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I have a variable Y that can assume a range of values, and an auxiliary variable X that assumes value 1 if it possesses a characteristic, 0 otherwise.

My first assumption here is that I can substitute mean by proportion for X, since $\frac1n\sum x_i=\frac{m_1}{n}$ here, where $m_1$ is the number of individuals possessing the characteristic. Am I correct?

If yes, I want to use this data to create en estimator for Y in the form of $T=f(\bar{x},\bar{y})$, for example, $T=a\bar{x}+\bar{y}$, say.

Now, usually, we apply the transformations $\bar{y}=\bar{Y}(1+e_0), \bar{x}=\bar{X}(1+e_1)$, where $e_i$ are errors.

Is this transformation valid in this case too?

If yes, when I have to use correlation coefficient between X and Y, I will use the point biserial correlation coefficient, right?

Thank you for your help! I'm a researcher in sampling theory, so any paper relevant to this would also be highly appreciated. I have not come across any such research in my literature review so far, and I'm genuinely interested to learn!

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    $\begingroup$ Does it make sense in the application of interest? So, say one is performing a sample estimation of IQ for the general population with a dummy variable if a person is a college graduate or say obtained a high GPA in high school. Likely would reduce sampling error in such an application, but the data has to be available (like total number of college graduates) in the parent population to be of most value. $\endgroup$ – AJKOER Oct 22 at 0:35
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To answer your question: "Can dichotomous variable be used as auxiliary variable in conventional way to design an estimator?", the answer here is likely yes, as there are studies advocating the general use of auxiliary variables, to quote from a study out of Sweden, for example:

Also Holmberg (2003) has investigated the combination of ${\pi}$ps and GREG and argues that adding auxiliary information to a survey will highly improve the quality of the estimators. According to him, using an estimator outside the GREG family may probably not reduce the variance.

where GREG refers to the employment of a generalized regression estimator. Also, citing a recommendation based on prior work in an economic context:

Their conclusion is that the most efficient method of the strategies considered is a stratified sample consisting of enterprises with the largest value of the auxiliary variables in each stratum and simple ratio estimation.

Now, for optimal application of a stratified random sampling scheme, knowledge of the strata weights is important especially with respect to bias, to quote:

To do a correct allocation, I need investments data for the whole population. Since I only have investment data from the enterprises in the sample, this thesis will also include a prediction of the investments for the enterprises outside the sample (but in the frame).

So this last point suggests the need for more modeling in the sampling frame, to assess strata weights, and not per se direct application of a sample implied weighting scheme, which may be an issue in cases of significant under-represented sampling units.

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