Document 6530025

MACROS FOR SYSTEMATIC SAMPLE SELECTION AND VARIANCE ESTIMATION FROM ORDERED FRAMES
Josefina Lago, Westat, Inc.
1.
Introduction
This paper discusses both
equal probability and PPS systematic sample selection, as well
as the basic formulation of vari-
Systematic sampling is one
of the most commonly used methods
of sample selection, particularly
at the second and latter stages
-ance estimation by "successive
differences". Also described are
the three MACROS developed to implement the sample selection and variance
of selection of a multi-stage design. Systematic sampling's
greatest advantage is its sim-
i
estimation of an estimated total.
The use of the MACROS--SYSSAMPl,
SYSSAMP2, and SYSVAR--is illustrated
by means of an example.
plicity. Another advantage is
that, under certain conditions,
systematic sampling variances are
often smaller than those from
alternative designs.
(As survey
practitioners well know, it is
safe to use systematic sampling
only when one is sufficiently
acquainted with the data to determine when systematic selection is
not appropriate, as for example,
with data that are periodic in
relation to the order of the
listing and the selection interval
is equal to or a multiple of the
period.)
2.
Equal Probability Systematic Selection
To draw an equal probability systematic sample we first compute
the "selection interval," k:
k =N/n
where
n = desired sample size; and
N
The major shortcoming of
systematic sampling--with a single random start--is that it does
not yield an unbiased estimator
of variance. A systematic sample
may be viewed as a sample of one
cluster, where a sample of size
two or more is generally needed
to construct an unbiased estimator
of variance. However, several
biased estimators are available
that provide satisfactory variance estimates for many situations where systematic sampling
is used in practice.
total number of units
in the frame.
Of course, the value of k is not
necessarily an integer. Typically,
the value of k is rounded off to
one or two decimal places when an
exact sample size is required.
Next a random number, referred to
as the "random start" (r), is chosen
between 0 and k (excluding 0 but
including k). The sample units are
those having positions on the list
corresponding to the integer portion
of r, r+k, r+2k, ... r+(n-l)k.
An unbiased estimator of a
population total, Ysy , estimated
from an equal' probabIlity systematic sample is given by:
This paper focuses on
systematic sample selection and
variance estimation when a frame
has been ordered on the basis of
an auxiliary variable presumed
related to the population characteristics of interest. A systematic sample selected from such
ordered list provides a kind of
implicit stratification with
equal or unequal sampling fractions, depending on whether equal
n
't sy
kEy.
i=1 1
n
1: y!
1=1 1
where
k
y~
1
probability or probability proportional to size (PPS) systematic selection is used.
In this
situation, a variance estimator
commonly referred to as the
"successive differences" estimator may be constructed by regarding each sample unit as selected
at random from a stratum.
the inverse of the
inclusion probability
of each unit; and
weighted value of the
ith sample observation of
variable Y.
MACRO SYSSAMPI implements an
equal probability systematic selection as described above, and MACRO
SYSVAR may be used to estimate the
variance of a total estimated from
such a sample.
764
3.
The random start, r, in PPS
selection is a random number between
o and k (defined above). The n'
selection numbers are then: r, r+k,
r+2k, •.• , r+(n'-l)k.
The unit to
be drawn into the sample corresponding to a given selection number
is the first unit on the list for
which the cumulative size,
is
greater than or equal to the selection number.
PPS Systematic Selection
The PPS systematic sampling scheme is widely used when
one expects a proportionality
between a size variable (or "measure of size"), Xi, and the characteristics of interest. With
this sampling scheme the inclusion probability of the ith unit
is proportional to Xi.
Mi,
An unbiased estimator of a
population total estimated from a
PPS systematic sample is given by
the Horvitz-Thomson estimator, YHT:
Assuming the units in the
frame have been arranged in the
desired sequence, the PPS selection is carried on as follows.
Pirst, a cumulative measure of
size, Mi' is calculated for each
unit in the frame.
This cummulative size is simply the measure
of size of the ith unit, say Xi'
added to the measure of size of
all preceding units in the list,
M(i-l)'
n
L
i=l
bility of the ith
unit in the sample
(i.e., 1 for certainty selections
and Xi/k for noncertainties) :
i
J
and the total of all measures of
size MN is given by:
n
N
=
1
total sample size;
inclusion proba-
n
'i
E x.
