08 Series homogenity tests

Series homogeneity tests

General

Dependent on the type of analysis series must fulfil one or more of the following requirements:

stationarity: i.e. the properties or characteristics of the series do not vary with time;
homogeneity: i.e. all elements of a series belong to the same population;
randomness: i.e. series elements are independent.
hymosincludes following statistical tests to investigate series' stationarity, homogeneity or randomness:
1. Median run test : a test for randomness by calculating the number of runs above and below the median;
2. Turning point test : a test for randomness by calculating the number of turning points;
3. Difference sign test : a test for randomness by calculating the number of positive and negative differences;
4. Spearman rank correlation test : the Spearman rank correlation coefficient is computed to test:
the existence of correlation between two series,
the significance of serial rank correlation, and
the significance of a trend;
5. Spearman rank trend test
6. Arithmetic serial correlation coefficient: a test for serial correlation;
7. Wilcoxon-Mann-Whitney U-test: a test to investigate whether two series are from the same population;
8. Student t-test: a test on difference in the mean between two series;
9. Wilcoxon W-test: a test on difference in the mean between two series;
10. Linear trend test: a test on significance of linear trend by statistical inference on slope of trend line;
11. Rescaled adjusted range test: a test for series homogeneity by the rescaled adjusted range.
12. Shapiro-Wilk W test
13. Rosner's Test for outliers
14. Sign Test: a test on the positive difference in pairs between two series;
15. Wilcoxon Rank Sum Test: a non-parametric test for independent data sets;
16. Wilcoxon Signed Rank Test
17. Seasonal Kendall Slope Estimator and Test
18. Dixon's Test for outliers
19. Lettenmaier Trend Test

notes
1. The Spearman rank correlation test may be used as a single or two series test; in the single series mode it tests the significance of correlation with time (Spearman rank trend test).
2. Tests nrs. 6, 7 and 8 (Wilcoxon-Mann-Whitney U-test, Student t-test and Wilcoxon W-test) are basically two-series tests; however, the test can also be used for a single series by means of the split-sample approach, where a series is divided into two parts, which are mutually compared.

Series codes
Series can be selected by clicking the checkbox of the series, for one series test only one series can be selected, for two series tests two series can be selected.

Test Option
Choose a statistical test from the list box, after pressing <Execute> the result of the test will be presented behind the test option.

Hypothesis testing
A statistical hypothesis is an assumption about the distribution of a statistical parameter. The assumption is stated in the null-hypothesis H₀ and is tested against an alternative formulated in the H₁ hypothesis. For easy reference the parameter under investigation is usually presented as a standardised variety, called test statistic . Under the null-hypothesis the test statistic has some standardised sampling distribution, e.g. a standard normal, a Student t-distribution, etc. For the null-hypothesis to be true the value of the test statistic should be within the acceptance region of the sampling distribution of the parameter under the null-hypothesis. If the test statistic does not lie in the acceptance region, the null-hypothesis is rejected and the alternative is assumed to be true. Some risk, however, is involved that we make the wrong decision about the test:
Type I error, i.e. rejecting H₀ when it is true, (producer's risk), and
Type II error, i.e. accepting H₀ when it is false, (consumer's risk).
The probability of making a Type I error is equal to the significance level of the test a. When a test is performed at a 0.05 or 5% level of significance it means that there are about 5% chance that the null-hypothesis will be rejected when it should have been accepted. This probability represents the critical region at the extreme end(s) of the sampling distribution under H₀ . Note, however, the smaller the significance level is taken, the larger becomes the risk of making Type II error and the less is the discriminative power of the test.
Depending on the type of alternative hypothesis H₁ one- or two-sided tests are considered. This is explained by the following example. Let in a test for randomness the correlation coefficient r be the test statistic. The null-hypothesis reads H₀ : r=0 against one of the following alternatives:
1. H₁ : r>0, i.e. a right-sided test
2. H₁ : r<0, i.e. a left-sided test
3. H₁ : r¹0, i.e. a two-sided test
Let the tests be performed at a significance level aand let the theoretical value of r under the null-hypothesis is denoted by r, then H₀ will not be rejected in:
1. a right-sided test, if: r£r(1-a)
2. a left-sided test, if: r³r(a)
3. a two-sided test, if: r(a/2)£r£r(1-a/2)
For a symmetrical distribution the last expression may be replaced by
½r½£r(1-a/2)

