1. 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?

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

3. 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...

4. A scatterplot is the best graph for showing relationships between two quantitative variables In a scatterplot, one axis is used to represent each of the variables, and the data are plotted as points on the graph. StudentBeersBAC 150.1 220.03 390.19 670.095 730.07 930.02 1140.07 1350.085 480.12 530.04 850.06 1050.05 1260.1 1470.09 1510.01 1640.05

5. Explanatory (independent) variable: number of beers Response (dependent) variable: blood alcohol content x y Explanatory and response variables A response variable measures or records an outcome of a study. An explanatory variable explains changes in the response variable. Typically, the explanatory or independent variable is plotted on the x axis, and the response or dependent variable is plotted on the y axis.

6. 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. Note the identical responses... Figure 2.4 Introduction to the Practice of Statistics, Sixth Edition © 2009 W.H. Freeman and Company

7. Form and direction of an association Linear Nonlinear No relationship

8. 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

9. 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. No relationship: X and Y vary independently. Knowing X tells you nothing about Y.

10. This is a very strong relationship. The daily amount of gas consumed can be predicted quite accurately for a given temperature value. This is a weak relationship. For a particular state median household income, you can’t predict the state per capita income very well. Strength of the relationship or association...

11. 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...

12. Categorical variables in scatterplots Often, things are not simple and one-dimensional. We need to group the data into categories to reveal trends. Lurking Variable! What may look like a positive linear relationship is in fact a series of negative linear associations. Plotting different habitats (the lurking variable) in different colors allows us to make that important distinction.

13. 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.

14. 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

15. 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 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. What association? What relationship?

16. 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