NONPARAMETRIC TESTS FOR TRENDS IN WATER … · Mann-Whitney-Wilcoxon rank sum test and slope...
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See discussions, stats, and author profiles for this publication at: https://www.researchgate.net/publication/277719484 Nonparametric Tests for Trends in Water-Quality Data Using the Statistical Analysis System (SAS) Technical Report · January 1983 CITATIONS 12 READS 157 3 authors, including: Some of the authors of this publication are also working on these related projects: Publication: Long-Term Changes in Sediment and Nutrient Delivery from Conowingo Dam to Chesapeake Bay: Effects of Reservoir Sedimentation View project Dissolved Oxygen Modeling of Indiana Streams View project Charles G Crawford United States Geological Survey 66 PUBLICATIONS 1,080 CITATIONS SEE PROFILE Robert M. Hirsch United States Geological Survey 86 PUBLICATIONS 10,121 CITATIONS SEE PROFILE All content following this page was uploaded by Charles G Crawford on 05 June 2015. The user has requested enhancement of the downloaded file.
Transcript of NONPARAMETRIC TESTS FOR TRENDS IN WATER … · Mann-Whitney-Wilcoxon rank sum test and slope...
See discussions, stats, and author profiles for this publication at: https://www.researchgate.net/publication/277719484
Nonparametric Tests for Trends in Water-Quality Data Using the Statistical
Analysis System (SAS)
Technical Report · January 1983
CITATIONS
12
READS
157
3 authors, including:
Some of the authors of this publication are also working on these related projects:
Publication: Long-Term Changes in Sediment and Nutrient Delivery from Conowingo Dam to Chesapeake Bay: Effects of Reservoir
Sedimentation View project
Dissolved Oxygen Modeling of Indiana Streams View project
Charles G Crawford
United States Geological Survey
66 PUBLICATIONS 1,080 CITATIONS
SEE PROFILE
Robert M. Hirsch
United States Geological Survey
86 PUBLICATIONS 10,121 CITATIONS
SEE PROFILE
All content following this page was uploaded by Charles G Crawford on 05 June 2015.
The user has requested enhancement of the downloaded file.
NONPARAMETRIC TESTS FOR TRENDS IN WATER-QUALITY DATA USING THE STATISTICAL ANALYSIS SYSTEM
by Charles G. Crawford, James R. Slack, and Robert M. Hirsch
U.S. GEOLOGICAL SURVEY Open-File Report 83-550
1983
U.S. DEPARTMENT OF THE INTERIOR
JAMES G. WATT, Secretary
GEOLOGICAL SURVEY
Dallas L. Peck, Director
For additional information write to: Copies of this report can bepurchased from:
Chief HydrologistU.S. Geological Survey, WRD Open-File Services Section 410 National Center Western Distribution Branch Restion, Virginia 22092 U.S. Geological Survey
Box 25425, Federal Center Lakewood, Colorado 80225 (Telephone: [303] 234-5888)
TABLE OF CONTENTS
Page
Abstract.................................................................. 1
Introduction.............................................................. 1
Retrieving data from the WATSTORE daily values and water quality files.... 2
Determining the relationship between water-quality constituents and streamf1ow.............................................................. 10
Plotting water-quality data as a time series.............................. 44
Statistical procedures to test for trends in water-quality time series.... 49
Appendix A. PROC SEASKEN user's guide with examples...................... 56
Appendix B. PROC SEASRS user's guide with examples....................... 69
Appendix C. Source code for SAS procedures............................... 82
References................................................................ 101
ILLUSTRATIONS
Figure 1. Example input for WATSTORE retrieval of water-quality data by the QWRETR and QWSAS procedures.................................... 4
2.--Example input for WATSTORE retrieval of streamflow data by theDVRETR and DVINPUT procedures.................................. 6
3.--Example input and output of a SAS procedure to do flow adjust ment regressions-abbreviated version........................... 14-21
4.--Example input and output of a SAS procedure to do flow adjust ment regressions-extended version.............................. 23-42
5.--Example input and output of SAS statements t plot water-qualitydata and regression residuals as time series................... 45-48
6. Example input and output of SAS statements to do the Seasonal Kendall test and slope estimator procedure on a water-quality data time series............................................... 51-52
7.--Example input and output of SAS statements to do the Seasonal Mann-Whitney-Wilcoxon rank sum test and slope estimator pro cedure on a water-quality data time series..................... 54-55
m
CONTENTS (continued)
Figure Al.--Example input and output using PROC SEASKEN to test fora trend in annual mean streamflow............................. 63-64
A2.--Example input and output using PROC SEASKEN to test fortrends in water-quality constituents.......................... 65-68
Bl.--Example input and output using PROC SEASRS to test for astep in annual mean streamflow................................ 76-77
B2.--Example input and output using PROC SEASRS to test for steptrends in water-quality constituents.......................... 78-81
Cl.--Parsing module for the Seasonal Kendall test and slopeestimator procedure........................................... 83
C2.--Procedure module for the Seasonal Kendall test and slopeestimator procedure........................................... 84-90
C3.--Parsing modeule for the Seasonal Mann-Whitney-Wilcoxon ranksum test and slope estimator procedure........................ 91
C4.--Procedure module for the Seasonal Mann-Whitney-Wilcoxon ranksum test and slope estimator procedure........................ 92-100
iv
TESTING FOR TRENDS IN WATER-QUALITY DATA USING THE STATISTICAL ANALYSIS SYSTEM
By Charles G. Crawford, James R. Slack, and Robert M. Hirsch
ABSTRACT
Two nonparametric procedures to test for trends in water-quality data
(SEASKEN AND SEASRS) have been developed for the Statistical Analysis System*
(SAS). The procedure SEASKEN tests for a monotonic trend in time by a modified
form of Kendall's tau, the Seasonal Kendall test. The procedure SEASRS tests
for a step trend between two different periods in a time series using a modi
fied form of the Wilcoxon (Mann-Whitney) rank sum test, the Mann-Whitney-
Wilcoxon rank sum test for seasonal data. Examples are presented using the
two procedures. The source code and user's guide for each of the two procedures
are also presented.
Procedures for flow adjusting water-quality data by the SAS procedures
REG and SYSREG and techniques for plotting water-quality data as a time series
by the SAS procedure PLOT are presented.
Additionally, examples are presented to demonstrate the use of the U.S.
Geological Survey procedures QWRETR, DVRETR, QWSAS, and DVINPUT to retrieve
data from the Geological Survey WATSTORE system and make it available to SAS.
INTRODUCTION
Increased public concern over the quality of the Nation's rivers in the
past several decades has led Federal, State, and local officials to implement
or greatly expand existing water-quality monitoring programs. Examples of
such programs are the Geological Survey's Benchmark and NASQAN networks
(see Briggs, 1978). Most of these monitoring programs have as a goal the
detection of trends in water quality.
* Use of brand and firm trade names in this report is for identification pur poses only and does not constitute endorsement by the U.S. Geological Survey.
Techniques of trend detection in water-quality data have correspondingly
received much attention recently in the literature (see, for example, Fuller
and Tsokos, 1971; Lettenmaier, 1976; and Hirsch and others, 1982). Most of
the trend detection methods presented to date are based on classical (para
metric) hypothesis testing. However, because of the nature of water-quality
data (typically skewed, serially correlated, and showing seasonality), many
of the assumptions underlying classical hypothesis tests are not met, rendering
them inappropriate. (For a discussion of the problems associated with these
tests, the reader is referred to Smith and others, 1982.) Recently, however,
thinking has begun to shift toward distribution-free (nonparametric) tests
that have less restrictive assumptions than their classical counterparts and
are therefore less sensitive to the distribution of the water-quality time
series.
This report describes procedures in the Statistical Analysis System (SAS)
that can appropriately be used to detect trends in water-quality data. The
report describes the use of both the standard procedures provided with SAS and
two additional procedures, SEASKEN and SEASRS. A working knowledge of SAS by
the reader is assumed. The report was written primarily for user's of the
U.S. Geological Survey's WATSTORE data base and Amdahl computer system; however,
the source code for the SAS macros and procedures are included in the appendices.
The user's guide for the SEASKEN and SEASRS procedures are also included in the
appendices.
RETRIEVING DATA FROM THE WATSTORE DAILY VALUES AND WATER QUALITY FILES
In order to use SAS on water-quality data, it is first necessary to get the
data into a SAS data set. A SAS data set is a collection of data observations
addressable by SAS procedures. Each observation has one or more variables asso
ciated with it. For more information about SAS data sets, see SAS Institute, Inc.
2
(1979) or SAS Institute, Inc. (1982a). Data can be entered into a SAS data set
from data cards in the job stream or by reading data stored on disk or tape
files. When using data from the WATSTORE file, one of the standard Survey
retrieval procedures must first be used. One of two Survey applications programs
(PROC QWSAS or the DVINPUT macro) is then called to convert the standard retrieval
output into a SAS data set. PROC QWSAS is a SAS procedure that produces a SAS
data set from the standard water quality file retrieval procedure QWRETR.
PROC QWSAS is described in detail in the WATSTORE user's guide, volume 3,
chapter IV, section R. DVINPUT is a SAS macro that converts output from the
WATSTORE daily values file retrieval, DVRETR, to a SAS 4 ata set. Use of the
DVINPUT macro is described in the WATSTORE message SAS documentation section
(member WRD06). The standard retrieval procedures, DVRETR and QWRETR, are
discussed in the WATSTORE user's guide, volumes 1 and 3, respectively.
Figure 1 shows example input using the QWRETR and QWSAS procedures. This
example retrieves twelve parameters - two streamflow parameters (00060 & 00061)
and ten water-quality constituents - for three stations from the water quality
file. Both daily mean streamflow (parameter code 00060) and instantaneous stream-
flow (00061) should be retrieved. For purposes of this test, the mean streamflow
may be used if the instantaneous streamflow is missing. PROC QWSAS creates a
SAS data set named DATA1 containing the station identification number and the
eleven parameters requested in the QWRETR procedure. The parameter values are
stored in variables named Pnnnnn where nnnnn is the WATSTORE parameter codes
given in the retrieval list. In addition, the variables YEAR, MONTH, DAY, DATE,
DECTIME, and SNAME were requested in the QWSAS statement. The variable DATE is
the day on which the sample was collected, in days since January 1, 1960.
DECTIME is a decimal number representing the time the sample was collected, in
1 //F1 JOB
2 // CLASS
3 /*SETUP
118545/H
4 //PROCLIB 00 DSNsWRD.PROCLIB/DISP=SHR
5 // EXEC QWRETR/VOL1=118545
6 //HDR.SYSIN 00
*7
M3
1968100119810930
8 XQWSASX
9 R000600006100410006300066500915009250093000940009450095070300
10
0000600006100410006300066500915009250093000940009450095070300
11
0 03276500
12
0 03374100
13
0 03378500
14
/*
15
// EXEC WROSAS
16
//TREND DO DSN = USERI 0.FILENAME/DISP = OLD
17
//SYSIN DO
*18
PROC Qh
/SAS
YEAR MONTH DAY DATE DECTIME SNAME;
19
DATA;SET;IF
P00061=. THEN pooo6i=P0006o;oROp P0
0060
;20
PROC
SO
RT;a
Y STATION
YEAR MONTH
DAY;
21
PROC
PR
INT;
BY ST
ATIO
N;22
VA
R YE
AR MONTH
DAY
DECT
IME
DATE
P0
0061
P0
0410
P0
0630
P0
0665
P0
0915
P0
0925
P009
23
30 P00940 P00945 P00950 P70300;
24
DATA TREND.MONTHLY;SET;
25
/*26
//
Fig
ure
1. E
xam
ple
inpu
t fo
r W
ATST
ORE
retr
iev
al
of w
ater
-qu
alit
y
data
by
the
QW
RETR
and
QW
SAS
proc
edur
es.
years. These two forms of sample collection time are useful in plotting the
data as time series. Additionally, the variable DATE may be formatted in
several different ways (see SAS Institute, Inc., 1982a, p. 409). SNAME is the
variable containing the station name.
The variable DECTIME is required to use the SAS procedures SEASKEN and
SEASRS. For data sets that do not already include the variable DECTIME, it
can be easily generated using the following statement in a SAS data step:
DECTIME = YEAR(DT)+(JULDATE(DT)-YEAR(DT)*1000)/365;, (1)
where DT is the date the observation was made.
An example use of this is shown in figure 2.
The example in figure 1 also sorts and prints the information retrieved
by station. Finally, the data is stored as file TREND.MONTHLY in the data set
USERID.FILENAME on a direct access device for later analysis. Line 16 describes
the existing file to be used for data storage, and line 24 copies the data to
the disk file. For information on creating disk data sets on the USGS Amdahl
computer system, see the USGS Computer Users Manual, Chapter 5.
To adapt this example input to a specific application, the user will need
to (1) substitute the six digit volume number of the appropriate water quality
back file tape for 118545 in lines 3 and 5; (2) substitute the appropriate re
trieval dates on the master control card on line 7; (3) change the retrieval and
output list as desired on lines 9 and 10; (4) substitute the desired station
numbers in lines 11 through 13 (and delete or add D cards as required);
(5) substitute an appropriate data set name in the DSN = field on line 16;
and (6) change the variables list in lines 22 and 23 to agree with the parameters
listed in the retrieval list and PROC QWSAS optional variables.
1 Iff2 JOB
2 II CLASS
3 /*SETUP
115620/H
4 //PROCLId DD DSN=WRD.PROCLIB,DISP=SHR
5 // EXEC DVRETR,AGENCY*USGS*VOL1=115620
6 //HOR.SYSIN DD
*7
M3
1970100119800930
& R000608015480155
9 F00003
10
D 03365500
11
/*12
// EXEC WRDSAS,MACRO=DV*DSNs'&&8KREC',DSNl=NULLFILE,DSN2=NULLFILE
13
//TREND DD DSN=USERID.FILENAME*DISP=OLD
U
//SYSIN DD
*15
DVINPUT _SNAME
16
DATA DATAA<RENAME=<VALUE = P00060» DATAB(RENAME=(VALUE=P80154)) DATAC(RENAME=(VAL
17
UE=P80155»;SET;
18
IF PA
RMCO
DE*6
0 TH
EN OUTPUT DATAA/'
19
IF PA
RMCO
DE=8
0154
THEN OU
TPUT
DA
TA8,
'20
IF PARM
CODE
*801
55 TH
EN OU
TPUT
DATAC;
21
DROP PA
RMCO
DE;
22
PROC
SORT DA
TA=D
ATAA
;BY
STATION
DATE;
23
PROC
SORT DA
TA=D
ATAB
;BY
STATION
DATE;
24
PROC
SORT OATA = DA
TAC;
t3Y
STATION
DATE;
25
DATA
TE
MP/'
flER
GE DATAA
DATA8
DATAC/'BY
STAT
ION
DATE
/'26
FORM
AT DA
TE YY
MMDD
8.;
27
JSJULDATE(DATE);Y=YEAR(DATE);
28
DECT
IME = Y
-KJ-Y*1000)/365;
29
LABEL
P801
54=S
USPE
NDED
SE
DIME
NT CO
NCEN
TRAT
ION
(MG/
L)30
P80155=SUSPENDED SE
DIME
NT DI
SCHA
RGE
(TONS/DAY)
31
P0006U»STREAMFLOU
(CFS);
32
PROC
SORT;aY
STATION
DATE
;33
PROC
PRINT;BY ST
ATIO
N;VA
R DATE DECTIME
P00060 P80154 P80155;
34
DATA
TR
END.
DAIL
Y;SE
T TEMP;
35
/*3o
//
Fig
ure
2.-
-Exa
mple
in
put
for
WAT
STO
RE re
trie
va
l o
f st
ream
-flo
w
data
by
th
e
DVRE
TR
and
DVI
NPU
T p
roce
du
res
The lines in the QWRETR example in figure 1 do the following:
Line 1. Job control.
2. Class (extension of line 1).
3. Instructs the computer operator to mount the specified backfile
tape.
4. Invokes the WRD cataloged procedure library.
5. Invokes the procedure QWRETR and specifies the mounted backfile
tape to be used in the retrieval.
6. Establishes that data for QWRETR follow.
7. Specifies that data from both the current and backfile are to be
retrieved for the period October 1, 1968, to September 30, 1981.
8. Specifies that QWSAS will be invoked as an application program.
9. Restricts the retrieval to the listed parameters only.
10. Restricts the output list to the listed parameters only.
11-13. Specifies the stations to be included in the retrieval.
14. Job control language step separator card (step delimiter).
15. Invokes the procedure WRDSAS (WRD modified version of SAS).
16. Defines existing direct access data set to be used for storage
of the retrieved data.
17. Establishes that SAS instructions follow.
18. Invokes the procedure QWSAS and requests the variables YEAR,
MONTH, DAY, DATE, DECTIME, and SNAME to be included in the data set
created in line 23 below.
19. Uses mean streamflow (parameter code 00060) if instantaneous
streamflow (00061) is missing, and discards the mean streamflow
parameter.
20. Sorts data set in order of variables listed in BY statement.
21. Prints data set.
7
22-23. Selects and orders variables to be included in print of data set.
24. Creates file TREND.MONTHLY containing the retrieved data and
stores it in the data set USERID.FILENAME.
£5. Step delimiter.
26. End of job.
Figure 2 shows example input using the DVRETR and DVINPUT macro procedures.
