Question

1. (55 points) The investigators are interested in asses the relationship between Systolic Blood Pressure (SBP) in mm Hg and Age in years among Hypertensive Patients. Specif- ically, whether a patient's SBP can be predicted from his or her age. They selected n=122 patients at random from a medical record database in a hospital. Assume that the simple linear regression model is appropriate. The following table shows regression output of a simple linear regression model relating the SBP to the predictor Age. A few items have been masked and replaced by 'xxxx'. Table 1: Least Squares Regression Results for the SBP Data Analysis of Variance (ANOVA) Table DF Mean Square F-Value p-value Source of Variation Regression Error Total Sum of Squares xxxx XXXX XXXX xxxx XXXX XXXX XXXX XXXX XXXX Parameter Estimates Table Variable Parameter Estimates Standard Error Intercept Age 131.16 XXXX 1.853 0.028 t-Value XXXX 15.27 p-value < 0.0001 <0.0001 sample size n=122, R2 = 0.6603, ô2=29.8447 Answer the following questions based on the information provided in Table 1. (a) (5 points) Let Y be SBP and X be Age of a patient, write out the simple linear regression model. What are the assumptions of the model?

Answer 1

First we completed we know that, these two tables. SECB) - . t for intercept is calculated as t = Bo Similarly 131.16 1.853 70.78 ta E B SE( 3 ) 0.028 = = 15. 2 7 x 0.42756 Thus parameter estimated table is Ivariable Intercept Age para. Estimate 131 16 0.42756 S. EE 1.853 0.028 f value 70.78 15.27 P-value. <0.0001 <0.0001 NOCO ASE) zo2 Degrees of freedom Fee Regression - Error = Total n (P-TEK) = (2-1)=1 n-up = 122-2 = 120 a = 122-13 121

62 = MSE = 29.8447 TMSE = SSE SSE SSE = = - MSE XSI I n MSE X Ch-2) 29.8447 x 120 358) 364 - R² = - SSE SS Total 1- R2 SSE SSTotal 1. SS Total E to SSE (I-R2) 3581.364 (1-0.6603) 10542.726 - SS Tot = SSReg + SSE 10542.726 = SSReg + 3581.364. 10542.726 - 8581.3641 6961.362

MS Reg : = 6961 362 6961.362 SSReg df F Reg MSE 6961.362 29.8447 233.25 P-value < 0.00001 ANOVA Table S.U Rearession Error Total dif 1 120 121 sis. 6961-362 3581.364 p-value 20.oool M.S Evalue 6961.362 233.25 29.8447 ® U Simple linear regression model. S $1 SBP = Bo + B, Age + Ei Least square estimate Intercept = Ro = 131.16 Slope = 3 = 0.42756 Interpretation For Bo when age is zero then SBP IS 131-16 For when age of person is Increased by 1 year

SBP is increased by o. 42756 units. d) Estimated Regression equation: SBP = 131.16 + 0.42756 Age Now Age = 70 then 70 SBP = = 131.16+0.42756 x 161.0892