathrm{,}{x}_1}^2$ and $r_{y\mathrm{,}{x}_2}^2$ are implicitly specified. For each of these six combinations, a population of size $N\mathrm{<mo>=</mo>}$ 50,000 was simulated by generating values from the distributions of the components of the vector $\mathrm{<mo\; stretchy="false">(</mo>}y\mathrm{,}{x}_1\mathrm{,}{x}_2\mathrm{<mo\; stretchy="false">)</mo>\; .}$ Four combinations of first-phase and second-phase sample sizes $(n_1\mathrm{,}{n}_2)$ with fixed $n_1$ and varying $n_2$ were specified, i.e., (3,000; 2,000), (3,000; 1,500), (3,000; 1,000), (3,000; 500), thus creating a total of 24 simulation settings.

Three different two-phase sampling designs were considered. Simple random sampling (SRS) without replacement was first used in both phases. For this sampling design, denoted by (SRS, SRS), the BLUE ${\widehat{t}}_y^B$ in (3.10) and its exact variance in (3.11) can be calculated. Using the fact that under SRS the correlation of the HT estimators for two totals is identical to the correlation coefficient of the associated variables, tedious but straightforward algebra gives the relative difference (RDV) of variances of the estimators ${\widehat{t}}_y^B$ and ${\tilde{t}}_y$ as

$$\frac{\text{Var}\mathrm{<mo\; stretchy="false">(</mo>}{\tilde{t}}_y\mathrm{<mo\; stretchy="false">)</mo>}-\text{Var}\mathrm{<mo\; stretchy="false">(</mo>}{\widehat{t}}_y^B\mathrm{<mo\; stretchy="false">)</mo>}}{\text{Var}\mathrm{<mo\; stretchy="false">(</mo>}{\tilde{t}}_y\mathrm{<mo\; stretchy="false">)</mo>}}\mathrm{<mo>=</mo>}\frac{N\mathrm{<mo\; stretchy="false">(</mo>}{n}_1-n_2\mathrm{<mo\; stretchy="false">)</mo>}}{n_1\mathrm{<mo\; stretchy="false">(</mo>}N-n_2\mathrm{<mo\; stretchy="false">)</mo>}}{r}^2+\frac{n_2\mathrm{<mo\; stretchy="false">(</mo>}N-n_1\mathrm{<mo\; stretchy="false">)</mo>}}{n_1\mathrm{<mo\; stretchy="false">(</mo>}N-n_2\mathrm{<mo\; stretchy="false">)</mo>}}{r}_{y\mathrm{,}{x}_1}^2\mathrm{.}$$

The percent RDV is the measure of the efficiency of the BLUE ${\widehat{t}}_y^B$ relative to the HT estimator ${\tilde{t}}_y.$ This exact maximum efficiency will serve to measure the closeness of the approximation of the BLUE by the optimal estimator, for the different sample sizes, as well as the efficiency of the other competing estimators relative to the HT estimator. Notice that as $n_2$ tends to zero, the RDV tends to $r^2,$ and as $n_2$ tends to $n_1,$ the RDV tends to $r_{y\mathrm{,}{x}_1}^2$ (the efficiency of the BLUE based on $s_1$ and $x_1).$ The second two-phase design, denoted by (STRSRS, SRS), was stratified simple random sampling (STRSRS) and SRS in the first and second phase, respectively. The simulated populations were stratified by the size of the variable $y,$ with three strata of sizes $N_1\mathrm{<mo>=</mo>}$ 30,000, $N_2\mathrm{<mo>=</mo>}$ 15,000, $N_3\mathrm{<mo>=</mo>}$ 5,000 and proportional allocation of the sample $s_1$ to the three strata ‒ giving equal inclusion probabilities in each of the two phases. For this design too, the BLUE ${\widehat{t}}_y^B$ and its exact variance can be calculated. The third two-phase design, denoted by (SRS, PPSS), involved SRS in the first phase and probability proportional to size systematic sampling (PPSS) in the second phase, using as size measure the simple transformation $z_2=15+0.5x_2$ of the variable $x_2;$ using $x_2$ as size would result in ${\widehat{t}}_{x_2}={\tilde{t}}_{x_2}.$ In this case the BLUE ${\widehat{t}}_y^B$ (and the optimal estimator ${\widehat{t}}_y^O)$ cannot be calculated, because of the unknown probabilities ${\pi }_{2kl}.$ However, GREG estimators can be calculated.