This example retrieves streamflow, suspended-sediment concentration, and suspended-
sediment discharge from the daily values file for one station. The DVINPUT
macro creates a SAS data set containing the station identification number,
parameter code, value, date, and the optional variable requested on the DVINPUT
statement, station name. The variable PARMCODE contains the values of the
WATSTORE parameter codes in the retrieval list. This format of the data set
is awkward since all the variables requested in the retrieval list are combined
into different observations of the variable PARMCODE. The statements in lines
16 through 31 convert the initial SAS data set into one containing the station
identification number, station name, date, P00060, P80154, and P80155. In
addition, lines 27 and 28 add the variables Y (year), J (Julian date), and
DECTIME for each observation, and lines 29 through 31 add variable labels for
suspended-sediment concentration, suspended-sediment discharge, and streamflow.
This example also sorts and prints the data contained in the modified
data set named TEMP. Finally, the data set is stored as the file TREND.DAILY
on the data set USERID.FILENAME on a direct access device.
Note that DVINPUT is a SAS macro and not a SAS procedure. It is not
preceded by the statement PROC or followed by a semicolon.
To adapt this example setup to a specific application, the user will need
to 1) substitute the six-digit volume number of the appropriate daily values
backfile tape for 115620 in lines 3 and 5; 2) substitute the appropriate
retrieval dates on the master control card on line 7; 3) change the retrieval and
output list as desired on lines 8 and 9; 4) substitute the desired station numbers
in line 10 (add or delete D cards as required); 5) change lines 18-20 and 29-31
appropriately; 6) substitute an appropriate data set name in the DSN = field
on line 13; and 7) change the variable list in line 33 to agree with the para
meters listed in the retrieval list and the DVINPUT optional variables.
The lines in the DVRETR example input of figure 2 do the following:
Line 1. Job control.
2. Class (extension of line 1).
3. Instructs the computer operator to mount the specified backfile
tape.
4. Invokes the WRD cataloged procedure library.
5. Invokes the procedure DVRETR with AGENCY=USGS being the agency
code and specifies the mounted backfile tape to be used in the
retrieval.
6. Establishes that data cards for DVRETR follow.
7. Establishes that data from both the current and backfile are to
be retrieved for the period October 1, 1970, to September 30, 1980.
8. Restricts the retrieval t@ the listed parameters only.
9. Restricts retrieval to listed statistics codes only.
10. Specifies the station to be included in the retrieval.
11. Step delimiter.
12. Invokes the procedure WRDSAS and defines the temporary data set
containing retrieval data and supplies the macro DV.
13. Defines existing direct access data set to be used for storage of
the retrieved data.
14. Establishes that SAS instructions follow.
15. Invokes the macro DVINPUT and requests that the variable SNAME
be included in the created SAS data set.
16-31. Converts the SAS data set.
32. Sorts data set in order of variables listed in BY statement.
33. Prints the variables in the VAR statement in the order they are
listed in the VAR statement.
34. Creates file TREND.DAILY containing the retrieved data and stores
it in the data set USERID.FILENAME.
35. Step delimiter.
36. End of job.
DETERMINING THE RELATIONSHIP BETWEEN WATER-QUALITY CONSTITUENTS AND STREAMFLOW
Quite frequently, concentrations of water-quality constituents are related
to streamflow. When a water-quality constituent and streamflow are related,
apparent trends in water quality may be due only to fluctuations in streamflow
rather than to changes in the processes that affect the introduction and fate
of a given constituent in the stream. For example, consider a stream where
dissolved solids and streamflow are negatively correlated. That is, as stream-
flow increases dissolved solids decrease and vice versa. During a period of
drought, high dissolved solids concentrations would be expected. If this period
of drought was followed by a period of wet weather, a decrease in dissolved
solids concentrations would be expected. If such a time series was tested
for trend in dissolved solids concentration, a significant downtrend would be
10
indicated. However, such a trend could be entirely attributable to the
fluctuation in streamflow during the period. In order to test for trends in
the processes affecting dissolved solids during the period, it would be necessary
to remove the effect of streamflow. Flow adjustment is an attempt to remove
a major source of variation in water quality (streamflow) which may be masking
those variations attributable to changes in the constituent inputs to the
stream or in the processes occurring in the stream.
Smith and others (1982) described a flow-adjustment procedure suitable for
this purpose. Their approach is to develop a time series of flow-adjusted
concentrations (FAC) and to test that series for trend. FAC is defined as theA
actual concentration (C) minus the expected concentration (C) predicted from
the discharge (Q) relationship. The FAC should be randomly distributed with a
mean of zero over the period of record if no change in the processes that
affect the water-quality constituent have occurred. Of course, for some con
stituents - e.g., biological - adjustment by some other variable - e.g., solar
radiation or air temperature - might be appropriate. Where some of the data
are reported as "less than" a detection limit, these approaches to flow adjust
ment are not valid. If only a very few values are reported as "less than," then
flow adjustment could be used provided some sensitivity analysis were done to
check that the choice of values to use in place of the "less than" was not very
influential in the overall results.
Some common models used for flow adjustment include the following:
(2) C = a+b Q linearA
(3) C = a+b In (Q) log-linearA
(4) C = a+b 1 hyperbolic, 3 a constant typically in the range1+ SQ 10-3 TH <. 3 j< 102 TJ-1, where TJ is mean discharge
(5) C = a+b 1 inverse
(6) C = a+bQ+bQ quadratic
11
A good guide to selecting a flow adjustment equation is the R 2 value, butA
one should check plots of the predicted (C) and observed (C) values versus Q
(a log Q scale is usually desirable), and plots of the residual versus the pre
dicted (Q) to confirm that the relationship fits well and is not excessively
heteroscedastic (e.g., variance increases as predicted concentrated increases).
For many constituents (particularly suspended constituents or biological ones
like bacteria or plankton), these models may be inadequate, because the constituents
are very heteroscedastic. In these cases, models based on fitting the log concen
trations may be preferred.
Candidate models include the following:
(7) In C = a+b In Q log-log
(8) In C = a+bj In Q+b 2 (In Q) 2 log-quadratic log
Deciding between a model based on concentration (equations 2-6) and one
based on log concentration (equation 7 or 8) should not be based on R 2 values.
Rather, the decision should be based on examination of residuals plots.
If none of the models considered results in a significant fit, as deter
mined from the probability values for the t statistics in the cases with one
explanatory variable (equations 2-5 or 7) or the probability value of the F
statistic (equations 6 or 8), then no flow adjustment should be performed.
If one of the linear models (equations 2-6) is used, the residuals (Flow/N
Adjusted Concentrations) are defined as C - C. If one of the log models
(equations 7 - 8) is used, the residuals are In C - In C. Note that in the
former case the residuals have the dimensions of C (typically mg/L), but in the
latter case they are dimensionless. This has important implications for the
interpretation of trend test results. If log models are used and a slope is
estimated in Proc SEASKEN, it must be transformed as follows: If B is the slope
12
value reported by Proc SEASKEN, then (eB-l)-100 is the change in percent per
year. If log models are used and a step change is being considered and 0 is the
difference (mean FAC of period 2 minus mean FAC of period 1), then the percentage
change from period 1 to period 2 would be (eD-l)-lOO.
The following section provides two examples of SAS jobs to search for
appropriate flow adjustment models. The first, using the macro ABREGMAC, simply
estimates the regression equation and computes the usual summary statistics;
the second, using the macro REGMAC, provides extensive diagnostics on the
results. These macros are intended to be illustrative of the kinds of analyses
one may want to run. Knowledge gained by working with a particular data set
should dictate variations of these that one may choose to pursue.
The SAS job listing and output in figure 3 follow the procedures used
by Smith and others (1982) but includes several other models as well. It
considers all of the models defined by equations 2-8. Plots of C versus Q
and log C versus log Q are produced to aid in model selection.
Simple regression is used to estimate the coefficients a and b of each
model, to calculate R 2 (the fraction of the variance of the dependent variable
explained by the given function of Q) and p (the probability of erroneously
rejecting the null hypothesis that b = 0). Bis equal to 10(- 2 « 5 - &*) where B*
is the interger part of log^o^I* where "Q" is the mean streamflow. Eight different
hyperbolic models are generated by incrementing the initial value of B by a
factor of 10°- 5 seven times.
The macro REGDATA (lines 6-43) calculates the necessary functions of Q
needed for the linear regressions. The macro ABREGMAC (lines 44-64) does
the linear regressions using the SAS regression procedure SYSREG (SAS Institute,
Inc., 1979, p. 403). Lines 65-67 create a data set named DATAA with only
sulfate and streamflow data for station 03374100. Line 68 sorts this newly
13
1 //F3 JOd
2 // CLASS
3 //PROCLld
DD DSN=dK D .
PROCL I
3, D I
SP = SHR
4 //A EXtC WRi)SAS,DSNl=NULLFILE,DSN2 =
5 //TREND OD DSN=USERID.FILENAME,DISP=OLD
6 //SYSIN UD
*Y
MACRO
RE60ATA
8 OPTIONS
NOMA
CROG
EN;
9 PROC MEANS
DA T A= F ILE NAME NOPRINT
MEAN
/'BY
ST
ATIO
N;VA
R P0
0061
,'1
0
OU
TP
UT
O
UT
= i1
EA
NQ
M
EA
N=
M0
00
6i;
11
PROC SORT OATA=MEANQ;SY
STATION;
12
DATA INPDATA;MERGE
MEANQ
FILE
NAME
;BY
STATION;
13
*GET
LOG
OF DE
PEND
ENT
VARI
ABLE
/*14
LN
DEPV
AR = LO
o(DE
PVAR
) ;
15
*GET
LOG
OF TH
E FL
OW;
16
L00061=LOG(P00061);
17
*BETA =
THE INTEGER PORTION OF THE BASE 10 LOG OF THE MEAN FL
OW/'
18
BETA*INT CLOG10(MOOU61 »;
19
*ON = DIFFERENT VALUES USED IN THE HYPERBOLIC FUNCTIONS/'
20
81=10**(-2.5-BETA) ;
21
82 = 81 *10**0.5/'
22
Q3 = B2*1 0**0.5/*
23
B4=B3*10**0.5;
24
B5=B4*1U**0.5;
25
86=85*10**J. 5;
26
07 = B6*10**0.5/'
27
88=87*1 U**0.5/'
26
*HYPERUNQ=DIFFERENT HYPERBOLIC FUNCTIONS/'
29
HYPERdlQ=1/C1+Bl*P00061)
30
HYPERB2Q = 1 /(1
+t32
* P00061 )
31
HYPERB3U=1/(1+U3*P00061
32
HYPERB4Q=1/(1+B4*P00061)
33
HYPERd5Q=1/(1+B5*POU061
34
HYPERB6U=1/(1+B6*P00061)
35
HYPERd7u=1/(1+B7*P00061
36
HYPERd8u=1/(1+B8*P00061
37
*INVQ* THE INVERSE OF THE FLOW/'
38
INVQ = 1/POOJ61/'
39
*Q2=SQUARE OF THE FLOW;
40
Q2 = P00061 **2
/'41
*LQ2 = SQUARE OF THE LOG OF THE FL
OW/'
42
LQ2 = L00061 **2
/'43
PROC SORT DAT A* I
NPD AT A;
BY STATION/'
44
%45
MACRO AdREGMAC
46
OPTIONS NOilACROGEN/'
47
PROC SYSKEb DATA=INPDATA;8Y STATION;
/. n
i T MC
AQ
MO
n( '
i
49
I. OGI. IN: MOO EL
Fig
ure
3.-
-Exa
mple
in
pu
t an
d o
utp
ut
of
a SA
S pro
cedure
to
do
flo
w a
dju
stm
en
t re
gre
ssio
ns'
a
bb
revi
ate
d
ve
rsio
n.
51
HYPER2:MOD£L DEPVAR=HYPERB2Q;
52
HYPER3:MOOtL OEPVAR = HYPER83Q ;
53
HYPER4:MODEL DEPVAR=HYPER343 ;
54
HYPER5:MOOEL DEPVAR=HYPERB5Q ;
55
HYPER6:MODEL DEPVAR=HYPERB6Q I
56
HYPER7:MODEL DEPVAR = HYPERB7Q ;
57
HYPERb:MODEL OE
P\l A R=H YPE R88Q ;
5tt
IN\/ERS:MODEL DEPV AR= I
NVQ ;
5V
QUAD:
MODbL DE
P\l AR = P00061 «2;
60
LOGLO(,:MOUEL LNO EPVAR = I_00061 ;
61
LO&QAO:HODEL LNOEPVAR=L00061
62
PROC PLOT OATA=INPOATA;BY STATION,
63
PLOT DEPVAR*P00061 ;
64
PLOT LNOEPVAH*L00061;
65
%66
DATA UA
TAA;
SET
TREN
D.MO
NTHL
Y;67
IF ST
ATIO
NS'
0337
4100
;
6tt
KEEP STATION SNAME P00945 P00061;
69
PROC SO
RT;B
Y STATION;
70
MACRO
FILE
NAME
DA
TAAX
71
MACR
O OE
PVAR
P00945%
72
MACRO
LNOE
PVAR
LO
Q9A5
X7.
3 R
EC
iOA
TA
7A
AO
RE
GM
AC
75
/*76
//
ST
AT
IS
TIC
AL
A
NA
LY
SIS
S
YS
TE
M
ST
AT
ION
ID
EN
TIF
ICA
TIO
N
NU
rBE
R*D
33
74
10
0T
HU
RS
DA
Y,
AU
GU
ST
IB,
HCDE
L:
LINE
AR
DtP
V
AR
: P
00
94
5
SU
LFA
TE
VA
RIA
BL
E
INT
ER
CE
PT
P
00
06
1
MODE
L: LO
GLIN
DE
P
VA
R*
P00945
SU
LFA
TE
VA
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BL
E
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L
00
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AR
: P
00945
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MODE
L: HY
PER*
PF
P
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R:
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E
INT
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PT
H
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82
0
SS
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OFE
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SF
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SO
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ED
<
MG
/L
DF 1 1
PA
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ME
TE
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ES
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AT
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60
.28
88
24
-0
.0005675
SS
E
DFE
M
SE
DIS
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LV
ED
C
MG
/L
OF 1 1
PA
RA
ME
TE
R
ES
TIM
AT
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136*4
55261
-9.3
173B
4
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E
DFE
M
SE
DIS
SO
LV
ED
(M
G/L
DF 1 1
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ES
TIM
AT
E
-1771.4
7
18
31
.05
*
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SE
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(M
G/L
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TE
R
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AT
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-54
4.0
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24
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67673
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96
10
3.9
77
62
9
AS
S
04
)
ST
AN
DA
RD
E
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OR
1.4
18811
.000067133B
5
7440.
6P
2
96
77
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71
01
A
S
S04
1
STA
ND
AR
D
ER
RO
R
7.4
95
37
6
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21471
9943.3
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96
103.5
76787
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AN
DA
RD
E
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21
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AN
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69.5
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(C
FS
)
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0.0
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727
VA
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BL
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OB
MT
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9
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>|T
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A
NA
LY
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TE
M
ST
AT
ION
ID
EN
TIF
ICA
TIO
N
NU
MB
FR
*03374100
15:41 THURSDAY, AUGUST IB, 1983
MO
DEL
: H
YPFR
3
DFP
VA
R:
P00
945
SU
LFA
TF
VA
RIA
BLE
INT
ER
CE
PT
H
YP
ER
B30
MODE
L: HY
PER*
OF
F
VA
R:
P00945
SU
LFA
TE
VA
RIA
BLE
INT
ER
CE
PT
H
YP
ER
B40
MBD
EL:
HYPE
RS
OFF
V
AR
: P
00945
SU
LFA
TE
VA
RIA
BL
E
INT
ER
CE
PT
H
YP
ER
B5Q
MP
DE
L:
HY
PE
R6
PE
P
VA
R:
P00945
SU
LFA
TE
VA
RIA
BLE
INT
ER
CE
PT
H
YP
ER
B60
SS
E
DFE
M
SE
DIS
SO
LV
ED
(M
G/L
OF 1 1
PA
RA
ME
TE
R
ES
TIM
AT
E
-155.2
41972
216.3
7B
395
SSE
DFE
M
SE
DIS
SO
LV
ED
(M
G/L
DF 1 1
PA
RA
ME
TE
R
ES
TIM
AT
E
-30.5
63720
93.2
18221
SS
E
DFE
M
SE
DIS
SO
LV
ED
(M
G/L
DF 1 1
PA
RA
ME
TE
R
ES
TIM
AT
E
11
.60
35
*7
5*
.32
06
65
SS
E
DFE
M
SE
DIS
SO
LV
ED
(M
G/L
DF 1 1
PA
RA
ME
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R
ES
TIM
AT
E
2B
. 67
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*6
44
. 30
35
6?
964*.
