ANOVA2(NR) Test Examples

Analyze data from a two-factor experiment that has no replication

The sample data are contained in a (6 rows by 4 columns) 2D wave dataWave1 where Factor A corresponds to rows and Factor B corresponds to columns:

4.045.082.833.06
4.914.683.172.71
2.941.91.612.02
5.955.264.854.45
3.94.852.692.93
5.725.144.393.29

To run the test execute the command:

StatsANOVA2NRTest/T=1 dataWave1

The results of the operation are displayed in the table ANOVA2(NR):

SSDFMSFFcAccept
Total36.503923
Factor B11.288133.762714.4273.287380
Factor A21.303654.2607216.33642.901290
Remainder3.91216150.260811

The assumption in this test is that there is no interaction between Factor A and Factor B so the operation computes the results for both factors. H0 provides that there is no difference in the means of the populations. SS are the sum of the squares, DF are the degrees of freedom, MS are the mean square, F are the computed statistics, Fc are the critical F values for the corresponding degrees of freedom. Evidently in this case H0 is rejected for both rows and columns.

Analysis of data of the same dimensions but having less variation as shown in dataWave2

4.64.924.394.26
4.5255.514.63
5.065.75.725.43
4.515.354.434.86
5.744.855.385.17
4.894.744.665.4

To repeat the test for these data use the command:

StatsANOVA2NRTest/T=1 dataWave2

The resulting table is:

SSDFMSFFcAccept
Total4.7129323
Factor B0.13783330.04594440.3035083.287381
Factor A2.3044350.4608873.044612.901290
Remainder2.27067150.151378

In this case we can't reject H0 for Factor B (columns), i.e., the the mean reading is the same for all columns. The situation is different for rows where F>Fc at least for the default 0.05 significance level. To test with relaxed significance (0.01) execute the command:

StatsANOVA2NRTest/T=1/ALPH=0.01 dataWave2

This results in:

SSDFMSFFcAccept
Total4.7129323
Factor B0.13783330.04594440.3035085.416961
Factor A2.3044350.4608873.044614.555611
Remainder2.27067150.151378

As expected, the change in significance was enough to drive F,sub>c above the value of F for Factor A and so at this significance level we can't reject H0 for Factor A.

Handling One Missing Value

In the following examples we use the same data as dataWave1 but we introduce one missing value. To do so we set (row 3, column 2) to NaN.

StatsANOVA2NRTest/T=1/FOMD dataWave3

The operation prints in the history:

Single missing value computed to be: 4.34667
SS bias estimate: 0.572033

The result table contains the following values:

SSDFMSFFcAccept
Total35.738722
Factor B11.343833.7812612.23753.343890
Factor A20.069154.0138212.99012.958250
Remainder4.32585140.30899

The computed datum 4.34667 is used as a substitute for 4.85 which we erased from the original data. The test results are essentially unchanged.