Download Generalized Program for Stratification & Strata Deterioration Analysis-GPSSD..

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GENERALIZED PROGRAM FOR STRATIFICATION & STRATA DETERIORATION ANALYSIS - GPSSD
Bonnie Brown Jacobson, Northeast Utilities Service Company
Where:
Most utilities use load research data collected
from load research studies of various subgroups
of their customer population. It is hoped that
the results of these studies will yield accurate
profiles of the demand patterns for these subgroups for use in ratemaking. forecasting and
load management.
,t
numb~r
of slrata.
t is the Student's t-value associated
with the desired confidence interval.
a~
is the population standard deviation
for the ith stratum.
This program is a generalized routine for the
calculation of the required sample size needed
to satisfy the confidence limits of 90% ± 10%
and 95% ± 5% [or rive separate sampling designs
all of which are currently utilized load research designs. Each design is further explored
through the calculation of the approximate deterioration of data for each stratum. Summary
tables are also generated for quick reference.
e. is the
per~entage of the mean for the
ifh stratum relating to the fiducial
limits.
OPTIMUM ALLOCATION:
This design requires a simple random sample to
be drawn wi thin e"'c.h str",tum utilizing the most
cost effective method for acquiring data relating to the population mean. Thus, the design
yields statistically reliable data (for the
chosen confidence limits) relating to the mean
of the population. An estimate of r_eliability
is made by the program for the individual
stratum, but this reliability is for the most
part much more conservative than the overall
population mean reliability. The formula for
the sample size is:
The sample sizes required [ur Lhe appropriate
operation of each of the following sample
designs are automatically calculated in GPSSD,
based on the assumption that the population size
is large.
OVERALL SRS:
This design requires a simple random sample to
be drawn from the total population without
regard to strata boundaries. This would yield
data relating to the means that ure statistically reliable (for the chosen confidence
limits) for the total popUlation. No estimate
of reliability can be made for any individual
strata before the sample is chosen. The formula
used for the sample size is:
j
n
E
i=1
Where: j is the total number of strata.
N. is the population total within the ith
shatum.
Where: t is the Student'S t-value associated
with the desired confidence interval.
N is LIte total population.
o. is the population standard deviation
f6r the itb stratum.
a is the standard deviation of the
population.
t is the Student's t-value associated
with the desired confidence interval.
e is the percentage of the mean relating
to the fiducial limits.
e is the percentage of the population
mean relating to the fiducial limits.
SRS WITHIN STRATA:
SRS WITHIN STRATA-STABLE:
This design requires a separate simple random
sample to be drawn from each of the pre-assigned
stratum. The design would yield data relating
to the mean that are statistically reliable (for
the chosen confidence limits) for the individual
strata as well as a more rigorous reliability
for the total population. The formula for the
sample size is:
This design is identical to the aforementioned
SRS within strata. N is raised to 30 fur any
stratum with an N of less than 30. This has
been determined to be the least number of load
research meters that allow a stable result.
Thus, the sample yields statistically reliable
data overall and per strata.
j
n
OPTIMUM ALLOCATION-STABLE:
= 1:
i=1
Similar to the above design. this design is
identical to the aforementioned optimum allocation. Again, N is substituted by 30 if it is
calculated to be less than 30. This design
ff
,t
is the total
592
statement is required.
yields statistically reliable data ovt:!rall and
within strata (although somewhat more conservative than its SRS counterpart).
THE SUBDES
PROC SPECIFICATIONS:
SUEDES variable;
GPSSD is invoked by a PROe statement and controlled by the following other statements;
The SUBDES statement identifies the variable
name containing the subpopulation description.
I f the SUBDESe statement is omit-ted. the subpopulation description is equal to the population description.
PRoe GPSSD options;
MEAN variable;
STD variable;
N variable;
DESe variable;
SUBDESC variable;
STRDES variable;
STATEME~T:
THE STRDES STATEMENT:
STRDES variable;
THE PROe GPSSD STATEMENT:
The STRDES statement identifies the variable
name containing the stratification variable
description. This statement is required.
These options may appear in the PROe GPSSD
statement:
THE INPUT DATA SET:
ThE> input data sP-t must contain hoth the t_otal
population data and each of the population
stratum data and optionally, the total subpopulation data and each of the subpopulation
stratum data. The input data set must take the
following general form:
STRATA "" n or
STRAT = n
or
S : n: specifies the number of strata desired
for all sample designs, between 1 and 5.
It the STRATA = option is omitted from
the PROC statement, the default strata
value is S. If more than 5 strata are
required, run the PROe several times.
DATA
Line 1: (Total population) mean, standard
deviation, size, label, stratification
variable label, (optionally) Bubpopulation label.
data_set: specifies the SAS data set
containing the population parameters to
be utilized in the sample design. If the
DATA = option is omitted from the PROC
statement, the most recently created SAS
data set is used. See the section concerning the input data set for specific
rl~t~
RPt
Line 2: (Stratum 1) mean, standard deviation,
Size, label, stratification variable
label, (optionally) subpopu1ation label.