02
9
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1
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55
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4)
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AN
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23
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P0563
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90932
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S
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4.0
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5.3
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96
8
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AS
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04)
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55140
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F R
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UA
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T R
AT
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9
B.7
92
4
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AT
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OB
>F
R
-SQ
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AT
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-3.4
21
29
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33
F R
AT
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PR
OB
>F
R
-SO
UA
RE
T R
AT
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2.B
B3
B
10
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48
F R
AT
IO
PR
OB
>F
R
-SO
UA
RF
T R
AT
IO
12.1
736
10
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01
77
.31
0.0
001
0.4
461
PR
OB
>|T
|
0.0
001
0.0
001
86
.55
0.0
001
0.4
74
1
PR
OB
>|T
|
0.0
009
0.0
00
1
10
2.1
10
.00
01
0.5
154
PR
08
>lT
l
0.0
049
0.0
00
1
115.9
9
0.0
001
0.5
47
2
PR
HP
>|T
|
0.0
00
1
0.0
001
VA
RIA
BLE
LA
BE
L
VA
RIA
BLE
L
AB
EL
VA
RIA
BLE
LA
BE
L
VA
RIA
BLE
L
AB
EL
ST
AT
ION
ID
EN
TIF
ICA
TIO
N
NU
MO
DE
L*
HY
PF
R7
DFP
VAR
: P0
0945
SU
LFA
TE
VA
RIA
BL
E
INT
ER
CE
PT
H
YP
ER
B70
MO
DEL:
HY
PERB
DFP
VAR
: P0
0945
SU
LFA
TE
VA
RIA
BL
E
INT
ER
CE
PT
H
YP
ER
B80
MOD
EL:
INVE
RS
DE
P
VA
R:
P00945
SU
LFA
TE
VA
RIA
BL
E
INT
ER
CE
PT
IN
VO
MO
DEL:
QU
AD
DE
P
VA
R:
P00945
SU
LFA
TE
VA
RIA
BL
E
INT
ER
CE
PT
P
00061
02
SSE
DFE
U
SE
D
ISS
OL
VE
D
(MG
/L
DF 1 I
PA
RA
ME
TE
R
ES
TIM
AT
E
36.9
01791
49.3
06521
SSE
DFE
U
SE
D
ISS
OL
VE
D
(MG
/L
DF 1 1
PA
RA
ME
TE
R
ES
TIM
AT
E
41. 5
62157
72
.20
66
30
SS
E
DFE
M
SE
DIS
SO
LV
ED
(M
G/L
OF 1 1
PA
RA
ME
TE
R
ES
TIM
AT
E
46
.11
72
72
27004.9
2
SS
E
DFE
M
SE
DIS
SO
LV
ED
(M
G/L
DF 1 1 I
PA
RA
ME
TE
R
ES
TIM
AT
E
64
.40
55
30
-0
.00
11
76
88
1.0
6125E
-08
78B
9.6
06
96
P
2.1
83394
AS
504)
ST
AN
DA
RD
E
RR
OR
1.6
78
29
*54.5
80650
8576.0
29
96
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33636
AS
S
04
)
ST
AN
DA
RD
E
RR
OR
1.4
21870
7.2
68
46
0
10538.9
7
96
109.7
80918
AS
S
04
I
ST
AN
DA
RD
E
RR
OR
1.2
96412
3514.1
16
PB
25
.4P
2
95
5
2.8
99
80
8
AS
504)
ST
AN
DA
RD
E
RR
OR
1.7
77655
0.0
00184011
3.0
0799E
-09
F R
AT
IO
PR
3B>
F
R-S
QU
AR
F
T R
AT
IO
21.9
B77
10.7
641
F R
AT
IO
PR
OB
>F
R-S
OU
AR
E
T R
AT
IO
29.2
306
9.9
45
2
F R
AT
IO
PR
OB
>F
R-S
QU
AR
E
T R
AT
IO
35.5
730
7.9
124
F R
AT
IO
PR
OB
>F
R-S
QU
AR
E
T R
AT
IO
36.2
306
-6.3
957
3.5
281
11
5.8
7
0.0
001
0.5
46
9
PR
OB
>|T
|
0.0
00
1
0.0
00
1
98.9
1
0.0
00
1
0.5
07
5
PR
OB
XT
I
0.0
00
1
0.0
00
1
62
.61
0.0
00
1
0.3
947
PR
OB
>|T
|
0.0
00
1
0.0
00
1
46
.21
0
.00
01
0
.49
31
PR
OB
>|T
|
0.0
00
1
0.0
00
1
0.0
00
6
15
:41
T
HU
RS
DA
Y,
AU
GU
ST
IS
, 1983
VA
RIA
PL
E
LA
BF
L
VA
RIA
BL
E
LA
BF
L
VA
RIA
BL
E
LA
BE
L
VA
RIA
BL
E
LA
BF
L
ST
RE
AM
FL
OW
, IN
ST
AN
TA
NE
OU
S
(CF
S)
ST
AT
IS
TIC
AL
A
NA
LY
SIS
S
Y S
T F
ST
AT
ION
ID
EN
TIF
ICA
TIO
N
NU
MB
FR
=0337*1
00
THURSDAY, AUGUST
Jflt
1963
KCDE
L:
LDCI
OG
DfP
V
AR
: L00945
VA
RIA
BLE
INT
ER
CE
PT
L
00
06
1
MCD
EL:
LOGQ
AD
Dtp
VAR
: L0
0945
VA
RIA
BL
E
INT
ER
CE
PT
L
C0
06
1
LO?
DF 1 1
DF 1 1 1
SSE
OFE
U
SE
PA
RA
ME
TE
R
ES
TIM
AT
E
5.5
P2
35
8
-0.1
83890
SS
E
DFE
M
SE
PA
RA
ME
TE
R
ES
TIM
AT
E
4.4
07
20
3
0. 0
82
7*9
-0
.01
*90
1
3.5
7709B
96
0.0
37
26
1
ST
AN
DA
RD
E
RR
OR
0.1
6*3
43
O
.D1
80
12
3.5
30233
95
0.0
37160
ST
AN
DA
RD
E
RR
OR
1.0
59222
0.2
38112
0.0
13269
F R
AT
IO
PR
OB
>F
R
-SQ
UA
RE
T R
AT
IO
33.9
676
-10.2
096
F R
AT
IO
PR
OB
>F
R
-SO
UA
PE
T R
AT
IO
4.1
60
B
0.3
47
5
-1.1
23
0
10
4.2
4
0.0
00
1
0.5
206
PR
On
>|T
|
0.0
00
1
0.0
00
1
52
.89
0
.00
01
0
.52
6B
PR
OB
>lT
l
0.0
00
1
0.7
29
0
0.2
64
3
VA
R1
AP
LE
LA
BF
L
VA
RIA
PL
E
LA
BF
L
ST
AT
ION
ID
EN
TIF
ICA
TIO
N
NU
KB
FR
*03
37
4 1
00
90.0
80
.0S u L F A
70.0
T r D I 60.0
S S 0 L V
50.0
F.rx
: D
o( M
40.0
G / L A 3
0.0
S S 0 4 2
0.0
<
1
10.0
0.0
<
PLO
T O
F P
00945*P
00061
LE
GE
ND
: A
« 1
DB
S,
B =
? O
BS
, E
TC
.
t-
A AA
A AA A
t A
AA
BA
BB
A A
A A
AB
A A
A
AA
A A
A
AA
AA
AB
B A
B A
AA
A
A A
AA
AA
A
AA
A A
A A
A A
A
A A
A
A A
A
AA
AA
AA
A
AA
A A
A A
AA
A A
AA
A
A A
AA
A A
A
A
\>
0.0
0
9000.0
0
1BO
OO
.OO
P
70
00
.00
3
60
00
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45000.0
0
54
00
0.0
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63090.0
0
7?000.0
0
PIO
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Nfl
TE
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PB
S
HA
D
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SlK
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VA
LU
ES
STRFAMFLOW, INSTAMT»NFDUS (CF5)
ro
10
09
45
4.4
4.2
4.0
3.8
3.6
3.4
3.2
3.0
2.8
2.6
* I
ST
AT
IS
TIC
AL
A
NA
LY
SIS
S
YS
TE
M
ST
AT
ION
ID
EN
TIF
ICA
TIO
N
NU
MB
ER
=0
3 3
74
130
PL
OT
O
F
L0
09
45
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00
61
L
EG
EN
DS
A
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S,
B *
2 D
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TC
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US
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A A
AA
A
9 A
A A
A A
AA
A A
A A
A A
A AA
A A
A A
A A
AA
A
AA
AA
A
A
AA
A
A A
AA
A
A A
AA
A
A A
AA
6.0
6.5
7.0
7.5
n.o
8.5
9.0
L00061
9.5
10.0
10.5
1
1.0
1
1.5
12.0
NPTF
: 5
OBS
HAD
MIS
SIN
G V
ALUE
S
created data set by station and SNAME (sorting by station and SNAME are required
by REGDATA and ABREGMAC). The macros REGDATA and ABREGMAC are written with
dummy names for the data set and dependent variables. Lines 69-71 substitute
the desired data set and variable names into these macros. In this example,
the flow-adjustment procedure will be done on the data set DATAA for the variable
P00945 (sulfate). The macro LNDEPVAR inserts a name for the natural logarithm
of the dependent variable. The argument of this macro may be any unused variable
name the user chooses for the log of the variable given in line 70. The procedures
could have been done on any number of variables by repeating the statements in
lines 69-73 and making changes in the data set and dependent variables. (For
more information on the use of a macro to substitute a variable for dummy
names, see SAS Institute, Inc., 1979, p. 12.) The statement in line 72 executes
the macro REGDATA and the statement in line 73 executes the macro ABREGMAC.
SYSREG (line 46) outputs for each of the models: (1) the error sum of
squares (SSE); (2) the error degrees of freedom (DFE); (3) the error mean square
(MSE); (4) the model F ratio and Prob >F (test and significance probability
that all parameters except the intercept are zero); (5) the model R-square
(R 2 ); and (6) the degrees of freedom, estimate, standard error, and T ratio
and Prob >|T| for parameters in the model. The R-square and Prob >|T| for the
function of Q in the model correspond to the R 2 and p value in the Smith and
others (1982) flow-adjustment procedure. Plots of streamflow versus sulfate,
in both arithmetic and log form, are shown in figure 3.
Figure 4 shows a listing and output of a sample SAS job that does the
flow-adjustment procedure using the macro REGMAC (line 44). In addition to
providing R 2 and p for each model, the program provides several diagnostic
22
1 //F4 JOB
2 // CLASS>
3 //PROCLIB
DD DSN=WRD.PROCLIB*DISP=SHR
4 //A
EXfc
C WKDSAS,DSN1=NULLFILE*DSN2=NULLFILE
5 //TREND DD DSN=USERID.FILENAME,DISP=OLD
6 //SYSIN DD
*7
MACRO
REGDATA
8 OPTIONS
NOMACROGEN;
9 PROC MEANS
DAT A=
F ILENAME
NOPRINT
MEAN/'BY STATION/'VAR P00061/'
10
OUTP
UT OU
T=ME
ANQ
MEAN
=M00
061;
11
PROC
SORT DATA=MEANU;BY
STAT
ION;
12
DATA
INPDATA;MERGE
MEANQ
FILE
NAME
^Y STATION;
13
*GET LOG
OF DE
PEND
ENT
VARI
ABLE
;14
LNDEPV
AR=L
OG(D
EPVA
R);
15
*GET
LOG
OF THE
FLOW
;16
L00061=LOG(P00061>;
17
*3ETA =
THE INTEGER PORTION OF THE BASE 10 LOG OF THE MEAN FLOW/'
18
BETA=INT(LOb10(M00061));
19
*dN=DIFFERENT VALUES USED IN THE HYPERBOLIC FUNCTIONS;
20
Bl=10**<-2.5-BETA);
21
B2=B1*10**0.5;
22
B3=B2*10**0.5;
23
B4=B3*10**0.5;
24
B5=B4*10**0.5;
25
B6=B5*10**0.5;
26
B7=
B6*1
0**
0.5
;^
2
7
B8=
B7*1
0**
0.5
;00
2
8
*HY
PE
Rd
NQ
= i)I
FF
ER
EN
T
HY
PE
RB
OL
IC
FU
NC
TIO
NS
;29
HY
PE
R61U
=1/(
1+
81*P
00061>
;3
0
HY
PE
RB
2U
= 1
/ (1
+b
2*P
00
06
1);
31
HY
PE
RB
3Q
=1/(
1+
d3*P
00061);
32
HY
PE
RtJ
4Q =
1 /
(1+
84*P
00061 )
;3
3
HY
PE
RB
5Q
=1
/(1
+d
5*P
00
06
1);
34
H
YP
ER
B6Q
=1/(
1+
B6*P
00061);
35
HY
PE
Rd
7Q
=1/(
1+
67*P
00061>
;3
b
HY
PE
RU
8U
=1/(
1+
38*P
00061);
37
*I
NV
Q=
T
HE
IN
VE
RS
E
OF
TH
E
FL
OW
;3
8
INV
Q=
1/P
00061;
39
*U2=
SQ
UA
RE
O
F T
HE
F
LO
W;
40
Q2
=P
00
06
1**
2;
41
*LQ2=SQUAR6 OF THE LOG OF THE FLOW;
42
LQ2=L00061**2;
43
PROC SORT
0 AT A = I NPDAT A; 3Y STATION/'
44
%45
MACRO REGMAC
4b
OPTIONS NOMACROGEN NOOVP/*
47
PROC REG 0ATA = INPDAT A;BY STATION;
48
ID P00061;
49
TITLE OUTPUT FROM MODLABEL:
Y = FLOW/'
50
MOOLAbEL:MODEL
Y =
FLOW
/ R;
Fig
ure
4.--
Exa
mpl
e in
put
and
outp
ut o
f a
SAS
proc
edur
e to
do
flow
adj
ustm
ent
regr
essi
ons
exte
nded
ver
sion
.
51
OJTPUT UUT = REGOAT1
P = P
R=R STODENT = SR COOKO = COOKD/*
52
PROC SORT
D A
' A = R EGDA T 1
/'BY STATION/*
53
PROC PLOT i)ATA=-REGOAT1
/'BY
STATION/*
54
PLOT P*LOOU61='P' Y*L00061='0'
/ OVERLAY/*
55
PLOT R*P
/ \/REF = 0/*
56
PROC UNIVARIATE OATA = REGOAT1 PLOT NORMAL/'VAR
R/*
57
X5a
DATA DA
TAA;
SET
TREN
O.MO
NTHL
Y;59
IF
ST
ATIONS'
03374100
;60
KEEP
STATION
SNAM
E P0
0945
P00061/*
61
PROC soRTr-av STATION;
62
MACRO FILENAME DATAAX
63
MACRO OEPVAR P00945/.
64
MACRO LNDEPVAR L00945X
65
REGDATA
66
MACRO MOOLABEL LINEARX
67
MACRO
Y DE
Pv/A
R/.
68
MACRO FLOW P00061X
69
REGMAC
70
MACRO MODLAdEL LOGLINX
71
MACRO
Y DEPVARX
72
MACRO FLOW L00061X
73
REGMAC
74
MACRO MOOLABEL LOGLOGX
75
MACRO
Y LNOEPVARX
76
MACRO FLOW L00061X
77
REGMAC
78
/*
79
//
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01
0.0
02
0.0
02
0.0
02
0.0
00
0.0
?5
0.0
00
0.0
00
0.0
55
OUTPUT FROM LOGLIN: P00945= LOOOM
STATION IDENTIFICATION NUMBFS=03374100
16806 THURSDAY, AUGUST 1R, 1993
DB
S
ID
31
23
20
.00
32
1
86
0.0
033
29
30
.00
34
83
79
.9B
35
11099.9
73
6
46
60
0.0
03
7
46
59
9.9
138
12699.0
739
37
59
.99
40
4899.9
941
3709.9
942
4099.9
943
10
40
.00
44
2790.0
045
19
60
.00
46
5
14
.00
47
57
4.0
04
8
25000.0
04
9
31200.0
05
0
8090.0
051
3100.0
05
2
3099.9
95
3
3440.0
054
16B
O.O
O5
5
1810.0
05
6
9860.0
057
2840.0
05B
33000.0
05
9
10700.0
06
0
5620.0
061
9220.0
06
2
19
00
0.0
06
3
17000.0
064
7320.0
06
5
82
80
.00
66
11
00
0.0
067
3770.0
06
8
3180.0
06
9
31
60
0.0
07
0
32200.0
071
8600.0
07
2
76000.0
073
30400.0
074
1
37
00
.00
75
9350.0
07
6
6820.0
077
6000.0
07
8
68400.0
07
9
41100.0
08
0
74
10
.00
81
10200.0
082
54900.0
08
3
5440.0
0
AC
TU
AL
65
.00
07
7.0
00
66.0
00
60
.00
05
1.0
00
33.0
00 .
53
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06
5.0
00
50.0
00
46
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06
0.0
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63.0
00
70
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2.0
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86.0
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83
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5.0
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38
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0.0
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56.0
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65.0
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56.0
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56
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04
2.0
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66
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037.0
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54
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4.0
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63
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051.0
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57.0
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63
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9.0
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33
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62
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54
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030.0
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41.0
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050.0
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45.0
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49
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026.0
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34.0
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45
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31 .
00
040.0
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PR
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ICT
S
TD
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RR
VA
LUE
P
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DIC
T
64
.25
26
6.3
11
62.0
77
52
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0.8
49.6
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0.9
36.2
99
36.2
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2
0.9
59.7
53
57.2
86
59.8
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58
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67
1.7
28
62.5
33
65.7
28
76
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477.2
65
42
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240.0
37
52.6
14
O.B
61
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16
1.5
51
60
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267.2
59
66.5
65
50.7
70
0 .8
62.3
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39.2
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95
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50.0
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956.0
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91
13
744.6
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45.6
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1.0
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53.5
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54
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59.7
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61.3
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39.9
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52.0
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1.9
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40.2
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554.2
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32.7
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1.9
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37.4
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4.4
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11.6
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0.6
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31.9
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2.2
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B.6
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8.7
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8.7
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9.7
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1.8
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tools which aid the user in selecting the most appropriate regression model.