M
1
where
That is:
j=l
y./n.
weighted value of Y
for the ith sample
unit.
y!
1
E Xi
i=l
MACRO SYSSAMP2 implements a
PPS systematic sample selection.
MACRO SYSVAR may be used to estimate
the variance of an estimated total
once sample observations have been
properly weighted.
Next, the selection interval, k, is calculated by:
N
k
( I
X.)
In
MN/n.
i=l 1
Obviously, k is not necessarily
an integer and, as mentioned
earlier, it is usually rounded to
one or two decimal places.
4.
When sampling systematically
from an ordered list, as described
above, one can view the units associated with the first k ~ossible
selection numbers as constituting a
first stratum, those associated
with k+l to 2k as constituting a
second stratum and so on, and finally
those associated with (n-l)k+l to
nk as a last stratum. An estimate
of variance may be obtained by regarding each unit in the sample as
selected at random from a stratum,
and grouping all possible pairs of
sample observations from contiguous
strata.
The "successive differences"
estimator of the variance of an
estimated total based on the overlapping differences is given by:
It -is possible that some
units in the list have a measure
of size greater than the selection interval k.
These units are
selected with certainty and are
deleted from the list prior to
sample selection.
No bias in
survey estimates is introduced
because of the prior selection of
certainty units (n c )' as long as
selection probabilities of the
noncertainty units are adjusted
to reflect the excluded certainty
units.
Hence, k is redefined as:
k
= ~,/n'
n'
=
where
MN ,=
variance Estimation by
Successive Differences
noncertainty sample
size; and
total cumulative measur~
of size after the nc units
have been excluded.
Vsy
765
n
2(n 1)
n-1
I
-i=l
(yl
_ y.1
1+1
)2
It must be remembered that before
using MACRO SYSVAR, values of the
variables for which vaeiances are
requested must be properly weighted.
where
y!
the weighted value
of the ith observation in the
sample; and
n
the effective
sample size (i.e.,
number of noncertainty units in
the sample).
1
6.
5.
To illustrate the use of
MACROS SYSSAMPl, SYSSAMP2 and
SYSVAR, consider the following
example. A state's department of
education wants to estimate chaeacteristics of private schools in
the state, such as total employment (EMPL), square feet (SQFEET),
electricity (ELECBTU), and total
energy consumption (TOTBTU). A
sample of 80 schools is to be
selected by two alternative
selection procedures: equal
probability, and PPS. The
measure of size to be us-ed is the
previous year1s reported student
enrollment (ENRLMNT). For illustration purposes, let's assume
the list of schools is also to be
sorted by reported enrollment
(ENRLMNT), to achieve a size
stratification effect.
Implementation
To use MACRO SYSSAMPI the
following user-defined MACROS are
required:
MACRO _STRVAR:
MACRO
MACRO
SAMPSZ:
DSFRAME:
MACRO
DSSAMP:
variable to be used
in implicit stratification
sample size desired
input data set containing the sampling frame
output data set
containing the
selected sample.
Since the universe size (N) is
determined by the number of observations in the frame, any observation with a missing value for
the auxiliary variable ( STRVAR)
should be deleted prior to sample
selection.
Exhibits 1, 2, and 3
illustrate the set-up, listing,
and output of the three MACROS,
as required for the example described above.
References
To use MACRO SYSSAMP2, a
user-defined macro must be included
to identify the measure of size
variable (MOS) , in addition to
the MACROS required by SYSSAMPl.
That is:
MACRO _MEASRSZ:
name of the MOS
variable to be used
in the.PPS selection.
(1)
Cochran, William G., 1977.
Sampling Techniques. John Wiley
and Sons, New York, Chapter 5.
(2)
Hansen, Hurwitz and Madow,
1953. Sample Survey Methods and
Theory, John Wiley and Sons, New
York, pp. 502-15.
Units with measure of size
greater than .75 times the selection
interval are identified as certainties. Then, the selection
interval k is redefined and the
noncertainty sample n~ selected.
MACRO SYSVAR requires the
following user-defined macros:
MACRO
DSNAME:
MACRO
VARLIST:
MACRO __SAMPSEQ:
Illustration
input data set
containing the
sample data:
list of variables
for which variances
are to be computed;
sort variable to be
used to order the
file with the sample data in the same
oeder as in sample
selection.
