Chapter 11--Regression and Correlation Methods

# 5234 06082 sigma sqrtsumresidualsrf231 2 sigma 1

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Unformatted text preview: pt) estriol 21.5234 0.6082 &gt; sigma&lt;-sqrt(sum(residuals(rf)^2)/(31-2)) &gt; sigma [1] 3.82111 &gt; pre&lt;-data.frame(estriol=c(10,13,20)) &gt; predict(rf,newdata=pre) 1 2 3 27.60533 29.42990 33.68724 Chapter 11: Regression and Correlation Methods Stat 491: Biostatistics Introduction Least Square Estimates of the Parameters Inference about the Parameters Prediction Assessing Adequacy of Fit Correlation Multiple Regression Example Cont’d ... b. Interpreting parameter estimates: ˆ From β1 = 0.6082, we conclude that for a 1mg/24 hr increase in the estriol level, there is an estimated expected increase of 60.82 grams in birth weight. ˆ Interpreting β0 would be extrapolation. The standard deviation around the line (the residual standard deviation) of sε = 3.8211 indicates that about 95% of the prediction errors should be between ±2(3.8211) = ±7.6422. c. Since, = 21.5234 + 0.6082(10) = 27.605 we expect the birth weight would be 2.76kg for a women who had estriol level of 10 mg/24 hr. Chapter 11: Regression and Correlation Methods Stat 491: Biostatistics Introduction Least Square Estimates of the Parameters Inference about the Parameters Prediction Assessing Adequacy of Fit Correlation Multiple Regression ˆ ˆ Standard Errors of β0 and β1 Standard Error of an estimate indicates how accurately one can estimate the correct population value. ˆ ˆ The standard errors of β1 and β0 denoted, respectively, by σβ1 ˆ and σβ1 are given by ˆ σε σβ1 = √ ˆ Sxx and σβ0 = σε ˆ 1 x2 ¯ + . n Sxx ˆ Therefore, the quality of estimation of β1 is inﬂuenced by the 2 and the amount of variation in the error variance σε independent variable Sxx . The ideal situation for estimating β0 is when x = 0. Why? ¯ Chapter 11: Regression and Correlation Methods Stat 491: Biostatistics Introduction Least Square Estimates of the Parameters Inference about the Parameters Prediction Assessing Adequacy of Fit Correlation Multiple Regression Inference about β1 The hypothesis x...
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