The program uses the macro REGDATA to establish the data. The macro REGMAC
uses the REG procedure (SAS Institute, Inc., 1982b, p. 39) rather than the
SYSREG procedure to generate regression results. In addition to the information
provided by SYSREG, the REG procedure in this application prints (1) the observed
and predicted value for each observation; (2) the residual (FAC), the standard
error of the residual and the studentized residual (the residual divided by
its standard error); and (3) Cook's D, a measure of the change to the parameter
estimates that would result from deleting each observation. This program also
plots for each model the observed and predicted values of the selected dependent
variables against the log of streamflow, and the FAC (residuals) against the
predicted value. In these plots, the log of the flow is used on the x-axis to
provide better resolution in the low discharge range. Finally, the macro does
a residuals analysis with the Univariate procedure (SAS Institute, Inc., 1979,
p. 427). This procedure does univariate statistics on the residuals, tests
for normality by the Shapiro-Wilk W statistic (for sample sizes _<50) or a
modified version of the Kolmogorov-Smirnov D Statistic (for sample sizes >50)
and provides a stemleaf, box plot, and probability plot of the residuals.
For a discussion of regression analysis and the use of these dianostic tools,
the reader is referred to Chatterjee and Price (1977), Belsley and others
(1980), Daniel and Wood (1980), and Draper and Smith (1981).
The procedure is invoked three times in the example, shown in figure 4,
once for the linear model of equation 2 (lines 65-68), once for the log-linear
model of equation 3 (lines 69-72), and for the log-log model of equation 7
(lines 73-76). In each instance three variables are set - the model label
(MODLABEL) in lines 65, 69, and 73, the independent variable (Y) in lines 66,
70, and 74, and the explanatory variables (FLOW) in lines 67, 71, and 75 -
43
then the macro REGMAC is called (lines 68, 72, and 76) to do the computation.
In the example shown in figure 4, the log-linear (equation 3) was selected
to flow adjust the sulfate concentrations. The linear model (equation 2) is
clearly deficient. Both plots demonstrate this quite clearly. However, the
log-log model (equation 7) is not manifestly better or worse than the log-linear
model. The pattern seen on the residuals plots for these two models are very
similar, both showing some modest lack of fit at the low end and a slight amount
of heteroscedasticity. The choice betwen these two models, in this case, is
largely a matter of preference. The log-linear model has the advantage that the
residuals are in units of mg/L.
PLOTTING WATER-QUALITY DATA AS A TIME SERIES
It is now possible to examine the flow-adjusted concentrations for trend.
A good exploratory method for detecting trends is to plot the data as a time
series. Figure 5 shows an example SAS program that generates time series
plots for streamflow, sulfate concentration, and the flow-adjusted sulfate con
centration for the data used in figure 3. Note that PROC REG (lines 12-14)
is used to calculate the FAC and generate the data set used by PROC PLOT (lines
16-19) (SAS Institute, Inc., 1979, p. 343). Note also that the variable
DECTIME (lines 17-19) is used as the time variable in these plots.
PROC PLOT produces line printer plots as shown in figure 5. More elaborate
plots of SAS data can be made using PROC GPLOT (SAS Institute Inc., 1981b,
p. 69). The procedure GPLOT is a part of the SAS/GRAPH system which is a
computer graphics system for producing figures on terminal screens and plotters.
A list of compatible graphics devices is given in the SAS/GRAPH User's Guide
(SAS Institute, Inc., 1981b, p. 2).
44
1 //F5 JOB
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STATISTICAL PROCEDURES TO TEST FOR TRENDS IN WATER-QUALITY TIME SERIES
Two nonparametric tests for detecting trends in time series have been
added to SAS at the U.S. Geological Survey Amdahl computer facility. PROC
SEASKEN does the Seasonal Kendall test and slope estimator developed by
Hirsch and others (1982). This procedure is suitable for detecting monotonic
trends in time series with seasonality, missing values, or values reported
as "less than." The procedure is not, however, robust against serial correla
tion. Additional information about the Seasonal Kendall test and slope esti
mator is given in the User's Guide presented in Appendix A.
PROC SEASRS tests for differences in the location parameters (mean, median,
etc.) of two separate periods in a time series. The procedure uses a version
of the Wilcoxon rank-sum test (Mann-Whitney test) modified to handle seasonality,
This test is appropriate for detecting step trends (changes) in water quality
before and after events such as construction of a dam or sewage treatment plant
or an abrupt land-use change (construction, mining, clear cutting). Additional
information about the modified Wilcoxon rank-sum test is given in the User's
Guide presented in Appendix B.
In many cases, it may be appropriate to apply one of these PROC's to the
concentration data as well as to the flow-adjusted concentration (FAC) data.
Trends or changes in FAC may be interpreted as an indication that the stream-
flow concentration relationship has changed; that is, for a given discharge
one may expect different concentration values today versus some time in the
past. In cases where there have been substantial changes in the flow fre
quency distribution (due to changing regulation, diversion, or consumption),
it may be very useful to look for trends or changes in raw concentration data.
This will be indicative of the combined effects of changes in human inputs to
49
the stream and changes in the flow frequency distribution on the frequency
distribution of concentrations.
The presence of a large number of missing values or "less than" values
(censoring) in the records being tested does not substantially affect the
significance of the tests, although their power (ability to detect actual
trends) may be reduced. PROC SEASKEN has been tested with as much as 50
percent of the seasonal values missing or 50 percent of them censored without
any problem in significance.
The consequences of using these tests when data are serially correlated
is that the actual significance of the test becomes higher than the nominal
significance level. For monthly data, serial correlations are generally in a
range such that actual significance may be about twice as high as the nominal.
In other words, a p-level reported as 0.05 may more accurately be as high as 0.10.
The problem becomes severe where the measurements are from a large body of water
with a long residence time (by comparison with the sampling frequ y). Aquifers,
large lakes, or reservoirs, or the streams draining large lakes or reservoirs,
may present problems. Keeping the parameters SEASON (see appendicies A and B)
small, typically no larger than 12, will help avoid serious problems due to
serial correlation. When the correlation problem is thought to be severe,
reducing SEASON (reducing the number of values per year used in the test) should
prove to be helpful.
Figure 6 shows a program listing and output of a SAS job that uses
PROC SEASKEN to test for time trends in the streamflow, sulfate concentration,
and the flow-adjusted sulfate concentrations used in the regression example
shown in figure 3. The flow-adjusted concentrations were calculated by the
REG procedure using the LOGLIN model shown in figure 3 and are included in the
output data set generated by PROC REG.
50
1 //F6 JOB
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17
VAR
OECT
IME
P00061 P0
0945
FA
C/'
16
/*
19
//
Figu
re 6
.--E
xamp
le in
put
and
outp
ut o
f SAS
statements to
do th
e Se
ason
al Kendall
test a
nd slope
estimator
procedure
on a
wat
er-q
uali
ty d
ata
time
ser
ies.
ST
AT
IS
TIC
AL
A
NA
LY
SIS
S
YS
TE
M
ST
AT
ION
ID
EN
TIF
ICA
TIO
N
NU
MB
ER
=0337*1
00
SE
AS
ON
AL
K
EN
DA
LL
T
FS
T
AN
D
SL
OP
E
ES
TIM
AT
OR
FO
R
TR
EN
D
MA
GN
ITU
DE
WA
TER
Y
EA
RS
1973-1
9P
1
lftt
l?
1H
JP
SD
AY
, A
UG
US
T IB
, 1993
VA
RIA
BL
EN
OB
SN
VA
LS
TAU
P L
EV
EL
NO
BS
IS
T
HE
N
UM
BE
R
OF
NO
N-M
ISS
IMG
O
BS
ER
VA
TIO
NS
IN
TH
E
OR
IGIN
AL
D
AT
A.
NV
AL
S
IS
TH
E
NU
MB
ER
O
F
NO
N-M
ISS
ING
S
EA
SO
NA
L
VA
LU
ES
C
ON
ST
RU
CT
ED
.
SLO
PE
r ooo
6i
P00945
FAC
ST
RE
AM
FLO
H,
INS
TA
NT
AN
EO
US
(C
FS
)
SU
LFA
TE
D
ISS
OL
VE
D
CM
G/L
A
S
SD
«)
RE
SID
UA
LS
103
98
98
98 96
96
-0.0
20
-0.1
82
-0.2
79
0.8
37
0.0
3?
0.0
01
-60
.00
-1.3
33
-1.0
66
en
ro
The results of the SEASKEN test show that no significant trend is evident
in streamflow but that both the sulfate and flow-adjusted sulfate concentrations
exhibited highly significant decreasing trends (p-level(p)<0.05) during the
period of record used in the test.
This trend can be further examined using PROC SEASRS. The data used in
this example are from a station draining a watershed containing considerable
surface coal mining. The regulations promulgated under the Surface Mine
Control and Reclamation Act of 1977 (PL 95-87) were implemented toward the
latter third of the period of record for this station (May 1979). Figure 7
shows a program listing and output of a SAS job that uses PROC SEASRS to
determine if differences exist in streamflow, sulfate concentration, and the
flow-adjusted sulfate concentration (using the LOGLIN model of figure 3) before
and after this date. The results of the SEASRS test show that no significant
step trend is evident in streamflow (p-level>0.05) but that sulfate concentration
and the flow-adjusted sulfate concentration exhibit a highly significant trend
(p-level<0.05).
The reader should be aware that both the Seasonal Kendall test and the
Seasonal Wilcoxon rank-sum test are exploratory in nature. The fact that the
Seasonal Wilcoxon rank-sum test indicates a difference between the periods in
the time series prior to and following the implementation of the surface mine
regulations does not necessarily imply causality, nor does it imply that it will
continue into the future.
53
01
1 //F
7
JOb
2 //C
LA
SS
3 //
PR
OC
LId
0
0
DS
N=
WR
D.P
RO
CL
IB*D
ISP
=S
HR
4 //A
E
XE
C
*IR
US
AS
,DS
N1
=N
UL
LF
ILE
,DS
N2
=5
//T
RE
ND
D
O
I>S
N=
US
ER
ID.F
IL
EN
AM
E*D
ISP
=O
LD
6 //
SrS
IN
DD
*
7 DATA OATAA/SET
T REND. MONT HLY ,'
8 IF
STATION*
1 03374100
';9
L00061=LOG(P00061>;
10
KEEP ST
ATIO
N SN
AME
P009
45 P0
0061
L00061 DECTIME;
11
PRJC SO
RT OATA=DATAA;BY
STAT
ION;
M
PROC
RE
6 OA
TA = DA
TAA
NOPRINT/'aY
STATION;
13
MODEL P00945»L00061;
14
OUTPUT OUTsFACDATA P = p
R=FAC»*
15
PROC SORT DATA=FACDATA;Br STATION;
16
PROC SEASRS DATA=FACDATA SEASON=12 DATE=1979.33;8Y STATION;
17
VAR DECTIME P00061 P00945 FAC;
18
/*
19
//
Fig
ure
7.-
-Exa
mple
in
put
and
outp
ut
of
SAS
stat
emen
ts
to
do
the
Sea
sona
l M
ann-
Whi
tney
-Wilc
oxon
ra
nk
sum
and
sl
ope
est
ima
tor
proc
edur
e on
a
wa
ter-
qu
alit
y d
ata
time
series.
ST
AT
ION
ID
EN
TIF
ICA
TIO
N
VA
RIA
BL
E
PC
006I
P009*5
F*C
MA
NN
-WH
ITN
EY
-HIL
CO
XA
NR
AN
K
SU
M
FIR
ST
S
ER
IES
(1
97
3-1
97
9)
NO
BS
N
VA
LS
ST
RE
AM
FIO
W,
INS
TA
NT
AN
EO
US
(C
FS
) 73
SU
LF
AT
C
DIS
SO
LV
ED
(N
G/L
A
S
S04)
69
RE
SID
UA
LS
69
69
6B
68
NU
PB
ER
'03
37
41
00
3 T
HU
RS
DA
Y,
AU
GU
ST
IB
.
TE
ST
F
OR
S
EA
SO
NA
L
DA
TA
CU
TO
FF
1979
1979
1979
DA
TE
.33
.33
.33
SF
CO
NO
S
ER
IES
(1
979-1
981)
NO
BS
N
VA
LS
30
29
29
29 29 28
P L
FV
EL
O.«
>0
7
0.0
06
0.0
06
ST
EP
10
50
.
-7.
OD
D
-5.3
50
NOBS IS THE NUMBER OF NON-HISSING OBSERVATIONS IN THE ORIGINAL DATA.
NVALS IS THE NUMBER OF NON-HISSING SEASONAL VALUES CONSTRUCTED.
en
en
APPENDIX A: PROC SEASKEN user's guide with examples
This procedure tests for monotonic trends in time series using a modified
form of Kendall's tau (Kendall, 1975) derived by Hirsch and others (1982).
The null hypothesis for this test is that the probability distribution of the random variable does not change over time. In the Kendall's tau test, all
possible pairs of data values are compared. If a later value (in time) is
higher, a plus is recorded, if the later value is lower, a minus is recorded.
If no trend exists in the data, the probability of a later value being higher
or lower than any previous value is 0.50. In this case, the number of pluses
should approximately equal the number of minuses. If the number of pluses
greatly exceeds the number of minuses, the values later in the series are more
often higher than those earlier in the series, indicating an uptrend. If the
number of minuses greatly exceeds the number of pluses, a downtrend is indicated.
In this modified Kendall's tau, the problem of seasonality is avoided by comparing
only observations from the same season of the year. Thus, for monthly data with
seasonality, January data is compared only with January data, and so on.
Trend magnitude is determined using the seasonal Kendall slope estimator
(Hirsch and others, 1982). The slope estimate is taken to be the median of
the slopes of the ordered pairs of data values compared in the Seasonal Kendall
test:
A more complete discussion of the Seasonal Kendall test and Seasonal
Kendall slope estimator and its use are given in Smith and others (1982).
SPECIFICATIONS
The following statements may be used in the SEASKEN procedure:
PROC SEASKEN SEASON = n options;
BY variables;
VAR variables;
56
The PROC SEASKEN statement invokes the procedure. The VAR statement is
used to specify the numeric variables for the analysis. The BY statement is
optional. The observations must be in chronological order. This can be done
by using PROC SORT and sorting by the variables in the BY statement followed
by DECTIME. For example:
PROC SORT; BY variable DECTIME;
PROC SEASKEN statement
The PROC SEASKEN statement has the form:
PROC SEASKEN SEASON = n options;
SEASON = n gives the number of seasonal values per year. For
example, SEASON = 12 would be used for data collected monthly
and SEASON = 52 would be used for data collected weekly.
SEASON = 1 may be used for annual summary data, such as
average annual streamflow. Each water year will be divided
into n equal length seasons. Note that for SEASON=12, for
example, the 12 seasons will not correspond exactly to the
12 calendar months since months are not of equal length.
For each season of each year, there may be no observations
(a missing value will be assumed as the seasonal value), one
observation (that value will be used as the seasonal value),
or several (not more than 400) observations (the median of
the values will be used as the seasonal value).
This statement is not optional and must be included.
The options that may be used in the PROC SEASKEN statement follow.
DATA = SASdataset gives the name of the SAS dataset to be used by
PROC SEASKEN. If it is omitted, the most
recently created SAS data set is used.
57
DL1 - DL15 = detection limit
gives the detection limit of the analytical
procedures used to determine the constituent
values in the time series. The number associated
with each call of the detection limit option
refers to the order of the constituents on the
VAR statement. The use of the detection limit
option is most important when several different
analytical procedures with different degrees of
sensitivity were used to determine values of the
same constituent in the time series. In this
case, the highest detection limit of all the
procedures should be used.
The detection limit option sets all values
less than or equal to the detection limit equal
to one-half the value of the detection limit.
This option would not be used when the test
is applied to Flow-Adjusted Concentrations (FAC).
VAR statement
VAR dectime variables;
The names of the variables to be tested for trends are listed in the
VAR statement. The first variable must contain the decimal ti;ne values
corresponding to the times the observations in the time series were made.
Dectime is in the form of decimal time; for example, 12 noon, June 1, 1975,
will be 1975.4178. Up to 15 additional numeric variables may be included.
The total number of seasonal values (number of seasons times the number of
water years) may not exceed 1200. The VAR statement is not optional and
must be included.
58
BY statement BY variables;
A BY statement may be used with PROC SEASKEN to obtain separate
analyses on observations in groups defined by the BY variables. The input
data set must be sorted in order of the BY variables.
DETAILS
Missing Values
Missing values may appear in a time series used in SEASKEN. The exist
ence of missing values presents no theoretical problem for applying the
seasonal Kendall test for trend. Comparisons of data pairs where one member
is missing are not included in the calculation of the test statistic. (Inter
nally, missing values are represented as -1E31.)
One or more seasons may be devoid of data; for example, one may use
SEASON=12 and have data only for the summer. In this case, all other months
will be set to missing values.
Formulas
The test statistic, Si, is:
ni-1 niSi = I I sgn (Xij-Xik)
k=l j=k+l
Where n-j are the number of annual values for season i,
X-jj the seasonal value for season i and year j,
X-j|< the seasonal value for season i and year k,
and
1 if e > 0sgn (0 ) = , 0 if 0 = 0
-1 if 0 < 0
59
The expected value of Si is 0, E [S n-] = 0 and its variance is:
n. (n,-l)(2n,+5) - ft. (t.-l)(2t.+5) ii i j ii i
Var [Si] = 18
where ti is the extent of a given tie (number of X's involved in tie)
for season i,
and I denotes the summation over all ties.
