Use the data set below Choose an observation index i eg i 3

Use the data set below. Choose an observation index i (e.g. i = 33, which corresponds to the outlying observation number 33) and create an indicator (dummy) variable Ui, where all the values of Ui are zero except for its ith value which is one. Now consider comparing the following models:

                  H0 : Time = B0 + B1Distance + B2Climb + E          (Model 4.27)

                  H1 : Time = B0 + B1Distance + B2Climb + B3Ui + E       (Model 4.28)

Let r*i be the ith externally standardized residual obtained from fitting Model 4.27. Show (or verify using an example) that:

The t-test for testing B3 = 0 in Model 4.28 is the same as the ith externally standardized residual obtained from Model 4.27, that is, t3 = r*i.

The F-test for testing Model 4.27 versus 4.28 reduces to the square of the ith externally standardized residual, that is, F = r*i^2.

Fit Model 4.27 to the data set without the ith observation.

Show that the estimates of B0, B1, and B2 in Model 4.28 are the same as those obtained in c. Hence adding an indicator variable for the ith observation is equivalent to deleting the corresponding observation.

Hill Race     Time     Distance     Climb
Greenmantle New Year Dash     965     2.5     650
Carnethy     2901     6     2500
Craig Dunain     2019     6     900
Ben Rha     2736     7.5     800
Ben Lomond     3736     8     3070
Goatfell     4393     8     2866
Bens of Jura     12277     16     7500
Cairnpapple     2182     6     800
Scolty     1785     5     800
Traprain Law     2385     6     650
Lairig Ghru     11560     28     2100
Dollar     2583     5     2000
Lomonds of Fife     3900     9.5     2200
Cairn Table     2648     6     500
Eildon Two     1616     4.5     1500
Cairngorm     4335     10     3000
Seven Hills of Edinburgh     5905     14     2200
Knock Hill     4719     3     350
Black Hill     1045     4.5     1000
Creag Beag     1954     5.5     600
Kildoon     957     3     300
Meall Ant-Suiche     1674     3.5     1500
Half Ben Nevis     2859     6     2200
Cow Hill     1076     2     900
North Berwick Law     1121     3     600
Creag Dubh     1573     4     2000
Burnswark     2066     6     800
Largo     1714     5     950
Criffel     3030     6.5     1750
Achmony     1257     5     500
Ben Nevis     5135     10     4400
Knockfarrel     1943     6     600
Two Breweries Fell     10215     18     5200
Cockleroi     1686     4.5     850
Moffat Chase     9590     20     5000

Solution