An article in Cancer Epidemiology, Biomarkers and Prevention...

An article in Cancer Epidemiology, Biomarkers and Prevention $(1996,$ Vol. $5,$ pp. $849-852)$ conducted a pilot study to assess the use of toenail arsenic concentrations as an indicator of ingestion of arsenic-containing water. Twenty-one participants were interviewed regarding use of their private (unregulated) wells for drinking and cooking, and each provided a sample of water and toenail clippings. The table below showed the data of age (years), sex of person $(1=$ male, $2=$ female), proportion of times household well used for drinking $(1 \leq 1 / 4,2=1 / 4,3=1 / 2,4=3 / 4,5 \geq 3 / 4),$ proportion of times household well used for cooking $(1 \leq 1 / 4,2=1 / 4,3=$ $1 / 2,4=3 / 4,5 \geq 3 / 4),$ arsenic in water $(\mathrm{ppm}),$ and arsenic in toenails (ppm) respectively. (a) lit a multiple linear regression model using arsenic concentration in nails as the response and age, drink use, cook usc, and arsenic in the water as the regressors. (b) Estimate $\sigma^{2}$ and the standard errors of the regression coefficients. (c) Use the model to predict the arsenic in nails when the age is $30,$ the drink use is catcgory $5,$ the cook use is catcgory $5,$ and arsenic in the water is $0,135 \mathrm{ppm}$.

An article in Cancer Epidemiology, Biomarkers and Prevention $(1996,$ Vol. $5,$ pp. $849-852)$ conducted a pilot study to assess the use of toenail arsenic concentrations as an indicator of ingestion of arsenic-containing water. Twenty-one participants were interviewed regarding use of their private (unregulated) wells for drinking and cooking, and each provided a sample of water and toenail clippings. The table below showed the data of age (years), sex of person $(1=$ male, $2=$ female), proportion of times household well used for drinking $(1 \leq 1 / 4,2=1 / 4,3=1 / 2,4=3 / 4,5 \geq 3 / 4),$ proportion of times household well used for cooking $(1 \leq 1 / 4,2=1 / 4,3=$ $1 / 2,4=3 / 4,5 \geq 3 / 4),$ arsenic in water $(\mathrm{ppm}),$ and arsenic in toenails (ppm) respectively. (a) lit a multiple linear regression model using arsenic concentration in nails as the response and age, drink use, cook usc, and arsenic in the water as the regressors.
(b) Estimate $\sigma^{2}$ and the standard errors of the regression coefficients.
(c) Use the model to predict the arsenic in nails when the age is $30,$ the drink use is catcgory $5,$ the cook use is catcgory $5,$ and arsenic in the water is $0,135 \mathrm{ppm}$.

Key Concepts

Multiple Linear Regression

Multiple linear regression is a statistical method used to model the relationship between one dependent variable and two or more independent variables. It helps in understanding how the dependent variable changes when any one of the independent variables is varied, while the others are held fixed. This technique is widely used for both explanatory analysis and prediction, providing insights into the effect of multiple factors simultaneously.

Least Squares Estimation

Least squares estimation is a fundamental method used to estimate the parameters in a regression model. By minimizing the sum of the squares of the differences between the observed and predicted values, it provides the best linear unbiased estimates under certain assumptions. This estimation method is pivotal in constructing the regression equation that accurately represents the underlying data.

Estimation of Error Variance (?²)

The estimation of the error variance, denoted as ?², is crucial in regression analysis as it quantifies the variability of the response variable that is not explained by the independent variables. An unbiased estimator of ?² is typically derived from the residual sum of squares divided by the degrees of freedom, and it serves as a key component in assessing the overall fit of the model and in further inferential procedures.

Standard Errors of Regression Coefficients

The standard errors of the regression coefficients measure the precision of the estimated parameters in the regression model. They are derived from the variance-covariance matrix of the estimators and are used to construct confidence intervals and perform hypothesis tests for each predictor. Smaller standard errors suggest more precise estimates, contributing to more reliable interpretation of the relationships modeled.

Prediction in Regression Analysis

Prediction in regression analysis involves using the fitted model to estimate the outcome for new observations by plugging in the predictor values into the regression equation. This process not only provides a point estimate for the response variable but, along with standard errors, also allows for the assessment of prediction uncertainty, typically through prediction intervals.

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