The composite statistic of the seasonal statistics, Si, is S 1 :
season E [S 1 ] =J E [Si] = 0
and the variance of S 1 is:
season season n . (n.-l)(2n.+5)Var [S 1 ] = I Var [Si] = I J 1 n
i=l i=l 18
The normal approximation with a continuity correction of 1 (toward zero)
is used to estimate p = Prob [|S'|>_s] (the probability that S 1 will depart
from zero by the amount s or more). The standard normal deviate, Z, is
calculated by:
f S' - 1(Var S 1 ) 172 if S 1 > 0
Z = < 0 if S 1 = 0
S 1 + 1(Var S 1 ) 172 if S 1 < 0
The p-level of the test is determined from Z using the standard normal
distribution. The p-level is the probability of obtaining a Z value that is
as large or larger in absolute value than the one obtained if the null hypo
thesis were true (i.e., there was actually no trend). If one pre-selects a
significance level a (the risk of rejecting the null hypothesis when it is
60
actually true), then one should reject whenever the p-level is less than a.*
A typical value for a is 0.05.
The statistic T (tau) is:
season cT=
1=1 ni( ni
Note that T may only take on values between -1 and +1. Negative values
indicate downwards trend, positive values indicate upwards trend. It is a
type of rank correlaton coefficient between the variable and time.
The seasonal Kendall slope estimator is the median value of
d ijk = ( x ij - x ik) /(J-k ) for a 11 (Xlj. *ik) Pa^5
where d-jj^ is the slope between seasonal values for season i, year j and
season i, year k with j>k.
The slope estimate is valid only when all of the data are reported as
above the limit of detection. If there are only a few "less thans" it can be
viewed as a reasonable approximation.
Printed output
For each variable SEASKEN prints (see figures Al and A2)
1. the variable name (VARIABLE)
2. the variable label, if any
3. the total number of non-missing observations in the input data (NOBS)
4. the number of seasonal values constructed from the observations (NVALS)
5. the statistic tau (T) (TAU)
6. The significance probability (p-level) of the trend, two-sided (P LEVEL)
7. The estimate of trend magnitude, in units per year (SLOPE)
61
EXAMPLES
Example 1: Annual Mean Streamflow
In this example (figure A-l), PROC SEASKEN is called to test for a
trend in the annual mean streamflow observed at a U.S. Geological
Survey streamflow gaging station. The data was entered from card input
(lines 9-19).
Example 2: NASQAN Data
In this example (figure A-2), PROC SEASKEN is called to test for
trends in 10 water quality constituents at three U.S. Geological Survey
NASQAN stations. The data was previously stored as a SAS data set on a
direct access storage device. Note that the SAS data set, TREND.MONTHLY
was sorted by STATION (line 6) so that the BY statement option (line 8)
could be used with SEASKEN.
62
en CO
1 /
/ A
i2
//
CLA
iS3
//P
RO
CL
ld
OU
D
SN
=W
RD
.PR
OC
LI8
/DIS
P4
//A
E
XE
C
Wrt
OS
AS
/DS
Nl=
NU
LLF
1LE
/DS
N2
5 //
SY
SIN
D
O
*6
OA
TA
SF
LO
*/;
7 IN
PU
T
WY
EA
K
AM
U
iol/'L
IST
;8
DECT
IME=
WYEA
R-0.
25;
9 LA
BEL
AMQ=ANNUAL ME
AN STREAMFLOW;
10
CARD
S;11
1925
801U 1926
12
1932
4660 1933
13
1939
7720 1940
14
1946
7660 1947
15
1953
6300 1954
16
1960
6340 1961
17
1967
6050 1968
18
1974
9930 1975
19
1981
5540
20 21
PR
OC
S
EA
SK
EN
S
EA
SO
N=
1;
22
V
AR
D
EC
TIM
E
AM
U,'
23
/*24
//
= SHR
=NULLFILE
4760
5930
3570
3430
2840
4650
7450
7180
1927
1934
1941
1948
1955
1962
1969
1976
10800
5630
2410
6930
5100
8330
7280
8180
1928
1 935
1942
1949
1956
1963
1970
1 977
10300
3430
3650
8370
5410
3880
6840
3080
1929
1936
1943
1950
1957
1964
1971
1978
4950
5230
5950
10400
3520
2700
7130
6440
1930
1937
1944
1951
1958
1965
1972
1979
10500
6960
6880
9440
8690
5160
4450
6870
1931
1938
1 945
1952
1959
1966
1973
1980
2200
7140
4910
8290
10200
4280
11200
6470
Figu
re A
l.--
Exam
ple
input
and
outp
ut us
ing
PROC SEASKEN
to test fo
r a
tren
d in
an
nual
me
an streamflow
STATISTICAL ANALYSIS SYSTEM
J5:3B THURSDAY, AUGUST IB, 1983
SEASONAL KEKOAll TEST AND SLOPE ESTIMATOR FOR TREND MAGNITUPF
WATER YEARS 1925-19P1
VARIABLE
MOBS
NVALS
TAU
P LEVFL
SLOPE
APC
ANNUAL FEAN STREAMFLOH
57
57
0.0&
0.670
9.B39
NOBS IS THE NUMBER OF NON-MISSING OBSERVATIONS IN THE ORIGINAL DATA.
NVALS IS THE NUMBER OF NON-MISSING SEASONAL VALUES CONSTRUCTED.
cr»
1 //A3 JO
dI
II CLASS
3 //
PR
OC
LId
t>
0 D
SN
=W
RD
.PR
OC
LI3
/DIS
P=
SH
R4
//A
E
Xh
C
»)R
I>$
AS
/DS
N1
=N
UL
LF
ILE
/DS
N2
=5
//T
RE
ND
O
D
DS
N=
US
ER
IO.F
ILE
NA
ME
* 0
ISP
=O
L0
6 //S
rS
lND
D*
7 PROC SURT UATA=TREND.MONTHLY;BY STATION,*
8 PHOC SEASKtN
0 ATA = T«END.MONTHLY SEASON=12 DL3 = 0.1 OL4 = 0.01 OL7 = 0.1
y BY
ST
ATIO
N;10
VAR OECTIME P70300 P00410 P00630 P00665 P00915 P00925 P00930 P00940 P00945 P0095
1 1
0 ;
1 2
/*13
//
Fig
ure
A
2.~
Exa
mp
le
input
and
outp
ut
usi
ng
PR
OC
SEAS
KEN
to
test
for
tre
nd
s in
w
ate
r-q
ua
lity
constitu
ents
.
ST
AT
IS
TIC
AL
A
NA
LY
SIS
S
TA
TIO
N
lOF
NT
H R
AT
ION
N
tlH
^E K
=S
Y S
I
SE
AS
ON
AL
K
EH
OA
LL
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ES
T
AN
D
SL
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ES
TIM
AT
OR
FO
R
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E^
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MA
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WA
TER
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1975-1
981
8:r
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FK
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9B
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04
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P00639
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09
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AS
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AL
V
AL
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S
CO
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TR
UC
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STATISTICAL
A N
A
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S 1
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T F
M
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IItJ
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33
74
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OR
FO
R
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D
MA
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WA
TE
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1973-1
981
VA
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.
Appendix B: PROC SEASRS user's guide with examples
The SEASRS procedure tests for differences in the location parameters of
two separate periods in a time series using a modified version of the Wilcoxon
rank-sum test (Bradley, 1968, p. 105). This test is equivalent to the
Mann-Whitney test described in Conover (1971, p. 224). The null hypothesis
for this test is that two populations comprised by data from two separate
periods in a time series are identical. This test assumes that sampling was
random.
If the null hypothesis is true, no distinction can be made between the n
observations in the first period and the m observations in the second period
of the time series. In effect, all were taken from a common population. There
fore, each of the possible combinations of the n+m observations taken from
the common population were equally likely to have become the two samples
actually collected. For each of these possible combinations exists a value of
the test statistic, W. This statistic is the sum of the ranks of the n
observations within the combined (n+m observations) sample. (The smallest
value in the combined sample receives a rank of 1, the next smallest a rank of
2, etc.). The null hypothesis is rejected if the value of the test statistic
W differs from the expected value of W by a preselected value corresponding to
a desired probability.
For a more thorough discussion of the Wilcoxon rank-sum test, and examples
of its use, the reader is referred to Bradley (1968, p. 105) and Hollander and
Wolfe (1973, p. 68).
SPECIFICATIONS
The following statements may be used in the SEASRS procedure:
PROC SEASRS SEASON = n DATE = x options;
BY variables;
VAR variables.
69
The PROC SEASRS statement invokes the procedure. The VAR statement is
used to specify the numeric variables for the analysis. The BY statement is
optional. The observations must be in chronological order. This can be done
by using PROC SORT and sorting by the variables in the BY statement followed
by DECTIME. For example:
PROC SORT; BY variable DECTIME;
PROC SEASRS statement
The PROC SEASRS statement has the form:
PROC SEASRS SEASON = n DATE = x options;
SEASON = n gives the number of seasonal values per year.
For example, SEASON = 12 would be used for
data collected monthly and SEASON = 52 would be
used for data collected weekly. SEASON = 1 may
be used for annual summary data, such as average
annual streamflow. Each water year will be
divided into n equal length seasons. Note that
for SEASON=12, for example, the 12 seasons will
not correspond exactly to the 12 calendar months,
since months are not of equal length. For
each season of each year, there may be no
observations (a missing value will be assumed as
the seasonal value), one observation (that
value will be used as the seasonal value),
or several (not more than 400) observations
(the median of the observations will be used as
the seasonal value).
This statement is not optional and must be
included.
70
DATE = xxxx gives the time separating the two periods of the
time series. DATE is in the form of decimal
time, for example, 12 noon, June 1, 1975, will be
1975.4178. If the two periods in the time series
are widely separated, any date in the gap will
suffice. An observation occurring at precisely
time DATE is placed in the second period.
This statement is not optional and must be
included.
The options that may be used in the PROC SEASRS statement follow.
DATA = SASdataset gives the name of the SAS dataset to be used by
PROC SEASRS. If it is omitted, the most recently
created SAS data set is used.
DL1 - DL15 = detection limit
gives the detection limit of the analytical
procedures used to determine the constituent
values in the time series. The number associated
with each call of the detection limit option
refers to the order of the constituents on the
VAR statement. The use of the detection limit
option is most important when several different
analytical procedures with different degrees of
sensitivity were used to determine values of the
same constituent in the time series. In this
case, the highest detection limit of all the
procedures should be used.
71
The detection limit option sets all values
less than or equal to the detection limit equal
to one-half the value of the detection limit.
This option would not be used when the test is
applied to Flow-Adjusted Concentration (FAC).
VAR statement
VAR dectime variables;
The names of the variables to be tested are listed in the VAR
statement. The first variable must contain the decimal time values
corresponding to the times the observations in the time series were made.
Dectime is in the form of decimal time; for example, 12 noon, June 1,
1975, will be 1975.4178. Up to 15 additional numeric variables may be
included. The total number of seasonal values (number of seasons times
the sum of the number of water years in the first period plus the number
of water years in the second period) may not exceed 1200. The VAR statement
is not optional and must be included.
BY statement
BY variables;
A BY statement may be used with PROC SEASRS to obtain separate
analyses on observations in groups defined by the BY variables. The input
data set must be sorted in order of the BY variables.
DETAILS
Missing values
Missing values may be included in time series used in SEASRS. The
existance of missing values presents no theoretical problem for applying the
Mann-Whitney-Wilcoxon rank-sum test for seasonal data. Ranks of observations
with missing values are not included in the calculation of the test statistic.
(Internally, missing values are represented as -1E31.)
72
One or more seasons may be devoid of data; for example, one may use
SEASON=12 and have data only for the summer. In this case all other months
will be set to missing values.
Formulas The test statistic, W-j, is:
niWi = I Rn
j=l
where n-j is the number of annual values for season i in the first period
in the time series,
and Rn are the ranks of the seasonal values for season i in the first
period of the time series.
The expected value of W-j is:
E [Wi] = [ ni (ni + mi + 1) ] / 2
where mi is the number of annual values for season i in the second
period in the time series,
and its variance is:
Var [Wi] = [ni mi (ni + mi +1) ] / 12.
The composite statistic of the seasonal statistic, Wi, is W and is
defined as:
seasonW = I W .
i = l i
The expectation of W is:
seasonE [W] = I E [W ]
1 = 1 i
and its variance is:
seasonVar [W] = I [W ].
1=1 i
73
The normal approximation is used to estimate p = Prob [|W - E [W]|>w]
(the probability that W will depart from its expected value by the amount w or
more). The standard normal deviate, Z, is calculated by:
W - E[W] Z = (Var[W]) 1/2 .
The p-level of the test is determined from Z using the standard normal distri
bution. The p-level is the probability of obtaining a Z value that is as large or
larger in absolute value than the one obtained if the null hypothesis were true
(i.e., there was actually no trend). If one pre-selects a significance level a (th
risk of rejecting the null hypothesis when it is actually true), then one should
reject whenever the p-level is less than a. A typical value for a is 0.05.
An estimate of the magnitude of the step trend is taken as the median of
the difference between all pairs of seasonal values, one from each period but
of the same season.
The step trend estimate is valid only when all of the data are reported
as above the limit of detection. If there are only a few "less thans" it can
be viewed as a reasonable approximation.
Printed Output
For each variable SEASRS prints (see figures Bl and B2):
1. the variable name (VARIABLE)
2. the variable label, if any
3. the number of original observations (NOBS) and number of constructed
seasonal values (NVALS) in the first series
4. the CUTOFF DATE value
5. the number of original observations and number of constructed seasonal
values in the second series
6. the significance probability of the difference, two-sided (P LEVEL)
7. the estimate of the step trend (STEP)
74
EXAMPLES
Example 1: Annual Mean Streamflow
In this example (figure B-l), PROC SEASRS is called to test for differences
in annual mean streamflows observed at a U.S. Geological Survey Streamflow
gaging station before and after construction of a series of reservoirs in the
drainage basin. The data was entered from card input (lines 9-19).
Example 2: NASQAN Data
In this example (figure B-2), PROC SEASRS is called to test for differences
in 10 water-quality constituents at three U.S. Geological Survey NASQAN stations
before and after implementation of the Surface Mining Control and Reclamation
Act of 1977. The second and third stations are located in watersheds mined
for coal. The first station is not and serves as a control. Note that the
SAS data set, TREND.MONTHLY was sorted by STATION (line 6) so that the BY
statement option (line 8) could be used with SEASRS.
75
1 2 3 4 5 6 7 8 910 1
112 13 14 15 16 1 7
18 19 20
21 22 23 24
//U1 JOd
// CLASS
//PROCLId
OD DSN = «IRD.PROCLIl3*DISP = SHR
//A EXEC WRDSAS/DSN1 =NU LL F I
L E/ D SN2 = NU L LF
I L E
/ / S Y S
I N
DO
*DATA SFLO
W;IN
PUT
WYEA
R AM
Q (JJ/'LIST;
DECUME = WYEAR-0.25;
LABE
L AM
U=AN
NUAL
ME
AN ST
REAM
FLOW;
CARD
S;1925
1932
1939
1946
1953
1960
1 967
1974
1981
8010
4660
7720
7660
63UO
6340
6050
9930
5540
1926
1933
1940
1947
1954
1961
1968
1975
4760
5930
3570
3430
2840
4650
7450
7180
1927
1934
1941
1948
1955
1962
1 969
1976
10800
5630
2410
6930
5100
8330
7280
8180
1928
1 935
1942
1949
1956
1963
1 970
1977
10300
3430
3650
8370
5410
3880
6840
3080
1929
1936
1943
1950
1957
1964
1971
1978
4950
5230
5950
10400
3520
2700
7130
6440
1930
1937
1944
1951
1958
1965
1972
1979
10500
6960
6880
9440
8690
5160
4450
6870
1931
1938
1945
1952
1959
1966
1973
1980
2200
7140
4910
8290
10200
4280
11200
6470
PROC SEASRS SEASON=1 DATE=1963.75;
VAR DECTIME AHQ;
/*
Fig
ure
B
l. E
xam
ple
in
pu
t an
d outp
ut
usi
ng
PROC
SE
ASRS
to
te
st
for
a st
ep
in
annu
al
mea
n st
rea
mflo
w
ST
AT
IS
TIC
AL
A
NA
LY
SIS
S
TS
TE
M
MA
NN
-WH
ITN
EY
-WK
CO
XA
N
RA
NK
SU
M
TE
ST
F
OR
S
EA
SO
NA
L
DA
TA
THURSDAY, AUGUST in, 1983
VARIABLE
FIRST SERIES
(1925-1963)
MOBS
NVALS
CUT3FF DATE
SECOMD SERIES
(1964-1981)
NOBS
NVALS
P LFVEL
STEP
AFO
ANNUAL HEAN STREAKFLOH
39
39
1963.75
0.770
750.0
NOBS
IS THE NUMBER OF NON-MISSIHG OBSERVATIONS IN THE ORIGINAL DATA.
NVALS IS
THE NUMBER OF MON-MISSIMG SEASONAL VALUES CDMSIRUCTEO.