Median run test
In the median run test the number of runs N_r of series A_i , (i=1,N) above and below the median is counted. A run is defined as an excursion above or below the median, denoted by A_m :
a positive run is bounded by an upcrossing and downcrossing,
a negative run is bounded by a downcrossing and an upcrossing.
Note that values equal to A_m are deleted from the series. If the number of values above and below the median are denoted by m and n respectively (where m = n is not necessarily true due to possible deletion), the quantity N_r for a random series is asymptotically normally distributed with N(m_r ,s_r ):
m_r = 2mn/(m+n) + 1
s_r ² = 2mn{2mn-(m+n)}/{(m+n)² (m+n-1)}
The normal approximation holds for m and n>20. For smaller values of m and n Table VII-5.1 may be used to obtain critical values of N_r at a 5% significance level. The differences with the normal approximation are, however, small.
hymosconsiders the following hypothesis:
H₀ : series A_i is random, and
H₁ : series is not random, with no direction for the deviation of randomness; hence, a two-tailed test is performed
and the absolute value of the following standardised test statistic is computed:
½u½= ½N_r + c - m_r ½/s_r
where: c = a continuity correction: c=0.5 if N_r <m_r , and c=-0.5 if N_r >m_r
For the interpretation of the test results reference is made to the section about Hypothesis testing.
Test conditions:
N ³10
N-(m+n) £N/2
Reference:
Kreyzig, E.: Introductory Mathematical Statistics, Principles and Methods. John Wiley & Sons, Inc, New York, 1970
Siegel S. and N.J. Castellan: Non-parametric statistics for the behavioural sciences. Mc Graw-Hill Book Company, 2^nd ed.,1988, pp. 58-64
note
Any observed value of N_r , which is less than or equal to the smaller value, or is greater than or equal to the larger value is significant at the 0.05 significance level

Turning point test
In a series A_i ,(i=1,N) a turning point is defined whenever
A_(i-1) < A_(i) > A_(i+1) and A_(i-1) > A_(i) < A_(i+1) ,
hence whenever a peak or a trough occurs.
In an independent stationary series of length N the number of turning points N_t is asymptotically normally distributed with N(m_t ,s_t ):
m_t = 2(N-2)/3
s_t ² = (16N-29)/90
hymosconsiders the following hypothesis:
H₀ : series A_i is random, and
H₁ : series is not random, with no direction for the deviation of randomness; hence, a two-tailed test is performed
and the absolute value of the following standardised test statistic is computed:
½u½= ½N_t - m_t ½/s_t
For the interpretation of the test results reference is made to the section about Hypothesis testing.
Test condition:
N³10
Reference:
Yevjevich, V: Stochastic Processes in Hydrology Water Resources Publications, Fort Collins, Colorado, 1972, pp. 215

Difference sign test
The difference-sign test counts the number of positive differences N_p and of negative differences N_n between successive values of series A_i ,(i=1,N): A_(i+1) -A_(i) . Let the maximum of the two be given by N_ds :
N_ds = max (N_p ,N_n )
For an independent stationary series of length N_eff (N_eff = N - zero differences) the number of negative or positive differences is asymptotically normally distributed with N(m_ds ,s_ds ):
m_ds = (N_eff -1)/2
s_ds ² = (N_eff +1)/12
hymosconsiders the following hypothesis:
H₀ : series A_i is random, and
H₁ : series is not random, with no direction for the deviation of randomness; hence, a two-tailed test is performed
and the absolute value of the following standardised test statistic is computed:
½u½= ½N_ds - m_ds ½/s_ds
For the interpretation of the test results reference is made to the section about Hypothesis testing.
Test condition:
N³10
Reference:
Yevjevich, V: Stochastic Processes in Hydrology Water Resources Publications, Fort Collins, Colorado, 1972, pp. 215

