7.1. In an experiment involving one dependent variable (y) and four explanatory

Question

7.1. In an experiment involving one dependent variable (y) and four explanatory variables x1,x2,xy, and x4, all possible regressions are fit to a data set consisting of n 13 cases. A constant term is routinely included in all models. The results are summarized as follows: Residual Sum of Squares 4,073.6 1,898.5 1,359.5 2,909.1 1,325.8 86.9 1,840.6 112.1 623.1 1,303.3 263.6 72.2 72.0 76.2 110.7 71.8 Regressors in Model one X4 xi, x4 x3, x4 xi, x2, x3 i. What model will result from automatic backward elimination with significance level(alpha to drop)0.05?

Explanation / Answer

The F statistic is given as,

F = [(RSSR - RSSUR) / q] / [RSSUR / (n-k)]

where RSSUR and RSSR are residual sum of squares of unrestricted and restricted model

n is number of samples . n = 13

q is number of coeffcients removed from restricted model

k is number of coefficients in the unrestricted model including intercept.

The Full model with x1, x2, x3 and x4 have RSS = 71.8

After removing , a variable, the minimum RSS is of the model x1, x2, x4 with RSS = 72

So,   RSSUR = 71.8 and RSSR = 72, q = 1, k = 5, n = 13

F = [(72-71.8) / 1] / [71.8 / (13-5)] = 0.0222

Critical value of F for df = q, n-k = 1 , 8 significance level as 0.05 is 5.32

As, Observed F is less than the critical value, we fail to reject the null hypothesis and conclude that the removed variable x3 is not significant.

So, the new model is x1, x2, x4

Now in next iteration, The Full model is x1, x2 and x4 have RSS = 72

After removing , a variable, the minimum RSS is of the model x1, x2 with RSS = 86.9

So,   RSSUR = 72 and RSSR = 86.9, q = 1, k = 4, n = 13

F = [(86.9-72) / 1] / [72 / (13-4)] = 1.8625

Critical value of F for df = q, n-k = 1 , 9 significance level as 0.05 is 5.12

As, Observed F is less than the critical value, we fail to reject the null hypothesis and conclude that the removed variable x4 is not significant.

So, the new model is x1, x2

Now in next iteration, The Full model is x1, x2 have RSS = 86.9

After removing , a variable, the minimum RSS is of the model x2 with RSS = 1359.5

So,   RSSUR = 86.9 and RSSR = 1359.5, q = 1, k = 3, n = 13

F = [(1359.5-86.9) / 1] / [86.9 / (13-3)] = 146.4442

Critical value of F for df = q, n-k = 1, 10 and significance level as 0.05 is 4.96

As, Observed F is greater than the critical value, we reject the null hypothesis and conclude that the removed variable x2 is significant. So, x2 variable cannot be removed.

So, the new model is x1, x2

So, the final model will contain the variables x1 and x2 and the eliminated variables as x3 and x4.

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