6.5.5. Quade Two-Way ANOVA

Data entry is in matrix format (see 6.0.5. Tests with Matrix Data). Columns selected for this test must have equal number of rows and rows containing at least one missing value are omitted.

6.5.5.1. Quade ANOVA Test Results

In the analysis of several related variables a more powerful alternative to the Friedman Two-Way ANOVA is Quade Two-Way ANOVA by ranks. First the ranks (R_ij, i = 1, …, N, j = 1, …, M) and the range of data in each row are found and range values are also ranked (Q_i, i = 1, …, N). The test statistic is:

  

where:

  

  

  

  

The test statistic displayed is corrected for ties. The output includes the rank sum and mean rank for each variable and the correction for ties. The one-tail probability is reported using the F-distribution with M – 1 and (N – 1)(M – 1) degrees of freedom.

6.5.5.2. Quade ANOVA Multiple Comparisons

If the null hypothesis is rejected as result of the Quade’s test, then a multiple comparison can be run to find out which column effects are different. Nonparametric Multiple Comparisons are performed in a way similar to the Tukey-HSD test using rank sums and the t-distribution. The standard error is:

  

6.5.5.3. Quade ANOVA Example

Example 2 on p. 375, Conover, W. J. (1999). A researcher wants to test the null hypothesis that “the treatments in blocks (i.e. columns) have identical effects” at a 95% confidence level.

Open NONPARM1, select Statistics 1 → Nonparametric Tests (Multisample) → Quade Two-Way ANOVA and include Brand A, Brand B, Brand C, Brand D and Brand E, (C35 to C39) in the analysis by clicking [Variable] to obtain the following results:

Quade Two-Way ANOVA

	Cases	Rank Sum	Mean Rank
Brand A	7	21.5000	3.0714
Brand B	7	15.0000	2.1429
Brand C	7	15.0000	2.1429
Brand D	7	26.0000	3.7143
Brand E	7	27.5000	3.9286
Total	35	105.0000	3.0000

 

Number of Columns =	5
Number of Rows =	7
F(4,24) =	3.8293
Right-Tail Probability =	0.0152

 

Since the right tail probability is less than 5%, reject the null hypothesis. Therefore, proceed with the Multiple Comparisons to find out which treatments are different.

Multiple comparisons with t-distribution

Method: 95% t interval.

** denotes significantly different pairs. Vertical bars show homogeneous subsets.

A pairwise test result is significant if its q stat value is greater than the table q.

 

Group	Cases	Rank Sum	Brand B	Brand C	Brand A	Brand D	Brand E
Brand B	7	-38.0000				**	**	|
Brand C	7	-14.0000					**	||
Brand A	7	-9.5000					**	||
Brand D	7	23.5000	**					||
Brand E	7	38.0000	**	**	**			|

 

Comparison	Difference	Standard Error	q Stat	Table q	Probability
Brand E – Brand B	76.0000	22.0586	3.4454	2.0639	0.0021
Brand D – Brand B	61.5000	22.0586	2.7880	2.0639	0.0102
Brand A – Brand B	28.5000	22.0586	1.2920	2.0639	0.2087
Brand C – Brand B	24.0000	22.0586	1.0880	2.0639	0.2874
Brand E – Brand C	52.0000	22.0586	2.3574	2.0639	0.0269
Brand D – Brand C	37.5000	22.0586	1.7000	2.0639	0.1021
Brand A – Brand C	4.5000	22.0586	0.2040	2.0639	0.8401
Brand E – Brand A	47.5000	22.0586	2.1534	2.0639	0.0416
Brand D – Brand A	33.0000	22.0586	1.4960	2.0639	0.1477
Brand E – Brand D	14.5000	22.0586	0.6573	2.0639	0.5172

 

Comparison	Lower 95%	Upper 95%	Result
Brand E – Brand B	30.4732	121.5268	**
Brand D – Brand B	15.9732	107.0268	**
Brand A – Brand B	-17.0268	74.0268
Brand C – Brand B	-21.5268	69.5268
Brand E – Brand C	6.4732	97.5268	**
Brand D – Brand C	-8.0268	83.0268
Brand A – Brand C	-41.0268	50.0268
Brand E – Brand A	1.9732	93.0268	**
Brand D – Brand A	-12.5268	78.5268
Brand E – Brand D	-31.0268	60.0268

 

Homogeneous Subsets:
Group 1:	Brand B Brand C Brand A
Group 2:	Brand C Brand A Brand D
Group 3:	Brand D Brand E

 

The overall conclusion is that Brand B & E, B & D, C & E, and A & E are different.