Spearman rank correlation test
The association between two series of length N is measured by means of the Spearman rank correlation coefficient r_s . The observations on each series are ranked from 1 to N. Tied observations are assigned the average of the tied ranks. The sum of squares of rank differences is calculated:

where:
A_i = first ranked vector
B_i = second ranked vector
N = length of series
A correction factor for ties is obtained:
T_A = S(t³ - t)/12 over series A
T_B = S(t³ - t)/12 over series B
where: t = number of observations tied for a given rank.
The Spearman rank correlation coefficient r_s is then computed for the following two cases:
(a) if T_A and T_B are zero,
r_s = 1 - 6D/(N³ - N)
(b) if T_A and/or T_B are not zero,
r_s = (X + Y - D)/2Ö(XY)
where:
X = (N³ - N)/12 - T_A
Y = (N³ - N)/12 - T_B
The statistic used to measure the significance of r_s is:
½t½ = ½r_s ½Ö{(N - 2)/(1 - r_s ² )}
Under the null-hypothesis of zero correlation the theoretical value of the test statistic has a Student t -distribution with N_df = N-2 degrees of freedom for N > 10.
For the interpretation of the test results reference is made to the section about Hypothesis testing.
The Spearman rank correlation test is basically a two-sample test. However it can be used as a one-sample test as well in two ways:
1. to measure the significance of serial correlation between series A_1i ,(i=1,N-1) and series A_2i ,(i=2,N). The second series in the test is equal to the first series, only shifted one time step.
2. to test the significance of a trend where series A_i is compared with time: B_i ,(i=1,N) = 1,2,3,....,N. In the Series homogeneity test list box this function is shown as the Spearman rank trend test.
Test condition:
N³10
Reference:
Siegel, S. and N.J. Castellan: Non-parametric statistics for the behavioural sciences. Mc. Graw-Hill Book Company, 2^nd ed., 1988, pp.235-244

Spearman rank trend test
The calculations in this function are explained in the Spearman rank correlation test function. However the significance of a trend is computed where the series is compared with time (see one sample test in Spearman rank correlation test).

Arithmetical serial correlation test
To test for absence of serial dependence in series A_i ,(i=1,N) the arithmetical serial correlation coefficient r₁ is computed:

with: m_A = mean of A_i
The statistic used to measure the significance of r₁ is:
½t½ = ½r₁ ½Ö{(N - 3)/(1 - r₁ ² )}
Under the null-hypothesis of zero correlation the theoretical value of the test statistic has a Student t -distribution with N_df = N-3 degrees of freedom for N > 10.
For the interpretation of the test results reference is made to the section about Hypothesis testing.
Test condition:
N³10

Wilcoxon-Mann-Whitney U-test
The Wilcoxon-Mann-Whitney U-test investigates whether two series of length m and n respectively (which may be split-samples of one series) are from the same population. The data of both series are ranked together in ascending order. Tied observations are assigned the average of the tied ranks. The sum of ranks in each of the series, S_m and S_n , is then calculated.
The U statistic is then computed as follows:
U_m = mn + m(m+1)/2 - S_m
U_n = mn - U_m
U = min(U_m ,U_n )
Correction for ties are incorporated as follows:
1. tied observations are given the average rank, and
2. the variance of U is corrected by a factor F₁ :
F₁ = 1 - S(t³ - t)/(N³ - N)
where:
t = number of observation tied for a given rank
If the elements in the two series belong to the same population then U is approximately normally distributed with N(m_U ,s_U ):
m_U = mn/2
s_U ² = F₁ .mn(N+1)/12
where:
N = m+n
hymosconsiders the following hypothesis:
H₀ : series are from the same population, and
H₁ : series are not from the same population; a two-tailed test is performed
and the absolute value of the following standardised test statistic is computed:
½u½= ½U + c - m_U ½/s_U
where: c = a continuity correction; c=0.5 if U<m_U , and c=-0.5 if U>m_U
For the interpretation of the test results reference is made to the section about Hypothesis testing.
Test conditions:
N ³20, m³5 and n³5
Reference:
Siegel, S. and N.J. Castellan: Non-parametric statistics for the behavioural sciences. Mc. Graw-Hill Book Company, 2^nd ed., 1988, pp.235-244

