Use the data in Table 9.5 (for Example 9.3) to test the following set of contrasts. (The subscripts correspond to Table 9.7.)
(a) Verify that each of these is a legitimate contrast and that each pair is orthogonal.
(b) Interpret the meaning of L4
and L5
in simple language.
(c) Create five independent variables, each of which has values corresponding to the coefficients for one of the contrasts. For example, the independent variable corresponding to L1
would have value 1 if CYLINDER = 6 and 21 if CYLINDER = 4.
Carry out a multiple regression of MPG on these five independent variables. Verify that the test for the model corresponds to the overall ANOVA given in the top of Table 9.6.
Verify that the t test for L3
corresponds to that computed directly from the cell means for the comparison of MULTI versus GASMISER in Section 9.4.
(d) Compute the t test for L5
using the formula based on the cell means (Section 9.4) and verify that it corresponds to the t test from the regression.
Example 9.3
To illustrate the computations for the analysis of a two-factor factorial experiment we assume that the two motor oil experiments were actually performed as a single 2
3 factorial experiment. In other words, treatments correspond to the six combinations of the two engine types and three oils in a single completely randomized design. For the factorial we define
factor A: type of engine with two levels: 4 and 6 cylinders, and
factor C: type of oil with three levels: STANDARD, MULTI, and GASMISER.
The data, together with all relevant means are given in Table 9.5.