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Title | Single degree-of-freedom tests after ANOVA using nonresidual error | |

Author | Kenneth Higbee, StataCorp |

How can I compute single degree-of-freedom tests after ANOVA and use some
other term besides residual error in the denominator of the resulting
*F* tests?

Testing terms or combinations of terms from an ANOVA model that use
something other than residual error for the denominator of the
*F* test is easily handled directly with the
test
command. For instance, as shown in the section titled “Testing
effects” in
[R]
**anova postestimation**, after

. anova output machine operator|machine

we can test the machine term using operator|machine as the error with

. test machine / operator|machine

We could have gotten this test directly in the ANOVA with

So, testing a term in the ANOVA with a nonresidual error term is easy. The question here is how to apply a nonresidual error term to a single degree-of-freedom test.. anova output machine / operator|machine /

I will illustrate by showing how you obtain the pairwise comparison tests
between the five different levels of the machine term in the ANOVA of the
machine data used in
[R] **anova**.

. anova output machine / operator|machine /Number of obs = 57 R-squared = 0.8661 Root MSE = 1.47089 Adj R-squared = 0.8077

Source | Partial SS df MS F Prob > F | ||

Model | 545.822288 17 32.1071934 14.84 0.0000 | ||

machine | 430.980792 4 107.745198 13.82 0.0001 | ||

operator|machine | 101.353804 13 7.79644648 | ||

operator|machine | 101.353804 13 7.79644648 3.60 0.0009 | ||

Residual | 84.3766582 39 2.16350406 | ||

Total | 630.198947 56 11.2535526 |

If you have not read the section titled “Testing coefficients
and contrasts of margins”
in [R] **anova postestimation** and the FAQ
“How can I form various tests
comparing the different levels of a categorical variable after anova or
regress?” please do so. The additional twist here is that we wish
to use something other than residual as our error term for our tests.

We are interested in the 10 pairwise comparison tests involving the five levels of machine (1 versus 2, 1 versus 3, 1 versus 4, 1 versus 5, 2 versus 3, 2 versus 4, 2 versus 5, 3 versus 4, 3 versus 5, and 4 versus 5). The appropriate error term for testing machine is operator|machine. We wish to also use this as our error term when testing the 10 single degree-of-freedom pairwise comparison tests.

The first step is to place the sums of squares (SS) and degrees of freedom
(df) for the operator|machine term into some scalars that we can use later.
The **test** command will return the values we need.

. test operator|machine

Source | Partial SS df MS F Prob > F | |

operator|machine | 101.353804 13 7.79644648 3.60 0.0009 | |

Residual | 84.3766582 39 2.16350406 |

The command
return list shows which items have been left behind by the
**test** command.

. return listscalars: r(ss) = 101.3538042240784 r(df) = 13 r(rss) = 84.37665820758086 r(df_r) = 39 r(F) = 3.603619995522845

**r(ss)** and **r(df)** are the SS and df for operator|machine (we
wish these to be used in the denominator of the *F*
tests we are about to perform). **r(rss)** and **r(df_r)** are the SS
and df for residual error. We will place all four values into scalars for
our later use.

. scalar den_ss = r(ss) . scalar den_df = r(df) . scalar res_ss = r(rss) . scalar res_df = r(df_r)

I prefer forming single degrees-of-freedom tests after a complicated ANOVA by using the equivalent cell means ANOVA model. We first generate one variable corresponding to the cells of our more complicated ANOVA model.

. egen z = group(machine operator) . table machine operator, c(mean z)

five | operator nested in | |

brands of | machine | |

machine | 1 2 3 4 | |

1 | 1 2 3 4 | |

2 | 5 6 7 8 | |

3 | 9 10 11 | |

4 | 12 13 14 15 | |

5 | 16 17 18 | |

Here, there are 18 cells (of a possible 20 for this design). Machine brands 3 and 5 each have a missing cell. This table indicates, for example, level 1 of machine corresponds to the first four cells (a value of z of 1, 2, 3, or 4).

We use the **noconstant** option of
**anova** and our cell indicator variable z to obtain
the cell means ANOVA.

. anova output bn.z, noconstantNumber of obs = 57 R-squared = 0.9912 Root MSE = 1.47089 Adj R-squared = 0.9871

Source | Partial SS df MS F Prob > F | ||

Model | 9512.17345 18 528.454081 244.26 0.0000 | ||

z | 9512.17345 18 528.454081 244.26 0.0000 | ||

Residual | 84.3766582 39 2.16350406 | ||

Total | 9596.55011 57 168.360528 |

This ANOVA table is not really of interest to us. However, the single degrees-of-freedom tests that we wish to perform are, in my opinion, easier to form from this ANOVA.

