ASSOCIATION: CONTINGENCY, CORRELATION, AND REGRESSION Chapter 3

3.1 The Association between Two Categorical Variables

Response and Explanatory Variables  Response variable (dependent, y) outcome variable  Explanatory variable (independent, x) defines groups  Response/Explanatory 1. Grade on test/Amount of study time 2. Yield of corn/Amount of rainfall

Association Association – When a value for one variable is more likely with certain values of the other variable Data analysis with two variables 1. Tell whether there is an association and 2. Describe that association

Contingency Table  Displays two categorical variables  The rows list the categories of one variable; the columns list the other  Entries in the table are frequencies www1.pictures.fp.zimbio.com

Contingency Table  What is the response (outcome) variable? Explanatory?  What proportion of organic foods contain pesticides?Conventionally grown?  What proportion of all sampled foods contain pesticides?

Proportions & Conditional Proportions

Side by side bar charts show conditional proportions and allow for easy comparison Proportions & Conditional Proportions

If no association, then proportions would be the same Proportions & Conditional Proportions Since there is association, then proportions are different

3.2 The Association between Two Quantitative Variables

Internet Usage & GDP Data Set

Scatterplot Graph of two quantitative variables:  Horizontal Axis: Explanatory, x  Vertical Axis: Response, y

Interpreting Scatterplots  The overall pattern includes trend, direction, and strength of the relationship  Trend: linear, curved, clusters, no pattern  Direction: positive, negative, no direction  Strength: how closely the points fit the trend  Also look for outliers from the overall trend

Used-car Dealership What association would we expect between the age of the car and mileage? a) Positive b) Negative c) No association

Linear Correlation, r Measures the strength and direction of the linear association between x and y

Correlation coefficient: Measuring Strength & Direction of a Linear Relationship  Positive r => positive association  Negative r => negative association  r close to +1 or -1 indicates strong linear association  r close to 0 indicates weak association

3.3 Can We Predict the Outcome of a Variable?

Regression Line  Predicts y, given x:  The y-intercept and slope are a and b  Only an estimate – actual data vary  Describes relationship between x and estimated means of y farm4.static.flickr.com

Residuals  Prediction errors: vertical distance between data point and regression line  Large residual indicates unusual observation  Each residual is:  Sum of residuals is always zero  Goal: Minimize distance from data to regression line

msenux.redwoods.edu Least Squares Method  Residual sum of squares:  Least squares regression line minimizes vertical distance between points and their predictions

Regression Analysis Identify response and explanatory variables  Response variable is y  Explanatory variable is x

Anthropologists Predict Height Using Remains?  Regression Equation:  is predicted height and x is the length of a femur, thighbone (cm) Predict height for femur length of 50 cm Bones

Interpreting the y-Intercept and slope  y-intercept: y-value when x = 0  Helps plot line  Slope: change in y for 1 unit increase in x  1 cm increase in femur length means 2.4 cm increase in predicted height

Slope Values: Positive, Negative, Zero

Slope and Correlation  Correlation, r:  Describes strength  No units  Same if x and y are swapped  Slope, b:  Doesn’t tell strength  Has units  Inverts if x and y are swapped

 Proportional reduction in error, r 2  Variation in y-values explained by relationship of y to x  A correlation, r, of.9 means  81% of variation in y is explained by x Squared Correlation, r 2

3.4 What Are Some Cautions in Analyzing Associations?

Extrapolation  Extrapolation: Predicting y for x-values outside range of data  Riskier the farther from the range of x  No guarantee trend holds Neil Weiss, Elementary Statistics, 7 th Edition

Outliers and Influential Points  Regression outlier lies far away from rest of data  Influential if both: 1. Low or high, compared to rest of data 2. Regression outlier www2.selu.edu

Correlation Does Not Imply Causation Strong correlation between x and y means  Strong linear association between the variables  Does not mean x causes y Ex. 95.6% of cancer patients have eaten pickles, so do pickles cause cancer?

Lurking Variables & Confounding 1. Ice cream sales & drowning => temperature 2. Reading level & shoe size => age  Confounding – two explanatory variables both associated with response variable and each other  Lurking variables – not measured in study but may confound

Simpson’s Paradox Example Probability of Death of Smoker = 139/582 = 24% Probability of Death of Nonsmoker = 230/732 = 31% Simpson’s Paradox:  Association between two variables reverses after third is included

Break out Data by Age Simpson’s Paradox Example

Associations look quite different after adjusting for third variable Simpson’s Paradox Example