Question

1. Tamoxifen is a drug often used to treat breast cancer patients. One effect of the drug is to change the levels of cortisol-binding globulin (CBG). One study attempted to see if the effect of Tamoxifen is age dependent Yis the change in CBG and the covariate X is age. The following summary statistics reported on the patients age. The response variable were Σε 1613,yi= 281.9, S 3756.96, Syy 465.34, Sy = -757.64 n = 26 (a) Find the correlation coefficient and explain in the context of the problem (b) Find the least squares estimates of the intercept and slope. (c) Find the MSE (d) Please conduct a test to confirm the utility of this linear model

Answer 1

The following summary statistics are: 281.9, Sex = 3756.96, S., = 465.34, and S = -757.64. The formula for calculating the correlation coefficient is resory Mesa) (S) -757.64 (3756.96)(465.34) -757.64 71748264 -757.64 1322.219 = -0.57301 rs -0.57301 Interpretation: Therefore, the pearson product-moment correlation coefficient is rs -0.57301 The pears on correlation coefficient explains the negative linear relationship between the two quantitative variables (x) and (y).

(b) The least-square estimates of the intercept and slope: With these values, we compute 5: -757.64 3756.96 =-0.20166 b -0.20166 The formulae for calculating the mean values of the variables X and Y: SY Y - 281.9 26 = 10.84231 Y - 10.84231 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - ΣΧ, X = _1613 26 = 62.03846 X = 62.03846 With these values, compute by: Bp = 7 - 63 = 10.84231-(-0.20166) 62.03846 = 10.84231+12.5109 = 23.35317 by = 23.35317 The equation of the simple linear regressi on line is ů = bo +bX û = 23.35317 -0.20166X)

() The values of total sum of squares is SST = S. = 465.34 SST = 465.34 The value of the sum of squares due to residual is SSR = (b)s =(-0.20166) (-757.64) = 152.7857 SSR = 152.7857 The formula for calculating the MSE and it is obtained by using the values of SSR and SST. We get the required value of MSE is MSESSE n-2 SST - SSR -2 1 1 1 465.34-6) n-2 465.34-(-0.20166) (-757.64) 26-2 465.34-152.7857 26-2 312.5543 1 24 = 13.0231 MSE 13.0231 Therefore, the mean sum of squares due to erroris MSE 13.0231 - - - - - - - - - - - -

To, test whether the overall fit of the linear model is significant or not using the F-test at the 5% level of significance. Step 1: The null and alternative hypotheses are as follows: H,: The overall model is not significant versus H,: The overall model is significant. Step 2: The level of significance is a =0.05. Step 3: The formula for calculating the F-test statistic is FMSE - MSR The values of MSR = 152.7857, MSE = 13.0231 in the equation (1), we get _152.7857 13.0231 = 11.7319 F# 11.7319 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Step 4: If the p-value <a = 0.05 We reject the null hypothesis H. Otherwise, we do not reject the null hypothesis H. Step5: The critical values are: F-0.05.21-18-24 = 4.259677 The P-value approach: P(F>F)=1-P(F F ) =1-P(F <11.7319) = 1-0.997784 0.002216 P(F>F) -0.002216] <<=0.05). We reject the null hypothesis Ho. Conclusion : Since, the test statistic is F = [11.7319 > F. = 14.25968 We reject the null hypothesis . Since, the P-value = 0.000 <a=0.05. We reject the null hypothesis H. There is sufficient evidence that the overall model is significant