Association between 2 variables We've described the distribution of 1 variable - but what if 2 variables are measured on the same individual? Examples? How could you describe the association between the two? Our descriptions will depend upon the types of variables (categorical or quantitative): categorical vs. categorical - Examples? categorical vs. quantitative - Examples? quantitative vs. quantitative - Examples?

Figure 2.1 © 2009 W.H. Freeman and Company Introduction to the Practice of Statistics, Sixth Edition © 2009 W.H. Freeman and Company

Explanatory variable vs. Response Variable One common task is to show that one variable can be used to explain variation in the other. Explanatory variable vs. Response Variable (sometimes these are called independent vs. dependent variables) These associations can be explored both graphically and numerically: begin your analysis with graphics find a pattern & look for deviations from the pattern look for a mathematical model to describe the pattern But again we do the above depending upon what type variables we have… we'll start with quantitative vs. quantitative ...

direction: positive, negative or flat (no direction) Describe the pattern of the relationship between the two variables in a scatterplot by its direction, strength, and form. direction: positive, negative or flat (no direction) strength: strong, weak, moderately strong, etc. form: linear, curved (non-linear), clusters, no pattern See example 2.8 on page 91(2.1, 4/5) Note the identical responses ... Figure 2.4 Introduction to the Practice of Statistics, Sixth Edition © 2009 W.H. Freeman and Company

Form and direction of an association Linear No relationship Nonlinear

Positive association: High values of one variable tend to occur together with high values of the other variable. Negative association: High values of one variable tend to occur together with low values of the other variable. The scatterplots below show perfect linear associations

No relationship: X and Y vary independently No relationship: X and Y vary independently. Knowing X tells you nothing about Y. One way to think about this is to remember the following: Imagine a line through the data points.. the equation for that line is y = 5. x is not involved.

What if there are categorical variables involved What if there are categorical variables involved? either as the explanatory variable or as a “lurking variable”? A scatterplot sometimes can help by indicating the categories of the lurking variable with different plotting symbols or colors... Often though the best way to see the pattern if the explanatory variable is categorical is to draw side-by-side boxplots. Put the categorical variable on the horizontal axis, and draw a boxplot for each category, side-by-side. Here are some some examples of various explanatory, lurking, and response variables...

Comparison of men and women racing records over time. Each group shows a very strong negative linear relationship that would not be apparent without the gender categorization. Relationship between lean body mass and metabolic rate in men and women. Both men and women follow the same positive linear trend, but women show a stronger association. As a group, males typically have larger values for both variables.

The next slide is tricky... Look at Figure 1.23 on page 52 - Note the ordinal scale of the explanatory variable education level. Are these two variables associated ? Why? The next slide is tricky... Figure 1.23 Introduction to the Practice of Statistics, Sixth Edition © 2009 W.H. Freeman and Company

Example: Beetles trapped on boards of different colors Beetles were trapped on sticky boards scattered throughout a field. The sticky boards were of four different colors (categorical explanatory variable). The number of beetles trapped (response variable) is shown on the graph below. Blue White Green Yellow Board color ? What association? What relationship? Blue Green White Yellow Board color  Describe one category at a time. When both variables are quantitative, the order of the data points is defined entirely by their value. This is not true for categorical data.

HW: Read the Introduction to Chapter 2 and section 2.1 Do #2.6-2.9, 2.11, 2.13-2.15, 2.18, 2.19, 2.21, 2.26 (use JMP to draw all scatterplots - Analyze -> Fit Y by X - (Y is the response & will go on the vertical axis, X is the explanatory & will go on the horizontal axis) Look ahead to correlation and regression in sections 2.2 and 2.3