Solution file for additional exercise 10.4
------------------------------------------
(see exercise 10.1 for discussion of model, design and notation)

MTB > WOpen "h:\vhm\vhm802\data_csv\hs10_4.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\hs10_4.csv'
Worksheet was saved on 03/04/2011

MTB > Name c5 "SRES1" c6 "TRES1"
MTB > GLM  'pH' = strain litter( strain);
SUBC>   Random 'litter';
SUBC>   Brief 2 ;
SUBC>   EMS;
SUBC>   Means strain;
SUBC>   SResiduals 'SRES1';
SUBC>   TResiduals 'TRES1';
SUBC>   GFourpack;
SUBC>   RType 2 .
General Linear Model: pH versus strain, litter 

Factor          Type    Levels  Values
strain          fixed        2  pHH, pHL
litter(strain)  random      14  1, 2, 3, 4, 5, 6, 7, 1, 2, 3, 4, 5, 6, 7

Analysis of Variance for pH, using Adjusted SS for Tests

Source          DF    Seq SS    Adj SS    Adj MS     F      P
strain           1  0.006645  0.006645  0.006645  1.26  0.283
litter(strain)  12  0.063093  0.063093  0.005258  2.22  0.028
Error           42  0.099375  0.099375  0.002366
Total           55  0.169113

S = 0.0486423   R-Sq = 41.24%   R-Sq(adj) = 23.05%

Unusual Observations for pH

Obs       pH      Fit   SE Fit  Residual  St Resid
  2  7.39000  7.47500  0.02432  -0.08500     -2.02 R
 31  7.63000  7.53250  0.02432   0.09750      2.31 R
 46  7.55000  7.44250  0.02432   0.10750      2.55 R

R denotes an observation with a large standardized residual.

Expected Mean Squares, using Adjusted SS

                   Expected Mean Square
   Source          for Each Term
1  strain          (3) + 4.0000 (2) + Q[1]
2  litter(strain)  (3) + 4.0000 (2)
3  Error           (3)

Variance Components, using Adjusted SS

                Estimated
Source              Value
litter(strain)    0.00072
Error             0.00237

Least Squares Means for pH

strain   Mean
pHH     7.477
pHL     7.455
 
Residual Plots for pH 

MTB > NormTest 'SRES1'.
Probability Plot of SRES1 
The P-value of the Anderson-Darling test of normality is 0.429.

Comments and answers to questions:
----------------------------------
The residuals of the error terms look okay, and the most extreme
residual has a corresponding deletion residual of 2.74, which is no
cause of concern in a dataset of this size. We discuss the residuals of
litter random effects below.

The mean strain levels are given above as least square means, and the
standard errors could be computed by the usual formulae except for using MS(Litt)
instead of MSE: SE = sqrt(MS(Litt)/28) = 0.014

The estimated variance component are listed above as well:
  sigma^2 (error):	0.00237
  sigma^2_B (litters):	0.00072
There appears to be much more variation among mice than among litters.

The ANOVA table shows weak effects of both litters and strains. In
particular, the F-statistic for testing no strain effect is only 1.26,
with a P-value of 0.28. Therefore, the breeding does not really seem to
have been successful (at least, the pHH strain has a higher level than
the pHL strain, but the message is that this could very well be caused
by random fluctuations).

Next we give commands to compute residuals at the litter level, or 
estimated random litter effects. Essentially, we aggregate the data
within litters, and analyze the litter means.

MTB > Name c7 "ByVar1" c8 "ByVar2" c9 "Mean1"
MTB > Statistics  'pH';
SUBC>   By 'strain' 'litter';
SUBC>   GValues 'ByVar1'-'ByVar2';
SUBC>   Mean 'Mean1'.
MTB > Name c10 "SRES2"
MTB > GLM  'Mean1' = ByVar1;
SUBC>   Brief 2 ;
SUBC>   Means ByVar1;
SUBC>   SResiduals 'SRES2';
SUBC>   GFourpack;
SUBC>   RType 2 .
General Linear Model: Mean1 versus ByVar1 

Factor  Type   Levels  Values
ByVar1  fixed       2  pHH, pHL

Analysis of Variance for Mean1, using Adjusted SS for Tests

Source  DF    Seq SS    Adj SS    Adj MS     F      P
ByVar1   1  0.001661  0.001661  0.001661  1.26  0.283
Error   12  0.015773  0.015773  0.001314
Total   13  0.017434

S = 0.0362551   R-Sq = 9.53%   R-Sq(adj) = 1.99%

Least Squares Means for Mean1

ByVar1   Mean  SE Mean
pHH     7.477  0.01370
pHL     7.455  0.01370
 
Residual Plots for Mean1 
MTB > NormTest 'SRES2'.
Probability Plot of SRES2 
The P-value of the Anderson-Darling test of normality is 0.485.

Comments:
---------
The ANOVA table gives the same F-test for strains as the full analysis
above (and this should be so). Also, the standard errors of the strain
least squares means are exactly the ones calculated above. The residuals
in this analysis show no particular problems - neither different variation
for the two strains, nor departures from normality.