A note on regression estimation with unknown population size 4. SimulationsA note on regression estimation with unknown population size 4. Simulations

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We carried out two simulation studies. The first one used a dataset provided in the textbook of Rosner (2006) and the second one was based on an artificial population created according to a simple linear regression model. The first simulation assessed the performance of all of the estimators with respect to different sample schemes while the second simulation study focused on the impact of changing the intercept value in the model.

The parameter of interest for these two simulations is the total of the variable of interest $y:$ $Y\mathrm{<mo>=</mo>}{\displaystyle {\sum }_{i\in U}}{y}_i.$ All estimators were used $({\widehat{Y}}_{\text{GREG}},{\widehat{Y}}_{\text{OPT}},{\widehat{Y}}_{\text{POPT}},{\widehat{Y}}_{\text{SREG}},{\widehat{Y}}_{\text{KREG}}$ and ${\widehat{Y}}_{\text{KOPT}})$ with the available auxiliary data. Table 4.1 summarizes the auxiliary data and the variance structure of the errors (when applicable) associated with the estimators used in the two studies.

The performance of all estimators was evaluated based on the relative bias, the Monte Carlo relative efficiency and the approximate relative efficiency. Expressions of these quantities as shown below.

• 1.   Relative bias:

$$\text{RB}\left({\widehat{Y}}_{\text{EST}}\right)\mathrm{<mo>=</mo>}\frac{100}{R}{\displaystyle \sum _{i\mathrm{<mo>=</mo>\; 1}}^R}\frac{\left({\widehat{Y}}_{\text{EST}\left(r\right)}-Y\right)}{Y}\mathrm{,}(4.1)$$

• where ${\widehat{Y}}_{\text{EST}\left(r\right)}$ represents one of the estimators presented in Table 4.1 as computed in the $r^{\text{th}}$ Monte Carlo sample.
• 2.   Monte Carlo Relative efficiency

$$\text{RE}\left({\widehat{Y}}_{\text{EST}}\right)\mathrm{<mo>=</mo>}\frac{{\text{MSE}}_{\text{MC}}\left({\widehat{Y}}_{\text{EST}}\right)}{{\text{MSE}}_{\text{MC}}\left({\widehat{Y}}_{\text{GREG2}}\right)}\mathrm{,}(4.2)$$

• where

$${\text{MSE}}_{\text{MC}}\left({\widehat{Y}}_{\text{EST}}\right)\mathrm{<mo>=</mo>}\frac{1}{R}{\displaystyle {\displaystyle \sum }}_{r\mathrm{<mo>=</mo>\; 1}}^R{\left({\widehat{Y}}_{\text{EST}\left(r\right)}-Y\right)}^2.$$

• The $\text{RE}$ measures the relative efficiency of the estimator ${\widehat{Y}}_{\text{EST}}$ with respect to ${\widehat{Y}}_{\text{GREG2}}.$
• 3.   Approximate Relative efficiency

$$\text{AR}\left({\widehat{Y}}_{\text{EST}}\right)\mathrm{<mo>=</mo>}\frac{{\text{AV}}_p\left({\widehat{Y}}_{\text{EST}}\right)}{{\text{AV}}_p\left({\widehat{Y}}_{\text{GREG2}}\right)}\mathrm{,}(4.3)$$

• where

$${\text{AV}}_p\left({\widehat{Y}}_{\text{EST}}\right)\mathrm{<mo>=</mo>}{\displaystyle {\displaystyle \sum }}_{i\in U}{\displaystyle {\displaystyle \sum }}_{j\in U}{\Delta }_{ij}\frac{E_i}{{\pi }_i}\frac{E_j}{{\pi }_i},$$

• is the approximate variance of ${\widehat{Y}}_{\text{EST}}$ with $E_i\mathrm{<mo>=</mo>}{y}_i-x_i^{{\rm T}}{B}_{\text{EST}}.$ The approximate relative efficiency $\left(\text{AR}\right)$ measures the relative gain in efficiency of ${\widehat{Y}}_{\text{EST}}$ with respect to ${\widehat{Y}}_{\text{GREG2}}$ using the population residual obtained by Taylor linearisation. It is expected that $\text{RE}$ and $\text{AR}$ give comparable results. However, as we will see, this may not be the case.

