Suppose a researcher wants to evaluate the effect of cholesterol on the blood pressure. The following data on serum cholesterol (in mg/dL) and systolic blood pressure (in mm/Hg) were obtained for 15 patients to explore the relationship between cholesterol and blood pressure:
| S.NO. | Cholesterol (mg/dL) | SBP (mm/Hg) |
| 1 | 300 | 150 |
| 2 | 410 | 270 |
| 3 | 380 | 210 |
| 4 | 530 | 310 |
| 5 | 570 | 350 |
| 6 | 490 | 310 |
| 7 | 340 | 210 |
| 8 | 320 | 150 |
| 9 | 280 | 110 |
| 10 | 550 | 320 |
| 11 | 340 | 220 |
| 12 | 350 | 170 |
| 13 | 410 | 260 |
| 14 | 390 | 230 |
| 15 | 450 | 270 |
(i) Fit a linear regression model using the method of least squares.
(ii) Construct the normal probability plot for the regression model fitted on serum cholesterol and systolic blood pressure.
(iii) Test the significance of the fitted regression model.
(i) Fit a linear regression model using the method of least squares.
Given data:
Cholesterol levels (x): 300, 410, 380, 530, 570, 490, 340, 320, 280, 550, 340, 350, 410, 390, 450
Systolic blood pressures (y): 150, 270, 210, 310, 350, 310, 210, 150, 110, 320, 220, 170, 260, 230, 270
Step 1: Calculate the sample means of cholesterol levels (x_bar) and systolic blood pressures (y_bar).
x_bar = (300 + 410 + 380 + 530 + 570 + 490 + 340 + 320 + 280 + 550 + 340 + 350 + 410 + 390 + 450) / 15 = 413.3333
y_bar = (150 + 270 + 210 + 310 + 350 + 310 + 210 + 150 + 110 + 320 + 220 + 170 + 260 + 230 + 270) / 15 = 242.6667
Step 2: Calculate the slope (β1) and intercept (β0) using the formulas:
β1 = (Σ(xi - x_bar)(yi - y_bar)) / (Σ(xi - x_bar)^2)
β0 = y_bar - β1x_bar
Σ(xi - x_bar)(yi - y_bar) = (300 - 413.3333)(150 - 242.6667) + (410 - 413.3333)(270 - 242.6667) + ... + (450 - 413.3333)(270 - 242.6667)
= 38,968.8889
Σ(xi - x_bar)^2 = (300 - 413.3333)^2 + (410 - 413.3333)^2 + ... + (450 - 413.3333)^2
= 80,000
β1 = 38,968.8889 / 80,000 = 0.4846
β0 = 242.6667 - (0.4846)(413.3333) = 51.2769
Therefore, the fitted linear regression model is:
SBP = 51.2769 + 0.4846 × Cholesterol
(ii) Construct the normal probability plot for the regression model fitted on serum cholesterol and systolic blood pressure.
To construct the normal probability plot for the regression model fitted on serum cholesterol and systolic blood pressure, we need to follow these steps:
Step 1: Calculate the residuals (observed values - fitted values) and arrange them in ascending order.
Residuals:
150 - (51.2769 + 0.4846 × 300) = -94.2231
270 - (51.2769 + 0.4846 × 410) = 70.7231
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Suppose a researcher wants to evaluate the effect of cholesterol on the blood pressure. The following data on serum cholesterol (in mg/dL) and systolic blood pressure (in mm/Hg) were obtained for 15 patients to explore the relationship between cholesterol and blood pressure:
| S.NO. | Cholesterol (mg/dL) | SBP (mm/Hg) |
| 1 | 300 | 150 |
| 2 | 410 | 270 |
| 3 | 380 | 210 |
| 4 | 530 | 310 |
| 5 | 570 | 350 |
| 6 | 490 | 310 |
| 7 | 340 | 210 |
| 8 | 320 | 150 |
| 9 | 280 | 110 |
| 10 | 550 | 320 |
| 11 | 340 | 220 |
| 12 | 350 | 170 |
| 13 | 410 | 260 |
| 14 | 390 | 230 |
| 15 | 450 | 270 |
State whether the following statements are true or false and also give the reason in support of your answer.
(i) We define three indicator variables for an explanatory variable with three categories.
(ii) If the coefficient of determination is 0.833, the number of observations and explanatory variables are 12 and 3, respectively, then the Adjusted R2 will be 0.84.
(iii) For a simple regression model fitted on 15 observations, if we have , then it is an indication to trace the leverage point in the regression model.
(iv) In a regression model if
is not rejected, then the variable X1 will remain in the model.
(v) The logit link function is log
For the data given in Question 2(b), obtain the followings:
(i) Diagonal of the hat matrix and also check the leverage points if any.
(ii) Cook’s Distances, DFFITS and DFBETAS. Also verify the influence points if any
The following data on diagnosis of coronary heart disease (where 0 indicating absence and 1 indicating presence), serum cholesterol (in mg/dl), resting blood pressure (in mmHg) and weight (in kg) were obtained for 80 patients to explore the relationship of coronary heart disease with cholesterol and weight:
| S. No. | Serum Cholesterol (mg/dl) | Weight (kg) | Number of Patients having CHD | Total Number of Patients | |||||
| 1 | 420 | 60 | 10 | 20 | |||||
| 2 | 450 | 68 | 15 | 30 | |||||
| 3 | 400 | 54 | 4 | 15 | |||||
| 4 | 510 | 74 | 2 | 10 | |||||
| 5 | 480 | 62 | 1 | 5 | |||||
(i) Fit a multiple logistic model for the dependence of coronary heart disease on the average serum cholesterol and weight considering and
as the initial vales of th parameters (solve only for one lteration ).
Write a short note on the problem of multicollinearity and autocorrelation.
A company conducted a study on its employees to see the relationship of several variables with an employ’s IQ. For this purpose, fifteen employees were selected and an IQ as well as five different personality tests were given to them. Each employ’s IQ was recorded along with scores on five tests. The data are shown in the following table:
| Employee | Test 1 | Test 2 | Test 3 | Test 4 | Test 5 | IQ | ||||
| 1 | 83 | 80 | 78 | 77 | 67 | 99 | ||||
| 2 | 73 | 85 | 67 | 80 | 63 | 92 | ||||
| 3 | 81 | 80 | 71 | 81 | 68 | 94 | ||||
| 4 | 96 | 86 | 82 | 83 | 56 | 99 | ||||
| 5 | 84 | 73 | 75 | 75 | 68 | 94 | ||||
| 6 | 72 | 74 | 71 | 67 | 59 | 79 | ||||
| 7 | 84 | 79 | 84 | 84 | 69 | 97 | ||||
| 8 | 54 | 86 | 61 | 69 | 53 | 92 | ||||
| 9 | 86 | 85 | 79 | 78 | 76 | 94 | ||||
| 10 | 42 | 71 | 60 | 80 | 56 | 86 | ||||
| 11 | 83 | 72 | 72 | 78 | 74 | 98 | ||||
| 12 | 63 | 86 | 65 | 85 | 56 | 83 | ||||
| 13 | 69 | 76 | 64 | 85 | 61 | 98 | ||||
| 14 | 81 | 84 | 65 | 79 | 64 | 96 | ||||
| 15 | 50 | 85 | 71 | 65 | 75 | 76 | ||||
Explain the assumptions underlying multiple linear regression model.