For each of these three two-phase designs, and all the 24 simulation settings, sampling was repeated 30,000 times, and each time we computed the estimators ${\tilde{t}}_y,$ ${\widehat{t}}_y^O,$ ${\widehat{t}}_y^{\text{GR}},$ ${\widehat{t}}_y^{\text{ES}}$ and ${\widehat{t}}_y^{\text{HS}},$  to obtain their empirical bias and variance. In all these cases, the simulation showed that the bias of all estimators was negligible, even for the smaller subsample sizes $n_2.$  Thus their comparison is based on their variances relative to the benchmark variance of the HT estimator ${\tilde{t}}_y.$  Specifically, the efficiency of each of the competing estimators ${\widehat{t}}_y^O,$   ${\widehat{t}}_y^{\text{GR}},$   ${\widehat{t}}_y^{\text{ES}}$  and ${\widehat{t}}_y^{\text{HS}}$  is assessed through the percent relative difference between its empirical variance and the empirical variance of the estimator ${\tilde{t}}_y;$  for example, for ${\widehat{t}}_y^{\text{GR}}$  the relative difference is $\left[\text{Var}\mathrm{<mo\; stretchy="false">(</mo>}{\tilde{t}}_y\mathrm{<mo\; stretchy="false">)</mo>}-\text{Var}\mathrm{<mo\; stretchy="false">(</mo>}{\widehat{t}}_y^{\text{GR}}\mathrm{<mo\; stretchy="false">)</mo>}\right]/\text{Var}\mathrm{<mo\; stretchy="false">(</mo>}{\tilde{t}}_y\mathrm{<mo\; stretchy="false">)</mo>}.$  The relative difference shows the reduction of the variance of the particular estimator relative to the variance of the basic estimator ${\tilde{t}}_y.$

In the (SRS, SRS) design, the exact efficiency of the BLUE ${\widehat{t}}_y^B,$ relative to the HT estimator ${\tilde{t}}_y,$ increases as $n_2$ decreases and as we move to higher values of $(r_{y\mathrm{,}{x}_2}^2\mathrm{,}{r}^2),$ confirming Remark 3.1; see column 2 of Table 7.1. It is also confirmed that the efficiency of ${\widehat{t}}_y^B$ tends to $r^2$ as $n_2$ decreases, faster for higher $r_{x_1\mathrm{,}{x}_2}^2.$ This maximum efficiency is closely approximated by the empirical efficiency of ${\widehat{t}}_y^O,$ even for the smaller subsample sizes $n_2;$ see column 3 of Table 7.1. For the (STRSRS, SRS) design the exact efficiency of ${\widehat{t}}_y^B$ is shown in column 6 of Table 7.1, exhibiting a pattern similar to that in the (SRS, SRS). This efficiency is closely approximated by the empirical efficiency of ${\widehat{t}}_y^O;$ see column 7 of Table 7.1. In both (SRS, SRS) and (STRSRS, SRS) the approximation of ${\widehat{t}}_y^B$ by ${\widehat{t}}_y^O$ is a little weaker in some settings involving the largest value of $n_2,$ for the reason given in Remark 4.1.

