Question

A recent study \(^{76}\) has found an association between elevation and lung cancer incidence. Incidence of cancer appears to be lower at higher elevations, possibly because of lower oxygen levels in the air. We are told that "for every one \(\mathrm{km}\) rise in elevation, lung cancer incidence decreased by 7.23 " where cancer incidence is given in cases per 100,000 individuals. (a) Is this a positive or negative association? (b) Which of the following quantities is given in the sentence in quotes: correlation, slope of regression line, intercept of regression line, or none of these? (c) What is the explanatory variable? What is the response variable?

Short answer

(a) The association is negative. (b) The sentence in quotes provides the slope of the regression line. (c) The explanatory variable is 'elevation' and the response variable is 'lung cancer incidence.'

Step-by-step solution

Identifying Type of Association

To identify whether the association is positive or negative, we need to analyze the direction of association. This association indicates that as the elevation (height from sea level) increases, the incidence of lung cancer decreases. This trend where one variable increases and the other decreases is referred to as a negative association.

Interpreting Sentence Quantities

The sentence 'for every one km rise in elevation, lung cancer incidence decreased by 7.23' describes the change in lung cancer incidence for each increase in kilometer in elevation. This does not represent correlation since correlation is a measure of the linear relationship between two variables and it is not represented in this way. It isn't an intercept either as intercept is the value of the response variable when explanatory variable is zero. It most aligns with the description of the slope of a regression line which describes the average change in the response variable for a unit change in the explanatory variable.

Identifying Explanatory and Response Variables

In this context, the explanatory variable (also known as the independent variable) is the elevation as it is the variable that we are using to predict or explain the variation in the response. The response variable (also known as the dependent variable) is the lung cancer incidence, as it is the outcome or the variable we are interested in predicting or explaining.