766
EXHIBIT 1
SAMPLE OUTPUT
IlIllfIIlIIHlllflltllflltIlIJlIIJIffll)llflll,nJffflf.ftUHIHHlH
II
TO
I~I.IOklE M~:RC SYS$~Mn
"
B~E
REQUIRED:
hRCRO _STRUHR
II
II
II
"~CRO
II
:H£
F~LLC1WC 1jSER-:'~f!:~£r ~flCP.OS
R IIJ!lWJL[ 10 BE USE] IN IPlPLlCIT
STRRTTrTCATlON I
_SAMPSZ
B (SAIWLE Sll[ D£SIRED I
II
11
D IUPIJT DATA SET CONTAINING THE
11
II
II
II
II
NIltRO _DSfRIlliE
II
HIltRO _ISSRIll'
EQUAL PROBABILITY SAMPLE SELECTION
THE UNIVERSE SIZE IS : 589
:HE SAmE SIZE IS : 88
THE SELECTION INTER~Rl IS : 6,36
RANDOM SHIRr IS: 3,86
I;
11
II
11
I;
fRf!ffE )
'H,
EQUAL PROSAlILl1'l' SRMPLE SELECTION
NUHEERS ORIGINRl UHIT HUmp. or SANPlE UNITS
AND VARIABLE USE] fOR INPLICIT STRATIfiCATION
SE~UENCE
II
I'
,
E IOUTPUT lATA SET .CONTAINING rH£ ..
SELECTElJ SAMPLE I
..
..
II
I;
II
II
II
II
HOTDSINeE THE UNIUElISE SIZE INI IS DETERMINE! BY THE NUNBER Of ..
OBSER'JATIONS IN THE fRAHE. ANY OBSERVATION iITH AHISSING n
VALUE FOR THE AUXILIARY VARIAIU _STRUHR SHOULD IE
U
DEllID BEfORE SAHPl[ SELECTION
II
_SAIIPsrg
1
3
4
S
6
7
B
IllnIHIIIIHIIHIHHHHlHHHIIHHHHHHlHHIIHIIHfHHlt;
OPTIONS MOIATE i
~.IItRO SYSSIIIII'!
ppoe SORT lATA ~. _ISfRIlliE OUT~IDIi
BY _STP-VAR i
mR DI1(t.[Ef'=t!UHIU) IlllKITPo _SmRR UNiLNOI
SoT III END~NDOr ;
UNIUO~_N_ ;
NUNn'~_H_ ;
OUTPUT IIIi
IT mOf ~ I THEN OUTPUT 111-;
D,Tf Dll; SET DI1 ;
,SA!P: _SR!PSl ;
69
Ii
71
72
73
74
75
mPlNT~Nl~IU/NSIIIIP;
SUPIMT;P'ErtJND ISKIPINT, ,111;
HlIlIST: ISKIPINT I UNIFO~:HI.8) I • ,8~ i
,·ItNIST:!OUND IRflHIST" 811 i
I,m _HULL ; SET DI~
FIlE PRINT NOTlillS;
PUT _PAGC /1/1 14B 'EQUAL PPOBAIILlH SRIIPLE SEl[CTION' 11111
1// m ' THE UNIVERSE sm IS : ' HUNIU /II
!18 ' THE SA!PLE SIZE IS : ' MSR!P /II
@l8 , THE smCTIOlt IHTEIi','Rl IS : ' SKIPINT III
m ' THE RANDOM START IS: ' RANIST II;
DAm 110 SET IDo
10 J;I TO NSR!P;
UNILNO=!HII RANDST' II HI I SKIPINT II
OUTPUTi
END;
KEEP liHIT JlO ;
DATA JlSHSRKP; HERGE 112 11!!:IN1) III 11«91l11
IF IHI RNI 1N1;
BY UNlT JlO ;
IF INI RNI IN1 THEN _SA!PSEQ' I;
PROC PRlhT; TITlE EQUAl PROBAIILIff smlE SELECTION;
TmE1 SEIlUEHCE 1U~IERS ORIGIHRl l'~1T r,U!l£F. Oi ,RN?lE U~ITS;
TITLE3 ~NI VARIAILE USED F':f: 1;;L1CIT SlF.P.TIFIC'TIOH ;
ID _SAMPSEl; V~R mUD _sm.'" ;
.