1 //d2
JOI3
2 // CLASS
3 //PROCL1B
00 DSN*WRO.PROCLIB/.DISP = SHR
A //A EXEC WRDSAS,DSN1 *NULLF1LE,DSN2=NULLFILE
5 //TRENO
l)D
OSN=USERID. F
ILEN AME/-0
I SP»OL D
6 //SYS1N DO
*7
PHOC
SOKT OATA=TREND.MONTHLY;BY
STAT
ION;
8 PR
OC SEASRS DA
TA=T
REND
.MON
THLY
SEASON=12
OATE
=197
9.A
OL3=0.1
OLA=
0.01
OL?s0.1
OL9
10=0
.I;BY
STATION;
10
VAR
DECTIME
P703
00 POOA1Q P0
0630
P0
0665
P0
0915
P0
0925
P00930 P009AO P0
09A5
P009
511
0;12
/*13
//
^,
Figu
re B2.--Example in
put
and
outp
ut u
sing PROC SE
ASRS
to te
st for
step
tr
ends
in w
ater-quality
00
cons
titu
ents
.
ST
AT
IS
TIC
AL
A
NA
LY
SIS
S
YS
TE
M
ST
AT
ION
ID
EN
TIF
ICA
TIO
N
NU
*«B
E**
D3
27
65
00
MA
NN
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ITN
EY
-HIL
CD
XA
N
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NK
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UM
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ES
T
FO
R
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15
:40
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HU
RS
DA
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AU
GU
ST
IB,
1983
VA
RIA
BL
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P7
03
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P00410
P0
06
30
P0
06
65
P00915
P00925
P0
09
30
P00940
P0
09
45
P0
09
50
FIR
ST
S
ER
IES
(1
97
5-1
97
9)
MD
BS
N
VA
LS
SO
LID
S,
RE
SID
UE
A
T
180
DE
C.
C D
ISS
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VE
D
ALK
ALIN
ITY
F
IELD
(M
G/L
A
S
CA
C03)
NIT
RO
GE
N,
NO
2 * N
O 3
T
OT
AL
(M
G/L
A
S
N)
PH
OS
PH
OR
US
, T
OT
AL
(M
G/L
A
S P
)
CA
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D
ISS
OLV
ED
H
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S C
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MA
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, D
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(MG
/L
AS
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SO
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(M
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A
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NA
)
CH
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LV
ED
(M
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A
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CD
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LF
AT
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DIS
SO
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(M
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A
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4)
FL
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(M
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A
S
F)
49
49
49
49
49
49
49
49
49
49
47
47
47
47
47
47
47
47
47
47
CU
T3
FF
D
AT
E
1979.4
0
1979.4
0
1979.4
0
1979.4
0
1979.4
0
1979.4
0
1979.4
0
1979.4
0
1979.4
0
1979.4
0
SE
CO
ND
S
ER
IES
(1
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9-1
99
1)
NO
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N
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LS
20
17
27
27
2B
2B 2fl
2B
2B 2B
27 16 26 26
27
27
27
27
27
27
P LE
VE
L
0.5
26
0.6
62
0.0
09
0.4
39
0.5
66
0.6
52
0.0
30
0.2
75
0.0
00
0.2
60
SIE
P
9.0
00
2.0
00
.3003
.1000E
-01
-1.0
00
.0
-1.1
00
-1.0
00
-5.0
00
.0
NO
BS
IS
T
HE
N
UM
BE
R
OF
N
ON
-MIS
SIN
G
OB
SE
RV
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ION
S
IN
TH
E
OR
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AL
D
AT
A.
NV
AL
S
IS
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E
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MB
ER
O
F
NO
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ING
S
EA
SO
NA
L
VA
LU
ES
C
ON
ST
RU
CT
ED
.
ST
AT
IS
TIC
AL
A
NA
LY
SIS
S
YS
TF
M
ST
AT
ION
ID
EN
TIF
ICA
TIO
N
NU
MB
ER
=0337*1
DO
MA
NN
-WH
ITN
EY
-HJ
LC
OX
AN
R
AN
K
SU
M
TE
ST
F
OR
S
EA
SO
NA
L
DA
TA
15M
O
TH
UR
SD
AY
. A
UG
US
T
18,
1983
VA
RIA
BL
E
FIR
ST
S
ER
IES
(1973-1
979)
MOBS
NVALS
CUTOFF DATE
SECOMD SERIES
(1979-19B1)
NOBS
NVALS
P LEVEL
STEP
P70300
P00410
P 0
0630
P0
06
65
P00915
P00925
P00930
P00940
P00945
§
P00950
SO
LID
S,
RE
SID
UE
A
T
10
0
DE
C.
C D
ISS
OL
VE
D
AL
KA
LIN
ITY
F
IEL
D
(HG
/L
AS
C
AC
03)
NIT
RO
GE
N*
N0
2+
N0
3
TO
TA
L I *G
/L
AS
N
)
PH
OS
PH
OR
US
* T
OT
AL
(HG
/L
AS
P
I
CA
LC
IUM
D
ISS
OLV
ED
(N
G/L
A
S
CA
)
MA
GN
ES
IUM
* D
ISS
OL
VE
D
(MG
/l
AS
M
G)
SO
DIU
M*
DIS
SO
LV
ED
(M
G/L
A
S
NA
)
CH
LO
RID
E*
DIS
SO
LV
ED
(M
G/L
AS
C
L1
SU
LFA
TE
D
ISS
OLV
ED
(M
G/L
A
S
SO
*)
FLU
OR
ID
E.
DIS
SO
LV
ED
(M
G/L
AS
F
)
70
70
65
64 70
70
70
70
70
70
69
69
6*
63
69
69
69
69
69
69
1979.
40
19
79
. 40
1979.4
0
19
79
.40
19
79
.40
1979.4
0
1979.4
0
1979.4
0
1979.4
0
19
79
.40
2B 16 ?n 28 2B 2B ?B 2B 2B 28
27 15 27
27 27 27 27
27
27
27
0.1
66
0.2
11
0.0
01
0.6
54
0.5
7B
0.2
45
0.5
7B
0.6
0B
0.0
06
0.7
46
-14.5
0
-14
.00
.5000
.1000E
-01
-2.0
00
-2.0
00
.6000
1.0
09
-7.0
00
.0
NC1BS
IS THE NUMBER OF NON-MISSING OBSERVATIONS IN THE ORIGINAL DATA.
NVALS IS
THE NUMBER OF NON-MISSING SEASONAL VALUES COMSTRUCTED.
ST
AT
ION
IDE
NT
IFIC
AT
ION
MA
NN
-WH
1T
NE
Y-H
1L
CO
XA
N
RA
NK
S
UM
VA
RIA
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E
P70300
P0
04
10
P0
06
30
P0
06
65
P00915
P00925
P0
09
30
P00940
P00945
P0
09
50
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LID
S,
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SID
UE
A
T
1PO
D
EC
. C
DIS
SO
LV
ED
AL
KA
LIN
ITY
F
IEL
D
(MG
/L
AS
C
AC
03 )
NIT
RO
GE
N,
N0
2*N
03
T
OT
AL
(MG
/L
AS
N
)
PH
OS
PH
OR
US
, T
OT
AL
(M
G/L
A
S
P)
CA
LC
IUM
D
ISS
OL
VE
D
(MG
/L
AS
C
A)
MA
GN
ES
IUM
, D
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D
(MG
/L
AS
M
G)
SO
DIU
M,
DIS
SO
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ED
(M
G/L
A
S
NA
)
CH
LO
RID
E,
DIS
SO
LV
ED
(M
G/L
A
S
CD
SU
LF
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E
DIS
SO
LV
ED
(M
G/L
A
S
S0
4I
FLU
OR
IDE
, D
ISS
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VE
D
(MG
/L
AS
F
)
FIR
ST
S
ER
IES
(1
97
5-1
97
9)
VDBS
NV
ALS
47
4
6
47
46^
46
4
6
46
46
45
45
46
45
46
45
47
46
47
4
6
47
46
TE
ST
FO
R
S Y
S T
F 3
3 7
9 530
SE
AS
ON
AL
DA
TA
M }*
>:**
SE
CO
MO
S
ER
IES
(1
97
9-1
9R
1)
CU
TO
FF
D
AT
E
NO
BS
N
VA
LS
19
79
.40
1979.4
0
1979.4
0
1979.4
0
1979.4
0
1979.4
0
1979.4
0
1979.4
0
1979.4
0
1979.4
0
25
16
29
29
27
27
27
29
27
27
?4 16 26
26 26
26 26 26 26
26
0 T
HU
RS
DA
Y,
P LE
VE
L
0.7
88
0.2
79
0.1
6B
0.1
33
O.P
42
0.1
39
0.5
47
0.5
12
O.O
B?
0.3
5?
AU
GU
ST
IP
,
ST
EP
-3.5
00
-10
.00
.30
00
.30
00
E-O
J
-1.0
00
-2.0
00
-1.0
00
-1.0
00
-8.0
00
.3
NO
RS
IS
TH
E
NU
MB
ER
OF
NO
N-M
ISS
IMG
O
BS
ER
VA
TIO
NS
IN
TH
E
OR
IGIN
AL
D
AT
A.
NV
ALS
IS
TH
E
NU
MB
ER
O
F N
ON
-MIS
SI1
G
SE
AS
ON
AL
VA
LU
ES
C
ON
ST
RU
CT
ED
.
Appendix C: Source code for SAS procedures
Two separate programs are required to add a procedure to the Statistical
Analysis System at a user's installation, a parsing module and a procedure
module. The parsing module acts as a control program for the procedure and
defines permissible options and parameters for the procedure. The procedure
module inputs the appropriate data and computes the test statistic. The parsing
module for Proc SEASKEN is given in figure C-l. The procedure module is given
in figure C-2. The parsing module for Proc SEASRS is given in figure C-3.
The procedure module is given in figure C-4. Information about SAS parsing
and procedure modules can be found in the SAS Programmer's Guide (SAS Institute,
Inc., 1981a). Only minimal JCL is shown since JCL is highly installation
dependent.
82
oo CO
1 2 3 A 5 6 7 8 910 1
112 13
1 A
15 16 17 18 19
//C1 JOb
// CLASS
// EXEC ASMSAS
//ASM.SYSIN DD
NOGEN
PROC
SASPROC NAME=SEASKEN,LOADMOD=SEASKEN2,DEFLIST=1
tDEFMODE=NUMERIC
PARMS
SASLIST
DL1,1,DL2,2,DL3,It
DL4,4,DL5,5,MODE=NUMER I
C
PARMS
SASLIST
DL11/11
tDL12t
12,DL13,13,DL14/U,MODE=NUMERIC
PARMS
SASLIST
DL15,15/SEASON,16/MODE=NUMER1C
LISTS
SASLIST
VARIABLESt
1tVAR
t1*MODE=NUMERIC
SASEND
END
//LKED.SYSIN DD
*SETSSI ACU00003
NAME SEASKEN(R)
/*
Figure
Cl.
--Pa
rsin
g module fo
r the
Seas
onal
Kendall
test
an
d slope
estimator
proc
edur
e
fD
0ro
i-a-sonfDexc-sfD
3OO-£1
fD
-bO-srt-rrfD
OOfDOJCOO3CU
7^fD3CLQ)
t
rt-fDtort-
&3 0.
to
O-afD
fD tort- _i.3 CUrt-O-s a-s0ofDO.C-sfD
Zooooc'<r>3^35c;o;ci-.» » > > O m o II m m m z as < r- r- -« a c » > > -«
»-» C co II » » o*- z z co T3O xococm»-'3>3>»^m 0 »113300110 XZZ»
Z«<-H>"T> ^33* > --» O Z - >. m 3 Z 3 -» ^ <-> | «N m -4
^- O -* ui Z "O
> o O m > 3» >> 2 0* 1- CO Z20 O > m T3-to o >. 03 r- o-< 3 " > m ON oo c s x r- v/i Nmo rn 3 ON ^ ZNO z > 1-1 v/» > r-
o oo > Z ZCD X oo -» -n cir~ -v ui o >O < 1 >^ 30 Z0 » -» 3 <A » > 3» H-, O Z 33
> m >-> o as L>4 TI >r- -» o zm ^« M -n 31-
>Z n O>Z n>Zc_Coo-4
XZ n»
Z nO>
n> Z
C
-H
CO
oomzmooo oc33>-l>?>23 S QJ
OC ff)»OOO| OC22;oomoo>22T32C:
o » r- x -tV.~va O»«^«^>«^l-.
>>»-»»-»3o o>-»-imSOJ «^-»3coC-tui>mr>ov^> mz>«-'-t:c
S^N >C-t co^«>
Z v/» m > ^ o Z23»-^ -lXi-i-»T3<-13 3>"DIVim^-<m ><»O«-«ZT3> T3m>-iO-»> me" m^oz^v^Z^J» si-i-t-i^oZ Q3 -< «M V/1 (JO-Dm 33 o ^ ooor- > o ^00 -< -^ -*> ^» > w»2 -* m ^r- o < > z ^ m zo z < > >z r-< 00J, /^X -»O U»
z >2 Z> c»3 >m -«> mZ 00r- «> <\jm ^m >r- -«> >z c n ^»O -»X V/13 ^> >Z
-n
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C Z CD OJQ
0c:-H
zm
-H
O
mj»c»
z
M cm
o03COmx<> Mi iozoo
>zo nOX)3
-4
Xm
oom»oooz J»r-
< j>r-c mCO
-* X 0O co m r~
»c -< o >
-H>a3>o>mo-H>-n<<:<m»>o of -"^ or-«-r-3r->zor- it -»-tor--t 3zoTJ| -Hr--Dr-O-tT3| ^Zfc-ll r»f-^-3>3 33
m c >-> Z<co22m -nr-oo-tO co-i»«-z <<:-* occoco-<T}zcooCOmmmZC » i-*<3oo» x »-*C> * m »<r>-»x>03 oooo3»~v»oo
C X X 3C« *-. » X X CO -> 1-i» m m »O z-» mx»<Ji>
m <^ c o v^ -tOO-HO 3-nO voo Cm> ff>OD > oo^vr-co-t om<-t i -l-t> *3 3> J» OZCO -< » TJ < OO O O i-" m >
-t -n > * * Z» -» Q3 -» > -1 O < 1 W Z1-1 » m »^ oOO 33 OO 03-t 1-1 001-1 » T> *+
0 03 50 -»oo r~ o >j\ m 3 ~
00 >00 Z
m » <.X 00 J»o r-r- ooc: ^xCO -»>-< v/»
m >Z
o c» n 3»
-4
-t mX 00m *^
<\j H '-'f-^ \
3 -4m j»
C< <^> ->33 Ut> 1 V^
> >CDf-m
<v;
c_0a
51
C SAVE LABEL.
52
DO 11
J=1 ,5
53
LA8EL(J,I )=NLABEL(J)
54
11 CONTINUE
55
N08S(I)=0
56
NVALSCI)=0
57
10 CONTINUE
58
SETUP*.FALSE.
59
C CYCLE TO GET EACH OBSERVATION.
60
20 CALL INPUT(IEND)
61
C WRAP UP IF
ALL OBSERVATIONS HAVE BEEN PROCESSED.
62
IF(IEND.EQ.1)GO TO 40
63
C G
ET
A
N
OB
SE
RV
AT
ION
.6
4
KO
BS
=K
OB
S+
16
5
CA
LL
tfA
RX
<1
/Y>
66
IF
(SE
TU
P)
GO
TO
8
067
C FIRST OBSERVATION* GET READY.
68
C CALCULATE TIME OF OCT.
1 OF WATER YEAR OF FIRST OBSERVATION.
69
YEAR=Y(1)+0.25
70
dASETM=IFIX(YEAR)-0,25
71
NDATESC1)=YEAR
72
C GET NUMBER OF
SEASONS.
73
PERYR=PARM(16)
74
DO 70 1=1,1200
75
HAVE(I)=.FALSE.
76
70 CONTINUE
77
LAST=1
78
SETUP*.TRUE.
79
80 CONTINUE
80
C PROCESS OBSERVATION,
81
M=CYC1)-BASETM)*PERYR+1.0
82
IF (M.GE.LAST) GO TO 200
83
WRITE (IPRINT,9001> KOBS
84
9001 FORMAT
(' OBSERVATION NUMBER
1,110,
85
& '
IS OUT OF CHRONOLOGICAL ORDER.
1)
86
STOP
87
200 CONTINUE
88
IF CM.LE.1200) GO TO 210
89
WRITE <IPRINT,9002) KOBS
90
9002 FORMAT
(' OBSERVATION NUMBER', 110,'
IS LATER THAN SEASON 1200.')
91
STOP
92
210 CONTINUE
93
C MULTIPLE OBSERVATIONS?
94
IF CHAVE(M)) GO TO 90
95
C IMO. SAVE.
96
HAVECM)=.TRUE.
97
K = 2
98
LAST=M
99
DO 30 1=1,NV
100
C PREPARE FOR MULTIPLE OBSERVATIONS.
