Question

Explain the assumptions underlying multiple linear regression model.

02 Apr 2024
Answer :
Word Count : 399

The multiple linear regression model is based on several key assumptions, and these assumptions must be satisfied for the model to produce valid and reliable results. The assumptions are as follows:

1. Linearity: The relationship between the dependent variable and the independent variables is linear. This assumption can be checked by plotting the residuals against the predicted values or the independent variables.

2. Independence of errors: The errors (residuals) are independent of each other. This means that the value ___ __________ _____ ____ ___ __________.
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Related Questions

Explain the assumptions underlying multiple linear regression model.

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  \hat{\beta}_{0}^{0}=1.279,\, \hat{\beta}_{1}^{0}=-0.035 and \hat{\beta}_{2}^{0}=0.172 as the initial vales of th parameters (solve only for one lteration ).

(ii) Test the significance of the fitted model using Hosmer-Lemeshow test at 5% level of significance.

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

 

Determine the most appropriate regression model for the employee’s IQ using stepwise approach at 5 % level of significance and interpret the results. Does the final regression model satisfy the linearity and normality assumptions?

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 and also check the leverage points if any.
(ii) Cook’s Distances, DFFITS and DFBETAS. 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

 (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.

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 h_{ii}=0.37, then it is an indication to trace the leverage point in the regression model.

(iv) In a regression model Y=\beta _0+\beta_1X_1+\beta_2X_2+\varepsilon , if H_0: \beta_1=0 is not rejected, then the variable X1 will remain in the model. 

(v) The logit link function is log \left [ -log(1-\pi) \right ].

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