4.1 Simulation 1

The population was the dataset (FEV.DAT) available on the CD that accompanies the textbook by Rosner (2006). The data file contains 654 records from a study on Childhood Respiratory Disease carried out in Boston. The variables in the file were: age, height, sex (male female), smoking (indicates whether the individual smokes or not) and Forced expiratory volume (FEV). Singh and Raghunath (2011) used the same data set. The parameter of interest is the total height $\left(y\right)$ of the population. The variable age $\left(x_1\right)$ was used as auxiliary variable in the regression. The variable FEV $\left(x_2\right)$ was chosen as the size variable to compute probabilities of selection for the sampling schemes that are considered in this simulation. The two variables sex and smoking were discarded from the simulation. Table 4.2 summarizes the central tendency measures of the three variables in the population. For each variable, the mean and median were similar. This indicates that the three variables have a symmetrical distribution.

Figure 4.1 displays the relationship between the variable of interest $y$ and the auxiliary variable $x_1.$ The relationship between Height $\left(y\right)$ and the age $\left(x_1\right)$ appears to be linear but does not go through the origin. The Pearson correlation coefficient between $y$ and $x_1$ was 0.79.

The objective of this simulation study was to evaluate the performance of the estimators presented in Table 4.1 using different sampling designs. We considered the Midzuno, the Sampford and the Poisson sampling designs. The variable $x_2$ were used as a size measure for the three sampling schemes to compute the inclusion probabilities. These sampling designs are as follows:

1. Midzuno sampling (see Midzuno 1952): The first unit is sampled with probability $p_i$ and the remaining $n-1$ units are selected as a simple random sampling without replacement from the remaining $N-1$ remaining units in the population. The probabilities of selection $p_i$ for unit $i$ is given by $p_i\mathrm{<mo>=</mo>}{x}_{2i}/{\displaystyle {\sum }_{i\in U}{x}_{2i}}.$ The first order inclusion probability for unit $i$ is given by ${\pi }_i\mathrm{<mo>=</mo>}{\left(N-1\right)}^{-1}\left[\left(N-n\right)p_i+\left(n-1\right)\right].$
2. Sampford sampling (see Sampford 1967): The algorithm for selecting the sample is carried out as follows. The first unit is selected with probability $p_i\mathrm{<mo>=</mo>}{x}_{2i}/{\displaystyle {\sum }_{i\in U}{x}_{2i}}$ and the remaining $n-1$ units are selected with replacement with probability ${\lambda }_i\mathrm{<mo>=</mo>}{\left(1-n{p}_i\right)}^{-1}{p}_i.$ If any of the units are selected more than once, the procedure is repeated until all elements of the sample are different. The probability of inclusion of the first order is given by ${\pi }_i\mathrm{<mo>=</mo>}n{p}_i.$
3. Poisson sampling: Each unit is selected independently, resulting in a random sample size. The probability of selecting unit $i$ is $p_i\mathrm{<mo>=</mo>}{x}_{2i}/{\displaystyle {\sum }_{i\in U}{x}_{2i}}.$ The inclusion probability associated with unit $i$ is ${\pi }_i\mathrm{<mo>=</mo>}n{p}_i.$ A good description of this procedure can be found in Särndal et al. (1992).

The total of $Y\mathrm{<mo>=</mo>}{\displaystyle {\sum }_{i\in U}}{y}_i$ was the parameter of interest. Based on each of these sampling schemes, we selected $R\mathrm{<mo>=</mo>}\text{2,000}$ Monte Carlo samples of size $n\mathrm{<mo>=</mo>\; 50.}$ Estimators in Table 4.1 were then computed for each sample. The performance of the estimators was then assessed using the Relative Bias, the Monte Carlo Relative Efficiency and the Approximate Relative Efficiency as described by the equations (4.1), (4.2) and (4.3) respectively.

4.2 Simulation 1 results

Simulation results are presented in Table 4.3. All estimators studied are approximately unbiased, and their relative bias is smaller than 1%. We discuss separately the approximate relative efficiency (AR) and the relative efficiency (RE) of the estimators when the population size $N$ is known and unknown.

Case 1: Population size $N$  is known

We compare the $\text{AR}$ and the $\text{RE}$ for the following estimators in Table 4.3: ${\widehat{Y}}_{\text{GREG2}},$ ${\widehat{Y}}_{\text{OPT2}},$ ${\widehat{Y}}_{\text{OPT3}}$ and ${\widehat{Y}}_{\text{POPT3}}$ for each of the three sampling designs. We can do so for almost all these estimators except for ${\widehat{Y}}_{\text{OPT3}}$ for the Midzuno and the Sampford sampling schemes. In this case, we cannot compute $B_{\text{OPT3}}$ for a similar reason as the one described in Remark 3.2.