Although the estimator ${\widehat{t}}_y^O$ can be calculated in the (SRS, SRS) and (STRSRS, SRS) designs, the performance of the more practical, and of general applicability, calibration (GREG) estimators ${\widehat{t}}_y^{\text{GR}}$ and ${\widehat{t}}_y^{\text{ES}}$ is of great interest. For (SRS, SRS), the empirical efficiencies of these estimators are shown in columns 4 and 5 of Table 7.1. The negative sign indicates loss of efficiency with respect to the HT estimator. The efficiency of ${\widehat{t}}_y^{\text{GR}}$ approximates closely the efficiency of ${\widehat{t}}_y^O,$ except for the four settings specified by $r_{x_1\mathrm{,}{x}_2}^2=$ 0, $r^2=$ 0.25, 0.75 and $n_2=$ 2,000; 1,500; in particular, when $n_2=$ 2,000 the estimator ${\widehat{t}}_y^{\text{GR}}$ is a little less efficient than the estimator ${\tilde{t}}_y.$ In contrast, the estimator ${\widehat{t}}_y^{\text{ES}}$ is less efficient than the estimator ${\tilde{t}}_y$ in six settings, when $r_{x_1\mathrm{,}{x}_2}^2=$ 0, $r^2=$ 0.25, 0.75 and $n_2=$ 2,000; 1,500; 1,000; substantially less efficient when $n_2=$ 2,000; 1,500. The highlight in columns 4 and 5 is that the estimator ${\widehat{t}}_y^{\text{GR}}$ is much more efficient than the estimator ${\widehat{t}}_y^{\text{ES}}$ in all settings, more so for higher values of $n_2$ and for the higher values of $(r_{y\mathrm{,}{x}_2}^2\mathrm{,}{r}^2);$ this indicates that ${\widehat{t}}_y^{\text{GR}}$ is more effective in using information from the complement of $s_2$ and in exploiting higher correlations of $y$ with $x_1$ and $x_2.$ The efficiency of the estimator ${\widehat{t}}_y^{\text{HS}}$ was virtually identical with that of ${\widehat{t}}_y^{\text{ES}},$ in all three designs, and hence is not reported in Table 7.1. For (STRSRS, SRS), the empirical efficiencies of the calibration estimators ${\widehat{t}}_y^{\text{GR}}$ and ${\widehat{t}}_y^{\text{ES}}$ are shown in columns 8 and 9 of Table 7.1. It should be noted that the correlations within the strata are much weaker than the correlations for the whole population (shown in Table 7.1). Also, the HT estimator ${\tilde{t}}_y$ is highly efficient because of the stratification, especially for the larger values of $n_2.$ The estimator ${\widehat{t}}_y^{\text{GR}}$ is less efficient than the estimator ${\tilde{t}}_y$ in 3 of the 24 settings, involving $n_2=$ 2,000, while for the rest its efficiency increases greatly as $n_2$ decreases, approaching the efficiency of ${\widehat{t}}_y^O.$ The estimator ${\widehat{t}}_y^{\text{ES}}$ is less efficient than the estimator ${\tilde{t}}_y$ in 12 settings. The estimator ${\widehat{t}}_y^{\text{GR}}$ is much more efficient than the estimator ${\widehat{t}}_y^{\text{ES}}$ in all settings, more so for higher values of $n_2$ and as we move from $r^2=$ 0.25 to $r^2=$ 0.75, and considerably more than in the (SRS, SRS) design.

For (SRS, PPSS), the empirical efficiencies of the calibration estimators ${\widehat{t}}_y^{\text{GR}}$ and ${\widehat{t}}_y^{\text{ES}}$ are shown in columns 10 and 11 of Table 7.1. The pattern of these efficiencies is very similar to that in the (SRS, SRS) design. This is particularly so for the efficiency of ${\widehat{t}}_y^{\text{GR}}$ relative to ${\widehat{t}}_y^{\text{ES}},$ which is not included in Table 7.1 but can be easily derived using the displayed efficiencies of ${\widehat{t}}_y^{\text{GR}}$ and ${\widehat{t}}_y^{\text{ES}}$ relative to ${\tilde{t}}_y.$ The HT estimator ${\tilde{t}}_y$ itself is more efficient with this two-phase design, which explains why the efficiency of the two calibration estimators ${\widehat{t}}_y^{\text{GR}}$ and ${\widehat{t}}_y^{\text{ES}}$ relative to ${\tilde{t}}_y$ is somewhat lower than in the (SRS, SRS) and (STRSRS, SRS) designs.

