Solution file for additional exercise 2.5:
------------------------------------------

First part of the solution examines the 1-way ANOVA assumptions by
descriptive statistics for each group of origins:

MTB > WOpen "H:\VHM\VHM802\Data_csv\hs02_5.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\hs02_5.csv’
Worksheet was saved on 12/01/2011


Results for: hs02_5.csv

MTB > Describe 'calv1serv';
SUBC>   By 'origin';
SUBC>   Mean;
SUBC>   SEMean;
SUBC>   StDeviation;
SUBC>   QOne;
SUBC>   Median;
SUBC>   QThree;
SUBC>   Minimum;
SUBC>   Maximum;
SUBC>   Skewness;
SUBC>   Kurtosis;
SUBC>   N;
SUBC>   NMissing.
Descriptive Statistics: calv1serv 

Variable   origin   N  N*   Mean  SE Mean  StDev  Minimum     Q1  Median      Q3  Maximum
calv1serv  1       13   0   79.8     10.2   37.0     43.0   53.5    69.0   100.0    162.0
           2       38   0  91.76     7.58  46.71    41.00  63.00   77.00  102.00   221.00
           3       73   0  76.93     3.83  32.76    33.00  53.00   70.00   91.50   205.00

Variable   origin  Skewness  Kurtosis
calv1serv  1           1.28      0.75
           2           1.31      0.95
           3           1.85      4.48

MTB > PPlot 'calv1serv';
SUBC>   Normal;
SUBC>   Symbol;
SUBC>   FitD;
SUBC>   Grid 2;
SUBC>   Grid 1;
SUBC>   MGrid 1;
SUBC>   Panel 'origin'.
Probability Plot of calv1serv 


Comments on descriptive statistics:
-----------------------------------
Quite clearly a normal distribution does not describe these data well at
all, because they are markedly right-skewed (the computed skewness in the 3
groups range between 1.3 and 1.8). The normal plots are clearly curved, with a
similar shape in all three groups. The tests for normal distribution have P-values
below 0.019 (and much lower for the two groups of substantial size). Therefore,
the need of transformation is obvious. Note finally, that the standard
deviation is largest in the group with the largest mean, something that
we can hope to correct by transformation (although the major concern is
not with the homoscedasticity).

The Box-Cox transformation analysis can be carried out either under Regression in the 
Regression-Fit Regression Model menu or under ANOVA in the General Linear Model menu;
we show the results relating to the transformation of the former:

MTB > Regress;
SUBC>   Response 'calv1serv';
SUBC>   Nodefault;
SUBC>   Categorical 'origin';
SUBC>   Terms origin;
SUBC>   Constant;
SUBC>   Boxcox;
SUBC>   Tmethod;
SUBC>   Tanova;
SUBC>   Tsummary;
SUBC>   Tcoefficients;
SUBC>   Tequation;
SUBC>   TDiagnostics 0.
 
Regression Analysis: calv1serv versus origin 

Method

Categorical predictor coding  (1, 0)

Box-Cox transformation
Rounded lambda                -0.5
Estimated lambda              -0.670068
95% CI for lambda             (-1.09157, -0.261568)
...

Comments:
---------
The analysis gives the estimated optimal lambda value, with a 95% CI. In
addition, it suggests a "rounded" value of -0.5 and carries out the analysis 
for this lambda. The estimate for lambda agrees with the value produced by
Stata, but the CIs differ slightly. In practice, they are probably both ok 
to use if the intervals appear sensible. For our data, we  could base our 
evaluation of the model on inverse square-root transformed scale on the 
residuals obtained with the General Regression command, but we will instead 
use descriptive statistics in the same way as above. Note that we can do
this only because of the simplicity of the 1-way ANOVA model; in more
complex cases we would rely on the residuals.

MTB > Name C3 'invroot'
MTB > Let 'invroot' = -1/sqrt('calv1serv')
MTB > Describe 'invroot';
SUBC>   By 'origin';
SUBC>   Mean;
SUBC>   SEMean;
SUBC>   StDeviation;
SUBC>   QOne;
SUBC>   Median;
SUBC>   QThree;
SUBC>   Minimum;
SUBC>   Maximum;
SUBC>   Skewness;
SUBC>   Kurtosis;
SUBC>   N;
SUBC>   NMissing.
Descriptive Statistics: invroot 

Variable  origin   N  N*      Mean  SE Mean    StDev   Minimum        Q1    Median
invroot   1       13   0  -0.11904  0.00647  0.02333  -0.15250  -0.13726  -0.12039
          2       38   0  -0.11285  0.00394  0.02428  -0.15617  -0.12602  -0.11399
          3       73   0  -0.12010  0.00246  0.02098  -0.17408  -0.13736  -0.11952

Variable  origin        Q3   Maximum  Skewness  Kurtosis
invroot   1       -0.10139  -0.07857      0.33     -0.66
          2       -0.09906  -0.06727      0.13     -0.74
          3       -0.10454  -0.06984      0.17     -0.26

MTB > PPlot 'invroot';
SUBC>   Normal;
SUBC>   Symbol;
SUBC>   FitD;
SUBC>   Grid 2;
SUBC>   Grid 1;
SUBC>   MGrid 1;
SUBC>   Panel 'origin'.
Probability Plot of invroot 


Comments:
---------
The transformation worked well. The three groups have almost the same
standard deviations and distributions with no substantial departures
from normality (and non-significant P-values of the normality tests).