COX STUART TEST

COX STUART TEST

Analysis Command

Perform a Cox-Stuart test for trend for a univariate data set.

In many measurement processes, it is desirable to detect the prescence of trend (e.g., due to drift). That is, if the data are assumed to be independent observations, we are interested in knowing if there is in fact a time dependent trend (i.e., the observations are in fact not independent).

Given a set of ordered observations X₁, X₂, ..., X_n, let

 
c = n/2       if n even
  = (n+1)/2   if n odd
    

Then pair the data as X₁,X_1+c, X₂,X_2+c, ..., X_n-c,X_n. The Cox-Stuart test is then simply a sign test applied to these paired data.

<LOWER TAILED/UPPER TAILED> COX STUART TEST <y>
                        <SUBSET/EXCEPT/FOR qualification>
where <LOWER TAILED/UPPER TAILED> is an optional keyword that specifies either a lower tailed or an upper tailed test;
            <y> is a response variable;
and where the <SUBSET/EXCEPT/FOR qualification> is optional.

If neither LOWER TAILED or UPPER TAILED is specified, the Cox and Stuart test will return the results for the two-tailed case, the lower tailed case, and the upper tailed case. If LOWER TAILED is specified, then only the results for the lower tailed case will be printed. If UPPER TAILED is specified, then only the results for the upper tailed case will be printed.

<LOWER TAILED/UPPER TAILED> COX STUART TEST <y1> ... <yk>
                        <SUBSET/EXCEPT/FOR qualification>
where <LOWER TAILED/UPPER TAILED> is an optional keyword that specifies either a lower tailed or an upper tailed test;
            <y1> ... <yk> is a list of 1 to 30 response variables;
and where the <SUBSET/EXCEPT/FOR qualification> is optional.

This syntax will perform a Cox-Stuart test for each of the response variables. For example,

COX STUART TEST Y1 TO Y4

is equivalent to

COX STUART TEST Y1
COX STUART TEST Y2
COX STUART TEST Y3
COX STUART TEST Y4

COX STUART TEST Y
COX STUART TEST Y1 Y2 Y3
COX STUART TEST Y1 TO Y5
LOWER TAILED COX STUART TEST Y
UPPER TAILED COX STUART TEST Y

The COX STUART TEST will accept matrix arguments. If a matrix is given, the data elements in the matrix will be collected in column order to form a vector before performing the Cox-Stuart test.

Dataplot saves the following internal parameters after a Cox and Stuart test:

STATVAL:	the value of the test statistic
STATCDF:	the CDF of the test statistic
PVALUE:	the p-value for the two-sided test
PVALUELT:	the p-value for the lower tailed test
PVALUEUT:	the p-value for the upper tailed test
CUTLOW50:	the 50% lower tailed critical value
CUTUPP50:	the 50% upper tailed critical value
CUTLOW80:	the 80% lower tailed critical value
CUTUPP80:	the 80% upper tailed critical value
CUTLOW90:	the 90% lower tailed critical value
CUTUPP90:	the 90% upper tailed critical value
CUTLOW95:	the 95% lower tailed critical value
CUTUPP95:	the 95% upper tailed critical value
CUTLOW99:	the 99% lower tailed critical value
CUTUPP99:	the 99% upper tailed critical value
CUTLO999:	the 99.9% lower tailed critical value
CUTUP999:	the 99.9% upper tailed critical value

The run sequence plot can be used to graphically assess whether or not there is trend in the data. The 4-plot can be used to assess the more general assumption of "independent, identically distributed" data.

The paired data can also be analyzed using other techniques for comparing two response variables (e.g., t-test, bihistogram, quantile-quantile plot).

None

None

SIGN TEST	= Compute a sign test.
KENDALL TAU INDEPENDENCE TEST	= Compute an independence test based on the Kendall tau statistic.
RANK CORRELATION INDEPENDENCE TEST	= Compute an independence test based on the Kendall tau statistic.
T-TEST	= Compute a t-test.
4-PLOT	= Generate a 4-plot.
RUN SEQUENCE PLOT	= Generate a run sequence plot.
BIHISTOGRAM	= Generates a bihistogram.
QUANTILE-QUANTILE PLOT	= Generate a quantile-quantile plot.

Conover (1999), "Practical Nonparametric Statistics", Third Edition, Wiley, pp. 170-175.

Confirmatory Data Analysis

2011/6

 
. Purpose: Test the Cox Stuart Trend Test
.          Sample data from example 2 on page 171 of Conover.
.
let y = data  45.25 45.83 41.77 36.26 45.37 52.25 35.37 57.16 35.37 ...
              58.32 41.05 33.72 45.73 37.90 41.72 36.07 49.83 36.24 ...
              39.90
.
set write decimals 4
cox stuart test y
    

The following output is generated.

            Cox Stuart Test for Trend
            (Compare Observations < Midpoint to Those > Midpoint)
 
Response Variable:  Y
 
H0: There is No Trend
Ha: There is a Trend
 
Summary Statistics:
Number of Observations                                    19
Number of Observations After Matching:                     9
 
Summary Statistics for Points Below Midpoint:
Sample Mean:                                         43.8477
Sample Median:                                       45.2500
Sample Standard Deviation:                            7.5965
Sample Median Absolute Deviation                      7.0000
 
 
Summary Statistics for Points Above Midpoint:
Sample Mean:                                         40.2399
Sample Median:                                       39.8999
Sample Standard Deviation:                            5.0799
Sample Median Absolute Deviation                      3.6599
 
Test:
Number of Positive Differences:                            5
Number of Negative Differences:                            4
Number of Ties:                                            0
CDF Value for Positive Values:                        0.7460
CDF Value for Negative Values:                        0.5000
P-Value (2-tailed test):                              1.0000
P-Value (lower-tailed test):                          0.7460
P-Value (upper-tailed test):                          0.5000
 
 
            Two-Tailed Test
 
H0: P(+) = P(-); Ha: P(+) <> P(-)
---------------------------------------------------------------------------
                                        Lower          Upper           Null
   Significance           Test       Critical       Critical     Hypothesis
          Level      Statistic      Value (<)      Value (>)     Conclusion
---------------------------------------------------------------------------
          50.0%              5              3              6         ACCEPT
          80.0%              5              3              6         ACCEPT
          90.0%              5              2              7         ACCEPT
          95.0%              5              2              7         ACCEPT
          99.0%              5              1              8         ACCEPT
          99.9%              5              0              9         ACCEPT