Student t-test
With the Student t-test differences in mean values of two series A_i ,(i=1,m) and B_i ,(i=1,n) are investigated. Let m_A and m_B denote the sample values of population means of A and B: m_A and m_B .
hymosconsiders the following hypothesis:
H₀ : m_A = m_B , and
H₁ : m_A ¹ m_B ,
Hence a two-tailed test is performed and the absolute value of the following standardised test statistic is computed:
½t½= ½m_A - m_B ½/s_AB
Under the null-hypothesis of equal population means the theoretical value of the test statistic has a Student t -distribution with N_df = m+n-2 degrees of freedom for N = m + n > 10.
For the interpretation of the test results reference is made to the section about Hypothesis testing.
The way the standard deviation s_AB is computed depends on whether the series A and B have the same population variance. For this a Fisher F-test is performed on the ratio of the variances.
hymosconsiders the following hypothesis:
H₀ : s_A ² = s_B ² , and
H₁ : s_A ² ¹ s_B ² , hence a two-tailed test is performed.
Following test statistic is considered:
Q = s_A ² /s_B ²
For the interpretation of the test results reference is made to Section VII.5.2.
The standard deviation s_AB is computed from:
1. in case of equal variances:

2. in case of unequal variances:

and the number of degrees of freedom N_df is given by:

with y:

Practically it implies that N_df becomes less than in the equal variance case, so the discriminative power of the test diminishes somewhat.
Test conditions:
N³10, m³5 and n³5
Reference:
Hald, A.: Statistical theory with engineering applications, John Wiley, New York, 1952

Wilcoxon W-test
The Wilcoxon test considers two series A_i ,(i=1,m) and B_i ,(i=1,n) and tests the significance of differences in the mean.
All values A_i are compared with all B_j , where w_i,j is defined by:
A_i < B_j : w_i,j = 2
A_i = B_j : w_i,j = 1
A_i > B_j : w_i,j = 0
The Wilcoxon statistic W is formed by:

In case m_A = m_B the W-statistic is asymptotically normally distributed with N(m_W ,s_W ):
m_W = mn
s_W ² = mn(N+1)/3
where: N = m+n
hymosconsiders the following hypothesis:
H₀ : m_A = m_B , and
H₁ : m_A ¹ m_B , hence, a two-tailed test is performed
and the absolute value of the following standardised test statistic is computed:
½u½= ½W - m_W ½/s_W
For the interpretation of the test results reference is made to the section about Hypothesis testing.
Test conditions:
m³10 and n³10

Linear trend test
The slope of the trend line of series A_i ,(i=1,N) with time or sequence is investigated. The linear trend equation reads:
A_i = b₁ + b₂ .i + n_i with n_i »N(0,s_n ² ), i = 1, N
where:
b₁ = m_A - b₂ .m_i
b₂ = cov(A,i)/s_i ²

with: m_A = sample mean of series A_i
m_i = mean of series i=1,2,...,N = = (N+1)/2
hymosconsiders the following hypothesis:
H₀ : no trend, i.e. m_b2 = 0, and
H₁ : significant trend, i.e. m_b2 ¹0, hence a two-tailed test is performed
and the absolute value of the following standardised test statistic is computed:
½t½= ½b₂ ½/s_b2

Under the null-hypothesis of no trend, the theoretical value of the test statistic has a Student t -distribution with N_df = N-2 degrees of freedom for N > 10.
For the interpretation of the test results reference is made to the section about Hypothesis testing.
Test condition:
N³10
Reference:
Yevjevich, V.: Stochastic Processes in Hydrology, Water Resources Publication, Fort Collins, 1972

Range test
The range test investigates the homogeneity of series.
The test statistic _a R_N /ÖN is computed, where N is number of elements in series A and _a R_N is the rescaled adjusted range with:
_a R_N = R_N /s_A
where:
R_N = S⁺ - S^-
s_A ² = S(A_i - m_A )² /N
S⁺ = max(S⁰ , S₁ ,....,S_N ) with S₀ = 0
S^- = min(S⁰ , S₁ ,....,S_N ) with S₀ = 0

and: m_A = sample mean of A_i
The null-hypothesis that series A_i is homogeneous is not rejected at a significance level a, using a one-tailed test, if:
_a R_N /ÖN < R_a
where R_a is the critical value derived from the asymptotic distribution of _a R_N given by Feller (1951), available for 0.01£a£0.1 and N³10.
Test condition:
N ³10
References:
Feller, W.: The asymptotic distribution of the range of sums of independent random variables. Ann. Math. Stat., 22, 1951, pp. 427-432
Wallis, J.R. and P.E. O'Connell: Firm reservoir yield. How reliable are historic hydrological records?, Hydrological Science Bulletin, XVIII, 347-365, 1973.
Buishand, T.A.: Some methods for testing the homogeneity of rainfall records. KNMI, Memorandum FM-81-15, June, 1981.

