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 $=0.135$. Obtain a point estimate and a $95 \%$ Cl for the relative risk of breast cancer comparing women whose total fat intake differs by $10 \mathrm{g} /$ day.

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 $=0.135$.
Obtain a point estimate and a $95 \%$ Cl for the relative risk of breast cancer comparing women whose total fat intake differs by $10 \mathrm{g} /$ day.

Key Concepts

Confidence Interval Estimation

A confidence interval (CI) for an odds ratio provides a range of values within which the true odds ratio is expected to lie with a certain degree of confidence, typically 95%. The CI is usually calculated by first determining the CI for the log odds coefficient by adding and subtracting a margin (e.g., 1.96 times the standard error for a 95% CI) from the estimate, and then exponentiating the endpoints of this interval. This transformation yields the CI for the odds ratio, allowing one to assess the precision and statistical significance of the estimated effect.

Odds Ratio

The odds ratio is a measure derived from logistic regression that represents the change in odds of the outcome for a one-unit change in the predictor variable. It is obtained by exponentiating the regression coefficient. An odds ratio greater than 1 indicates an increased odds of the event, while an odds ratio less than 1 indicates a decreased odds. This measure is particularly useful because it translates the model's log odds information into a more interpretable multiplicative effect on the probability scale.

Interpretation of Regression Coefficients

In logistic regression, each regression coefficient represents the change in the log odds of the outcome per unit increase in the corresponding predictor, holding other variables constant. This interpretation is central to understanding the effect of predictors on the probability of the outcome. For instance, a positive coefficient implies an increase in the log odds, and by exponentiating it, one obtains an odds ratio, which quantifies the multiplicative change in odds.

Logistic Regression

Logistic regression is a statistical technique used to model a binary outcome variable by estimating the probability that the outcome occurs as a function of one or more predictor variables. It uses a logistic function to ensure that the predicted probabilities lie between 0 and 1, making it well-suited for dichotomous response data. This method is widely used in fields such as epidemiology, social sciences, and machine learning to assess the impact of various predictors on the likelihood of an event.

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