Analysis of Fixed Effects ANOVA
Models for Factorial Experiments with
No Data in Some Cells
Michael H. Kutner, Ph.D.
Rollins Professor
Department of Biostatistics and
Bioinformatics
Rollins School of Public Health
Emory University
Abstract
Most standard statistical software programs provide
F-tests to test main effect hypotheses and
interactions for fixed effects in factorial experimental
models when some of the cells do not contain data.
Simple examples using SAS
will be given
documenting the difficulties with properly dealing
with cells without data. SAS claims that the results
may not be meaningful. We will see that the tests for
main effects are not invariant to simple permutations
of the rows and columns using data from a twofactor 3X3 factorial experiment. The hypotheses
that are actually being tested will be delineated. The
balanced incomplete block (BIB) design will be
shown to be a special case.

Data: Example 1
y
19
18
17
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
1
2
3
a
b
2,1 and 1,2 missing
1
2
3
Source
DF
Sum of
Squares Mean Square F Value Pr > F
Model
6 271.1428571
45.1904762
Error
14 280.0000000
20.0000000
Corrected Total
20 551.1428571
Source DF
2.26
0.0980
Type IV SS Mean Square F Value Pr > F
a
2 *
23.6431227
11.8215613
0.59
0.5670
b
2 * 196.7286245
98.3643123
4.92
0.0241
a*b
2
2.8996283
0.14
0.8663
2,1 and 1,2 missing
5.7992565
Data: Example 2
y
19
18
17
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
1
2
3
a
b
3,1 and 2,2 missing
1
2
3
Source
DF
Sum of
Squares Mean Square F Value Pr > F
Model
6 271.1428571
45.1904762
Error
14 280.0000000
20.0000000
Corrected Total
20 551.1428571
Source DF
2.26
0.0980
Type IV SS Mean Square F Value Pr > F
a
2 * 105.1935484
52.5967742
2.63
0.1072
b
2 *
23.6431227
11.8215613
0.59
0.5670
a*b
2
5.7992565
2.8996283
0.14
0.8663
3,1 and 2,2 missing
Data: Example 3
y
19
18
17
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
1
2
3
a
b
1,3 and 2,3 and 3,2 missing
1
2
3
Source
DF
Sum of
Squares Mean Square F Value Pr > F
Model
5 311.0833333
62.2166667
Error
12 169.4166667
14.1180556
Corrected Total
17 480.5000000
Source DF
4.41
0.0164
Type IV SS Mean Square F Value Pr > F
a
2 * 73.33333333
36.66666667
2.60
0.1156
b
2 * 37.46666667
18.73333333
1.33
0.3016
a*b
1
0.12254902
0.01
0.9273
0.12254902
1,3 and 2,3 and 3,2 missing