LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

M.Sc. DEGREE EXAMINATION – STATISTICS

AC 28

FIRST SEMESTER – APRIL 2006

                                            ST 1811 – APPLIED REGRESSION ANALYSIS

 

 

Date & Time : 27-04-2006/1.00-4.00 P.M.   Dept. No.                                                       Max. : 100 Marks

SECTION – A

Answer ALL the Questions                                                                     (2´10 = 20 marks)

1. Define ‘Residuals’ of a linear model.
2. What is Partial F- test.
3. What are the two scaling techniques for computing standardized regression coefficients.
4. Define ‘Externally Studentized Residuals’.
5. Stae the variance stabilizing tramsformation if V(Y) is proportional to [E(Y)]³.
6. What is F_OUT in Backward selection process.
7. How is the multicollinearity trap avoided in regression models with dummy variables.
8. State any one method of detecting multicollinearit.
9. Give an example of a polynomial regression model.
10. Give the motivation for Generalized Linear Models.

SECTION – B

Answer any FIVE Questions                                                                   (5´8 = 40 marks)

11. Fill up the missing entries in the following ANOVA for a regression model with 5 regressors and an intercept:

Source	d.f	S.S	Mean S.S.	F ratio
Regression Residual	? 14	? ?	40.5 ?	13.5 ——-
Residual	?	?	——–	——-

Also, test for the overall fit of the model.

 

12. The following table gives the data matrix corresponding to a model
    Y = b₀+b₁X₁+b₂X₂+b₃X₃. Suppose we wish to test H₀: b₂ = b₃. Write down the restrcited model under H₀ and the reduced data matrix that is used to build the restricted model.

1    2   -3    4

1   -1    2    5

1    3    4    -3

1   -2   1     2

X =     1    4    5   -2

1   -3    4    3

1    2    3     1

1    1    2     5

1    4   -2    2

1   -3    4    2

13. Explain how residual plot are used to check the assumption of normality of the errors in a linear model.

 

14. Discuss ‘Generalized Least Squares’ and obtain the form of the GLS estimate.

 

15. Explain the variance decomposition method of detecting multicollinearity and derive the expression for ‘Variance Inflation Factor’.
16. Discuss ‘Ridge Regression’ and obtain the expression for the redge estimate.

 

17. Suggest some strategies to decide on the degree of a polynomial regression model.

 

18. Describe Cubic-Spline fitting.

SECTION – C

Answer any TWO Questions                                                                 (2 ´ 20 = 40 marks)

19. Build a linear regression model with the following data and test for overall fit . Also, test for the individual significance of X₁ and of X₂.

Y:  12.8    13.9    15.2     18.3     14.5     12.4

X₁:    2          3        5          5          4          1

X_2:       4          2        5          1          2          3

 

20. (a)Decide whether “Y =b₀ + b₁X” or “Y² = b₀ + b₁X” is the more appropriate model for the following data:

X:    1      2       3      4

Y:  1.2   1.8    2.3   2.5

 

(b)The starting salary of PG students selected in campus interviews are given below

along with the percentage of marks they scored in their PG and their academic

stream:

Salary  (in ‘000 Rs)	Stream	Gender	% in PG
12 8 15 12.5 7.5 6 10 18 14	Arts Science Commerce Science Arts Commerce Science Science Commerce	Male Male Female Male Female Female Male Male Female	75 70 85 80 75 60 70 87 82

It is believed that there could be a possible interaction between Stream and % in

PG and between Gender and % in PG. Incorporate this view and create the data

matrix. (You need not build the model).                                                      (10+10)

21. Based on a sample of size 16, a model is to be built for a response variable with four regressors X₁, …,X₄. Carry out the Forward selection process to decide on the significant regressors, given the following information:

SS_T = 1810.509, SS_Res(X₁) = 843.88, SS_Res(X₂) = 604.224, SS_Res(X₃) = 1292.923, SS_Res(X₄) = 589.24, SS_Res(X₁, X₂) = 38.603, SS_Res(X_1,X₃) = 818.048,           SS_Res(X_1,X₄) = 49.84, SS_Res(X₂,X₃) = 276.96, SS_Res(X_2,X₄) = 579.23,          SS_Res(X_3,X₄) = 117.14, SS_Res(X_1,X₂,X₃) = 32.074, SS_Res(X_1,X₂, X₄) = 31.98, SS_Res(X_1,X₃,X₄) = 33.89, SS_Res(X_2,X₃,X₄) = 49.22, SS_Res(X_1,X₂,X₃,X₄) = 31.91.

 

22. (a) Obtain the likelihood equation for estimating the parameters of a logistic regression model.

(b) If the logit score (linear predictor) is given by –2.4 + 1.5 X₁ + 2 X₂, find the estimated P(Y = 1) for each of the following combination of the IDVs:

X₁:  0       1.5        2       3       -2      -2.5

X₂:  1         0       1.5     -1        2       2.5                                    (12+8)

 

 

 

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