%
.
"P.CROS-
MACRO _S!RI,lAR ENRIJIlT %
~p'cR~ _DSFRA~[ IHIt. SCHOOL
NIltRO _,AHPSZ 81 %
NACRO .BSSIIIiP OUII %
SYSSAMPl
1/
71
79
8e
%
HACRO SYSSAHP I --
3
IB
16
11
29
35
42
48
[HRL!NT
133
134
13.
134
235
236
237
237
436
441
449
783
783
783
461
46B
474
78+
73S
185
785
487
785
897
SS6
m
m
m
4~9
Sij£
373
275
EXHIBIT 2
JUJUtUlttflltffllllllltHfllltllUl!llifl.IIHIUllllllfillUJiUII
II
II
II
II
II
II
TO W,IQI<E
~RCfO
m {:[QUIREI:
SYSSA!P1 THE FBLLOWING USER-!EflNEI NACROS
II
.
NACRO _STRUHR
A I'JRRIAIL[ TO IE USE] IN IHPLICIT
STRATIFICATION I
II
HOCRO _SAHPSZ
B ISRHPLE mE BESIRED I
II
!ACRO _ISFfI1l1E
D ( DI\1& SIT NANE CONmlNING THE
II
';
II
FRI\!iE )
II
MACRO
_DSSA!~
II
II
MACRO _HEASRSl
n
II
';
II
I( ..
Ij II
I;
II
II
II
u---II~IMIHG R[~I)IF..~!1
ll~lHVOK1NG
16
77
UIlIUll
E IOIJTPlJT !AlA SET CONTRINING THE n
II
SELECTED SAHPLr I
I;
f INERSURE or SIZE IJARll!lIlI 10
H
IE USED IN PPS SELECTION)
'(
Ii
u 'OI[:SINCE THE UNIUERSE SIZE II.I IS lETERHIIlEJ IY THE IIJHIER Of
II
OBSER'JATIGNS IN THE fRR~G ANY OBSERVATION WITH AHISSING
'JALUE FOR THE AUXILIARY 'JHR1ABLE _SWJAR SHIJUIJ! BE
BEllITl. BEfORE SAHPLE SELECT! ON
IT
U
U
II
II
II
'i"
'( " ,OT[: THE C[RTAINT'I CUTOff IS .7SI (SELECTION INTER\'Rl IKI)
u
I;
I;
~
ttfilHHIHHIHHIHlllHElJl-HiHlUHHIHIHHHHHHlHBHHi
767
fI--fROGRRlI CODUOR !ffCRO SYSSfiltPMACRO SYSSRlif'~
II
••,
••,
PROC SORT IAT~ ~ _DSfRflME OUT~DDl;
IY _STF,~RR ;
DAm DI2ItEEP=TOTHOSI ; SET 111 EHI'i:NIOr;
IOIMOS + ~~[flSRSl ;
IF E~]OF ~ 1 TII8', OUTPur ;
),TA IIIIOROP ~ wnm, TOTMOS GERDIOSI D13; MERGE DDI DDo
RETAIN GEm,OS ;
UNILNO= _H_ ;
IF _H_ ~ 1 THO! 10 ;
CEP.T!OS.JiOUNI( ITDT~OSI _SAIIPSl I , .BI i
1I---'EiUIREI USEP.-lEFlNEU HRCRO$-
••,
II
~HCfO _DSFRAME A %
~fiCRO _Sfit!?SZ 88 %
I,ACRO _m~AR
~RCRO DSSRXP OUT! %
smslZE
%
!HCp.e _mSRSZ ENf:L!NT ),
II
••
iI---IN'!OKING RACRO S'tSSR!PI .n
SYS5R:,\P2
'1
END;
IF J\EASRSZ GT .71ICERll\OS lllEN DO ;
CERTRIN ~ 1;
OUTPUT 013 ;
DiD;
ELSE OUTl'UT DDI ;
PROC PRINT DATfl~I'3 ;
IIllE PPS SfiltPLE SELECTION;
1IlLE2 CERTAINTY CASES; II mmlNi
VAR UHITYO _~[ASPSZ ;
n~TR'
DD3(Y.EEP:;(HTPIN1;
IF
E!llJOF~1
ltAfP. H4
~:[T
SAMPLE OUTPUT
PPS SA~f'lE ~:[lECTIOH
CERTAINTY CASES
If _N_
~
j
331
332
333
1695
269B
3267
PPS SAIPLE SELECTION
THE r.U!BER OF HOH-(;E£TRINTY CASES IN FRAHE IS
