Descriptive measures of the strength of a linear associationTranslating a research question into a statistical procedureWhere does this topic fit in?Situation #1 A very weak linear relationshipSituation #2 A fairly strong linear relationshipCoefficient of determination r2Interpretation of r2R-sq in Minitab fitted line plotR-sq in Minitab regression outputPearson correlation coefficient rFormulas for the Pearson correlation coefficient rWhat do we learn from the formulas for r?Interpretation of Pearson correlation coefficient rPearson correlation coefficient r in MinitabHow strong is the linear relationship between Celsius and Fahrenheit?How strong is the linear relationship between # of stories and height?How strong is the linear relationship between driver age and see distance?How strong is the linear relationship between height and g.p.a.?Caution #1Example of Caution #1Clarification of Caution #1Caution #2Example of Caution #2Caution #3Example of Caution #3Slide 26Caution #4Example of Caution #4Caution #5Example of Caution #5Slide 31Slide 32Caution #6Caution #7Using the sample correlation r to learn about the population correlation ρSlide 36Slide 37Is there a linear correlation between husband’s age and wife’s age?Slide 39The formal t-test for correlation coefficient ρSlide 41When is it okay to use the t-test for testing H0: ρ = 0?The three tests will always yield similar results.The three tests will always yield similar results.Which results should I report?Descriptive measures of the strength of a linear associationr-squared and the (Pearson) correlation coefficient rTranslating a research question into a statistical procedure•How strong is the linear relationship between skin cancer mortality and latitude?–(Pearson) correlation coefficient r–Coefficient of determination r2Where does this topic fit in?•Model formulation•Model estimation•Model evaluation•Model use10 9 8 7 6 5 4 3 2 1 0605040xyS = 7.81137 R-Sq = 6.5 % R-Sq(adj) = 3.2 %y = 54.4758 - 0.764016 xRegression Plot 6.182712niiyySSTO 5.1708ˆ12niiiyySSE 1.119ˆ12niiyySSRyyˆSituation #1A very weak linear relationship0 1 2 3 4 5 6 7 8 9 101020304050607080xyy = 75.5458 - 5.76402 xS = 7.81137 R-Sq = 79.9 % R-Sq(adj) = 79.2 %Regression Plot 3.6679ˆ12niiyySSR 5.1708ˆ12niiiyySSE 8.848712niiyySSTOyyˆSituation #2A fairly strong linear relationshipCoefficient of determination r2SSTOSSESSTOSSRr 12•r2 is a number (a proportion!) between 0 and 1.•If r2 = 1:–all data points fall perfectly on the regression line–the predictor x accounts for all of the variation in y•If r2 = 0:–the fitted regression line is perfectly horizontal–the predictor x accounts for none of the variation in yInterpretation of r2•r2 ×100 percent of the variation in y is reduced by taking into account predictor x. •r2 ×100 percent of the variation in y is “explained by” the variation in predictor x.R-sq in Minitab fitted line plot30 40 50100150200Latitude (at center of state)MortalityMort = 389.189 - 5.97764 LatS = 19.1150 R-Sq = 68.0 % R-Sq(adj) = 67.3 %Regression PlotR-sq in Minitab regression outputThe regression equation is Mort = 389.189 - 5.97764 Lat S = 19.1150 R-Sq = 68.0 % R-Sq(adj) = 67.3 %Analysis of VarianceSource DF SS MS F PRegression 1 36464.2 36464.2 99.7968 0.000Error 47 17173.1 365.4 Total 48 53637.3Pearson correlation coefficient r2rr •r is a (unitless) number between -1 and 1, inclusive.•Sign of coefficient of correlation–plus sign if slope of fitted regression line is positive–negative sign if slope of fitted regression line is negativeIf r2 is represented in decimal form, e.g. 0.39 or 0.87, then:Formulas for the Pearson correlation coefficient r niniiiniiiyyxxyyxxr1 1221 11212byyxxrniiniiWhat do we learn from the formulas for r?•The correlation coefficient r gets its sign from the slope b1.•The correlation coefficient r is a unitless measure.•The correlation coefficient r = 0 when the estimated slope b1 = 0 and vice versa.Interpretation of Pearson correlation coefficient r•There is no nice practical interpretation for r as there is for r2.•r = -1 is perfect negative linear relationship.•r = 1 is perfect positive linear relationship.•r = 0 is no linear relationship.•For other r, how strong the relationship between x and y is deemed depends on the research area.Pearson correlation coefficient r in MinitabCorrelations: Mort, LatPearson correlation of Mort and Lat = -0.825Correlations: Lat, MortPearson correlation of Lat and Mort = -0.825How strong is the linear relationship between Celsius and Fahrenheit? 0 10 20 30 40 50 30 40 50 60 70 80 90100110120CelsiusFahrenheitFahrenheit = 32 + 1.8 CelsiusS = 0 R-Sq = 100.0 % R-Sq(adj) = 100.0 %Regression PlotPearson correlation of Celsius and Fahrenheit = 1.000How strong is the linear relationship between # of stories and height?105 95 85 75 65 55 45 35 25 151200 700 200STORIESHEIGHTS = 58.3259 R-Sq = 90.4 % R-Sq(adj) = 90.2 %HEIGHT = 90.3096 + 11.2924 STORIESRegression PlotPearson correlation of HEIGHT and STORIES = 0.951How strong is the linear relationship between driver age and see distance?80706050403020600500400300DrivAgeDistanceS = 49.7616 R-Sq = 64.2 % R-Sq(adj) = 62.9 %Distance = 576.682 - 3.00684 DrivAgeRegression PlotPearson correlation of Distance and DrivAge = -0.801How strong is the linear relationship between height and g.p.a.?75706560432heightgpaS = 0.542316 R-Sq = 0.3 % R-Sq(adj) = 0.0 %gpa = 3.41021 - 0.0065630 heightRegression PlotPearson correlation of height and gpa = -0.053Caution #1•The correlation coefficient r quantifies the strength of a linear relationship. •It is possible to get r = 0 with a perfect curvilinear relationship.Example of Caution #1 5 0-540302010 0xyS = 13.4907 R-Sq = 0.0 % R-Sq(adj) = 0.0 %y = 14 - 0.0000000 xRegression PlotPearson correlation of x and y = 0.000yˆClarification of Caution #1 5 0-540302010 0xyS = 0 R-Sq = 100.0 % R-Sq(adj) = 100.0 %y = 0.0000000 - 0.0000000 x + 1 x**2Regression PlotPearson correlation of x and y = 0.000Caution
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