On the basis of both $\text{AR}$ and $\text{RE},$ the pseudo-optimal estimator ${\widehat{Y}}_{\text{OPT3}}$ is the most reliable estimator regardless of the sampling scheme. It is close to the optimal estimator ${\widehat{Y}}_{\text{OPT2}}$ only in terms of $\text{AR}.$ Both the $\mathrm{RE}$ and the $\text{AR}$ of the optimal estimator ${\widehat{Y}}_{\text{OPT2}}$ were not as close as expected under the Midzuno sampling design. The poor behaviour of the $\text{RE}$ of the optimal estimator ${\widehat{Y}}_{\text{OPT2}}$ has also been observed by Montanari (1998). Figure 4.2 explains what is happening. We observe that most estimates obtained for the optimal estimator ${\widehat{Y}}_{\text{OPT2}}$ for the 2,000 Monte Carlo samples are close to the mean. However, in some samples, the estimates are quite far from it. This is in contrast to ${\widehat{Y}}_{\text{POPT3}}$ where the values are tightly centered around the mean: note that the associated $\text{RE}$ and $\text{AR}$ are quite close to one another.

The optimal estimator ${\widehat{Y}}_{\text{OPT3}}$ is equivalent to the pseudo-optimal estimator ${\widehat{Y}}_{\text{POPT3}}$ in the case of Poisson sampling scheme. Recall that the optimal estimator ${\widehat{Y}}_{\text{OPT2}}$ used $x_i\mathrm{<mo>=</mo>}{\left(\mathrm{1,}{x}_{2i}\right)}^{{\rm T}}$ as auxiliary data. The optimal estimator ${\widehat{Y}}_{\text{OPT3}}$ used $x_i\mathrm{<mo>=</mo>}{\left(\mathrm{1,}{\pi }_i\mathrm{,}{x}_{2i}\right)}^{{\rm T}}$ as auxiliary data. The addition of the ${\pi }_i$ has significantly improved the efficiency of the optimal estimator for the Poisson sampling scheme.

Singh and Raghunath (2011) used ${\widehat{Y}}_{\text{SREG1}}$ when $N$ was known, but did not include it as a control count. Nonetheless, they observed that ${\widehat{Y}}_{\text{SREG1}}$ was quite comparable to ${\widehat{Y}}_{\text{GREG2}}$ in terms of $\text{AR}$ and $\text{RB}$ for the Midzuno sampling design. The reason for this is that this sampling scheme is quite close to simple random sampling without replacement. However, using these two measures, ${\widehat{Y}}_{\text{SREG1}}$ is by far the worst estimator for the other two sampling schemes.

Case 2: Population size $N$  is unknown

Five estimators are reported in Table 4.3 for this case. However, as ${\widehat{Y}}_{\text{KREG2}}$ is quite close to ${\widehat{Y}}_{\text{KOPT2}}$ and ${\widehat{Y}}_{\text{POPT2}},$ we comment on the results obtained for ${\widehat{Y}}_{\text{SREG1}},$ ${\widehat{Y}}_{\text{OPT1}}$ and ${\widehat{Y}}_{\text{KREG2}}.$ Estimators ${\widehat{Y}}_{\text{SREG1}},$ ${\widehat{Y}}_{\text{OPT1}}$ and ${\widehat{Y}}_{\text{KREG2}}$ were very similar in terms of relative efficiency and approximate relative efficiency for the Midzuno sampling design. For the Sampford sampling scheme, ${\widehat{Y}}_{\text{OPT1}},$ ${\widehat{Y}}_{\text{KREG2}}$ and ${\widehat{Y}}_{\text{POPT2}}$ were comparable and slightly better than ${\widehat{Y}}_{\mathrm{SREG}1}.$ Under the Poisson sampling scheme, ${\widehat{Y}}_{\text{OPT1}}$ and ${\widehat{Y}}_{\text{KREG2}}$ outperformed ${\widehat{Y}}_{\text{SREG1}}.$ We can also see that ${\widehat{Y}}_{\text{SREG1}}$ was very inefficient with an $\text{RE}$ at least 10 times larger than those associated with ${\widehat{Y}}_{\text{KREG2}}$ or ${\widehat{Y}}_{\text{POPT2}}.$ Note that ${\widehat{Y}}_{\text{KREG2}}$ was better than ${\widehat{Y}}_{\text{OPT1}}\text{}:$ this is reasonable as ${\widehat{Y}}_{\text{KREG2}}$ uses two auxiliary variables whereas ${\widehat{Y}}_{\text{OPT1}}$ uses the single auxiliary variable $x_{2i}.$