THE NON~;ERTRINTY SR!PLE Sill IS 77
THE SElECTION INTEP.'JAL IS 1109.13
THE RRNDO! STRRr IS 1613.889
338
PPS SA!PLE SELEtTlON
NINCEliTAINTY SR!PLE SELEtTTOIIS
UNIUO llumms TJj[ LINE HUM OF THE SR!PLE UNIT IN THE
_S~PSEil
I
1
1
4
5
6
7
8
9
MEf'GE PD5 !lD4 j
11
IB
PI'T _PReL I I /1/ !69 'PPS SA!PLE SELECTl ON ' III II
11; 'THE NIJHBEP. Of H['N-tERTAINTY CRSES IN FRAHE IS 'NOHCEP.T I f l
m 'THE NON-{UTRINr( SAMPLE SIZE IS' NSA!P Ifl
m 'Til[ SELECTIOh IHTERl'f!l IS ' INITP.URL /II
!JB ' ThE RANIOH START IS 'RANDOH;
'P,)[ PRINT DATA~IJ' ; TITLE PPS SAMPLE SELECTION;
'!TLE1 hONeERTAINIY SAmE SELECTIONS;
TlTm UHILNO lDENTIFlE'3 THE LINE HU! OF THE SftHPlE UNIT IN THE FRAIi£;
I) _SA~PSEQ; 'JftR _!ERSRSZ IJNILNO CUH!OS ;
I'm Ai SET INA. SCHOOL
SC~L812[; EHRL~Hr j
I;
11
11
13
11
IS
1f
7B
71
71
73
71
7S
I
7.
77
~
~
,t
1
3
tISH~F') j
'I L[ ?W.T NOTl!LES ;
!
i,
!
EHRLMNT
Bn n1D=EHx,r;
(r[EP:TOT~OS ~[;HCERT
_HULL
UNIT JlO
meN [uTPUTi
SET 111 ENI=l:HIOF ;
TOTIOS + _~EASRSZ i
HOHCERT + I;
IF ENIOF = 1 THEN OUTPUT ;
lATA mil. ; meE III mill ;
RElAIN RfiHDOI INTER'JAL ;
IF _N_ ~ I THEN DO ;
IF CERTAIN HE • THEN "SA!P~ _SR!PSZ - CERTRIN ;
ELSE nSA!P~ _IAr-PSI ;
INTEP.VAL = ROUND I TOT!OS/NSA!P, ,ell ;
y.AHIO~ = I INTERVAL I
UnIFORHIBI + .i9 ) i
CUmAS ~ INT ( RANIa!
I j
,QUTPUT DIS ;
ENI;
W!HEAS + _HERSPSZ j
!IV~ INT ICUM!ERS IIHTEP.l'AI. I ;
CU'HEAS ~ !OD ( CtH!EAS, IHTERUAL I
C"!OS + _MEASRSl ;
If DIU GT e THEN DO ;
_SRIPS[Q + 1 ;
QUTF'UT libi .
EHD;
I'~TP
mTRIN
768
ENRl!NT
UNIUO
133
23113.
237
1
18
19
17
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43
51
m
254
255
15.
263
161
165
265
166
267
167
276
BBI
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824
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867
873
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315
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EXHIBIT 3
UIUIIIIUIIH fttill J"I 1111i'Ji'UUIII HI If n l t' fli JIU '1IIJHIUn
H
I;
n TO !H'JG~E ~PCR(I USti~~ T~[ rOLLClo!J~G USE~:-D~tI~n ~HCROS
n
m REQUIRED:
..
MHCRO
I;
II
_!SIl~!E
R( INPUT !ATA SET
DITR I
u
CO~IIAINmG
THE
II
{lRT~
.
or UARIRILES
FUR WHIC~
'IHRIRNCES ARE TO tE CIl!PUTED)
F lUST
_'.IPRUST
:
..
C ( S~RT l,!ARlfiBLE TO BE USED TO
H
ORDER THE fiLE AS IN
n
SELECTION
..
