Ch 2 and 9.1 Relationships Between 2 Variables. More than one variable can be measured on each individual. Examples: Gender and Height Size and Cost Eye color and Major We want to look at the relationship among these variables. Is there an association between these two variables?
Overall TotalTwo-Way Tables
Shows the percentages
for the joint, marginal,
and conditional distributions.
___________AssociationRelationships between 2 numeric variables
Response Variable (y-axis)
Explanatory Variable (x-axis)
AssociationRelationships between 2 numeric variables
r = 1
r = 0
r = -1
r = 0.04
r = -0.84
r = 0.76
r = 0.21
It is possible for there to be a strong relationship between two variables and still have r ≈ 0.
where y is the point and is the predicted point.
How much of the variation is explained
by the least squares line of y on x? ______
What is the correlation coefficient? ______
Horsepower = -10.78 + 0.04*weight (Equation of the line.)
__________: y-value or response (horsepower) when line crosses the y-axis.
_______: increase in response for a unit increase in explanatory variable.
So if weight increases by one pound, horsepower increases by 0.04 units (on average).
Lurking Variable: A variable that is not among the explanatory or response variables in a study and yet may influence the interpretation of relationships among those variables.
Simpson’s Paradox: An association or comparison that holds for all of several groups can reverse direction when the data are combined to form a single group. This reversal is called Simpson’s Paradox. This can happen when a lurking variable is present. Please see Examples 9.9 and 9.10 in the text.
Child 18 is an outlier in the x direction. Because of its extreme position on the age scale, this point has a strong influence on the position of the regression line.
r2 is also affected by the influential observation. With Child 18, r2 = 41%, but without Child 18, r2 = 11%. The apparent strength of the association was largely due to a single influential observation.
The dashed line was calculated leaving out Child 18. The solid line is with Child 18.