The whole simulation study was repeated with the simulated population for the vector $\mathrm{<mo\; stretchy="false">(</mo>}y\mathrm{,}{x}_1\mathrm{,}{x}_2\mathrm{<mo\; stretchy="false">)</mo>}$ generated from a trivariate lognormal distribution with the specified correlation structures. For all three designs (SRS, SRS), (SRS, PPSS) and (STRSRS, SRS), the results (not shown here) were very similar to those based on the linear model for $y$ used above.

It is of interest to consider the setup of auxiliary variables in which the scalar variable $x_1$ is augmented to $\mathrm{<mo\; stretchy="false">(</mo>1,}{x}_1),$ with known totals $\mathrm{<mo\; stretchy="false">(</mo>}N\mathrm{,}{t}_{x_1}\mathrm{<mo\; stretchy="false">)</mo>.}$ Then in the (SRS, SRS) design, in which construction of the BLUE ${\widehat{t}}_y^B$ and the optimal estimator ${\widehat{t}}_y^O$ is feasible, using the complete setup $\mathrm{<mo\; stretchy="false">(</mo>1,}{x}_1\mathrm{,}{x}_2\mathrm{<mo\; stretchy="false">)</mo>}$ in calibration gives the same ${\widehat{t}}_y^B$ and practically the same ${\widehat{t}}_y^B$ as when using $\mathrm{<mo\; stretchy="false">(</mo>}{x}_1\mathrm{,}{x}_2\mathrm{<mo\; stretchy="false">)</mo>.}$ It would also convert the regression estimator ${\widehat{t}}_y^{\text{GR}}$ to ${\widehat{t}}_y^O$ (using the same adjustment $1/{\pi }_{2k}$ of $Y_1$ as in ${\widehat{t}}_y^O),$ and the regression estimator ${\widehat{t}}_y^{\text{ES}}$ estimator to the pseudo-optimal estimator ${\widehat{t}}_y^{\text{PSO}}$ (defined in Section 6). These properties are derived from known theory, see for example Merkouris (2004, 2015), more directly for ${\widehat{t}}_y^{\text{ES}}$ and the second regression term of ${\widehat{t}}_y^{\text{GR}}$ and the optimal ${\widehat{t}}_y^O,$ irrespective of any specific functional relationship of $y$ with $\mathrm{<mo\; stretchy="false">(</mo>1,}{x}_1\mathrm{,}{x}_2\mathrm{<mo\; stretchy="false">)</mo>.}$ Then, the three sample-based estimators would show virtually identical empirical behavior. This follows from Proposition 1, which gives the condition (satisfied by specific designs, including (SRS, SRS)) under which the pseudo-optimal regression estimator ${\widehat{t}}_y^{\text{PSO}}$ is asymptotically equivalent to the proposed optimal estimator ${\widehat{t}}_y^O.$ Experimental calculations have confirmed this equivalence. In the (STRSRS, SRS) design too, using $\mathrm{<mo\; stretchy="false">(</mo>1,}{x}_1\mathrm{,}{x}_2\mathrm{<mo\; stretchy="false">)</mo>}$ gives the same ${\widehat{t}}_y^B$ and ${\widehat{t}}_y^O$ as when using $\mathrm{<mo\; stretchy="false">(</mo>}{x}_1\mathrm{,}{x}_2\mathrm{<mo\; stretchy="false">)</mo>},$ and converts the ${\widehat{t}}_y^{\text{GR}}$ and ${\widehat{t}}_y^{\text{ES}}$ estimators to the ${\widehat{t}}_y^O$ and ${\widehat{t}}_y^{\text{PSO}}$ estimators, respectively. However, by Proposition 1 the equivalence of the latter two estimators, and hence of ${\widehat{t}}_y^{\text{GR}}$ and ${\widehat{t}}_y^{\text{ES}},$ does not hold in this sampling design.

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