SA~FLE
)
URP.lBL:: RLBl i
Ii
Ii
c"lEF _'JAR 'ReU ;
':lfP HU!_OBS ESmATE UARIAHC[
:'UTPUTi
[J
II
iDlE: I:EFOF;[ 1!3IHG THIS FROC£DURE THE TIATA HUST SE
PP.CPERD' WEll;HED
[J
JlLILLi. SET DM j
F'LL PRINT HQTlTLES ;
:r _KA THEH PUT _PACL III
H1 'SUCmmE DIFFERENCES UA'IR~CE [SmATION PROCEDURE'
,38 'NU!BER OF' !6! 'ESmATEII' 1115 'COEFfICIENT OF' I
['Ar~
~~(RG SYS~AP.
r'DU SET _DSNAKE i
f£EP )J8P.LIST
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~s '1,IRRIASlE LABEL'
:~!18 'UflP.HtTIOI'f' l
j
~ ~X SO~T
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TIflTF=!IDl; R'f _SF,XPSEQ ;
DDI; 3ET ~Dl ;
~5
!'
",'AY REST J'RRllST;
~fJAl ,DIF I'lFHIF2S
DO D'JER Am;
,IIF' IIIIF!AESTl "1)
~'~1
@62 'TOTRl'
@92
J~IARIAHC[,
14['_' @36 141 ,., ~59 111 I_! @98 111'.' !115 141-'.'
!Ie VARLBL E4e NUH_OBS 161 [smAlE
I
l.!ARIANCE @128 COELVHR 4.2 j
"
R j SET (tU.CmlPUAI?S ;
I' L LT !se ~ND ELE(BTU HE • ; ;
I"~TP
;'~RRV
REST _VARUST ;
~~'RHY
~~,~:AY
RSIGK S!GI'iI-SIGX25 ;
,
,
II
II
C:UTPllT CUT =TIE3 n=H l-N25 SUM=Dlf1- U F2S i
:;r~ ['n; ~ERGE DII2 IID3 i
I'
I'
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II
II
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ARRAY R[:U C'!! -(V1S ;
110 OIJER fiESTi
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IF REST'9 nEH Rev, e ;
EL~E ACi,I = 5QF:TlR51G~ l lHESI ;
ENE;
F'Roe PI.!NT ;
~·p'OC r;~U:IX ;
FETCH X 1,Al~;; I ~;: !:.~ ~?',~~E =;;?~ I~.i
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.:
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r1_;fPUT GUT=III'2 SIJ~ =_I,iARLIST j
~'P'::(' ~,EA~S N su~ DRTR=[I[I\ HOF'R!HT ;
:,:fR nIFl-illF2Si
OUTPUT I.iARE:
VRRLBL ;
COEUR~:
,~D;
U
H
~
;IU'iIHIIIUIIHUIIJIHIUIIIUIIUIUIHIIIHHHfIlIIlHIlIlllllI,
!'~T~
E=:UP(IP:~Q~l;
ESmRTEoAEST ;
IjARIRHCE,ASICK ;
u
II
rtflCRO _SR~P-SEQ
DD3
A~WI;
NUK_OBS=
I;
II
u
r.u:;~[
ARRRY REST _~ARUSf ;
ARRAY ASIGli :IG!1-SIGlI1) ;
ARRRY Reu CIJI-CU2~ ;
ARRRY RLBL $ B COLHOL1S;
DO OVER REST
I;
~RCRO
II
II
In::
H,~A( F~l!~ ~1-N(5j
~~'['[<~~R;
SAMPLE OUTPUT
SlI(CES)lt!E DIFFERENCES IJARIANtE
HI.l~BEP.
i.iRR1ABlE LP.EEL
or
S9FEET
mem
10mu
PROCEDURE
CT~H
ESTl~ATE:'
:jBS£F.I)RTIO~S
------------
Enf'L
EST:~m"~
67
13
13
13
TOTAL
IJRF:!f-f1CE
----------
--------
9694,5
4,~?,c;~q
,t:~
769
,.
2917144
%E.2132B!4Ba
618. ,'i, I
~9t53,25
1582. G
539659.5
,~
[(LENT OF
i.iF.FIATICIK
---------
B.18
8.11
9.1a
1.46