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Four gasoline additives (deicers) were studied to compare their
ability to prevent carburetor icing (Robert Hader and H. Lucas,
personal communication).  Four carburetors appropriately equipped
were available. The engineers could never be sure that all of the
fuel additive from one test had been cleaned off the metal surfaces
in the carburetor before a second test could be performed. The
Latin square, balanced for carryover effects was used.

The SAS code for the analysis follows, as well as selected parts
of the output. Note in particular that the carryover effects must
be coded in the DATA step, rather than using the CLASS statement
in PROC GLM, and are tested using a CONTRAST statement in the GLM
step.  The treatment, carburetor, and period effects can be set up
using the coding generated by the CLASS statement.
******************************************************************;

 data one ;
    do period = 1 to 4 ;
     do carburetor = 1 to 4 ;
       input deicer $ prev_d $ y @@ ;
       output ;
       end ;
     end ;
    cards ;
  A N 88.0 B N 78.0 C N 87.5 D N 90.5
  B A 76.0 D B 94.0 A C 95.5 C D 78.5
  C B 88.0 A D 90.0 D A 95.5 B C 82.5
  D C 92.0 C A 90.0 B D 87.5 A B 94.5
 data two ; set one ;
  carryo_A = 0 ;
  carryo_B = 0 ;
  carryo_C = 0 ;
  if (prev_d = 'A') then carryo_A = 1 ;
  if (prev_d = 'B') then carryo_B = 1 ;
  if (prev_d = 'C') then carryo_C = 1 ;
  if (prev_d = 'D') then do ;
              carryo_A = -1 ;
              carryo_B = -1 ;
              carryo_C = -1 ;
                   end ;
*******************************************************
The code to obtain the analysis of variance is follows.
*******************************************************;
 proc glm data = two ;
  class period carburetor deicer ;
  model y = period carburetor carryo_A
                carryo_B carryo_C deicer;
  contrast 'carryover' carryo_A 1,
       carryo_B 1, carryo_C 1;
  lsmeans deicer / stderr;
  estimate 'A vs B' deicer 1 -1 0 0;
 run;
******************************************************
Selected output
******************************************************;
 Source      DF  Sum of Square Mean Square
 Model       12     564.40       47.03
 Error        3      21.60        7.20
 Total(Corr )15     586.00

 Source      DF  Type I SS     Mean Square
 PERIOD       3      72.00       24.00
 CARBURETOR   3      74.00       24.67
 CARRYO_A     1       6.55        6.55
 CARRYO_B     1     128.24      128.24
 CARRYO_C     1       7.76        7.76
 DEICER       3     275.85       91.95

 Source     DF Type III SS Mean Square F Value Pr > F
 PERIOD      3     72.00      24.00      3.33  0.1746
 CARBURETOR  3    101.31      33.77      4.69  0.1183
 CARRYO_A    1      2.13       2.13      0.30  0.6241
 CARRYO_B    1     26.13      26.13      3.63  0.1529
 CARRYO_C    1      2.13       2.13      0.30  0.6241
 DEICER      3    275.85      91.95     12.77  0.0325

 Contrast DF  Contrast SS  Mean Square F Value  Pr > F
 carryover 3      42.40       14.13      1.96   0.2968

                     Least Squares Means

            DEICER    Y      Std Err
                    LSMEAN    LSMEAN
               A     91.80     1.39
               B     81.70     1.39
               C     86.20     1.39
               D     92.30     1.39

                      T for H0:  Pr > |T| Std Error of
 Parameter  Estimate  Parameter=0            Estimate
  A vs B     10.10       5.08     0.0148       1.99
******************************************************************
The estimated means (estimated by ordinary least squares) for the
four fuel additives are 91.8, 81.7, 86.2, and 92.3 for additives A-D,
respectively. The standard error of the difference between two means
is 1.99. The analysis of variance table is

    Source       df  Partial SS  F-ratio  p-value
    -------------------------------------------------
    Carburetor    3     101.31     4.69     0.12
    Period    	  3      72.00     3.33     0.17
    Deicer        3     275.85    12.77     0.03
    Carryover     3      42.40     1.96     0.30
    Error         3      21.60

Notice that the sequential and partial sums of squares for periods
coincide, because periods are orthogonal to all other factors in
the model. In contrast, the sequential sums of squares differ from
the partial sums of squares for all other factors (except the
deicing treatment, Why?).
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