Supplementary Exercises 13.15 and 13.16 of IPS7e
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13.15:
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Minitab commands for interaction plot:

MTB > WOpen "H:\VHM\VHM801\Datasets\Minitab\Chapter 13\ex13_015.mtw".
Retrieving worksheet from file: ‘H:\VHM\VHM801\Datasets\Minitab\Chapter
13\ex13_015.mtw’
Worksheet was saved on 21/11/2014

MTB > Interact 'Gender' 'Group';
SUBC>   Response 'Mean_score';
SUBC>   Full.
Interaction Plot for Mean_score 

Comments:
---------
There is no clear indication of an overall effect of gender on the mean
scores, but there is a clear difference between LU and PG students, with
the PG students scoring higher regardless of gender. There apperas to also
be an interaction, because the gender effect is different, actually
reversed, in the two groups of students. For PG students, females scores
higher, whereas for LU students males scored (slightly) higher. In order
to put numbers to these findings, we generate a table with marginal means:

MTB > Table 'Gender' 'Group'; 
SUBC>   Layout 1 1; 
SUBC>   DMissing 'Gender' 'Group';
SUBC>   Means 'Mean_score';
SUBC>   Counts.
Tabulated statistics: Gender, Group 

Rows: Gender   Columns: Group

             LU     PG    All

Female    24.94  29.25  27.09
              1      1      2

Male      25.34  27.56  26.45
              1      1      2

All       25.14  28.41  26.77
              2      2      4

Cell Contents:  Mean_score  :  Mean
                               Count

Comments:
---------
The main effect of student group is represented by the two means: 25.14
for LU and 28.41 for PG, showing the higher average score for PG. The
main effect of gender is represented by the two means: 27.09 for females
and 26.45 for males, showing a fairly similar average score (slightly
higher for females). The interaction can be represented by conditional
distributions of any of the two factors, here we will look at
conditional distributions given student group. The differences F-M
between the two student groups are

  Group LU: 24.94-25.34 = -0.40
  Group PG: 29.25-27.56 = 1.69
  Overall : 27.09-26.45 = 0.64

As already noted, the gender effects are switched across the two student
group. In the PG group, females score 1.7 units higher, whereas in the
LU group they score 0.4 units lower.


13.16:
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(a) The completed ANOVA table is as follows:

Source of	Degrees of	Sum of		Mean	  F	  P	
variation	freedom		squares		square
-----------------------------------------------------------------------
Gender          1                  62.40	  62.40   2.73	  0.10
Group		1		 1599.03	1599.03	  69.9	  <0.001
Interaction	1		  163.80	 163.08	  7.13	  0.008
Error		596		13633.29	  22.875
Total		599		15458.52

Note that the SS for the interaction is computed by subtracting from the total 
the sum of terms from the other rows. The P-values for the F-tests were computed
in Minitab (using in all cases (1,596) as the degrees of freedom).

(b) The F-value for testing the interaction (more precisely, the null hypothesis
that there is no interaction) is 7.13. Its reference distribution is, as already
noted, the F(1,596) distribution. The best (conservative) approximation in Table F
of PSLS is F(1,200), which has 99% and 99.9% percentiles of 6.76 and 11.15, 
respectively. From this we conclude that 0.001<P<0.01. An exact calculation in
Minitab (below) shows that P=0.008. Our conclusion is a clear significance 
for the presence of an interaction betwen group and gender. That is, the effect 
of group does depend on gender, and vice versa.

(c) With a clearly significant interaction, the main effects of gender
and group are of less interest (because we have already established that
they both play a role for interpretation of the other effect). In any case, 
the test for the main effect of Group (F=69.9) is strongly significant, 
whereas the one for Gender (F=2.73) is non-significant; exact P-values
are also computed below. Because the interaction is significant, we conclude that
both factors are important and base our conclusions on the 2*2=4 combined groups.

The LSD-value at confidence level 95% is t(.975,596)*4.783*sqrt(2/150)=1.08,
using a t-value of 1.964 (from Minitab). Comparisons of the four means using
this value shows that the two PG groups are significantly different and also
significantly larger than both of the LU groups, which in turn are not
significantly different. In particular, the gender effect is only
present in the PG group.

(d) The pooled within-group variance can be read off the ANOVA table as
s^2 = MSE = 22.875. Hence, s = sqrt(22.875) = 4.783.

(e) The summary has already been provided above under (c).

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Minitab commands to compute P-value for interaction test and tstar value for 95% CI:

MTB > CDF 2.73;
SUBC>   F 1 596.
Cumulative Distribution Function 

F distribution with 1 DF in numerator and 596 DF in denominator

   x  P( X <= x )
2.73     0.900994

MTB > CDF 69.9;
SUBC>   F 1 596.
Cumulative Distribution Function 

F distribution with 1 DF in numerator and 596 DF in denominator

   x  P( X <= x )
69.9            1

MTB > CDF 7.13;
SUBC>   F 1 596.
Cumulative Distribution Function 

F distribution with 1 DF in numerator and 596 DF in denominator

   x  P( X <= x )
7.13     0.992213

MTB > InvCDF 0.975;
SUBC>   T 596.
Inverse Cumulative Distribution Function 

Student's t distribution with 596 DF

P( X <= x )        x
      0.975  1.96395