PAGE TEST

PAGE TEST

Analysis Command

Perform a Page test that k treatments have identical effects against the alternative of ordered treatment effects.

The Page test is a non-parametric test for analyzing randomized complete block designs. It is derived from a Spearman's rho rank correlation of the Friedman within block rankings and the specified ordering of the treatments. The Page test statistic is actually a monotonic function of Spearman's rho when there is no ties in the data.

The distinction between the Friedman test and the Page test is that the Page test is used when you want to test for ordered effects. By ordered effects, we mean that treatment 1 is less than or equal to treatment 2 which is less than or equal to treatment 3 and so on (and at least one of these should be strictly less than).

As with the Friedman test, assumes that there are k experimental treatments (k >= 2). The observations are arranged in p 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(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. The ranks are summed to obtain

\( R_{j} = \sum_{i=1}^{b}{R(X_{ij})} \)

Then the Page test is

H0:	The treatment effects have identical effects
Ha:	Ha: \( \mu_1 \le \mu_2 \le \ldots \le \mu_k \)
Test Statistic:	The Page test statistic is \( \mbox{T4} = \sum_{j=1}^{k}{j R_{j}} \) In order to compare this to a standard normal distribution, this statistic is transformed to \( \mbox{T5} = \frac{T4 - bk(k+1)^2/4} {\sqrt{b(k^3 - k)^2/(144(k-1))}} \)
Significance Level:	\( \alpha \)
Critical Region:	\( \mbox{T5} > \Phi^{-1}(\alpha) \) where \( \Phi^{-1} \) is the normal percent point function.
Conclusion:	Reject the null hypothesis if the test statistic is in the critical region.

PAGE 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.

PAGE TEST Y BLOCK TREATMENT
PAGE TEST Y X1 X2
PAGE TEST Y BLOCK TREATMENT SUBSET BLOCK > 2

Dataplot formulates the test as a "less than or equal to" test. That is, the treatment that is hypothesized to be the smallest should be coded as 1, the treatment that is hypothesized to be the second smallest should be coded as 2, and so on. If your treatments are coded in the opposite order, you can reverse them with the command
LET TREAT2 = FLIP TREAT

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 Y_k, 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
PAGE TEST Y BLOCK TREAT

The response, ranked response, block, and treatment are written to the file dpst1f.dat in the current directory.

The Page 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.

The following statistics are also supported:
LET A = PAGE TEST Y X1 X2
LET A = PAGE TEST NORMALIZED Y X1 X2
LET A = PAGE TEST CDF Y X1 X2
LET A = PAGE TEST PVALUE Y X1 X2


The PAGE TEST returns the T4 value described above and the PAGE TEST NORMALIZED returns the T5 value described above.

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

None

None

FRIEDMAN TEST	= Perform a Friedman test.
QUADE TEST	= Perform a Quade 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	= Perform a Kruskall Wallis test.
BLOCK PLOT	= Generate a block plot.
DEX SCATTER PLOT	= Generates a dex scatter plot.
DEX ... PLOT	= Generates a dex plot for a statistic.

Conover, "Practical Nonparametric Statistics," Third Edition, Wiley, 1999, pp. 380-383.

Analysis of Variance

2013/2

 
.  Test Page test as given in
.  Conover (1999), "Practical Non-Parametric Statistics"
.  Third Edition, pp. 380-381.
.
read y treat block
79   1    1
78   1    2
72   1    3
66   1    4
75   1    5
64   1    6
68   1    7
70   1    8
76   2    1
76   2    2
74   2    3
65   2    4
75   2    5
66   2    6
70   2    7
72   2    8
77   3    1
76   3    2
77   3    3
68   3    4
72   3    5
65   3    6
68   3    7
78   3    8
84   4    1
80   4    2
78   4    3
72   4    4
74   4    5
69   4    6
74   4    7
76   4    8
82   5    1
80   5    2
75   5    3
65   5    4
77   5    5
68   5    6
70   5    7
77   5    8
end of data
.
let statval = page test             y block treat
let statva2 = page test normalized  y block treat
let cdf     = page test cdf         y block treat
let pval    = page test pvalue      y block treat
.
set write decimals 4
page test y block treat
    

The following output is generated.

 
 PARAMETERS AND CONSTANTS--

    STATVAL --       401.5000
    STATVA2 --         2.9345
    CDF     --         0.9983
    PVAL    --         0.0017


            Page Two Factor Test
 
Response Variable: Y
Group-ID Variable (Block): BLOCK
Group-ID Variable (Treatment): TREAT
 
H0: u1 = u2 = ... = uk
Ha: u1 <= u2 <= ... <= uk
 
Summary Statistics:
Total Number of Observations:                        40
Number of Blocks:                                     8
Number of Treatments:                                 5
 
Test:
Page Test Statistic:                           401.5000
Page Normalized Test Statistic:                  2.9344
CDF of Test Statistic:                           0.9983
P-Value:                                         0.0016
 
 
Percent Points of the Normal Reference Distribution
-----------------------------------
  Percent Point               Value
-----------------------------------
           50.0    =          0.000
           75.0    =          0.674
           90.0    =          1.282
           95.0    =          1.645
           97.5    =          1.960
           99.0    =          2.326
           99.9    =          3.089
 
 
Conclusions (Upper 1-Tailed Test)
----------------------------------------------
  Alpha    CDF   Critical Value     Conclusion
----------------------------------------------
    10%    90%            1.282      Reject H0
     5%    95%            1.645      Reject H0
   2.5%  97.5%            1.960      Reject H0
     1%    99%            2.326      Reject H0