We first perform the test letting residual error be the error term for the test. Here is the test comparing machine 1 with 2:

. test (i1.z+i2.z+i3.z+i4.z)/4 == (i5.z + i6.z + i7.z + i8.z)/4( 1) .25*1bn.z + .25*2.z + .25*3.z + .25*4.z - .25*5.z - .25*6.z - .25*7.z - .25*8.z = 0 F( 1, 39) = 33.17 Prob > F = 0.0000

We could have obtained the same result with

. mat c12 = (1,1,1,1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,0,0) . test , test(c12)

This is not the *F* test or *p*-value that we
really want. However, we can grab some values that are returned after this
**test** command and use them with the **den_ss**, **den_df**,
**res_ss**, and **res_df** scalars that we saved earlier to obtain the
desired *F* test and *p*-value.

. return listscalars: r(p) = 1.11948557182e-06 r(df) = 1 r(F) = 33.16595383079808 r(df_r) = 39 r(drop) = 0. scalar num_df = r(df) . scalar num_ss = r(F)*num_df*res_ss/res_df . scalar newF = (num_ss/num_df)/(den_ss/den_df) . scalar newp = Ftail(num_df,den_df,newF)

The numerator degrees-of-freedom is simply 1 here, but I place it in a
scalar (**num_df**) so that if you apply these same steps to some other
situation where you are combining several tests (using the
**accum** option of
**test**, for instance), the following steps will
continue to work. The numerator sums-of-squares are obtained using some
algebra on the formula for an *F* test. An
*F* test is
(numerator_SS/numerator_df)/(denominator_SS/denominator_df). The
*F* test just produced has residual error in the
denominator. We solve for the numerator sums of squares and place that
result in the scalar **num_ss**. The new *F* test
value and *p*-value are then computed.

Many people think it is important to also perform a correction on the
*p*-values when performing multiple comparisons. Here, we are
interested in 10 single degree-of-freedom tests. We can apply the
Bonferroni correction by simply multiplying the *p*-value by 10. We
show the *F* test, *p*-value, and
Bonferroni-corrected *p*-value next.

. disp "F(" num_df "," den_df ") = " newF ", p-value = " newp>", Bonferroni p-value = " min(1,newp*10)F(1,13) = 9.2035103, p-value = .00959573, Bonferroni p-value = .09595729

We could gather the unadjusted *p*-values for all 10 tests into a
matrix and then use the
**_mtest**
command to compute the corrected *p*-values. We will show this later.

The test comparing machine 1 with machine 3 is obtained similarly. Refer to the table showing how the z variable corresponds to our original ANOVA design to see how we constructed this test.

. test (i1.z+i2.z+i3.z+i4.z)/4 = (i9.z+i10.z+i11.z)/3( 1) .25*1bn.z + .25*2.z + .25*3.z + .25*4.z - .3333333*9.z - .3333333*10.z - .3333333*11.z = 0 F( 1, 39) = 10.20 Prob > F = 0.0028

Again we ignore this output and compute the appropriate
*F* test and *p*-value(s).

. scalar num_df = r(df) . scalar num_ss = r(F)*num_df*res_ss/res_df . scalar newF = (num_ss/num_df)/(den_ss/den_df) . scalar newp = Ftail(num_df,den_df,newF) . disp "F(" num_df "," den_df ") = " newF ", p-value = " newp>", Bonferroni p-value = " min(1,newp*10)F(1,13) = 2.8306705, p-value = .11632831, Bonferroni p-value = 1

There are eight more pairwise tests of interest. They can be computed in the same way. I omit them for brevity. These steps could be done interactively or by writing a do-file or ado-file.

The **test()** and
**mtest()** options of **test**
after **anova** provide a shortcut. We create a matrix
with 10 rows corresponding to the tests of interest. There are many ways to
form such a matrix. Here is one example.

. mat m4 = J(1,4,1/4) . mat m3 = J(1,3,1/3) . mat z4 = J(1,4,0) . mat z3 = J(1,3,0) . // mach = 1 2 3 4 5 . mat c = (m4,-m4, z3, z4, z3 \ /// 1 vs 2 > m4, z4,-m3, z4, z3 \ /// 1 vs 3 > m4, z4, z3,-m4, z3 \ /// 1 vs 4 > m4, z4, z3, z4,-m3 \ /// 1 vs 5 > z4, m4,-m3, z4, z3 \ /// 2 vs 3 > z4, m4, z3,-m4, z3 \ /// 2 vs 4 > z4, m4, z3, z4,-m3 \ /// 2 vs 5 > z4, z4, m3,-m4, z3 \ /// 3 vs 4 > z4, z4, m3, z4,-m3 \ /// 3 vs 5 > z4, z4, z3, m4,-m3) // 4 vs 5

