Solution file for additional exercise 7.1
-----------------------------------------

Data: measurements of 6 male and 6 female rat's consumption (in g) of fat (fresh or
rancid). Notation:
  y_i = consumption for i'th rat, i=1,...,12,
or
  y_ijk = consumption for k'th rat in gender group i and fat type group j
  i=f,m; j=f,r; k=1,2,3.
  
The design would usually be considered a block design because gender is inherent
in the animals and cannot be randomised (thus, the randomisation is within gender
groups and consists in randomly allocating the 6 rats of each gender to the two
types of fat). Despite being a blocking factor, there is probably real interest in
gender and its interaction with fat type should be included in the full model.

The statistical model is
    y_i = mu + alpha_sex(i) + beta_fat(i) + gamma_sex*fat(i) + eps_i,
or
    y_ijk = mu + alpha_i + beta_j + gamma_ij + epsilon_ijk,
depending on the chosen notation.

MTB > WOpen "h:\VHM\VHM802\Data_csv\hs07_1.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_1.csv'
Worksheet was saved on 17/02/2011

MTB > Name c4 "SRES1" c5 "TRES1"
MTB > GLM 'consum' = gender| fat;
SUBC>   Brief 2 ;
SUBC>   Means gender|fat;
SUBC>   SResiduals 'SRES1';
SUBC>   TResiduals 'TRES1';
SUBC>   GFourpack;
SUBC>   RType 2 .
General Linear Model: consum versus gender, fat 

Factor  Type   Levels  Values
gender  fixed       2  f, m
fat     fixed       2  f, r

Analysis of Variance for consum, using Adjusted SS for Tests

Source      DF  Seq SS  Adj SS  Adj MS      F      P
gender       1    3781    3781    3781   2.59  0.146
fat          1   61204   61204   61204  41.97  0.000
gender*fat   1     919     919     919   0.63  0.450
Error        8   11667   11667    1458
Total       11   77570

S = 38.1881   R-Sq = 84.96%   R-Sq(adj) = 79.32%

Least Squares Means for consum

gender         Mean  SE Mean
f             580.0    15.59
m             615.5    15.59
fat
f             669.2    15.59
r             526.3    15.59
gender*fat
f          f  642.7    22.05
f          r  517.3    22.05
m          f  695.7    22.05
m          r  535.3    22.05

Residual Plots for consum 
MTB > GLM 'consum' = gender| fat; 
SUBC>   SMeans C4000; 
SUBC>   Brief 0; 
SUBC>   Interact 'gender' 'fat'.
MTB > GFInt 'gender' 'fat'; 
SUBC>   Responses 'consum'; 
SUBC>   FMeans C4000.
Interaction Plot for consum 
MTB > Erase C4000.

MTB > PPlot 'SRES1';
SUBC>   Normal;
SUBC>   Symbol;
SUBC>   FitD;
SUBC>   Grid 2;
SUBC>   Grid 1;
SUBC>   MGrid 1.
Probability Plot of SRES1 
The P-value for the Anderson-Darling test for normality is 0.775.

Comments:
---------
The analysis shows a clearly nonsignificant interaction (F=0.63),
an also nonsignificant main effect of gender (F=2.59,P=0.15), and a
strongly significant effect of fat type (F=42.0,P<0.001). The
residuals look quite good, there is no sign of heteroscedasticity,
and the normal plot is reasonably straight for only 12 observations
(and the P-value for the normality test is large). None of the residuals
are getting close to critical.

Thus, there may be slight indication of larger consumption
for males than females, but it is far from significant. There is clear
evidence of different consumption of rancid and fresh fat, we give
95% confidence intervals:
  fresh:  669.2 +- 2.306*15.59 = 669.2 +- 36.0
  rancid: 526.3 +- 2.306*15.59 = 526.3 +- 36.0
(the value 2.306 is the 97.5 percentile of the t(8) distribution)