101
VAL=FIXMIS(Y(I+1))
102
L(I)=0
103
IF (VAL.GT.XCHEK) L(I)=1
104
IF (VAL.GT.XCHEK) NOBS(I) = NOBS( I)+1
105
IF (VAL.GT.XCHEK) NVALS(I)=NVALS(I)+1
106
X(M*I)=VAL
107
SORT(1,I)=VAL
108
30 CONTINUE
109
GO TO 20
110
C MULTIPLE OBSERVATIONS
111
90 CONTINUE
112
00 110 1=1,NV
113
IF (K.EQ.2.AND.L(I).EQ.1) NVALS( I
) = NVALS(I)-1
114
VAL=FIXMIS(Y(I+1))
115
IF (VAL.LT.XCHEK) GO TO 110
116
NOBS(I)=NOBS(I)+1
117
L(I)=L(I)+1
118
SORT(L( I),I)=VAL
119
110 CONTINUE
120
CALL INPUT(IEND)
121
C CYCLE UNTIL NEW SEASON.
122
IF (IEND.EQ.1) GO TO 120
123
KOBS=KOBS+1
124
CALL VARX(1*Y)
125
NEWM*(Y(1)-BASETM)*PERYR+1.0
126
IF (NEWM.NE.M) GO TO 120
127
K=K+1
128
C LIMIT TO 400 OBSERVATIONS* MISSING OR NOT* IN ONE SEASON.
129
IF (K.LE.400) GO TO 90
130
WRITE (IPRINT*9005) KOBS
131
9005 FORMAT
( AT OBSERVATION NUMBER,M10,
132
& '
THERE WERE MORE THAN 400 OBSERVATIONS IN THAT SEASON.
1)
133
STOP
134
C GET MEDIANS.
135
120 CONTINUE
136
DO 130 1=1,NV
137
C ANY NON-MISSING OBSERVATIONS?
138
IF (L(I).EQ.O) GO TO 130
139
C YES.
140
NVALSU )=NVALS(I) + 1
141
IH=(L(I)+1)/2
142
EVEN=2*IH.EQ.L(I)
143
C FIND MEDIAN.
144
CALL VSRTA(SORT(1
,1 ),L(I) )
145
IF (.NOT.EVEN) X(M,I)= SORT(IH,I)
146
IF (EVEN)
X(M,I)=(SORT(IH,I)+SORT(IH+1,I))/2.0
147
130 CONTINUE
148
LAST=M
149
C CONTINUE PROCESSING IF
END-OF-FILE NOT ENCOUNTERED.
150
IF (IEND.NE.1 )
GO TO SO
151
40 IF (M.GT.1) GO TO 50
152
WRITE
( IPRINT/9003)
153
9003 FORMAT
(' THERE ARE FEWER THAN
2 SEASONS.
1)
154
STOP
155
50 CONTINUE
156
N=LAST
157
NDATES(2)=Y(1)+0.25
158
IF (LAST*(LAST/PERYR+1,0).LE.60000.0) GO TO 60
159
vJRI
TE (IPRINT/9004)
160
9004 FORMAT
(' THE COMBINATION OF
YEARS AND SEASONS EXCEEDS
1/
161
& '
AN INTERNAL LIMIT.
1)
162
STOP
163
60 CONTINUE
164
DO
1 50
M = 1/LAST
165
IF (HAVECM)) GO TO 150
166
DO 140
1 = 1,NV
167
X(M/I)=XMISS
168
140 CONTINUE
169
150 CONTINUE
170
RETURN
171
END
172
C PERFORM KENDALL TEST.
173
SUBROUTINE SEAS
174
COMMON /STATS/ NV/N/NOBS(15)/NVALS(15),NDATES(
2),TAU(15)/
175
& ALPHA(15)/SLOPE(15)
176
COMMON /DATA/ X(1200,15>
177
REAL YC60000)
00
178
REAL*8 PARM
-J
179
REAL XCHEK/-1.0E30/
180
LOGICAL ODD
181
LOGICAL*1 WASTIEC1200)
182
NPER=PARM(16)
183
PERYR=NPER
184
DO
1 J=1/NV
185
C BYPASS DETECTION LIMIT PROCESSING IF
NONE SPECIFIED.
186
IF (PARMCJ).EQ.O.ODO) GO TO 50
187
DLIMIT=PARM(J)
188
DO 100 I=1/N
189
IF (X(I/J).LE.DLIMIT.AND.XCI,J).GT.O.O)
X( I/J )=0.5 *DL IMIT
190
100 CONTINUE
191
50 CONTINUE
192
C ZERO OUT THE COUNTERS.
193
DO 60 1=1,N
194
WASTIECI)=.FALSE.
195
60 CONTINUE
196
NPLUS=0
197
NMINUS=0
198
NCOMPT=0
199
VARTOT=0.0
200
INDEX=0
201
FIXVAR=0.0
202
C LOOP THROUGH THE SEASONS.
203
DO 10 ISEAS*1,NPER
204
NCOMP=0
205
N1=N-NPER
206
C LOOP THROUGH YEARS FOR VALUES IN SEASON ISEAS.
207
DO 20 ISTART=ISEAS,N1,NPER
208
C VALID VALUES?
209
IF
(X(ISTART,J).LE.XCHEK) GO TO 20
210
C VALUE IS ALWAYS TIED WITH ITSELF.
211
NTIE=1
21
2
N2=
IST
AR
T+
NP
ER
21
3
C T
RY
E
AC
H
LA
TE
R
SE
AS
ON
.214
DO
3
0
IEN
D=
N2
*N,N
PE
R215
C VALID VALUE?
216
IF (X(IEND,J).LE.XCHEK) GO TO 30
217
C COMPARE.
218
NCOMP=NCOMP+1
219
INDEX=INDEX+1
220
YY=(X(IEND,J)-X(ISTART,J))/«IEND-ISTART)/PERYR)
221
IF (YY.GT.0,0) NPLUS=NPLUS+1
222
IF (YY.LT.0.0) NMINUS=NMINUS+1
223
IF (YY.EQ.0.0) NTIE=NTIE+1
224
C MARK VALUES THAT ARE TIED.
225
IF (YY.EQ.0.0) WAST IE(I END) = .TRUE.
226
C SAVE ADJUSTED DIFFERENCES.
227
Y(INDEX)=YY
228
30 CONTINUE
00
229
C UPDATE VARIANCE CORRECTION IF TIES OCCURED AND TIES
230
C WERE NOT COUNTED BEFORE.
231
IF (WTIE.NE.1.AND..NOT.WASTIE(ISTART))
FI XVAR=FI XVAR+NT IE*(NT IE-
232
& 1.0)*(2.0*NTIE+5.0)/18.0
233
20 CONTINUE
234
C ACCUMULATE THIS MONTH'S RESULTS.
235
NCOMPT=NCOMPT+NCOMP
236
NMONTH=(1.0+SQRT(1,0+8.0*NCOMP))/2.0
237
VARTOT*VARTOT+NMONTH*(NMONTH-1.0)*(2.0*NMONTH+5.0)/18.0
238
10 CONTINUE
239
C DONE COMPARING.
240
S=NPLUS-NMINUS
241
VARTOT=VARTOT-FIXVAR
242
C WERE THERE ANY VALID COMPARISONS AND AT
LEAST TWO DIFFERENT VALUES?
243
IF (NCOMPT.GT.O.AND.VARTOT.GT.O.O) GO TO 40
244
C NONE. GO HOME EMPTY.
245
TAU(J)=0.0
246
ALPHA(J)=1.0
247
S'LOPE(J)=0.0
248
GO TO
1249
40 CONTINUE
250
C CALCULATE THE STATISTICS.
251
TAU(J)=S/NCOUPT
252
C CONTINUITY CORRECTION.
253
IF (S.GT.0.0)
S = S-1
254
If (S.LT.0.0) S=S*1
255
Z=S/SQ*T(VARTOT)
256
C COMPARE TO THE STANDARD NORMAL DISTRIBUTION.
257
IF (Z.LE.0.0) ALPHA(J)=2.0*C0FN(Z)
258
IF (Z.GT.0.0) ALPHA(J)=2.0*(1.0-CDFN(Z))
259
C SORT THE DIFFERENCES. VSRTA
IS AN IMSL ROUTINE.
260
CALL V/SRTA (Y,INDEX)
261
C PICK THE MEDIAN.
262
ODO = MOD (INDEX,2) .EQ.1
263
IF (ODD) YMED=Y((INDEX+1)/2)
264
IF (.NOT.ODD) YMED=0.5*(Y(INDEX/2)+Y((INDEX/2)+1))
265
SLOPEU )=YMED
266
IF (SLOPEU).NE.0.0) GO TO
1267
C ADJUST FOR THE FACT THAT TAU AND ALPHA MAY SAY THERE
IS A
268
C SIGNIFICANT TREND BUT THE ESTIMATE OF THE SLOPE
IS269
C ZERO DUE TO TIES.
270
IF (NMINUS.GT.NPLUS) SLOPE(
J )=-1.0E-30
271
IF (NMINUS.LT.NPLUS) SLOPE(
J )=1.0E-30
272
1 CONTINUE
273
RETURN
274
END
275
C REPORT RESULTS.
276
SUBROUTINE OUTPUT
277
COMMON /LABCM/ NAME,LABEL
278
REAL*8 NAME(15),LABEL(5,15>
C5
279
COMMON /STATS/ NV,N,NOBS(15),NVALS(1 5),NDATES(2),TAU(1 5)*
280
& ALPHAM 5),SLOPE(1 5)
281
COMMON /IOUNIT/ IPRINT
282
C STANDARD SAS HEADING.
283
CALL STITLECO,LINES)
284
WRITE
( IPRINT,6001) NDATES
285
6001 FORMAT (/26X,'SEASONAL KENDALL TEST AND SLOPE ESTIMATOR FOR'/
286
& '
TREND MAG NI TUD E '
/ MOX, '
W AT E R YEARS
' ,
I 4, '
- f
, U / / )
287
WRITE (IPRINT,6002)
288
6002 FORMAT C VARIABLE*
,50X, NOBS
NVALS
TAU
P',
289
& '
LEVEL
SLOPE'/1X,107( '-')/)
290
00 10 1=1,NV
291
WRITE (IPRINT,6003)
NAME(I),(LABEL(J/I),J=1,5),NOBS(I),NVALS(I)
292
£ TAU( I),ALPHA(I),SLOPE(I)
293
6003 FORMAT (1X,A8/3X,5A8,2I10,2F10.3/G15.4/)
294
10 CONTINUE
295
WRITE
( IPRINT,6004)
296
6004 FORMAT (/" NOBS IS
THE NUMBER OF NON-MISSING OBSERVATIONS',
297
& '
IN THE ORIGINAL DATA.'/
298
S *
NVALS IS
THE NUMBER OF NON-MISSING SEASONAL VALUES',
299
& ' CONSTRUCTED.')
300
RETURN
301
EN
D30
2 c
CUM
ULAT
IVE
DIS
TRIB
UTI
ON
FU
NCTI
ON
FOR
THE
NORM
AL
DIS
TRIB
UTI
ON
.303
C PRIMARY REFERENCE
IS ABRAMOWITZ AND STEGUN/
304
C N8S HANDBOOK OF MATHEMATICAL FUNCTIONS/ EQ. 26.2.19.
305
FUNCTION CDFN(X)
306
IF
(X) 10/20/30
307
10 CONTINUE
308
IF (X.LT.-6.0) GO TO 40
30V
T=-X
31
0
CD
FN
=0
.5/(
1.0
+0
.04
98
67
34
70
*T+
0.0
21
U1
00
61
*T**
2+
0.0
03
27
76
26
3*T
**3
31 1
X
+ 0.3
80036E
-4*T
**4+
0..488906E
-4*T
**5 +
0.5
3830E
-5*T
**6)*
*16
31
2
RE
TU
RN
313
20 CONTINUE
314
CDFN=0.5000
315
RETURN
316
30 CONTINUE
317
IF (X.GT.6.0) GO TO 50
318
CDFN=1.0-0.5/(1.0+0.0498673470*X+0.0211410061*X**2+0.0032776263*X
319
X**3+0.380036E-4*X**4+0.488906E-4*X**5+0.53830E-5*X**6)**16
320
RETURN
321
40 CONTINUE
322
CDFN'0.0000
323
RETURN
324
50 CONTINUE
325
CDFN=1.0000
326
RETURN
327
END
328
C CONVERT
A SAS MISSING VALUE TO ONE RECOGNIZED BY THESE ROUTINES.
329
FUNCTION FIXMIS(X)
330
REAL*8 X/VALUE
331
EQUIVALENCE(VALUE/M1SSX)
332
VALUE=X
333
IF (VALUE.EQ.O.ODO.AND.MISSX.NE.O) VALUE=-1.OD31
334
FIXMIS=VALUE
335
RETURN
336
END
337
//LKED.SYSIN DD
*338
ENTRY ENTRY
THIS STATEMENT MUST BE PRESENT
339
SETSSI ADOOOOOO
340
NAME SEASKEN2(R>
341
/*
342
//
1 //C3 JOB
2 //
CLASS
3 // EXEC AS
rtSA
S4
//ASM.SYSIN DD
*5
NOG EN
6 PROC
SASPROC NAME=SEASRS,LOADMOD=SEASRS2,DEFLIST=1
,7
DEFHODE=NUMERIC
8 PARMS
SASLIST
9 PARMS
SASLIST
10
PARMS
SASLIST
11
PARMS
SASLIST
DL15,15
,SEASON,16
,DATE
,17,MODE=NUMERIC
12
LISTS
SASLIST
13
SASEND
1 I*
END
15
//LKED.SYSIN DO
*16
SETSSl ACU00003
17
NAME SEASRSCR)
18
/*19
//
Fig
ure
C
3.-
-Pars
ing
mod
ule
for
the
Sea
sona
l M
ann-W
hitn
ey-
Wilc
oxo
n
rank
sum
te
st
and
slope
estim
ato
r p
roce
du
re.
1 //C4 JOB
2 // CLASS
3 // EXEC FORTSAS
4 //C.SYSIN DU
*5
C MOTIF* SAS.
6 CALL SASFHX
7 C
PROCESS THE DATA.
8 C
GET NUMBER OF VARIABLES FROM SAS.
9 NV1=NOVAR(1)
10
C NV IS THE NUMBER OF VARIABLES EXCLUSIVE OF THE TIME VARIABLE.
11
NV=NV1-1
12
IF (NV.GT.O) GO TO 10
13
CALL VARERR
14
STOP
15
10 CONTINUE
16
C READ IN
DATA.
17
CALL READ (NV1)
18
DO 20 J=1,NV
19
C COMPUTE RANK SUM FOR EACH VARIABLE.
20
CALL SEASRS (J)
21
20 CONTINUE
22
C WRITE OUT RESULTS.
23
CALL OUTPUT (NV)
24
STOP
25
END
26
C SUBROUTINE TO READ IN THE OBSERVATIONS AND FORM THE SEASONAL VALUES,
27
SUBROUTINE READ (NV1)
28
COMMON /DATA/ X(1200
115)
,NX1 ,NX2
,ALPHA(15)
DF(15)
,CUTOFF,
29
& NOBS1(15>,NOBS2<15>,NVALS1(15>,NVALS2(15>,NDATES<4)
30
REAL*8 CUTOFF
31
COMMON /LABINF/ NAME(15)*LABEL<5,15)
32
INTEGER NOBS<15)
,NVALS(15)
33
DIMENSION L(15)
34
COMMON /IOUNIT/ IPRINT
35
REAL*8 NAME/LA3EL/Y<16>*PARM,BASETM/PERYR
36
REAL SORT<400*15>
37
LOGICAL*1 HAVE(1200)t
SETUP,SECOND,EVEN
38
COMMON /NAMECM/ NTYPE
tNPOS,NLNG*NVARO,NNAME,NLABEL,NFORM,NIFORM,
39
& NFL,NFD*NF,NJUST
40
INTEGER*2 NTYPE,NPOS,NLNG,NVARO,NFL,NFD,NF
,NJUST
41
REAL*8 NNAME,NLABEL<5),NFORM,NIFORM
42
REAL XCHEK/-1.0E30//XMISS/-1.OE31/
43
M = 0
44
KOBS=0
45
NV=NV1-1
46
C GET SPECIFICTION OF EACH VARIABLE.
47
DO 10 1=1,NV
48
C CALL SAS TO LOAD COMMON BLOCK.
49
CALL NAMEVC1,1+1,NTYPE>
50
C SAVE NAME.
Fig
ure
C
4.-
-Pro
cedure
m
odul
e fo
r th
e
Sea
sona
l M
an
n-W
hitn
ey-
Wilc
oxo
n
rank
sum
te
st
and
slope
estim
ato
r p
roce
du
re.
51
NAME<I>=NNAME
52
C SAVE LABEL.
53
00
11 J=1,5
5A
LABELCJ/ I)=NLABEL<J)
55
11 CONTINUE
56
NOBS(I)=0
57
NVALS(I>=0
58
10 CONTINUE
59
SETUP=.FALSE.
60
SECONi)=.FALSE.
61
MBASd = 1
62
C CYCLE TO GET EACH OBSERVATION.
63
20 CALL INPUT(IEND)
64
C WRAP UP
IF ALL OBSERVATIONS HAVE BEEN PROCESSED.
65
IF (IENO.EQ.1) GO TO AO
66
C
GE
T
AN
O
BS
ER
VA
TIO
N.
67
K
OB
S=
K03S
+1
68
C
AL
L
VA
RX
d/Y
)69
IF
(SE
TU
P)
GO
TO
7
070
C FIRST OBSERVATION/ GET READY.
71
C CALCULATE TIME OF OCT.
1 OF WATER YEAR OF FIRST OBSERVATION,
72
YEAR=Y(1)+0.25
73
NDATESC1)=YEAR
74
BASETM=IFIX(YEAR)-0.25
75
C GET NUMBER OF
SEASONS.
