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Chapter 3 Association: Contingency, Correlation, and Regression Learn …. How to examine links between two variables Agresti/Franklin Statistics, 1 of 52 Section 3.2 How Can We Explore the Association Between Two Quantitative Variables? Agresti/Franklin Statistics, 2 of 52 Scatterplot Graphical display of two quantitative variables: • Horizontal Axis: Explanatory variable, x • Vertical Axis: Response variable, y Agresti/Franklin Statistics, 3 of 52 Example: Internet Usage and Gross National Product (GDP) Agresti/Franklin Statistics, 4 of 52 Positive Association Two quantitative variables, x and y, are said to have a positive association when high values of x tend to occur with high values of y, and when low values of x tend to occur with low values of y Agresti/Franklin Statistics, 5 of 52 Negative Association Two quantitative variables, x and y, are said to have a negative association when high values of x tend to occur with low values of y, and when low values of x tend to occur with high values of y Agresti/Franklin Statistics, 6 of 52 Example: Did the Butterfly Ballot Cost Al Gore the 2000 Presidential Election? Agresti/Franklin Statistics, 7 of 52 Linear Correlation: r Measures the strength of the linear association between x and y • • • • A positive r-value indicates a positive association A negative r-value indicates a negative association An r-value close to +1 or -1 indicates a strong linear association An r-value close to 0 indicates a weak association Agresti/Franklin Statistics, 8 of 52 Calculating the correlation, r 1 xx y y r ( )( ) n 1 sx sy Agresti/Franklin Statistics, 9 of 52 Example: 100 cars on the lot of a used-car dealership Would you expect a positive association, a negative association or no association between the age of the car and the mileage on the odometer? Positive association Negative association No association Agresti/Franklin Statistics, 10 of 52 Section 3.3 How Can We Predict the Outcome of a Variable? Agresti/Franklin Statistics, 11 of 52 Regression Line Predicts the value for the response variable, y, as a straight-line function of the value of the explanatory variable, x ˆ a bx y Agresti/Franklin Statistics, 12 of 52 Example: How Can Anthropologists Predict Height Using Human Remains? Regression Equation: yˆ 61.4 2.4 x yˆ is the predicted height and x is the length of a femur (thighbone), measured in centimeters Agresti/Franklin Statistics, 13 of 52 Example: How Can Anthropologists Predict Height Using Human Remains? Use the regression equation to predict the height of a person whose femur length was 50 centimeters ˆ 61.4 2.4(50) y Agresti/Franklin Statistics, 14 of 52 Interpreting the y-Intercept y-Intercept: • the predicted value for y when x = 0 • helps in plotting the line • May not have any interpretative value if no observations had x values near 0 Agresti/Franklin Statistics, 15 of 52 Interpreting the Slope Slope: measures the change in the predicted variable for every unit change in the explanatory variable Example: A 1 cm increase in femur length results in a 2.4 cm increase in predicted height Agresti/Franklin Statistics, 16 of 52 Slope Values: Positive, Negative, Equal to 0 Agresti/Franklin Statistics, 17 of 52 Residuals Measure the size of the prediction errors Each observation has a residual Calculation for each residual: ˆ y y Agresti/Franklin Statistics, 18 of 52 Residuals A large residual indicates an unusual observation Large residuals can easily be found by constructing a histogram of the residuals Agresti/Franklin Statistics, 19 of 52 “Least Squares Method” Yields the Regression Line Residual sum of squares: 2 ˆ (residuals) ( y y) 2 The optimal line through the data is the line that minimizes the residual sum of squares Agresti/Franklin Statistics, 20 of 52 Regression Formulas for y-Intercept and Slope Slope: b r( sy sx ) Y-Intercept: a y b( x ) Agresti/Franklin Statistics, 21 of 52 The Slope and the Correlation Correlation: • Describes the strength of the association • • between 2 variables Does not change when the units of measurement change It is not necessary to identify which variable is the response and which is the explanatory Agresti/Franklin Statistics, 22 of 52 The Slope and the Correlation Slope: • Numerical value depends on the units used to • • • measure the variables Does not tell us whether the association is strong or weak The two variables must be identified as response and explanatory variables The regression equation can be used to predict the response variable Agresti/Franklin Statistics, 23 of 52 Section 3.4 What Are Some Cautions in Analyzing Associations? Agresti/Franklin Statistics, 24 of 52 Extrapolation Extrapolation: Using a regression line to predict y-values for x-values outside the observed range of the data • Riskier the farther we move from the range • of the given x-values There is no guarantee that the relationship will have the same trend outside the range of x-values Agresti/Franklin Statistics, 25 of 52 Regression Outliers Construct a scatterplot Search for data points that are well removed from the trend that the rest of the data points follow Agresti/Franklin Statistics, 26 of 52 Influential Observation An observation that has a large effect on the regression analysis Two conditions must hold for an observation to be influential: Its x-value is relatively low or high compared to the rest of the data It is a regression outlier, falling quite far from the trend that the rest of the data follow Agresti/Franklin Statistics, 27 of 52 Which Regression Outlier is Influential? Agresti/Franklin Statistics, 28 of 52 Example: Does More Education Cause More Crime? Agresti/Franklin Statistics, 29 of 52 Correlation does not Imply Causation A correlation between x and y means that there is a linear trend that exists between the two variables A correlation between x and y, does not mean that x causes y Agresti/Franklin Statistics, 30 of 52 Lurking Variable A lurking variable is a variable, usually unobserved, that influences the association between the variables of primary interest Agresti/Franklin Statistics, 31 of 52 Simpson’s Paradox The direction of an association between two variables can change after we include a third variable and analyze the data at separate levels of that variable Agresti/Franklin Statistics, 32 of 52 Example: Is Smoking Actually Beneficial to Your Health? Agresti/Franklin Statistics, 33 of 52 Example: Is Smoking Actually Beneficial to Your Health? Agresti/Franklin Statistics, 34 of 52 Example: Is Smoking Actually Beneficial to Your Health? Agresti/Franklin Statistics, 35 of 52 Example: Is Smoking Actually Beneficial to Your Health? Agresti/Franklin Statistics, 36 of 52 Example: Is Smoking Actually Beneficial to Your Health? An association can look quite different after adjusting for the effect of a third variable by grouping the data according to the values of the third variable Agresti/Franklin Statistics, 37 of 52 Data are available for all fires in Chicago last year on x = number of firefighters at the fires and y = cost of damages due to fire Would you expect the correlation to be negative, zero, or positive? a. Negative b. Zero c. Positive Agresti/Franklin Statistics, 38 of 52 Data are available for all fires in Chicago last year on x = number of firefighters at the fires and y = cost of damages due to fire If the correlation is positive, does this mean that having more firefighters at a fire causes the damages to be worse? a. Yes b. No Agresti/Franklin Statistics, 39 of 52 Data are available for all fires in Chicago last year on x = number of firefighters at the fires and y = cost of damages due to fire Identify a third variable that could be considered a common cause of x and y: a. Distance from the fire station b. Intensity of the fire c. Time of day that the fire was discovered Agresti/Franklin Statistics, 40 of 52