Comments on “Statistical inference with non-probability survey samples” – Miniaturizing data defect correlation: A versatile strategy for handling non-probability samples
Section 5. Quasi-randomization and super-population implementations

• Previous
• Next

Once a joint model for $\mathrm{<mo>\{</mo>}{R}_i\mathrm{,}{y}_i\mathrm{<mo>\}</mo>}$ is set up, of course we can use it for estimating both ${\pi }_i$ and the regression function $m\mathrm{<mo\; stretchy="false">(</mo>}x\mathrm{<mo\; stretchy="false">)</mo>},$ each of which is made possible by the availability of the auxiliary probability sample, and the assumption of missing at random. But as shown before, correctly specifying and estimating one of them is sufficient for miniaturizing $c_{\tilde{R},z}.$ However, from (4.3), in order for the covariance/correlation to be zero, neither multiplicative correction to ${\pi }_I$ via $W_I$ nor the additive adjustment for $\text{E}\mathrm{<mo\; stretchy="false">(</mo>}{y}_I|x_I\mathrm{<mo\; stretchy="false">)</mo>}$ via $m\mathrm{<mo\; stretchy="false">(</mo>}{x}_I\mathrm{<mo\; stretchy="false">)</mo>}$ need to be correct. All we need is that, after the correction or adjustment, what is left would be uncorrelated with each other. The aforementioned framework of Collaborative TMLE was built essentially on this insight (e.g., see Section 3.1 of van der Laan and Gruber, 2009), though the heavy mathematical treatments in its literature might have discouraged readers to seek such intuitive understanding.

To provide a simple illustration, consider a finite population that is an i.i.d. sample from a super-population model:

$$\text{E}\mathrm{<mo\; stretchy="false">[</mo>}y|x\mathrm{<mo\; stretchy="false">]</mo>}={\displaystyle \sum _{k=0}^3}{\beta }_k{x}^k\mathrm{,}x~N\mathrm{<mo\; stretchy="false">(</mo>0,}\mathrm{1<mo\; stretchy="false">)</mo>.}(5.1)$$

The non-probability sample is generated by a mechanism $R$ such that $\mathrm{Pr}\left(R=1|y\mathrm{,}x\right)=\pi \left(\left|x\right|\right),$ that is, it is determined by the magnitude of $x$ only. Suppose we mis-specify the function form for $\pi$ (e.g., the divine model may not be monotone in $\left|x\right|,$ but the device model such as the conventional logistic link is), as well the regression model by choosing $m\mathrm{<mo\; stretchy="false">(</mo>}x\mathrm{<mo\; stretchy="false">)</mo>}=b_0+b_{1}x+b_2{x}^2.$ Since $x^2$ is uncorrelated with $x$ or $x^3$ under $x~N\mathrm{<mo\; stretchy="false">(</mo>0,}\mathrm{1<mo\; stretchy="false">)</mo>},$ we know that our least-square estimator for $b_2$ would still be valid for ${\beta }_2$ even under the mis-specified regression model. This turns out to be sufficient to ensure the asymptotic unbiasedness (as $N\to \infty )$ of the following “doubly robust” estimator for $\mu ={\overline{y}}_N,$ the finite-population mean,

$${\widehat{\mu }}_{+}=\frac{{\displaystyle {\sum }_{i=1}^N{R}_{i}w\left(\left|x_i\right|\right)\mathrm{<mo\; stretchy="false">(</mo>}{y}_i-\widehat{m}\mathrm{<mo\; stretchy="false">(</mo>}{x}_i\mathrm{<mo\; stretchy="false">)</mo><mo\; stretchy="false">)</mo>}}}{{\displaystyle {\sum }_{i=1}^N{R}_{i}w\left(\left|x_i\right|\right)}}+\frac{{\displaystyle {\sum }_{i=1}^N{R}_i^{*}\widehat{m}\mathrm{<mo\; stretchy="false">(</mo>}{x}_i\mathrm{<mo\; stretchy="false">)</mo>}}}{{\displaystyle {\sum }_{i\mathrm{<mo>=</mo>1}}^N{R}_i^{*}}}\mathrm{,}(5.2)$$

