Write a short note on the problem of multicollinearity and autocorrelation.
Multicollinearity and autocorrelation are two critical issues encountered in applied regression analysis, both of which can significantly affect the validity and reliability of regression models. These problems often distort the statistical inferences made from the model and can lead to misleading conclusions if not properly addressed.
Multicollinearity refers to a situation in multiple regression models where two or more independent variables are highly correlated with each other. This high degree of inter-correlation makes it difficult to isolate the individual effects of each predictor variable on the dependent variable. As a result, the estimated coefficients become unstable and highly sensitive to small changes in the model or data. One of the most common symptoms of multicollinearity is inflated standard errors of the estimated coefficients, which in turn leads to ____ _____ ______ __________ _______ _______.
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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 |
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 |
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.
Write a short note on the problem of multicollinearity and autocorrelation.
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)].
Explain the assumptions underlying the multiple linear regression model.
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).