Chapter 11Linear Regression and CorrelationLeast Squares Estimation of b0, b1Example - Pharmacodynamics of LSDLeast Squares ComputationsSlide 6SPSS Output and Plot of EquationInference Concerning the Slope (b1)Hypothesis Test for b1(1-a)100% Confidence Interval for b1Slide 11Confidence Interval for Mean When x=x*Prediction Interval of Future Response @ x=x*Correlation CoefficientSlide 15Slide 16Example - SPSS Output Pearson’s and Spearman’s MeasuresHypothesis Test for ryxAnalysis of Variance in RegressionAnalysis of Variance for RegressionSlide 21Slide 22Example - SPSS OutputChapter 11Linear Regression and CorrelationLinear Regression and Correlation•Explanatory and Response Variables are Numeric•Relationship between the mean of the response variable and the level of the explanatory variable assumed to be approximately linear (straight line)•Model:),0(~210 eNxY• 1 > 0 Positive Association• 1 < 0 Negative Association• 1 = 0 No AssociationLeast Squares Estimation of 0, 10 Mean response when x=0 (y-intercept)1 Change in mean response when x increases by 1 unit (slope)• 0, 1 are unknown parameters (like )• 0+1x Mean response when explanatory variable takes on the value x•Goal: Choose values (estimates) that minimize the sum of squared errors (SSE) of observed values to the straight-line: 211^0^12^1^0^^niiiniiixyyySSExyExample - Pharmacodynamics of LSDScore (y) LSD Conc (x)78.93 1.1758.20 2.9767.47 3.2637.47 4.6945.65 5.8332.92 6.0029.97 6.41• Response (y) - Math score (mean among 5 volunteers)• Predictor (x) - LSD tissue concentration (mean of 5 volunteers)• Raw Data and scatterplot of Score vs LSD concentration:LSD_CONC7654321SCORE80706050403020Source: Wagner, et al (1968)Least Squares Computations xxxyyyyyxyxxSSSSSEyySyyxxSxxS222 2212^21^0^21^nSSEnyysxySSxxyyxxniiixxxyeSummary CalculationsParameter EstimatesExample - Pharmacodynamics of LSD72.5001.910.8910.89)33.4)(01.9(09.5001.94749.224872.202333.4733.30087.50761.3502^1^0^1^esxyxyxyScore (y) LSD Conc (x) x-xbar y-ybar Sxx Sxy Syy78.93 1.17 -3.163 28.843 10.004569 -91.230409 831.91864958.20 2.97 -1.363 8.113 1.857769 -11.058019 65.82076967.47 3.26 -1.073 17.383 1.151329 -18.651959 302.16868937.47 4.69 0.357 -12.617 0.127449 -4.504269 159.18868945.65 5.83 1.497 -4.437 2.241009 -6.642189 19.68696932.92 6.00 1.667 -17.167 2.778889 -28.617389 294.70588929.97 6.41 2.077 -20.117 4.313929 -41.783009 404.693689350.61 30.33 -0.001 0.001 22.474943 -202.487243 2078.183343(Column totals given in bottom row of table)SPSS Output and Plot of EquationCoefficientsa89.124 7.048 12.646 .000-9.009 1.503 -.937 -5.994 .002(Constant)LSD_CONCModel1B Std. ErrorUnstandardizedCoefficientsBetaStandardizedCoefficientst Sig.Dependent Variable: SCOREa. Linear Regression1.00 2.00 3.00 4.00 5.00 6.00lsd_conc30.0040.0050.0060.0070.0080.00scorescore = 89.12 + -9.01 * lsd_concR-Square = 0.88Math Score vs LSD Concentration (SPSS)Inference Concerning the Slope (1)•Parameter: Slope in the population model (1)•Estimator: Least squares estimate:•Estimated standard error: •Methods of making inference regarding population:–Hypothesis tests (2-sided or 1-sided) –Confidence Intervals1^xxeSsE 1S^1Hypothesis Test for 1•2-Sided Test–H0: 1 = 0–HA: 1 0•1-sided Test–H0: 1 = 0–HA+: 1 > 0 or–HA-: 1 < 0|)|(2:value||:..1:..2,2/1^obsnobsxxeobsttPPttRRSstST)(:)(::..:..1:..2,2,1^obsobsnobsnobsxxeobsttPvalPttPvalPttRRttRRSstST(1-)100% Confidence Interval for 1xxeSstSEt12/1^2/1^^1• Conclude positive association if entire interval above 0• Conclude negative association if entire interval below 0• Cannot conclude an association if interval contains 0• Conclusion based on interval is same as 2-sided hypothesis testExample - Pharmacodynamics of LSD50.1475.22112.7SE475.2212.772.5001.971^1^xxeSsn• Testing H0: 1 = 0 vs HA: 1 0 571.2|:|..01.650.101.9:..5,025. ttRRtSTobsobs• 95% Confidence Interval for 1 :)15.5,87.12(86.301.9)50.1(571.201.9 Confidence Interval for Mean When x=x*•Mean Response at a specific level x* is•Estimated Mean response and standard error (replacing unknown 0 and 1 with estimates):•Confidence Interval for Mean Response: **)|(10xxyEy xxeySxxnsx21^0^^*1SE*^ xxenynySxxnstt22,2/^2,2/^*1SE^Prediction Interval of Future Response @ x=x*•Response at a specific level x* is•Estimated response and standard error (replacing unknown 0 and 1 with estimates):•Prediction Interval for Future Response: *10*xyyx xxeySxxnsxy21^0^^*11SE*^ xxenynSxxnstyty22,2/^2,2/^*11SE^Correlation Coefficient• Measures the strength of the linear association between two variables•Takes on the same sign as the slope estimate from the linear regression•Not effected by linear transformations of y or x•Does not distinguish between dependent and independent variable (e.g. height and weight)•Population Parameter: yx•Pearson’s Correlation Coefficient: 11 rSSSryyxxxyyxCorrelation Coefficient• Values close to 1 in absolute value strong linear association, positive or negative from sign• Values close to 0 imply little or no association• If data contain outliers (are non-normal), Spearman’s coefficient of correlation can be computed based on the ranks of the x and y values• Test of H0:yx = 0 is equivalent to test of H0:1=0• Coefficient of Determination (ryx2) - Proportion of variation in y “explained” by the regression on x:10)Total()Residual()Total()(222 rSSSSSSSSSESrryyyyyxyxExample - Pharmacodynamics of
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