Solution file for Additional exercise 7.2
-----------------------------------------

Data: measurements of 3 rabbits' blood glucose levels after injection with 3 different
doses of insulin and done on 3 different days. Notation:
  y_i = blood glucose conc. (mg per 100 ml blood) for i'th sample, i=1,...,9,
or
  y_ijk = blood glucose conc. (mg per 100 ml blood) for rabbit j, after having received
treatment i, at day k,
  i=A,B,C (0,1,2 doses); j=1,2,3; k=1,2,3.
  
The design is a 3x3 Latin square. As measurements with different treatments are
obtained from the same rabbits, the order of the treatments is an issue and is taken as
a blocking factor in the design. This is a safeguard against any spurious treatment 
effects caused by day effects, but does not really take into account carry-over effects
very well (because only certain combinations of treatments accur after each other). It is not
possible to do better in such a small design. 

The statistical model is
    y_i = mu + alpha_trt(i) + beta_rabbit(i) + gamma_day(i) + eps_i,
or
    y_ijk = mu + alpha_i + beta_j + gamma_k + epsilon_ijk,
depending on the chosen notation.

MTB > WOpen "h:\VHM\VHM802\Data_csv\hs07_2.csv";
SUBC>   FType;
SUBC>     CSV;
SUBC>   DecSep;
SUBC>     Period;
SUBC>   Field;
SUBC>     Comma;
SUBC>   TDelimiter;
SUBC>     DoubleQuote.
Retrieving worksheet from file: 'h:\VHM\VHM802\Data_csv\hs07_2.csv'
Worksheet was saved on 17/02/2011

MTB > Name c5 "SRES1" c6 "TRES1"
MTB > GLM 'glucose' = day rabbit insulin;
SUBC>   Brief 2 ;
SUBC>   Means day rabbit insulin;
SUBC>   GFourpack;
SUBC>   RType 2 .
General Linear Model: glucose versus day, rabbit, insulin 

Factor   Type   Levels  Values
day      fixed       3  1, 2, 3
rabbit   fixed       3  1, 2, 3
insulin  fixed       3  A, B, C

Analysis of Variance for glucose, using Adjusted SS for Tests

Source   DF  Seq SS  Adj SS  Adj MS      F      P
day       2   92.67   92.67   46.33   5.56  0.152
rabbit    2   96.00   96.00   48.00   5.76  0.148
insulin   2  416.67  416.67  208.33  25.00  0.038
Error     2   16.67   16.67    8.33
Total     8  622.00

S = 2.88675   R-Sq = 97.32%   R-Sq(adj) = 89.28%

Least Squares Means for glucose

day       Mean  SE Mean
1        41.67    1.667
2        47.00    1.667
3        49.33    1.667
rabbit
1        46.00    1.667
2        50.00    1.667
3        42.00    1.667
insulin
A        54.33    1.667
B        46.00    1.667
C        37.67    1.667
 
Residual Plots for glucose 

Comments:
---------
The analysis shows a significant effect of treatments and non-significant effects of
rabbits and days. Note that the power of the F-tests is rather low with only 2
degrees of freedom for error. Therefore, the significant effect is quite
satisfactory, despite a P-value of only 0.038. The treatment means indicate that
doses of insulin decrease the blood glucose level. 

One question is whether to refit the model without the non-significant factors.
Usually this is not of interest because the partial and sequential sum of
squares coincide (so that results would be unchanged). However, with only 2
degrees of freedom for error, one potential and important advantage could be to
increase the degrees of freedom for error (pooling). In this situation, I would 
not do that because the F-statistics are numerically large and pooling will 
therefore increase our estimate of model variance considerably.

We carry out a test of linearity ("lack of fit") for number of doses in the
full Latin square model.

MTB > Name c5 "dose"
MTB > Code ( "A" ) 0 ( "B" ) 1 ( "C" ) 2 'insulin' 'dose'
MTB > GLM 'glucose' = day rabbit dose;
SUBC>   Covariates 'dose';
SUBC>   Brief 2 .
General Linear Model: glucose versus day, rabbit 

Factor  Type   Levels  Values
day     fixed       3  1, 2, 3
rabbit  fixed       3  1, 2, 3

Analysis of Variance for glucose, using Adjusted SS for Tests

Source  DF  Seq SS  Adj SS  Adj MS      F      P
day      2   92.67   92.67   46.33   8.34  0.060
rabbit   2   96.00   96.00   48.00   8.64  0.057
dose     1  416.67  416.67  416.67  75.00  0.003
Error    3   16.67   16.67    5.56
Total    8  622.00

S = 2.35702   R-Sq = 97.32%   R-Sq(adj) = 92.85%

Term         Coef  SE Coef      T      P
Constant   54.333    1.242  43.74  0.000
dose      -8.3333   0.9623  -8.66  0.003

Comments:
---------
The (numerator of the) F-test statistic would now be obtained by subtracting SSE's 
and DFE's from the two model as usual, however in this case the SSE's are equal!
This is because the 3 treatment means are exactly on a straight line. Obviously,
the linear model is well supported by the data. A 95% confidence interval for
the impact of one added insulin dose on the glucose level is
   -8.33 +- 3.1824*0.9623 = -8.33 +- 3.06 = (-11.4,-5.3).