4.3 Simulation 2

The performance of the estimators was assessed for different values of the intercept in the model. We restricted ourselves to the Poisson sampling design to illustrate Remark 2.1 in Section 2: that is the efficiency of ${\widehat{Y}}_{\text{SREG}}$ deteriorates as the intercept gets bigger. The population was generated according to the following model

$$y_i\mathrm{<mo>=</mo>}a+x_i+e_i\mathrm{.}(4.4)$$

The $e_i$ values were generated from a normal distribution with mean 0 and variance ${\sigma }_i^2\mathrm{<mo>=</mo>\; 1.}$ The $x$ values were generated according to a chi-square distribution with one degree of freedom. Three populations of size $N=\text{5,000}$ were generated using (4.4) with different values of the intercept $a.$ Note that $x-$ values were re-generated for each population. The three populations were labelled as A, B and C depending on the intercept used. The intercept values were set to 3, 5 and 10 respectively for populations A, B and C. From each of these populations we drew $R\mathrm{<mo>=</mo>}\text{2,000}$ Monte Carlo samples with expected sample size $n\mathrm{<mo>=</mo>\; 50}$ using the Poisson sampling design. The first inclusion probability was set equal to ${\pi }_i\mathrm{<mo>=</mo>}n{z}_i/{\displaystyle {\sum }_{i\in U}}{z}_i$ for each unit $i.$ The $z$ values were generated according to the following model

$$z_i\mathrm{<mo>=</mo>\; 0.5}{y}_i+u_i\mathrm{,}$$

where $u_i$ was a random error generated according to an exponential distribution with mean $k$ equals to 0.5 or 1.

4.4 Simulation 2 results

Numerical results are given in Table 4.4 for $k\mathrm{<mo>=</mo>\; 1}$ and Table 4.5 for $k\mathrm{<mo>=</mo>\; 0.5.}$ All estimators are approximately unbiased with relative biases smaller than 1%.

Case 1: Population size $N$  is known

As expected, both optimal estimators ${\widehat{Y}}_{\text{OPT2}}$ and ${\widehat{Y}}_{\text{OPT3}}$ are more efficient than ${\widehat{Y}}_{\text{GREG2}}.$ The optimal estimator ${\widehat{Y}}_{\text{OPT2}}$ based on ${\left(\mathrm{1,}{x}_{2i}\right)}^{{\rm T}}$ is slightly better than ${\widehat{Y}}_{\text{GREG2}}.$ The inclusion of the additional variable ${\pi }_i$ resulting in ${\widehat{Y}}_{\text{OPT3}}$ yields significant gains in terms of $\text{RE}$ and $\text{AR}:$ these gains decrease as the intercept gets larger. Once more, ${\widehat{Y}}_{\text{SREG1}}$ is quite inefficient, and as noted in Remark 2.1, this inefficiency increases as the intercept gets larger. The previous observations are valid regardless of $k.$ The efficiency of both optimal estimators ${\widehat{Y}}_{\text{OPT2}}$ and ${\widehat{Y}}_{\text{OPT3}}$ decreases as $k$ gets smaller.

Case 2: Population size $N$  unknown

The most efficient estimator is ${\widehat{Y}}_{\text{KREG2}}.$ It outperforms ${\widehat{Y}}_{\text{OPT1}}$ as it uses more auxiliary variables. Estimator ${\widehat{Y}}_{\text{SREG1}}$ is by far the most inefficient one. As the intercept in the population model increases, the relative efficiency (both in terms of $\text{RE}$ and $\text{AR})$ is fairly stable for ${\widehat{Y}}_{\text{KREG2}}.$ On the other hand, the relative efficiencies associated with ${\widehat{Y}}_{\text{SREG1}}$ and ${\widehat{Y}}_{\text{OPT1}}$ deteriorate rapidly, as the intercept in the population model increases. The effect of $k$ on the efficiencies of the estimators is as described when the population size is known.

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