**c** is a 10 × 18 matrix with each of the rows providing one of
the 10 comparisons of interest.

. test , test(c) mtest(noadjust)( 1) .25*1bn.z + .25*2.z + .25*3.z + .25*4.z - .25*5.z - .25*6.z - .25*7.z - .25*8.z = 0 ( 2) .25*1bn.z + .25*2.z + .25*3.z + .25*4.z - .3333333*9.z - .3333333*10.z - .3333333*11.z = 0 ( 3) .25*1bn.z + .25*2.z + .25*3.z + .25*4.z - .25*12.z - .25*13.z - .25*14.z - .25*15.z = 0 ( 4) .25*1bn.z + .25*2.z + .25*3.z + .25*4.z - .3333333*16.z - .3333333*17.z - .3333333*18.z = 0 ( 5) .25*5.z + .25*6.z + .25*7.z + .25*8.z - .3333333*9.z - .3333333*10.z - .3333333*11.z = 0 ( 6) .25*5.z + .25*6.z + .25*7.z + .25*8.z - .25*12.z - .25*13.z - .25*14.z - .25*15.z = 0 ( 7) .25*5.z + .25*6.z + .25*7.z + .25*8.z - .3333333*16.z - .3333333*17.z - .3333333*18.z = 0 ( 8) .3333333*9.z + .3333333*10.z + .3333333*11.z - .25*12.z - .25*13.z - .25*14.z - .25*15.z = 0 ( 9) .3333333*9.z + .3333333*10.z + .3333333*11.z - .3333333*16.z - .3333333*17.z - .3333333*18.z = 0 (10) .25*12.z + .25*13.z + .25*14.z + .25*15.z - .3333333*16.z - .3333333*17.z - .3333333*18.z = 0 Constraint 1 dropped Constraint 2 dropped Constraint 3 dropped Constraint 6 dropped Constraint 7 dropped Constraint 10 dropped

F(df,39) df p | ||

(1) | 33.17 1 0.0000 # | |

(2) | 10.20 1 0.0028 # | |

(3) | 182.36 1 0.0000 # | |

(4) | 55.31 1 0.0000 # | |

(5) | 5.25 1 0.0274 # | |

(6) | 49.72 1 0.0000 # | |

(7) | 2.28 1 0.1393 # | |

(8) | 85.30 1 0.0000 # | |

(9) | 14.37 1 0.0005 # | |

(10) | 31.17 1 0.0000 # | |

all | 49.80 4 0.0000 | |

We send **test** our matrix using the
**test()** option and use
**mtest(noadjust)**
to obtain each of the unadjusted single degrees-of-freedom tests. These
tests used residual error.

We can follow similar steps as before starting with this result to obtain
the single degrees-of-freedom tests with the
*operator|machine* term as the error term.

Remember, we previously ran:

. test operator|machine . scalar den_ss = r(ss) . scalar den_df = r(df) . scalar res_ss = r(rss) . scalar res_df = r(df_r)

after running the standard overparameterized ANOVA. We now execute

. local rows = rowsof(noadj) . mat num_ss = J(`rows',1,.) . mat new = J(`rows',3,.) . mat colnames new = F df p . mat new[1,2] = noadj[1...,2] . forvalues i = 1/`rows' {2. mat num_ss[`i',1] = noadj[`i',1]*noadj[`i',2]*res_ss/res_df 3. mat new[`i',1] = (num_ss[`i',1]/noadj[`i',2])/(den_ss/den_df) 4. mat new[`i',3] = Ftail(new[`i',2],den_df,new[`i',1]) 5. }. mat list newnew[10,3] F df p r1 9.2035103 1 .00959573 r2 2.8306705 1 .11632831 r3 50.603981 1 7.890e-06 r4 15.348188 1 .00176614 r5 1.4577248 1 .24880133 r6 13.797842 1 .00259704 r7 .6320179 1 .44088968 r8 23.671917 1 .00030856 r9 3.9880057 1 .06720042 r10 8.6498183 1 .0114672

Matrix **new** contains the tests using
*operator|machine* as the error term.

We can obtain multiple comparison testing corrections by calling
**_mtest**.

. _mtest adjust new , mtest(bonferroni) pindex(3) append . mat new_adj = r(result) . mat list new_adjnew_adj[10,4] F df p adjustp r1 9.2035103 1 .00959573 .09595729 r2 2.8306705 1 .11632831 1 r3 50.603981 1 7.890e-06 .0000789 r4 15.348188 1 .00176614 .01766142 r5 1.4577248 1 .24880133 1 r6 13.797842 1 .00259704 .0259704 r7 .6320179 1 .44088968 1 r8 23.671917 1 .00030856 .00308561 r9 3.9880057 1 .06720042 .67200415 r10 8.6498183 1 .0114672 .11467197

Here we asked for Bonferroni-adjusted *p*-values.