Question

QUESTION 1: Regression and Forecasting 125 marks] Huka A recent 10-year study of senior Taupo residents conducted by a research team at the Falls Medical School the risk of strokes. Assume that the following data are from a portion of this stu interpreted as the probability (times 100) that the patient will have a stroke over the next i0- year period. For the smoking variable, a dummy variable with 1 indicates a smoker an was conducted to assess how age, blood pressure, and smoking relate to dy. Risk is d 0 indicates a non-smoker: Risk Age Pressure Smoker Smoker 12 57 152 No0 24 67 163 No 0 13 58 155 No0 59 189 Yes 56 86 177 Yes 1 28 59 196 No 51 76 189 Yes 18 56 155 Yes 31 78 120 No 0 37 80 135 Yes 1 15 78 98 No0 22 71 152 No 0 36 70 173 Yes 15 67 135 Yes 48 77 209 Yes 34 66 119 34 1 80 125-1-Wes- 15 60 199 No 0 60 119 No 36 82 119 Yes 1 8 66 166 No0 66 125N 3 62 117 No 37 59 207Yes In an attempt to measure the effect of blood pressure, age and smoking on the risk of having a stroke, the Manager of the research team used the Data Analysis Add-in to an Excel spreadsheet to analyse the relationships between these variables. The results of this analysis are presented below

Answer 1

1. Correlation Matrix :

The correlation matrix indicate how the variables, Risk , Age, Blood pressure and smoker dummy are related to each other. When the correlation is more than 0.5 it is strongly correlated to the other variables.

1. The age & risk, Smoking & risk are strongly & positively related.
2. Blood pressure is weakly related to Risk & Age but positive & negative respectively.
3. Smoking is somewhat weakly related to Age

1. Regression Equation

Risk, Y = -91.76 + 1.08 Age + 0.25 Blood Pressure + 8.74 smoker Dummy

1. Regression Coefficient

The coefficient gives the least squares of the respective variables. The coefficient of regression is multiplied with the respective variables value to estimate the forecast.

1. ‘Good fit’

Multiple R is the correlation coefficient. There is a positive relationship as the value is closer to 1. The Squared R conveys that 87.3% of the variation of risk around the mean are explained by the variables, Age, blood pressure and smoking.

1. Risk, Y = -91.76 + 1.08 Age + 0.25 Blood Pressure + 8.74 smoker Dummy

Age = 65, Blood Pressure = 164, Smoker dummy = 1 (Smoking)

Risk = -91.76 + (1.08*65) + (0.25*164) +(8.74*1)

Risk = 28.18

1. Confidence interval

The lower & upper confidence interval values for 95% defines the confidence interval for the variables. These are the confidence interval for the slope co-efficients.

Intercept (-124.03,-59.49)

Age (0.72, 1.43)

Blood Pressure (0.16, 0.35)

Smoker dummy (2.38, 15.1)

Using Lower co-efficient:

Risk = -124.03 + (0.72 *65) + (0.16*164) + (2.38*1) = -48.61

Using Upper co-efficient :

Risk = -59.49 + (1.43 *65) + (0.35*164) + (15.1*1) = 105.96

The risk estimate is the mean of upper & lower limit.

g) The regression model can be improved by having the level of smoking. The smoking dummy is binary in the current model which does not justifies the risk associated with it.

h) Strength & weakness of Regression.

1.For multi variable correlation, regression is the best solution as it is a straightforward method.

2. It assumes only the linearity of the data and eliminates the extreme data.