Shapiro and Wilk W-test
The Shapiro and Wilk W-test is a powerful test specifically designed for small sample numbers (n<=50). An aspect of the power of this test is its suitability for various data types and its freedom from restrictive requirements. Its main disadvantage is its limitation to sample sets of 50 or less.
First HYMOS checks the selected data set on missing values, if present they are removed from the set. After this filtering the data are arranged on ascending order and the sum of the squared data value deviations from the mean are computed:

Compute k , where:
k = n/2 if n is even
k = (n-1)/2 if n is uneven
The Shapiro and Wilk W-test makes use of W-test coefficients in statistical tables. These tables are also included in HYMOS, and used in the calculation.
The W statistic is computed by applying these coefficients to the ranges between the two endsof the order statistics:

The critical value for W_crit for significance level aand the number of data n is taken from a statistical table. This critical value W_crit is compared with the calculated value W.
Hymosconsiders the following hypothesis:
H₀ : series could come from a normal population (W < W_crit ), and
H₁ : series are not from a normal population (W > W_crit ).
If the test is rejected, the raw data would be better approximated by an alternative, non-normal distribution and are not suitable for parametric statistical analysis.
Reference:
Water Quality Assessments. A guide to the use of biota/sediments and water in environmental monitoring. D. Chapman(ed.)/1992. Chapman & Hall.

Rosner's Test
The Rosner's test is used to identify up to 10 outliers. The test assumes that the population has a normal distribution. If a lognormal distribution is more plausible, all computations should be performed on the logarithms of the data. Therefore a new series must be created and logarithmic values computed with the Series Transformation function. Rosner's approach is designed to avoid masking of one outlier by another. Masking occurs when an outlier goes undetected because it is very close in value to another outlier.
The maximum number of outliers detected is 10. The procedure repeatedly delete the value farthest from the mean and recompute the test statistic after each deletion. A table is used to evaluate the test statistic when n ³25 and n £5000.
Rosner's test is two-tailed since the procedure identifies either suspiciously large or suspiciously small data.
Hymosconsiders the following hypothesis:
H₀ : The entire data set is from a normal distribution, there are no outliers
H₁ : The data set contains outliers.
The following procedure is followed:
1. Compute the mean m and standard deviation s from the data set.
2. Compute the Rosner's test statistic R .

4. where x _i is the most outlying value.
5. Retrieve the tabled critical value l.
6. Compare the two test values and reject H₀ when R > l.
7. Remove the outlier from the data set and redo the total procedure starting from step 1 for a total of 10 most outlying values.
Reference:
Statistical Methods for Environmental Pollution Monitoring, R.O. Gilbert, 1987, John Wiley & Sons Inc..

Sign Test
The sign test is a two series two-sided test. The sign test statistic, B , is the number of pairs (x_1i , x_2i ) for which x_1i < x_2i , that is, the number of positive differences D _i . The magnitudes of the D _i . are not considered; only their signs are. If any D _i . is zero so that a + or - sign cannot be assigned, this data pair is dropped from the data set and n is reduced by 1. Also pairs for which one series has a missing value is dropped from the data set. The statistic B is used to test the null hypothesis:
H₀ : The median of the population of all possible differences is zero, that is, x_1i is likely to be larger than x_2i as x_2i , is likely to be larger than x_1i .
If the number + and - signs are about equal, there is little reason to reject H₀ .
The alternative hypothesis, H₁ is:
H₁ : The median difference does not equal zero, that is, x_1i is more likely to exceed x_2i than x_2i , is likely to exceed x_1i .
The number of valid pairs must be larger than 20, if so the following test procedure is used:
1. Compute

2. Reject H0 and accept H1 if Z_B £- Z_1- _a _/2 or Z_B ³- Z_1- _a _/2 .
Reference:
Statistical Methods for Environmental Pollution Monitoring, R.O. Gilbert, 1987, John Wiley & Sons Inc..

