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ASSOCIATION: CONTINGENCY, CORRELATION, AND REGRESSION Chapter 3

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3.1 The Association between Two Categorical Variables

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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

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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

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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

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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?

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Proportions & Conditional Proportions

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

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

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3.2 The Association between Two Quantitative Variables

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Internet Usage & GDP Data Set www.knitwareblog.com

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

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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

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

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

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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

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3.3 Can We Predict the Outcome of a Variable?

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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

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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 www.chem.utoronto.ca Goal: Minimize distance from data to regression line

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

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

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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 www.geektoysgamesandgadgets.com Bones

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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

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Slope Values: Positive, Negative, Zero

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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

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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

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3.4 What Are Some Cautions in Analyzing Associations?

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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

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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

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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?

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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

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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

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Break out Data by Age Simpson’s Paradox Example

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

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