A logistic-regression analysis similar to that presented in...

A logistic-regression analysis similar to that presented in Example 13.73 (on page 743 ) was run relating breast-cancer incidence in $1980-1984$ to calorie-adjusted total fat (heretofore referred to as total fat intake) as reported on a 1980 FFQ. In addition, age in 5 -year categories and alcohol in categories $(0,0.1-4.9,5.0-14.9,15+\mathrm{g} / \text { day })$ were also controlled for. The regression coefficient for a 10 -g/day increase in total fat intake was -0.163 with standard error $=0.135$. Use the data for total fat to fit the linear regression of DR total fat intake on FFQ total fat intake. Obtain the regression coefficient, standard error, and $p$ -value from this regression.

A logistic-regression analysis similar to that presented in Example 13.73 (on page 743 ) was run relating breast-cancer incidence in $1980-1984$ to calorie-adjusted total fat (heretofore referred to as total fat intake) as reported on a 1980 FFQ. In addition, age in 5 -year categories and alcohol in categories $(0,0.1-4.9,5.0-14.9,15+\mathrm{g} / \text { day })$ were also controlled for. The regression coefficient for a 10 -g/day increase in total fat intake was -0.163 with standard error $=0.135$.
Use the data for total fat to fit the linear regression of DR total fat intake on FFQ total fat intake. Obtain the regression coefficient, standard error, and $p$ -value from this regression.

Key Concepts

Controlling for Confounding Variables

Controlling for confounding variables involves including additional predictor variables in the regression model that may influence both the primary predictor and the outcome. This approach helps to isolate the effect of the main variable of interest by accounting for the influence of other variables, thereby providing a more accurate and valid estimate of the relationship between the main predictor and the outcome.

p-value

The p-value is a statistical metric used to determine the significance of the regression coefficient. It represents the probability of observing the data, or something more extreme, assuming that there is no real effect (null hypothesis). A low p-value suggests that the observed association is unlikely to be due to random chance, thereby providing evidence in favor of a true relationship.

Standard Error

The standard error measures the amount of sampling variability in the estimated regression coefficient. It indicates how much the estimated coefficient is expected to vary from one sample to another if the study were repeated. A smaller standard error relative to the coefficient suggests more precision in the estimate, while a larger value indicates more uncertainty.

Linear Regression

Linear regression is a modeling approach that estimates the relationship between a continuous dependent variable and one or more independent variables. The model assumes a linear relationship, which allows for the estimation of a regression coefficient that represents the expected change in the outcome for a one-unit change in the predictor, along with measures of variability such as the standard error.

Logistic Regression

Logistic regression is a statistical method used for modeling a binary outcome variable (such as the presence or absence of a disease) in relation to one or more predictor variables. It estimates the probability of an event occurring by fitting data to a logistic function, allowing for interpretation of the association between predictors and the odds of the outcome after adjusting for potential confounders.

Regression Coefficient

The regression coefficient quantifies the relationship between a predictor variable and an outcome variable. In linear regression, it represents the expected change in the outcome per unit change in the predictor variable, and in logistic regression it reflects the change in the log odds of the outcome. It is a key parameter for interpreting the strength and direction of the association between variables.

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