CSTRENDS Function
Performs the Cox and Stuart sign test for trends in location and dispersion.
Usage
result = CSTRENDS(x)
Input Parameters
x—One-dimensional array containing the data in chronological order.
Returned Value
result—One-dimensional array of length 8 containing the probabilities.
The first four elements of result are computed from two groups of observations.
0—Probability of Nstat(0) + Nstat(2) or more negative signs (ties are considered negative).1—Probability of obtaining Nstat(1) or more positive signs (ties are considered negative).2—Probability of Nstat(0) + Nstat(2) or more negative signs (ties are considered positive).3 —Probability of obtaining Nstat(1) or more positive signs (ties are considered positive).The last four elements of result are computed from three groups of observations.
4—Probability of Nstat(0) + Nstat(2) or more negative signs (ties are considered negative).5—Probability of obtaining Nstat(1) or more positive signs (ties are considered negative).6—Probability of Nstat(0) + Nstat(2) or more negative signs (ties are considered positive).7—Probability of obtaining Nstat(1) or more positive signs (ties are considered positive).Input Keywords
Double—If present and nonzero, double precision is used.
Dispersion—A one-dimensional array of length 2. If Dispersion is set, the Cox and Stuart tests for trends in dispersion are computed. Otherwise, as default, the Cox and Stuart tests for trends in location are computed. k = Dispersion(0) is the number of consecutive x elements to be used to measure dispersion. If ids = Dispersion(1) is zero, the range is used as a measure of dispersion. Otherwise, the centered sum of squares is used.
Fuzz—A nonnegative constant used to determine when elements in x are tied. If |x(i) – x(j)| is less than or equal to Fuzz, x(i) and x(j) are said to be tied. Fuzz must be nonnegative. Default: Fuzz = 0.0.
Output Keywords
Nstat—Named variable into which the one-dimensional array of length 8 containing the statistics below is stored:
0—Number of negative differences (two groups)1—Number of positive differences (two groups)2—Number of zero differences (two groups)3—Number of differences used to calculate result(0) through result(3) (two groups).
4—Number of negative differences (three groups)5—Number of positive differences (three groups)6—Number of zero differences (three groups)7—Number of differences used to calculate result(4) through result(7) (three groups).Nmissing—Named variable into which the number of missing values in x is stored.
Discussion
Function CSTRENDS tests for trends in dispersion or location in a sequence of random variables depending upon the usage of Dispersion. A derivative of the sign test is used (see Cox and Stuart 1955).
Location Test
For the location test (Default) with two groups, the observations are first divided into two groups with the middle observation thrown out if there are an odd number of observations. Each observation in group one is then compared with the observation in group two that has the same lexicographical order. A count is made of the number of times a group-one observation is less than (Nstat(0)), greater than (Nstat(1)), or equal to (Nstat(2)), its counterpart in group two. Two observations are counted as equal if they are within Fuzz of one another.
In the three-group test, the observations are divided into three groups, with the center group losing observations if the division is not exact. The first and third groups are then compared as in the two-group case, and the counts are stored in Nstat(4) through Nstat(6).
Probabilities in result are computed using the binomial distribution with sample size equal to the number of observations in the first group (Nstat(3) or Nstat(7)), and binomial probability p = 0.5.
Dispersion Test
The dispersion tests (when keyword Dispersion is set) proceed exactly as with the tests for location, but using one of two derived dispersion measures. The input value k = Dispersion(0) is used to define N_ELEMENTS(x)/k groups of consecutive observations starting with observation 1. The first k observations define the first group, the next k observations define the second group, etc., with the last observations omitted if N_ELEMENTS(x) is not evenly divisible by k. A dispersion score is then computed for each group as either the range (ids = 0), or a multiple of the variance (ids ≠ 0) of the observations in the group. The dispersion scores form a derived sample. The tests proceed on the derived sample as above.
Ties
Ties are defined as occurring when a group one observation is within Fuzz of its last group counterpart. Ties imply that the probability distribution of x is not strictly continuous, which means that Pr(x1 > x2) ≠ 0.5 under the null hypothesis of no trend (and the assumption of independent identically distributed observations). When ties are present, the computed binomial probabilities are not exact, and the hypothesis tests will be conservative.
Hypothesis tests
In the following, i indexes an observation from group 1, while j indexes the corresponding observation in group 2 (two groups) or group 3 (three groups).
H0 : Pr(Xi > Xj) = Pr(Xi < Xj) = 0.5 H1 : Pr(Xi > Xj) < Pr(Xi < Xj)
Hypothesis of upward trend. Reject if result(2) (or result(6))is less than the significance level.
H0 : Pr(Xi > Xj) = Pr(Xi < Xj) = 0.5 H1 : Pr(Xi > Xj) > Pr(Xi < Xj)
Hypothesis of downward trend. Reject if result(1) (or result(5)) is less than the significance level.
H0 : Pr(Xi > Xj) = Pr(Xi < Xj) = 0.5 H1 : Pr(Xi > Xj) ≠ Pr(Xi < Xj)
Two tailed test. Reject if 2 max(result(1), result(2)) (or 2 max(result(5), result(6)) is less than the significance level.
Assumptions
1. The observations are a random sample; i.e., the observations are independently and identically distributed.
2. The distribution is continuous.
Example
This example illustrates both the location and dispersion tests. The data, which are taken from Bradley (1968), page 176, give the closing price of AT&T on the New York stock exchange for 36 days in 1965. Tests for trends in location (Default), and for trends in dispersion (Dispersion) are performed. Trends in location are found.
x = [9.5, 9.875, 9.25, 9.5, 9.375, 9.0, 8.75, 8.625, 8.0, $
8.25, 8.25, 8.375, 8.125, 7.875, 7.5, 7.875, 7.875, $
7.75,7.75, 7.75, 8.0, 7.5,7.5, 7.125, 7.25, 7.25, 7.125, $
6.75,6.5, 7.0, 7.0, 6.75, 6.625, 6.625,7.125, 7.75]
k = 2
ids = 0
pstat = CSTRENDS(x, Nstat = nstat)
; PV-WAVE prints the following:
; % CSTRENDS: Warning: STAT_AT_LEAST_ONE_TIE
; At least one tie is detected between the samples.
PM, nstat, Title = ' NSTAT'
; PV-WAVE prints the following:
; NSTAT
; 0
; 17
; 1
; 18
; 0
; 12
; 0
; 12
PM, pstat, Title = ' PSTAT'
; PV-WAVE prints the following:
; PSTAT
; 0.999996
; 7.24792e-05
; 1.00000
; 3.81470e-06
; 1.00000
; 0.000244141
; 1.00000
; 0.000244141
pstat = CSTRENDS(x, Nstat = nstat, Dispersion = [k, ids])
% CSTRENDS: Warning: STAT_AT_LEAST_ONE_TIE
At least one tie is detected between the samples.
PM, nstat, Title = ' NSTAT'
; PV-WAVE prints the following:
; NSTAT
; 4
; 3
; 2
; 9
; 4
; 2
; 0
; 6
PM, pstat, Title = ' PSTAT'
; PV-WAVE prints the following:
; PSTAT
; 0.253906
; 0.910156
; 0.746094
; 0.500000
; 0.343750
; 0.890625
; 0.343750
; 0.890625