Wilcoxon Rank Sum Test
The Wilcoxon Rank Sum Test is a non-parametric test for independent data sets. The Wilcoxon rank sum test may be used to test for a shift in location between independent series, that is, the measurements from one series tend to be consistently larger (or smaller) than those from the other series. The rank sum test hat the advantage that the two data sets need not to be drawn from normal distributions, and the test can handle a moderate number of ND values by treating them as ties. The test assumes, however, that the distributions of the two series are identical in shape (variance), but the distributions need not be symmetric.
Hymosconsiders the following hypothesis:
H₀ : The populations from which the two series have been drawn have the same mean,
H₁ : The populations have different means.
The following procedure is followed:
1. Combine the two series and rank the m (= n ₁ + n ₂ ) data, assign rank 1 to the smallest value, rank two to the next, .., and the rank m to the largest value. If several data have the same value, assign them the midrank, that is, the average of the ranks that would otherwise be assigned to those data. When missing values are encountered the function is stopped. The number of values per series must be larger than 10.
2. Sum the Ranks assigned to the first series, W _rs .
3. If no ties (same values) are present, the large sample statistic is computed from:

4. If ties are present, the following function is used:

where g is the number of tied groups and tj is the number of tied data in the j ^th group.
5. For an alevel two-tailed test, H ₀ is rejected and H ₁ _{} is accepted if Z _rs £- Z _1- _a _/2 or if Z _rs ³ Z _1- _a _/2
6. For a one-tailed alevel test of H ₀ versus the H ₁ that the values from series 1 tend to exceed those from series 2, H ₀ is rejected and H ₁ _{} is accepted if Z _rs ³ Z _1- _a .
7. For a one-tailed alevel test of H ₀ versus the H ₁ that the values from series 2 tend to exceed those from series 1, H ₀ is rejected and H ₁ _{} is accepted if Z _rs £ Z _1- _a .
This test is almost similar to the Mann-Whitney-U test.
Reference:
Statistical Methods for Environmental Pollution Monitoring, R.O. Gilbert, 1987, John Wiley & Sons Inc..

Wilcoxon Signed Rank Test
The Wilcoxon signed rank test can be used instead of the sign test if the underlying distribution is symmetric, though it need not to be a normal distribution. To perform the Wilcoxon Signed Rank Test a series may not contain missing values. Hymosconsiders the following hypothesis:
H₀ : The populations from which the two series have been drawn have the same mean,
H₁ : The populations have different means.
The following procedure is followed:
1. Compute for each pair of values the difference.
2. Rank the absolute differences, assign rank 1 to the smallest value, rank two to the next, .., and the rank m to the largest value. If several data have the same value, assign them the midrank, that is, the average of the ranks that would otherwise be assigned to those data.
3. Sum the positive ranks T+ and the absolute value of the negative ranks T-.
4. Select the smallest value for T+ and T-.
5. Compute

where:

Reference:
Statistical Methods for Environmental Pollution Monitoring, R.O. Gilbert, 1987, John Wiley & Sons Inc..
Methodiek voor de evaluatie en optimalisatie van routine waterkwaliteitsmeetnetten, Deel II: Overzicht van technieken en methoden, Stowa, 1998 (16)

Seasonal Kendall Slope Estimator and Test
The seasonal Kendall test is a trend test to be used when seasonal cycles are present in the data. The test may even be used when missing data or tied data are present in the series. The validity of the test does not depend on the data being normally distributed. The test consists of computing the Mann-Kendall test statistic S and its variance VAR(S), separately for each season. These seasonal statistics are then summed, and a Z statistic is computed. The normal distribution is used to test for a statistically significant trend.
The Mann-Kendall statistic S is computed for each season with:

where:
n = number of years
l, k = years
x = data values
i = season number
If x_il > x_ik then the sign = 1, if x_il < x_ik then the sign = -1 else the sign = 0
The Variance of S is computed for each season as follows:

where:
g _i = the number of groups of tied data in season i
t _ip = the number of tied data in p th group of season i
h _i = the number of sampling times in season i that contain multiple data
u _ip = the number of multiple data in the q th time period of season i
When_S_ _i and VAR(S _i ) are computed, they are summed across the K seasons:

