PSLS.PPT.Ch23

PSLS.PPT.Ch23 - Inferenceforregression PSLS chapter 23 2009...

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Inference for regression PSLS chapter 23 © 2009 W.H. Freeman and Company
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Objectives (PSLS chapter 23) Inference for regression The regression model Confidence interval for the regression slope β Testing the hypothesis of no linear relationship Inference for prediction Conditions for inference
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Most scatterplots are created from sample data. If we observe a linear relationship between x and y there, would we also have found a linear relationship with a different sample? Is the observed relationship statistically significant (not entirely explained by chance events due to random sampling)? What is the population mean response μ y as a function of the explanatory variable x? μ y = α + β x Mating calls of Hyla avivoca y = 0.4799x - 4.8279 R 2 = 0.7489 2 3 4 5 6 7 8 9 16 18 20 22 24 26 28 Temperature (Celsius) Call frequency (notes per second) .
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The regression model The least squares regression line ŷ = a + bx is a mathematical model of the relationship between 2 quantitative variables: sample data = fit + residual” The regression line is the fit. For each data point in the sample , the residual is the difference (y - ŷ).
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At the population level, the model becomes y i = ( α + β x i ) + ( ε i ) with residuals ε i independent and normally distributed N(0, σ ). The population mean response μ y is μ y = α + β x ŷ unbiased estimate for mean response μ y a unbiased estimate for intercept α b unbiased estimate for slope β
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The regression standard error, s , for n sample data points is calculated from the residuals (y i ŷ i ): Regression assumes equal variance of Y ( σ is the same for all values of x). For any fixed x, the responses y follow a Normal distribution with standard deviation σ . 2 ) ˆ ( 2 2 2 - - = - = n y y n residual s i i
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Frogs mating calls A study recorded mating call frequency (in notes per second) and ambient temperature (in Celsius) in 17 bird-song tree frogs. Mating calls of Hyla avivoca y = 0.4799x - 4.8279 R 2 = 0.7489 2 3 4 5 6 7 8 9 16 18 20 22 24 26 28 Temperature (Celsius) Call frequency (notes per second) .
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Using software Regression Analysis: CallFreq versus Temperature The regression equation is CallFreq = - 3.59 + 0.401 Temperature Predictor Coef SE Coef T P Constant -3.585 1.507 -2.38 0.031 Temperature 0.40141 0.06501 6.17 0.000 S = 0.774686 R-Sq = 71.8% R-Sq(adj) = 69.9% MINITAB 2 ) ˆ ( 2 2 2 - - = - = n y y n residual s i i s: regression standard error , unbiased estimate of σ
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EXCEL s: regression standard error , unbiased estimate of σ 2 2 - = n residual s Regression Statistics Multiple R 0.847 R Square 0.718 Adjusted R Square 0.699 Standard Error 0.775 Observations 17 Coefficients Standard Error t Stat P-value Lower 95% Upper 95% Intercept -3.585 1.507 -2.379 0.031 -6.797 -0.373 Temperature 0.401 0.065 6.175 0.000 0.263 0.540
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Confidence interval for the slope  β Estimating the regression parameter β for the slope is a case of one- sample inference with σ unknown. Hence we rely on t-distributions.
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PSLS.PPT.Ch23 - Inferenceforregression PSLS chapter 23 2009...

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