| LOYOLA COLLEGE (AUTONOMOUS), CHENNAI 600 034
M.Sc. Degree Examination – Statistics I Semester – November 2006 ST 1811 – APPLIED REGRESSION ANALYSIS 02 / 11/ 2006 Time: 1.00. – 4.00 Max. Marks: 100 |
| SECTION – A |
Answer ALL the Questions (10 x 2 = 20 marks)
- Define ‘residuals’ and ‘residual sum of squares’ in a linear model.
- State the test for the overall fit of a linear regression model.
- Define Adjusted R2 of a linear model.
- Give an example of a relationship that can be linearized.
- What is the variance stabilizing transformation used when σ2 is proportional to E(Y)[1 – E(Y)]?
- State any one criterion for assessing and comparing performances of linear models.
- State any one ill-effect of multicollinearity.
- Illustrate with an example why both X and X2 can be considered for inclusion as regressors in a model.
- Define the logit link used for modeling a binary dependent variable.
- Define any one measure of performance of a logistic model.
| SECTION – B |
Answer any FIVE Questions (5 x 8 = 40 marks)
- Discuss “No-Intercept Model” and give an illustrative example where such a model is appropriate. State how you will favour such a model against a model with intercept. Indicate the ANOVA for such a model.
- A model (with an intercept) relating a response variable to four regressors is to be built based on the following sample of size 10:
| Y | X1 | X2 | X3 | X4 |
| 13.8
22.9 23.7 16.8 21.6 25.5 16.6 17.4 19.9 24.6 |
3
1 6 2 7 6 4 9 4 5 |
14
26 13 17 23 21 29 17 16 27 |
33
35 28 27 39 38 28 25 30 32 |
5
7 9 12 8 15 11 7 13 15 |
Write down the full data matrix. Also, if we wish to test the linear hypothesis H0: β4 = 2β1 + β2, write down the reduced model under the H0 and also the reduced data matrix.
- Give the motivation for standardized regression coefficients and explain anyone method for scaling the variables.
- The following residuals were obtained after a linear regression model was built:
0.17, – 1.04, 1.24, 0.48, – 1.83, 1.57, 0.50, – 0.32, – 0.77
Plot the ‘normal probability plot’ on a graph sheet and draw appropriate conclusions.
- Describe the Box-Cox method of analytical selection of transformation of the dependent variable.
- Discuss the role of dummy variables in linear models, explaining clearly how they are used to indicate different intercepts and different slopes among categories of respondents /subjects. Illustrate with examples.
- The following is part of the output obtained while investigating the presence of multicollinearity in the data used for building a linear model. Fill up the missing entries and point out which regressors are involved in collinear relationship(s), if
any:
| Eigen
Value (of X’X) |
Singular
value (of X) |
Condition
Indices |
Variance Decomposition Proportions
X1 X2 X3 X4 X5 X6 |
| 2.429 | ? | ? | 0.0003 0.0005 0.0004 0.0000 0.0531 ? |
| 1.546 | ? | ? | 0.0004 0.0000 ? 0.0012 0.0032 0.0559 |
| 0.922 | ? | ? | ? 0.0033 0.9964 0.0001 0.0006 0.0018 |
| 0.794 | ? | ? | 0.0000 0.0000 0.0002 0.0003 ? 0.4845 |
| 0.308 | ? | ? | 0.0011 ? 0.0025 0.0000 0.7175 0.4199 |
| 0.001 | ? | ? | 0.9953 0.0024 0.0001 ? 0.0172 0.0029 |
- Discuss ‘Spline’ fitting.
| SECTION – C |
Answer any TWO Questions (2 x 20 = 40 marks)
- (a)Depict the different possibilities that can arise when residuals are plotted against the fitted (predicted) values and explain how they can be used for detecting model inadequacies.
(b) Explain ‘partial regression plots’ and state how they are useful in model building. (13 + 7)
- The following data were used to regress Y on X1, X2, X3 and X4 with an intercept term and the coefficients were estimated to be β0^ = 45.1225, β1^ = 1.5894, β2^ = 0.7525, β3^ = 0.0629, β4^ = 0.054. Carry out the ANOVA and test for the overall significance of the model. Also test the significance of the intercept and each of the individual slope coefficients.
| Y(Heat in calories) | X1(Tricalium Aluminate) | X2(Tricalcium Silicate) | X3(Tetracalcium alumino ferrite) | X4(Dicalium silicate) |
| 78.5 | 7 | 26 | 6 | 60 |
| 74.3 | 1 | 29 | 15 | 52 |
| 104.3 | 11 | 56 | 8 | 20 |
| 87.6 | 11 | 31 | 8 | 47 |
| 95.9 | 7 | 52 | 6 | 3 |
| 109.2 | 11 | 55 | 9 | 22 |
| 102.7 | 3 | 71 | 17 | 6 |
| 72.5 | 1 | 31 | 22 | 44 |
| 93.1 | 2 | 54 | 18 | 22 |
| 115.9 | 21 | 47 | 4 | 26 |
The following is also given for your benefit:
| 15.90911472 | -0.068104115 | -0.216989375 | -0.042460127 | -0.165914393 |
| -0.068104115 | 0.008693142 | -0.001317006 | 0.007363424 | -0.000687829 |
| -0.216989375 | -0.001317006 | 0.003723258 | -0.001844902 | 0.002629903 |
| -0.042460127 | 0.007363424 | -0.001844902 | 0.009317298 | -0.001147731 |
| -0.165914393 | -0.000687829 | 0.002629903 | -0.001147731 | 0.002157976 |
(X’X)– 1 =
- Build a linear model for a DV with a maximum of four regressors using Stepwise Procedure, based on a sample of size 25, given the following information:
SST = 5600, SSRes(X1) = 3000, SSRes(X2) = 3300, SSRes(X3) = 3600,
SSRes(X4) = 2400, SSRes(X1,X2) = 2300, SSRes(X1,X3) = 2760,
SSRes(X1,X4) = 2100, SSRes(X2,X3) = 2600, SSRes(X2,X4) = 2040,
SSRes(X3,X4) = 1980, SSRes(X1, X2, X3) = 2000, SSRes(X1, X2, X4) = 1800,
SSRes(X1,X3, X4) = 1700, SSRes(X2,X3,X4) = 1500, SSRes(X1,X2,X3, X4) = 1440.
- (a) Briefly indicate the Wilk’s Likelihood Ratio Test and the Wald’s Test for testing the significance of a subset of the parameters in a Logistic Regression model.
(b) The following data were used to build a logistic model:
| DV | 1 | 1 | 0 | 1 | 0 | 0 | 1 | 0 | 1 | 1 | 1 | 0 | 0 | 1 | 0 | 1 | 1 | 0 | 0 | 0 |
| X1 | 2 | 4 | 1 | 0 | -1 | 3 | 5 | -2 | 3 | -2 | 3 | 0 | -4 | 2 | -3 | 1 | -1 | 3 | 4 | -2 |
| X2 | -2 | -4 | 2 | 0 | 4 | -2 | 1 | 3 | -4 | 2 | 1 | 3 | 0 | -2 | -4 | -3 | 1 | -1 | 2 | 0 |
The estimates were found to be β0 = 2.57, β1 = 3.78, β2 = – 3.2. Construct the Gains Table and compute KS Statistic. (8+12)
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