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SAS code for the rice/CO2/nitrogen experiment of
Example 7.1 using REML. The experiment was a
split-plot with whole-plots in an RCBD.
**************************************************;

proc mixed data=subplots;
 class block co2 nitrogen;
 model fertility = co2 nitrogen nitrogen*co2;
 random block block*co2;
 lsmeans co2*nitrogen;
 estimate 'amb vs elev CO2' co2 1 -1 /cl;
 estimate 'low N vs high N' nitrogen -1 1 0;
 estimate 'amb CO2: low vs high N'
           nitrogen -1 1 0 co2*nitrogen -1 1 0 0 0 0;
 estimate 'elev CO2: low vs high N'
           nitrogen -1 1 0 co2*nitrogen 0 0 0 -1 1 0;
 estimate 'low N: amb vs elev CO2'
           co2 1 -1 co2*nitrogen 1 0 0 -1 0 0;

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Proc Mixed output
***************************************************

The Mixed Procedure
                    Iteration History
Iteration    Evaluations -2 Res Log Like  Criterion
        0           1       72.59238890
        1           1       71.29681578   0.00000000
                 Convergence criteria met.

Covariance Parameter Estimates
Cov Parm      Estimate
Block          0.3965
Block*CO2      0.0566
Residual       1.6281

        Type 3 Tests of Fixed Effects
                 Num     Den
Effect            DF      DF   F Value   Pr > F
CO2               1       3      3.06    0.1783
Nitrogen          2      12     30.80    <.0001
CO2*Nitrogen      2      12      0.74    0.4982

                    Estimates
                                Std.      t
Label                 Estimate Error DF Value Pr > |t|
amb vs elev CO2         -0.958 0.547  3 -1.75  0.1783
low N vs high N          4.663 0.638 12  7.31  <.0001
amb CO2: low vs high N   4.300 0.902 12  4.77  0.0005
elev CO2:  low vs high N 5.025 0.902 12  5.57  0.0001
low N: amb vs elev CO2  -0.200 0.918 12 -0.22  0.8312

                   Least Squares Means
                                      Standard
Effect         CO2  Nitrogen Estimate    Error
CO2*Nitrogen   AMB   High    91.0500    0.7213
CO2*Nitrogen   AMB   Low     95.3500    0.7213
CO2*Nitrogen   AMB   Medi    94.1875    0.7213
CO2*Nitrogen   FACE  High    91.2500    0.7213
CO2*Nitrogen   FACE  Low     96.2750    0.7213
CO2*Nitrogen   FACE  Medi    95.9375    0.7213


*********************************************************
The rice/CO2/nitrogen example is balanced, so the
generalized least squares equations do not require
iteration; this is why the iteration history shows
only one iteration.   The "Type 3 Tests of Fixed
Effects" are Wald-type F-tests and coincide with
those of Table 7.8.

Note that in Proc Mixed the user may specify the
degrees of freedom for the standard errors of the
differences between whole-plot treatment means,
split-plot treatment means, and the interaction means.
The degrees of freedom for standard errors of
differences between whole-plot treatment levels at
specific split-plot treatment levels are an approximation
since no correct degrees of freedom parameter exists.
In the rice/CO2/nitrogen example Satterthwaite's
approximation would give df=13, which is larger than
the degrees of freedom of MSE(2), so we have used the
default MSE(2) degrees of freedom.
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