Precision Variables

Author

Affiliation

Alex Kaizer

University of Colorado-Anschutz Medical Campus

This page is part of the University of Colorado-Anschutz Medical Campus’ BIOS 6618 Recitation collection. To view other questions, you can view the BIOS 6618 Recitation collection page or use the search bar to look for keywords.

Precision Variables

Another role a predictor can play in our model is to be a precision variable.

The term precision refers to the size of an estimator’s variance, or equivalently, the narrowness of a confidence interval for the parameter being estimated.

The smaller the variance of the estimator, the higher the precision of the estimator:

\[ \frac{Var(\hat{\beta}_{adj})}{Var(\hat{\beta}_{crude})}=\frac{1-\hat{\rho}_{YZ | X}^2}{n-3}\left(\frac{n-2}{1-\hat{\rho}_{XZ}^2}\right) \]

where $Z$ is another independent variable, $\hat{\rho}_{YZ|X}$ is the partial correlation between $Y$ and $Z$ that controls for $X$, and $\hat{\rho}_{XZ}$ is the correlation between $X$ and $Z$.

A strong association between $Y$ and $Z$ has a beneficial effect upon the precision of $\hat{\beta}_{adj}$ (i.e., it decreases $SE(\hat{\beta}_{adj})$).

A strong association between $X$ and $Z$ has a detrimental effect on the precision of $\hat{\beta}_{adj}$ (i.e., it increases $SE(\hat{\beta}_{adj})$).

Thus, the precision of $\hat{\beta}_{adj}$ reflects the competing effects of the $Y$-$Z$ and $X$-$Z$ relationships. A precision variable improves the precision of the estimate of the PEV.

In our regression model, if we have ruled out the variable $Z$ as being a potential confounder, we can evaluate the change in the variability of our PEV’s beta coefficient to see if it may be a precision variable. If there is a strong association between $Z$ and $Y$, then we would expect our variance of $\hat{\beta}_{adj}$ to decrease (i.e., the presence of $Z$ helped address the variability in a way that improved our estimation).

However, if there is a strong(er) association between $Z$ and $X$, we may actually increase our variance of $\hat{\beta}_{adj}$. This may lead us to remove it from the model (assuming we have no strong biological/scientific reasons to keep the variable).

The consequence of increasing the variance of our beta coefficients is that it may lead to a smaller test statistic and larger p-value.

--- title: "Precision Variables" author: name: Alex Kaizer roles: "Instructor" affiliation: University of Colorado-Anschutz Medical Campus toc: true toc_float: true toc-location: left format: html: code-fold: show code-overflow: wrap code-tools: true --- ```{r, echo=F, message=F, warning=F} library(kableExtra) library(dplyr) ``` This page is part of the University of Colorado-Anschutz Medical Campus' [BIOS 6618 Recitation](/recitation/index.qmd) collection. To view other questions, you can view the [BIOS 6618 Recitation](/recitation/index.qmd) collection page or use the search bar to look for keywords. # Precision Variables Another role a predictor can play in our model is to be a precision variable. The term **precision** refers to the size of an estimator's variance, or equivalently, the narrowness of a confidence interval for the parameter being estimated. The smaller the variance of the estimator, the higher the precision of the estimator: $$ \frac{Var(\hat{\beta}_{adj})}{Var(\hat{\beta}_{crude})}=\frac{1-\hat{\rho}_{YZ | X}^2}{n-3}\left(\frac{n-2}{1-\hat{\rho}_{XZ}^2}\right) $$ where $Z$ is another independent variable, $\hat{\rho}_{YZ|X}$ is the **partial correlation** between $Y$ and $Z$ that controls for $X$, and $\hat{\rho}_{XZ}$ is the correlation between $X$ and $Z$. A strong association between $Y$ and $Z$ has a *beneficial* effect upon the precision of $\hat{\beta}_{adj}$ (i.e., it *decreases* $SE(\hat{\beta}_{adj})$). A strong association between $X$ and $Z$ has a *detrimental* effect on the precision of $\hat{\beta}_{adj}$ (i.e., it *increases* $SE(\hat{\beta}_{adj})$). Thus, the precision of $\hat{\beta}_{adj}$ reflects the competing effects of the $Y$-$Z$ and $X$-$Z$ relationships. A **precision variable** improves the precision of the estimate of the PEV. In our regression model, if we have ruled out the variable $Z$ as being a potential confounder, we can evaluate the change in the variability of our PEV's beta coefficient to see if it may be a precision variable. If there is a strong association between $Z$ and $Y$, then we would expect our variance of $\hat{\beta}_{adj}$ to decrease (i.e., the presence of $Z$ helped address the variability in a way that improved our estimation). However, if there is a strong(er) association between $Z$ and $X$, we may actually increase our variance of $\hat{\beta}_{adj}$. This may lead us to remove it from the model (assuming we have no strong biological/scientific reasons to keep the variable). The consequence of increasing the variance of our beta coefficients is that it may lead to a smaller test statistic and larger p-value.