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Chapter 3 Association: Contingency, Correlation, and Regression. Section 3.4 Cautions in Analyzing Associations. Extrapolation Is Dangerous. Extrapolation: Using a regression line to predict y -values for x -values outside the observed range of the data.
Cautions in Analyzing Associations
Figure 3.18 An Observation Is a Regression Outlier if it is Far Removed from the Trend that the Rest of the Data Follow. The top two points are regression outliers. Not all regression outliers are influential in affecting the correlation or slope. Question: Which regression outlier in this figure is influential?
Data are available for all fires in Chicago last year on x = number of
firefighters at the fire and y = cost of damages due to the fire.
1. Would you expect the correlation to be negative, zero, or positive?
2. If the correlation is positive, does this mean that having more
firefighters at a fire causes the damages to be worse? Yes or No?
3. Identify a third variable that could be considered a common cause
of x and y:
A lurking variable is a variable, usually unobserved, that influences the association between the variables of primary interest.
When two explanatory variables are both associated with a response variable but are also associated with each other, there is said to be confounding.
Lurking variables are not measured in the study but have the potential for confounding.
Is Smoking Actually Beneficial to Your Health?
Table 3.7 Smoking Status and 20-Year Survival in Women
Probability of Death of Smoker = 139/582= 24%
Probability of Death of Nonsmoker = 230/732= 31%
This can’t be true that smoking improves your chances of living!
What’s going on?!
Break out Data by Age
Table 3.8 Smoking Status and 20-Year Survival, for Four Age Groups
For instance, for smokers of age 18–34, from Table 3.8 the proportion who died was 5/(5 + 174) = 0.028, or 2.8%
Could age explain the association?
Table 3.9 Conditional Percentages of Deaths for Smokers and Nonsmokers, by Age
Figure 3.23 MINITAB Bar Graph Comparing Percentage of Deaths for Smokers and Nonsmokers, by Age. This side-by-side bar graph shows the conditional percentages
from Table 3.9.
An association can look quite different after adjusting for the effect of a third variable by grouping the data according to the values of the third variable (age).
Lurking variables can affect associations in many ways. For instance, a lurking variable may be a common cause of both the explanatory and response variable.
In practice, there’s usually not a single variable that causally explains a response variable or the association between two variables. More commonly, there are multiple causes . When there are multiple causes, the association among them makes it difficult to study the effect of any single variable.
When two explanatory variables are both associated with a response variable but are also associated with each other, confounding occurs.
It is difficult to determine whether either of them truly causes the response because a variable’s effect could be at least partly due to its association with the other variable.