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 as 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 of gender 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_gender(i) + beta_fat(i) + gamma_gender*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 "SRES".
MTB > GLM;
SUBC>   Response 'consum';
SUBC>   Nodefault;
SUBC>   Categorical 'gender' 'fat';
SUBC>   Terms gender fat gender*fat;
SUBC>   Means gender fat gender*fat;
SUBC>   TExpand;
SUBC>   TMethod;
SUBC>   TAnova;
SUBC>   TSummary;
SUBC>   TCoefficients;
SUBC>   TFactor;
SUBC>   TDiagnostics 0;
SUBC>   Rtype 2;
SUBC>  GFOURPACK;
SUBC>  SResiduals 'SRES'.
General Linear Model: consum versus gender, fat 

Method
Factor coding  (-1, 0, +1)

Factor Information
Factor  Type   Levels  Values
gender  Fixed       2  f, m
fat     Fixed       2  f, r

Analysis of Variance
Source        DF   Seq SS  Contribution   Adj SS   Adj MS  F-Value  P-Value
  gender       1   3780.7         4.87%   3780.8   3780.8     2.59    0.146
  fat          1  61204.1        78.90%  61204.1  61204.1    41.97    0.000
  gender*fat   1    918.8         1.18%    918.8    918.8     0.63    0.450
Error          8  11666.7        15.04%  11666.7   1458.3
Total         11  77570.3       100.00%

      S    R-sq  R-sq(adj)  PRESS  R-sq(pred)
38.1881  84.96%     79.32%  26250      66.16%

Coefficients
Term         Coef  SE Coef      95% CI      T-Value  P-Value   VIF
Constant    597.7     11.0  (572.3, 623.2)    54.22    0.000
gender
  f         -17.8     11.0  (-43.2,   7.7)    -1.61    0.146  1.00
fat
  f          71.4     11.0  ( 46.0,  96.8)     6.48    0.000  1.00
gender*fat
  f f        -8.8     11.0  (-34.2,  16.7)    -0.79    0.450  1.00

Means
            Fitted
Term          Mean  SE Mean
gender
  f          580.0     15.6
  m          615.5     15.6
fat
  f          669.2     15.6
  r          526.3     15.6
gender*fat
  f f        642.7     22.0
  f r        517.3     22.0
  m f        695.7     22.0
  m r        535.3     22.0

Residual Plots for consum 

MTB > FacPlot 'consum';
SUBC>   Factors gender fat;
SUBC>   GMain;
SUBC>   GInt;
SUBC>     Full.
Main Effects Plot for consum 
Interaction Plot for consum 

MTB > PPlot 'SRES';
SUBC>   Normal;
SUBC>   Symbol;
SUBC>   FitD;
SUBC>   Grid 2;
SUBC>   Grid 1;
SUBC>   MGrid 1.
Probability Plot of SRES 
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
(tests for equal variance are clearly nonsignificant; not shown),
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 t* value (97.5 percentile) of the t(8) distribution)