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Jonckheere-Terpstra Test (Jonckheere's Trend Test)

We are studying a mutant gene that we suspect has a quantitative effect on male fertility. We collect a number of males (3 or 4) having 2, 1, or 0 copies of the mutant allele and mate each to 3 females. The observation for each male is the total number of progeny obtained in the matings.
Number of          Total number
mutant alleles     of progeny
  2                16  8  6
  1                27 16 15
  0                31 29 18 42

Source: http://mcardle.wisc.edu/mstat/help/help/Notes-05.html


Data:

  2: { 16  8  6 }
  1: { 27 16 15 }
  0: { 31 29 18 42 }


Procedure:
  1. Nonparametric
  2. Three or More Samples...
  3. Jonckheere-Terpstra Test...
  4. Increasing order of medians, selected
  5. Stop the test if the medians are not arranged in order, checked
  6. Significance Level = 5%
  7. OK


Results:
DOT PLOTS

   +---+-----------+-----------+-----------+-----------+---+ 
   |                                                       | 
   |                                                       | 
   |                                                       | 
   |                                                       | 
   |                                                       | 
   |                                                       | 
   |                                                       | 
   |   *  *         *                                      | Group 1
   +---+-----------+-----------+-----------+-----------+---+ 
   |                                                       | 
   |                                                       | 
   |                                                       | 
   |                                                       | 
   |                                                       | 
   |                                                       | 
   |                                                       | 
   |               **              *                       | Group 2
   +---+-----------+-----------+-----------+-----------+---+ 
   |                                                       | 
   |                                                       | 
   |                                                       | 
   |                                                       | 
   |                                                       | 
   |                                                       | 
   |                                                       | 
   |                   *              * *              *   | Group 3
   +---+-----------+-----------+-----------+-----------+---+ 
    6.00       15.00       24.00       33.00       42.00


BOX-AND-WHISKER PLOTS

   +---+-----------+-----------+-----------+-----------+---+
   |                                                       |
   |    +-+----+                                           |
   |   || | *  |----|                                      | Group 1
   |    +-+----+                                           |
   |                +-------+                              |
   |               ||    *  |------|                       | Group 2
   |                +-------+                              |
   |                          +--------+--------+          |
   |                   |------|        *        |------|   | Group 3
   |                          +--------+--------+          |
   |                                                       |
   +---+-----------+-----------+-----------+-----------+---+
    6.00       15.00       24.00       33.00       42.00    


JONCKHEERE-TERPSTRA TEST (= JONCKHEERE TREND TEST)

    Ho: Median 1 = Median 2 = Median 3
    Ha: Median 1 <= Median 2 <= Median 3
        (with at least one strict inequality)
        (Significance Level = 0.05)


  -- Method 1 --

DATA TABLE

      No  Group 1  Group 2  Group 3
  ---------------------------------
       1       16       27       31
       2        8       16       29
       3        6       15       18
       4                         42
  ---------------------------------
  Median     8.00    16.00    30.00


  ORDERED CONTINGENCY TABLE

      Data   Group1  Group2  Group3      Ti 
    -------+------------------------+--------
         6 |     1                  |     1
         8 |     1                  |     1
        15 |             1          |     1
        16 |     1       1          |     2
        18 |                     1  |     1
        27 |             1          |     1
        29 |                     1  |     1
        31 |                     1  |     1
        42 |                     1  |     1
    -------+------------------------+--------
        Mi       3       3       4       10 


    P = 30
    Q = 2
    S = 30 - 2 = 28

    n = 10
    n ^ 2 = 100
    n ^ 3 = 1000

    Sum(Ti ^ 2) = 12
    Sum(Ti ^ 3) = 16

    Sum(Mi ^ 2) = 34
    Sum(Mi ^ 3) = 118

    Var(S) = 
     (2(1000 - 16 - 118) + 3(100 - 12 - 34) + 5(10)) / 18 +
     (16 - 3(12) + 2(10))(118 - 3(34) + 2(10)) / 9(10)(10 - 1)(10 - 2) + 
     (12 - 10)(34 - 10) / 2(10)(10 - 1)
     = 108.266667

    Critical Value: Z(0.05) = 1.644853
    Test Statistic: Z = 28 / (108.266667 ^ 0.5) = 2.690981
    P(>Z) = 0.003562

    Reject the null hypothesis at the 0.05 significance level.
    Median 1 <= Median 2 <= Median 3 (with at least one strict inequality)



  -- Method 2 --

DATA TABLE

      No  Group 1  Group 2  Group 3
  ---------------------------------
       1       16       27       31
       2        8       16       29
       3        6       15       18
       4                         42
  ---------------------------------
  Median     8.00    16.00    30.00


    U1,2 = 1.5 + 3 + 3 = 7.5
    U1,3 = 4 + 4 + 4 = 12
    U2,3 = 3 + 4 + 4 = 11

    U = 7.5 + 12 + 11 = 30.5


    Critical Value: Z(0.05) = 1.644853

    E(U) = 16.5
    Var(U) = 27.25
    Test Statistic: Z = (30.5 - 16.5) / 27.25 ^ 0.5 = 2.681914
    P(>Z) = 0.00366

    Reject the null hypothesis at the 0.05 significance level.
    Median 1 <= Median 2 <= Median 3 (with at least one strict inequality)

See also:

Jonckheere-Terpstra Test 1 - previous
Jonckheere-Terpstra Test 3 - next

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