where $R^{*}$ indicates the auxiliary sample (of $x$ only). Or equivalently,

$${\widehat{\mu }}_{+}-{\overline{y}}_N=\frac{{\text{Cov}}_I\left(R_{I}w\left(\left|x_I\right|\right)\mathrm{,}{y}_I-\widehat{m}\left(x_I\right)\right)}{{\text{E}}_I\left(R_{I}w\left(\left|x_I\right|\right)\right)}+\frac{{\text{Cov}}_I\left(R_I^{*}\mathrm{,}\widehat{m}\left(x_I\right)\right)}{{\text{E}}_I\left(R_I^{*}\right)}\mathrm{,}(5.3)$$

which makes it clearer that any bias in ${\widehat{\mu }}_{+}$ is controlled by the covariance (or correlation) involving $R,$ since the covariance involving $R^{*}$ is already miniaturized by the assumption that the auxiliary sample is probabilistic (which, for simplicity, is assumed to be a simple random sample).

Here $w\mathrm{<mo\; stretchy="false">(</mo>}x\mathrm{<mo\; stretchy="false">)</mo>}$ is any weight function such that ${\text{E}}_{\varphi }\left[{\left|x\right|}^{3}w\left(\left|x\right|\right)\right]<\infty ,$ where the expectation is with respect to $x~N\mathrm{<mo\; stretchy="false">(</mo>0,}\mathrm{1<mo\; stretchy="false">)</mo>},$ and $\widehat{m}\mathrm{<mo\; stretchy="false">(</mo>}x\mathrm{<mo\; stretchy="false">)</mo>}=b_0+b_{1}x+{\widehat{\beta }}_2{x}^2,$ with ${\widehat{\beta }}_2$ being the least-square estimator for ${\beta }_2$ from the biased sample, and $b_0$ and $b_1$ can be chosen arbitrarily. Because the finite-population covariance/correlation between $\pi \left(\left|x_I\right|\right)w\left(\left|x_I\right|\right)$ and $x_I^k$ is $O_p\mathrm{<mo\; stretchy="false">(</mo>}{N}^{-1/2}\mathrm{<mo\; stretchy="false">)</mo>},$ for $k=1$ and $k=3,$ the misfitted parts for $\pi$ or $m$ do not contribute to the ddc (asymptotically) since they are uncorrelated with each other under the super-population model, leading to further robustness going beyond “double robustness”. This of course does not mean that we can misfit a model arbitrarily and still obtain valid estimators, but it does imply that having at least one model being correct is a sufficient, but not necessary, condition for the validity of the doubly robust estimators.

It is also worth stressing that, in formatting the regression model, we do not necessarily need to invoke a device probability, e.g., a super-population regression model, because the FPI variable provides a finite-population regression via applying the least-squares method to regress $y_i$ on $x_i\mathrm{,}i\in N.$ This regression fitting itself says little about whether the resulting regression line $y=\widehat{m}\mathrm{<mo\; stretchy="false">(</mo>}x\mathrm{<mo\; stretchy="false">)</mo>}$ is a good fit to $\mathrm{<mo\; stretchy="false">(</mo>}{y}_i\mathrm{,}{x}_i\mathrm{<mo\; stretchy="false">)</mo>}$ or not. However, the example above indicates that, for the purpose of estimating the population average of $y,$ the lack of fit may not matter that much, as long as the “residual” $z_I=y_I-\widehat{m}\mathrm{<mo\; stretchy="false">(</mo>}{x}_I\mathrm{<mo\; stretchy="false">)</mo>}$ has little correlation with $W_I{\pi }_I,$ as two functions of the FPI variable $I.$ Indeed, as discussed in Section 3, we can consider including ${\widehat{\pi }}_I$ in the regression model $\widehat{m}\mathrm{<mo\; stretchy="false">(</mo>}{x}_I\mathrm{,}{\widehat{\pi }}_I\mathrm{<mo\; stretchy="false">)</mo>.}$ How effective this strategy is in general is a topic of further research.

• Previous
• Article main page
• Next