QUADE TEST

Name:

QUADE TEST Type:

Analysis Command Purpose:

Description:

The Quade test assumes that there are k experimental treatments (k ≥ 2). The observations are arranged in b blocks, that is

	Treatment
Block	1	2	...	k
1	X₁₁	X₁₂	...	X_1k
2	X₂₁	X₂₂	...	X_2k
3	X₃₁	X₃₂	...	X_3k
...	...	...	...	...
b	X_b1	X_b2	...	X_bk

Let R(X_ij) be the rank assigned to X_ij within block i (i.e., ranks within a given row). Average ranks are used in the case of ties.

Compute the range in each block (the maximum value - the minimum value for the original data) and then rank these:

Q_i

Then let

\( S_{ij} = Q_{i} (R(X_{ij}) - (k+1)/2) \)

and

\( S_{j} = \sum_{i=1}^{b}{S_{ij}} \hspace{.5in} j = 1, 2, \ldots , k \)

Then the Quade test is

H₀: The treatment effects have identical effects

H_a: At least one treatment is different from at least one other treatment

Test Statistic: The test statistic is

\( T3 = \frac{(b-1)B}{A2 - B} \)

where

\( A2 = \sum_{i=1}^{b}{\sum_{j=1}^{k}{S_{ij}^2}} \)
\( B = \frac{1}{b} \sum_{j=1}^{k}{S_{j}^2} \)

The T3 statistic is equivalent to performing a two-way analysis of variance of the S_ij. The A2 term is equivalent to the total sum of squares and B is equivalent to the treatment sum of squares.

Significance Level: \( \alpha \)

Critical Region: \( T_{2} > F_{(\alpha,k-1,(b-1)(k-1))} \)
where F is the percent point function of the F distributuion.

Conclusion: Reject the null hypothesis if the test statistic is in the critical region.

If the hypothesis of identical treatment effects is rejected, it is often desirable to determine which treatments are different (i.e., multiple comparisons). Treatments i and j are considered different if

\( |S_{i} - S_{j}| > t_{(1-\alpha/2,(b-1)(k-1))} \sqrt{\frac{2 b(A2 - B)}{(b-1)(k-1)}} \)

This is equivalent to the Fisher least significant difference computed on the S_ij rather than the data.

Syntax:

Examples:

Note:

Y	BLOCK	TREAT

X₁₁	1	1
X₁₂	1	2
...	1	...
X_1k	1	k
X₂₁	2	1
X₂₂	2	2
...	2	...
X_2k	2	k
...	...	...
X_b1	b	1
X_b2	b	2
...	b	...
X_bk	b	k

If your data are in a format similar to that given in the DESCRIPTION section (i.e., you have colums Y₁ to Y_k, each with b rows), you can convert it to the format required by Dataplot with the commands:

Note:

The treatment ranks and multiple comparisons are written to the file dpst2f.dat in the current directory. Comparisons that are statistically significant at the 95% level are flagged with a single asterisk while comparisons that are statistically significant at the 99% level are flagged with two asterisks.

Note:

The b rows are mutually independent. That means that the results within one block (row) do not affect the results within other blocks.
The data can be meaningfully ranked.
The data have at least interval scale so that the sample range may be determined within each block.

Note:

Enter HELP STATISTICS to see what commands can use these statistics.

Note:

For k = 2, the Friedman test is equivalent to a sign test while the Quade test is equivalent to a signed rank test.
According to Conover, the Quade test is typically more powerful for k < 5 while the Friedman test tends to become more powerful for k ≥ 5.
The Friedman test only requires ordinal scale data (i.e., the data can be ranked) while the Quade test requires at least interval scale data (the range within a block can be computed).

Default:

None Synonyms:

None Related Commands:

FRIEDMAN TEST	= Perform a Friedman test.
ANOVA	= Perform an analysis of variance.
DURBIN TEST	= Perform a Durbin test for two-way incomplete balanced block designs.
COCHRAN TEST	= Perform a Cochran test for two-way complete block designs (binary data).
KRUSKAL WALLIS TEST	= Perform a Kruskall Wallis one-factor test.
SIGN TEST	= Perform a sign test.
MEDIAN POLISH	= Performs a median polish (robust analysis of variance).
T TEST	= Perform a t-test.
RANK SUM TEST	= Perform a rank sum test.
SIGNED RANK TEST	= Perform a signed rank test.
DEX ... PLOT	= Generates a dex plot for a statistic.

Reference:

Practical Nonparameteric Statistics

Applications:

Analysis of Variance Implementation Date:

2011/7 Program:

SKIP 25
READ QUADE2.DAT Y X1 X2
SET WRITE DECIMALS 5
.
LET A1 = QUADE TEST        Y X1 X2
LET A1 = QUADE TEST CDF    Y X1 X2
LET A1 = QUADE TEST PVALUE Y X1 X2
PRINT A1 A2 A3
.
QUADE TEST Y X1 X2

            Quade Two Factor Test
 
Response Variable: Y
First Group-ID Variable: X1
Second Group-ID Variable: X2
 
H0: Treatments Have Identical Effects
Ha: Treatments Do Not Have Identical Effects
 
Summary Statistics:
Total Number of Observations:                            35
Number of Blocks:                                         7
Number of Treatments:                                     5
 
Test:
Quade Test Statistic:                               3.82925
Total Sum of Squares (A2):                       1366.50000
Treatment Sum of Squares (B):                     532.35714
CDF of Test Statistic:                              0.98481
P-Value:                                            0.01518
 
 
Percent Points of the F Reference Distribution
-----------------------------------
  Percent Point               Value
-----------------------------------
            0.0    =          0.000
           50.0    =          0.863
           75.0    =          1.445
           90.0    =          2.194
           95.0    =          2.775
           97.5    =          3.379
           99.0    =          4.217
           99.9    =          6.589
 
 
Conclusions (Upper 1-Tailed Test)
----------------------------------------------
  Alpha    CDF   Critical Value     Conclusion
----------------------------------------------
    10%    90%            2.194      Reject H0
     5%    95%            2.775      Reject H0
   2.5%  97.5%            3.379      Reject H0
     1%    99%            4.217      Accept H0