Finally the Z statistic is computed from:

Hymosconsiders the following hypothesis:
H₀ : The populations from which the series has been drawn has no trend,
H₁ : The populations has a trend.
Together with the seasonal Kendall test, the seasonal Kendall slope estimator is computed. The N _i individual slope estimates for the i th season is computed form:

This is done for each season. The individual slopes are ranked and the median values are computed for each season., this is the seasonal Kendall slope estimator. A confidence interval around the true slope is obtained by using the normal distribution, and:

The lower and upper confidence limits are the M ₁ th largest and the (M ₁ +1) th largest values of the N^' ordered slope estimates.
Reference:
Statistical Methods for Environmental Pollution Monitoring, R.O. Gilbert, 1987, John Wiley & Sons Inc..
Methodiek voor de evaluatie en optimalisatie van routine waterkwaliteitsmeetnetten, Deel II: Overzicht van technieken en methoden, Stowa, 1998 (16)

Dixon's test for outliers
The Dixon's test is testing the time series on outliers. This test can be used for small samples, from 3 values up to 25 values. From the series of data values missing values are first removed before the series is ordered according to magnitude. The ratio of the difference of an extreme value from one of it's neighbour values in the ordered series is then calculated, using a formula that varies with sample size. This ratio is then compared to a tabulated value and, if found greater, the extreme value is considered an outlier.
The confidence limits used are restricted to a finite set of values, i.e. 30%, 20%, 10%, 5%, 2%, 1% and 0.5%. Also the number of input values is restricted to a range of 3 to 25.
HYMOS uses the following formulas for computing the critical values for the smallest and highest values of the ordered series:

Under the null-hypothesis, the extreme values of the ordered series are not considered to be outliers.
The table for the test statistic is taken from the statistical tests book of Gopal Kanji.
Reference:
Gopal K. Kanji, 100 Statistical Tests, Sage Publications Ltd., 1999
ME.A.McBean and F.A.Rovers, Statistical procedures for analysis of environmental monitoring data and risk assessment, Prentice Hall PTR, 1998

Lettenmaier rank correlation test
This test is almost similar to the Spearman rank trend test, there is added a correction for the number of effective measurements. The computation procedure is also almost similar. The observations on the series is ranked from 1 to N. Tied observations are assigned the average of the tied ranks. The sum of squares of rank differences is calculated. Then the Spearman rank correlation coefficient r_s is computed:
r_s = 1 - 6D/(N³ - N)
where:
D = sum of ranks
N = number of measurements
The statistic used to measure the significance of r_s is:
½t½ = ½r_s ½Ö{(N' - 2)/(1 - r_s ² )}
where:
r_s = Spearman rank correlation coefficient
N' = number of effective measurements
N' is computed from:

where:
r= Lag-1 auto-correlation coefficient
Under the null-hypothesis of zero correlation the theoretical value of the test statistic has a Student t -distribution with N_df = N'-2 degrees of freedom for N' > 10.
For the interpretation of the test results reference is made to the section about Hypothesis testing.
Test condition:
N³10
Reference:
Methodiek voor de evaluatie en optimalisatie van routine waterkwaliteits meetnetten, Deel II: Overzicht van technieken en methoden, Stowa, 1998 (16)

Child pages

08 Series homogenity tests

Series homogeneity tests

General

Series codes

Test Option

Hypothesis testing

Median run test

Turning point test

Difference sign test

Spearman rank correlation test

Spearman rank trend test

Arithmetical serial correlation test

Wilcoxon-Mann-Whitney U-test

Student t-test

Wilcoxon W-test

Linear trend test

Range test

Shapiro and Wilk W-test

Rosner's Test

Sign Test

Wilcoxon Rank Sum Test

Wilcoxon Signed Rank Test

Seasonal Kendall Slope Estimator and Test

Dixon's test for outliers

Lettenmaier rank correlation test