Biostatistics

8.5 THE FACTORIAL EXPERIMENT 367
Row C1 C2 C3 Row C1 C2 C3
29 30 2 2 69 30 4 2
30 30 2 2 70 31 4 2
31 39 2 3 71 24 4 3
32 42 2 3 72 25 4 3
33 36 2 3 73 30 4 3
34 42 2 3 74 26 4 3
35 40 2 3 75 23 4 3
36 40 2 4 76 29 4 4
37 45 2 4 77 30 4 4
38 50 2 4 78 28 4 4
39 45 2 4 79 27 4 4
40 60 2 4 80 30 4 4
8. Statistical decision. The variance ratios are V:R: ðAÞ ¼997:5=
14:7 ¼ 67:86, V:R: ðBÞ ¼400:4=14:7 ¼ 27:24, and V:R: ðABÞ ¼
67:6= 14:7 ¼ 4:60. Since the three computed values of VR. are all
greater than the corresponding critical values, we reject all three null
hypotheses.
9. Conclusion. When H 0 : a 1 ¼ a 2 ¼ a 3 ¼ a 4 is rejected, we conclude
that there are differences among the levels of A, that is, differences in the
average amount of time spent in home visits with different types of
patients. Similarly, when H 0 : b 1 ¼ b 2 ¼ b 3 ¼ b 4 is rejected, we conclude
that there are differences among the levels of B, or differences in
the average amount of time spent on home visits among the different
nurses when grouped by age. When H 0 : ðabÞ ij
¼ 0 is rejected, we
conclude that factors A and B interact; that is, different combinations
of levels of the two factors produce different effects.
10. p value. Since 67.86, 27.24, and 4.60 are all greater than the critical
values of F :995 for the appropriate degrees of freedom, the p value for
each of the tests is less than .005. When the hypothesis of no interaction
is rejected, interest in the levels of factors A and B usually become
subordinate to interest in the interaction effects. In other words, we are
more interested in learning what combinations of levels are significantly
different.
Figure 8.5.4 shows the SAS ® output for the analysis of Example 8.5.2.
&
We have treated only the case where the number of observations in each cell is the same.
When the number of observations per cell is not the same for every cell, the analysis
becomes more complex.
In such cases the design is said to be unbalanced. To analyze these designs with
MINITAB we use the general linear (GLM) procedure. Other software packages such as
SAS ® also will accommodate unequal cell sizes.