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Dataplot Vol 1 Vol 2

QUADE TEST

Name:
    QUADE TEST
Type:
    Analysis Command
Purpose:
    Perform a Quade test that k treatments have identical effects.
Description:
    The Quade test is a non-parametric test for analyzing randomized complete block designs.

    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 ... X1k
      2 X21 X22 ... X2k
      3 X31 X32 ... X3k
      ... ... ... ... ...
      b Xb1 Xb2 ... Xbk

    Let R(Xij) be the rank assigned to Xij 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:

      Qi = rank of the range for the i-th block

    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
    Ha: 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

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

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

    The T3 statistic is equivalent to performing a two-way analysis of variance of the Sij. 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 Sij rather than the data.

Syntax:
    QUADE TEST <y> <block> <treat>
                            <SUBSET/EXCEPT/FOR qualification>
    where <y> is the response variable;
                <block> is a variable that identifies the block;
                <treat> is a variable that identifies the treatment;
    and where the <SUBSET/EXCEPT/FOR qualification> is optional.
Examples:
    QUADE TEST Y BLOCK TREATMENT
    QUADE TEST Y X1 X2
    QUADE TEST Y BLOCK TREATMENT SUBSET BLOCK > 2
Note:
    In Dataplot, the variables should be given as:

      Y BLOCK TREAT

      X11 1 1
      X12 1 2
      ... 1 ...
      X1k 1 k
      X21 2 1
      X22 2 2
      ... 2 ...
      X2k 2 k
      ... ... ...
      Xb1 b 1
      Xb2 b 2
      ... b ...
      Xbk b k

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

      LET NBLOCK = SIZE Y1
      LET BLOCKID = SEQUENCE 1 1 NBLOCK
      LET Y BLOCK TREAT = REPLICATED STACK Y1 Y2 Y3 Y4 Y5 BLOCKID
      QUADE TEST Y BLOCK TREAT
Note:
    The response, ranked response, block, and treatment are written to the file dpst1f.dat in the current directory.

    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 Quade test is based on the following assumptions:

    1. The b rows are mutually independent. That means that the results within one block (row) do not affect the results within other blocks.

    2. The data can be meaningfully ranked.

    3. The data have at least interval scale so that the sample range may be determined within each block.
Note:
    The following statistics are also supported:

      LET A = QUADE TEST Y X1 X2
      LET A = QUADE TEST CDF Y X1 X2
      LET A = QUADE TEST PVALUE Y X1 X2

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

Note:
    The Quade test is similar to the Friedman test. A few distinctions:

    1. For k = 2, the Friedman test is equivalent to a sign test while the Quade test is equivalent to a signed rank test.

    2. 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.

    3. 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: Reference:
    W. J. Conover (1999), "Practical Nonparameteric Statistics," Third Edition, Wiley, pp. 373-380.
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
        
    The following output is generated for the Quade test
                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
        

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Date created: 08/31/2011
Last updated: 11/04/2015

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