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.
Nice — I solved this manually (numeric steps shown) and produced the regression, diagnostics, and the numbers you can use to draw the normal probability plot yourself. --- ## Data summary (n = 15) $$ \begin{aligned} \sum x &=6110,\qquad \sum y =3540,\\ \sum x^2 &=2{,}608{,}100,\qquad \sum xy =1{,}529{,}500. \end{aligned} $$ --- ## (i) Least-squares linear regression We fit the model $y=\beta_0+\beta_1 x + \varepsilon$. Using the standard normal-equation formulas $$ \beta_1=\frac{n\sum xy-(\sum x)(\sum y)}{\,n\sum x^2-(\sum x)^2\,},\qquad \beta_0=\frac{\sum y-\beta_1\sum x}{n}, $$ compute numerator and denominator for $\beta_1$: $$ \begin{aligned} \text{num} &= n\sum xy-(\sum x)(\sum y)=15(1{,}529{,}500)-6110(3540)=1{,}313{,}100,\\ \text{den} &= n\sum x^2-(\sum x)^2=15(2{,}608{,}100)-6110^2=1{,}789{,}400. \end{aligned} $$ So $$ \beta_1=\frac{1{,}313{,}100}{1{,}789{,}400}=0.733821\quad(\text{approx}), $$ $$ \beta_0=\frac{3540-0.733821\cdot6110}{15}=-62.909914\quad(\text{approx}). $$ Estimated regression line $$ \boxed{\ \hat y = -62.9099 \;+\; 0.733821\,x\ } $$ (Interpretation: each 1 mg/dL rise in cholesterol is associated with ≈0.734 mmHg increase in SBP.) --- ## Goodness-of-fit and sums of squares Let $\bar y=236$, $\bar x=407.3333$. Calculated sums of squares: * Total sum of squares: $\displaystyle SST=\sum (y-\bar y)^2 =70{,}560.000$. * Regression sum of squares: $\displaystyle SSR=\sum(\hat y-\bar y)^2 =64{,}238.725$. * Error sum of squares: $\displaystyle SSE=\sum (y-\hat y)^2 =6{,}321.275$. Check: $SST = SSR + SSE$ (70,560 ≈ 64,238.725 + 6,321.275). Coefficient of determination: $$ ________ ________ _____ ___ ________ _______ _______ ___ ______.
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Explain the assumptions underlying the multiple linear regression model.
A company conducted a study on its employees to see the relationship of several variables with an employee’s IQ. For this purpose, fifteen employees were selected, and an IQ test and five different personality tests were given to them. Each employee’s IQ was recorded along with scores on five tests. The data are shown in the following table
| Employee | Test 1 | Test 1 | 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 | 85 | 64 | 96 |
Write a short note on the problem of multicollinearity and autocorrelation.
For the data given in Question 2(b), obtain the followings:
(i) Diagonal of the hat matrix. Also, check the leverage points, if any.
(ii) Cook’s Distances. Also, verify the influence points, if any.
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 |
The following data on the diagnosis of coronary heart disease (where 0 indicates absence and 1 indicates presence), serum cholesterol (in mg/dl), 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 as the initial values of the parameters (solve only for one Iteration).
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 h₁ = 0.37, then it is an indication to trace the leverage point in the regression model.
(iv) In a regression model is not rejected, then the variable X₁ will remain in the model.
(v) The logit link function is log [-log(1-π)].