76
PERYR=PARM(16)
77
C GET CUTOFF DATE.
78
CUTOFF=PARM(17)
79
DO 60 1=1/1200
80
HAVE(I) = .FALSE.
81
60 CONTINUE
82
LAST=1
83
SETUP=.TRUE.
84
C PROCESS OBSERVATION.
85
70 CONTINUE
86
IF (SECOND.OR. Yd ).LT. CUTOFF) GO TO 80
87
C FIRST OBSERVATION AFTER CUTOFF DATE.
88
NDATES(2)=YEAR
89
SECOND=.TRUE.
90
MBASE=LAST+1
91
NX1=LAST
92
DO 75 I=1/NV
93
NOBS1(I)=NOBS(I)
94
N03S(I)=0
95
NVALS1( I)=NVALS(I)
96
NVALS(I)=0
97
75 CONTINUE
98
YEAR=Y(1>+0.25
99
aASET,"l=IFIX(YEAR)-0.25
100
NDATES(3)=YEAR
10
1
80
CO
NT
INU
E102
YE
AR
=Y
(1)+
0.2
51
03
M
=(Y
d)-
8A
SE
TM
)*P
ER
YR
+M
8A
SE
104
IF
CM
.GE
.LA
ST
) G
O
TO
90
105
WR
ITE
(1
PR
INT
/90
01
) K
03
S106
90
01
F
OR
MA
T
('
OB
SE
RV
AT
ION
N
UM
BE
R1,H
O,
10
7
& '
IS
OU
T
OF
CH
RO
NO
LO
GIC
AL
O
RD
ER
.1
)108
ST
OP
109
90 CONTINUE
110
IF (M.LE.1200) GO TO 100
111
WRITE (IPRINT,9002) K09S
112
9002 FORMAT
( OBSERVATION NUMBER', 110,'
IS LATER THAN SEASON 1200.')
113
STOP
114
100 CONTINUE
115
C MULTIPLE OBSERVATIONS?
116
IF (HAVE(M) )
GO TO 110
117
C NO. SAVE.
118
HAVE(M)=.TRUE.
119
K=2
120
LAST=M
121
00 30 1=1,NV
122
C PREPARE FOR POSSIBLE MULTIPLE OBSERVATIONS.
1 23
VAL=FIXMIS (YU+1 )
)124
L(I)=U
125
IF (VAL.GT.XCHEK) L(I)=1
126
IF (VAL.GT.XCHEK) NOBS(I)=N08S(I)+1
127
IF (VAL.GT.XCHEK) NVALS(I)=NVALS(I)+1
128
SORTU, I)=VAL
129
30 X(M,I)=VAL
130
GO TO 20
131
C MULTIPLE OBSERVATIONS
132
110 CONTINUE
133
00 130 1=1,NV
134
IF (K.EQ.2.AND.L(I),EQ.1 )
NVALS ( I
)=NVALS(I)-1
135
VAL=FIXMIS(Y(I+1) )
136
IF (VAL.LT.XCHEK) GO TO 130
137
N08S(I)=NOaS(I)+1
138
L(I) = L< I)
+ 1
139
SORT(L( I)»I) = VAL
140
130 CONTINUE
141
CALL INPUT(IEND)
142
C CYCLE UNTIL NEW SEASON.
143
IF (IEND.EQ.1) GO TO 140
144
KOBSsKOBS+1
145
CALL VARX(1,Y)
146
NEWM=(Y(1)-BASETM)*PERYR+M8ASE
147
IF (NEWM.NE.M) GO TO 140
148
K=K+1
149
C LIMIT TO 400 OBSERVATION, MISSING OR NOT,
IN ONE SEASON.
150
IF (K.LE.400) GO TO 110
en
1 51
1 52
1 53
154
155
1 56
1 57
158
1 59
160
1 61
162
163
164
165
166
167
163
169
170
171
172
1 73
1 74
1 75
176
177
178
1 79
180
181
182
183
184
185
186
187
1 88
189
190
191
192
193
194
195
196
197
198
199
200
WRITE
( IPRINT,9005) K08S
9005 FORMAT
(' AT
OBSERVATION NUMBER
1, 110,
& ' THERE WERE MORE THAN 400 OBSERVATIONS
STOP
GET MEDIANS
140 CONTINUE
DO 150
1 = 1, NV
ANY NON-MISSING 09S E RVAT I
ONS ?
IF (L(I).EQ.O) GO TO
1 50
YES.
NVALS(I)=NVALS(I)+1
IN THAT SEASON.
1)
150
EVEN=2*IH.EQ.L(I)
FIND MEDIAN.
CALL VSRTA(SORT(1,I),L(I))
IF (.NOT.EVEN) X(M,I)= SORT(IH,I)
IF (EVEN)
X(M,I) = (SORT( IH,I )+ SORT(IH+1,I))/2.0
CONTINUE
LAST=M
CONTINUE PROCESSING IF
END-OF-FILE
(IEND.NE.1) GO TO 70
NOT ENCOUNTERED.
IF40 IF
(M.GT.1) GO TO 50
WRITE
( IPRINT,9003)
9003 FORMAT C THERE ARE FEWER THAN
2 SEASONS.
1)
STOP
50 CONTINUE
IF (SECOND) GO TO 160
WRITE
( IPRINT,9004)
9004 FORMAT C THERE ARE NO OBSERVATIONS AFTER TJHE CUTOFF DATE.
1)
STOP
160 CONTINUE
NDATES(4)=Y(1)+0.25
NX2=LAST-NX1
DO 170 1=1,NV
N08S2U )=NOBS( I)
NVALS2(I)=NVALS(I)
170 CONTINUE
I SET UNUSED CELLS TO MISSING.
DO 210 M=1,LAST
IF (HAVE(M)) GO TO 210
DO 200 1=1,NV
X(M,I)=XMISS
200 CONTINUE
210 CONTINUE
RETURN
END
! PERFORM RANK SUM TEST.
SUBROUTINE SEASRS (J)
COMMON /DATA/ X(1200,15),NX1,NX2,ALPHA(15),DF(15),CUTOFF,
& NOdS1(15),N08S2(15),NVALS1(15),NVALS2(15),NDATES(4)
201
REAL*8 CUTOFF
202
DIMENSION Y<1200)/R<1200)
203
REAL*8 PARM
204
REAL XCHEK/-1.OE30/
205
IF (NVALS1(J).EQ.0.OR.NVALS2<J).EQ.O) GO TO 700
206
NPER=PARM<16)
207
WS=O.U
208
EW=U.O
209
VW=U.O
210
LAST=NX1+NX2
211
C BYPASS DETECTION LIMIT PROCESSING
IF NONE SPECIFIED.
212
IF (PARMU).EQ.0.000) GO TO 20
213
ULIMIT=PARM(J)
214
DO 1J 1=1,LAST
215
IF
( X(I^J).LE.DLIMIT.AND.X<I,J) .GT.0.0) X(I
,J)=0.5*DL IM
I T
216
10 CONTINUE
217
20 CONTINUE
218
C LOOP THROUGH THE SEASONS.
219
DO 500 IM=1,NPER
220
C LOOP THROUGH YEARS FOR VALUES IN
SEASON IM.
221
C FIRST PERIOD.
222
NI
= 0
223
DO 60 I=IM/NX1/NPER
224
XX=X(I,J)
225
IF(XX.LT.XCHEK) GO TO 60
226
NI
a NI
+
1227
Y(NI)
= XX
228
60 CONTINUE
229
C SKIP SEASONS IF NO VALUES.
230
IF(NI.EQ.O) GO TO 500
231
C SECOND PERIOD.
232
MI
* 0
233
DO 70 I=IM,NX2,NPER
234
XX=X(I+NX1,J)
235
IF(XX.LT.XCHEK) GO TO 70
236
MI
= MI
+ 1
237
Y(MH-NI)
= XX
238
70 CONTINUE
239
C SKIP SEASON IF
NO VALUES.
240
IF (MI.EQ.O) GO TO 500
241
NN
= Ml
+ NI
242
CALL RANKCY/R/NN)
243
RSUM *
0.0
244
DO 130
I =
1/ NI
245
RSUM
= RSUM
+ R(I)
246
130 CONTINUE
247
RSSQ=0.0
248
DO 14Q 1=1,NN
249
RSSQ=RSSQ+R(I)**2
250
140 CONTINUE
251
C ACCUMULATE STATISTICS.
252
WS=WS+RSUM
253
EW=EvJ + NI*(NN+1 )/2.0
254
VW=VU+MI*N1*(RSSU/NN-<NN+1)**2/4.0>/(NN-1>
255
500 CONTINUE
256
C CLACULATE FINAL STATISTICS.
257
IF(VW.LE.O.O) GO TO 700
258
Z=<-<WS-EW)/SQRT(VW))
259
IF (Z.LE.0.0) ALPHA(J>=2.0*CDFN(Z>
260
IF (Z.GT.0.0) ALPHA(J)=2.0*(1,0-COFN(Z))
261
C GET ESTIMATE OF JUMP.
262
CALL SEARSD(X(1,J>,NX1,X(NX 1+1,J),NX2,NPER,OF(J>)
263
RETURN
264
700 CONTINUE
265
ALPHA(J)=1.0
266
OF(J)=0.0
267
RETURN
268
END
269
C CALCULATE AN ESTIMATE OF THE STEP TREND.
270
SUBROUTINE SEARSO(X1,NX1
,X2,NX2*NPER,0)
271
COMMON /IOUNIT/ IPRINT
272
REAL X1(NX1)
,X2(NX2)
273
REAL Y(12000)
274
REAL XCHEK/-1.OE30/
275
LOGICAL EVEN
276
NY1*( (NX1-1 >/NPER>+1
277
NY2 = «NX2-1 >/NPER> + 1
276
NMAX=NY1*NY2*12
279
IF (NMAX.LE.12000) GO TO 10
280
dRITE
( IPRINT,9001) NX1,NX2
281
9001 FORMAT
(' THE
Y ARRAY
IN SEARSO
IS TOO SMALL.
1/
282
& 'NX1=',I6/'NX2=',16)
283
STOP
284
10 CONTINUE
285
C CALCULATE THE DIFFERENCE BETWEEN EACH PAIR OF VALUES -
ONE IN
286
C THE FIRST SERIES/ THE OTHER OF THE SECOND, AND BOTH OF THE
287
C SAME SEASON.
288
K=0
239
DO 40 IM=1,NPER
290
DO 30 I=IM,NX1,NPER
291
XF=X1(I)
292
IF (XF.LT.XCHEK) GO TO 30
293
DO 20 J=IM,NX2,NPER
294
XL=X2(J)
295
IF (XL.LT.XCHEK) GO TO 20
296
K»K+1
297
Y(K)=XL-XF
298
20 CONTINUE
299
30 CONTINUE
300
40 CONTINUE
301
C GET MEDIAN.
302
CALL VSRTA(Y/K)
303
IH=« + 1)/2
304
EVEN=2*IH.EQ.K
305
IF (EVEN)
D=(Y(IH)+Y(IH+1))/2.0
306
IF (.NOT.EVEN) 0=Y(IH)
307
RETURN
308
END
309
C REPORT RESULTS.
310
SUBROUTINE OUTPUT (NV)
311
COMMON /DATA/ X(1200,15)/NX1/NX2/ALPHA(15)/DF(15)/CUTOFF/
312
& N03S1 (15)/N08S2(15)/NVALS1 (
1 5 )/NVA LS2 (
1 5 )/ NDATES (4
)313
REAL'S CUTOFF
314
COMMON /LABINF/ NAME(15)/LABEL(5/15>
315
REAL*8 NAME/LABEL
316
COMMON /IOUNIT/ IPRINT
317
C STANDARD SAS TITLE.
318
CALL STITLE (0/LINES)
319
WRITE (IPRINT,6001)
320
6001 FORMAT (/39X/ MANN-WHITNEY-WILCOXAN RANK SUM TEST FOR SEASONAL'/
321
& '
DATA'///)
322
WRITE (IPRINT/6002) NDATES
323
6002 FORMAT (53X/'
FIRST SERIES
'/16X/' SECOND SERIES
'/324
& 52X/'
('/I4/'-'/I4/» )
'/16X/'
('/14/ '
- '
/14/ ) '
/325
& '
VARIABLE'/
326
& 43X/'
NOBS
NVALS
'/'
CUTOFF DATE
'/327
& '
NOBS
NVALS'/
^
328
& 5X/'P LEVEL'/5X/'STEP'/1X/123('-')/)
00
329
DO 10 I=1/NV
330
WRITE (IPRINT/6003) NAME(I)/(LABEL(J/I)/J=1/5)/NOBS1(I)/NVALS1(I)/
331
& CUTOFF/NOBS2(I)/NVALS2(I)/ALPHA(I)/OF(I)
332
6003 FORMAT (1X/A8/3X/5A8/16/18/F13.2/ 11O/I8/F12.3/G15.4/)
333
10 CONTINUE
334
WRITE (IPRINT/6004)
335
6004 FORMAT (//' NOBS IS THE NUMBER OF NON-MISSING OBSERVATIONS
'/336
& 'IN THE ORIGINAL DATA.'/
337
& '
NVALS IS
THE NUMBER OF NON-MISSING SEASONAL VALUES'/
338
& ' CONSTRUCTED.
1)
339
RETURN
340
HNO
341
C CUMULATIVE DISTRIBUTION FUNCTION FOR THE NORMAL DISTRIBUTION.
342
C PRIMARY REFERENCE IS ABRAMOWITZ AND STEGUN,
343
C N8S HANDBOOK OF MATHEMATICAL FUNCTIONS/ EQ. 26.2.19.
344
FUNCTION CDFN(X)
345
IF
(X) 10/20/30
346
10 CONTINUE
347
IF (X.LT.-6.0) GO TO 40
348
T=-X
349
CDFN=d.5/(1.0+0.0498673470*T+0.0211410061*T**2+0.0032776263*T**3
350
& +0.380036E-4*T**4+0.488906E-4*T**5+0.53830E-5*T**6)**16
351
RETURN
352
20 CONTINUE
353
CDFN=U.50U
354
RETURN
355
30 CONTINUE
356
IF (X.GT.6.0) GO TO 50
357
CDFN=1.0-0.5/(1.0+0.0498673470*X+0.0211410061*X**2+0.0032776263*X
358
& **3+0.380036E-4*X**4+0.488906E-4*X**5+0.53830E-5*X**6> **16
359
RETURN
360
40 CONTINUE
361
COFN=0.000
362
RETURN
363
50 CONTINUE
364
CDFN=1.000
365
RETURN
366
END
367
C CO
NVER
T A
SAS
MISS
ING
VALUE
TO ON
E RECOGNIZED BY
THESE
ROUT
INES
.368
FUNC
TION
FIXMIS(X)
369
REAL
*8 X,
VALU
E37
0 EQUIVALENCE(VALUE^MISSX)
371
VALUE'X
372
IF (VALUE.EQ.O.ODO.AND.MISSX.NE.O) VALUE=-1.0D31
373
FIXMIS*VALUE
374
RETURN
375
END
376
SUBROUTINE RANK(XsRsN)
377
C COMPUTES THE RANKS OF
THE VALUES IN
VECTOR
XVD
378
C X
IS NOT RE-ARRANGED ON RETURN
4X3
379
C IN
CASES OF TIES/ MID-RANKS ARE USED
380
C MISSING VALUES ARE NOT ALLOWED
381
REAL X(N>, R(N>, Y(1200)
382
INTEGER ORDC1200)
383
C INITIALIZE ORD AND
Y384
00 10
I
= 1, N
385
ORD(I)
- I
386
Y(I)
= X(I)
387
10 CONTINUE
388
C REARRANGE
Y IN
ASCENDING ORDER
389
CALL VSRTR(Y,N,ORD>
390
C INITIAL RANKING
391
DO 1UO
I =
1*N
392
R(ORO(1»
* I
393
100 CONTINUE
394
C ADJUST RANKS FOR TIES
395
1=1
396
C VALUE ALWAYS TIED WITH ITSELF
397
IEND =
I398
150 IEND
= IEND +
1399
IF (IEND.LE.N.AND.Y(IEND).EQ.Y( I)) GO TO 150
400
C COMPUTE AVERAGE RANK
401
AVE
= <IEND*<IEND-1)-I*(I-1)>/<2.0*<IEND-I))
402
IENDM1
= IEND -
1403
DO 190
J =
It IENDM1
404
R<ORD(J»
= AVE
405
190 CONTINUE
406
I =
IEND
407
IF(I.LT.N) 60 TO 150
40°
RETURN
4 0 v
END
410
//LKED.SYSIN DD
*411
ENTRY ENTRY
THIS STATEMENT MUST BE PRESENT
412
SETSSI ADOJOOOO
413
NAME SEASRS2(R)
414
/*
415
//
O
O
REFERENCES
Belsley, D. A., Kuh, Edwin, and Welsch, R. E., 1980, Regression diagnostics:
New York, John Wiley and Sons, 292 p.
Bradley, 0. V., 1968, Distribution-free statistical tests: Englewood Cliffs,
N. J., Prentice-Hall, 388 p.
Briggs, J. C., 1978, Nationwide surface water quality monitoring networks of
the U.S. Geological Survey, in Establishment of Water-Quality Monitoring
Programs, San Francisco, June 1978, Proceedings: Minneapolis, American
Water Resources Association, p. 49-57.
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