Linl:! 2 is repeated (up Lo 4 times) ont:!
line per stratum.
This general population information may optionally be followed by any subpopulation information. Up to 4Y subpopulation data groups may
follow. The form is the same as the population
(i.e., first line is the general subpopulation
data followed by lines of stratum data). All
results from these groups will be weighted
eRtimatp_H whi ch Himlllatp prohHh1 e co11 ec_tion
data. The order of the information in each
line is arbitrary. Any value found missing is
treated as a "zero" and the computation of
sample sizes proceeds accordingly.
form~t.
THE MEAN STATEMENT:
MEAN variable;
The MEAN statement identifies the variable name
containing the population mean numbers. This
statement is required.
THE STD STATEMENT:
srD variable;
OUTPUT:
The STD statement identifies the variable name
containing the population standard deviation
numbers. This statement is required.
The first section of output indicates the
required size of the sample for the overall SRS
(90% + 10%, 95% + 5%, the SRS within strata
(90%
10%, 95%
5%) and the optimum allocation (90% ± 10%,-95% ± 5%). The SRS within
stratum and optimum allocation are further
explored. Each stratum sample size is listed
with the expected standard error of the mean
and the standard error as a percentage of the
mean [or l'ac\t stratified sample design. This
information is also printed for the total
sample for each stratified design. With this
information, the analyst can determine the
likely accuracy of the data to be collected from
each design.
+
THE N STATEMENT:
The N statement identifies the variable name
containing the population size numbers. This
statement is required.
THE DESC STATEMENT:
DEse variable;
The DEse statement identifies the variable name
containing the population description. This
593
+
The second section repeats this stratified
information (since the weight for each stratum
with regard to the original stratified population is equal to 1). If, for any of the above
stratified designs, the stratum sample size Is
less than 30, it is raised to 30 and re-evaluated. These analyses yield sample sizes for the
SKS within strata-stable (90% + 10%, 95% + 5%,
and the optimum allocation stable (90% + 10%,
95% ± 5%) designs.
-
ACKNOWLEDGEMENTS
I would like to thank the following people
for their help in the development of both
lhe PROC and this paper:
Ms. Karen E. Gree_Iey. Mr. James D. Oleksiw,
Mrs. Jean H. Ehle.
REFERENCES
Cochran, W. G., Sampling Techniques Third
Edition, c. 1977, John Wiley & Sons, Inc.,
New York. New York.
If additional sUbpopulation and/or stratification variable data cards are_ included in the
input data, the sample design process is repeated. Since all evaluation is done in
relation to the original strata boundaries,
these analyses can at best be only estimates of
the accuracy of the data to be collected.
Kish, L., Survey Sampling, c. 1965, John
Wiley & Sons, Inc., New York, New Yurko
For more information, please contact Bonnie B.
Jacobson, Consumer EconomiCS, Northeast
Utilities Service Company, P. O. Box 270,
Hartford, CT 06101 or call (203) 666-6911,
Ext. 5030.
The first He_cti on of the optional data analyses
is unweighed. It is only intended to give the
analyst an idea of the sample size required if
the subpopulation or new variable were the
original.
The second section of the optional data analyses
reflects the potential behavior of the suhpopulation and/or new variable within the
original population-stratification variable
framework. All sample sizes are weighted to
simulate the true sample size applicable to the
optional situation. The analyst should keep in
mind that this information is an estimate of
variable accuracy.
Following the optional analyses is a summary
table for each variable and population/subpopulation situation. Listed are the required
sample sizes for each stratum within each
sample design. Also listed is the value of 100
minus t times the standard error (expressed as
a percentage of the mean) for the "best" and
"worst" strata. This quantity is relative to
tht! mt!an in that it is a measure of how close to
the true mean the stratum should result. The
best possible stratum would result in a value of
100.00. Evaluation should be based on this
standard.
The final summary table lists the overall
slandard error times t expressed as a percentage
of the mean for e_ach sample design and for each
variable and population/subpopulation situation.
594
INT ._cotI1ERCIAL
SUBSTRATIFICATION VARIABLE; HAX._KW
SUBPOf'ULATlON: IN! ._COMtlERCIAL
SAt1PLE ·SIZE
WITH
T-VALIJE :;; 1.645
POPULATION
----------
POP MEMI
POP STO
7.(90)
7.(95)
UNSTRATIFIED
9839
169.000
166.000
.100
• 050
STRATAti
STR.l.TA 12
STRATA 13
STRATA #4
STRATA 15
4026
252&
1573
1064
65.000
llJ.OOO
198.000
349.000
671.000
10.000
1&.000
33.000
57.000
152.000
.100
.100
.100
.100
.100
.050
.050
.050
.050
• 050
...
STRATA
ACCURACY
----------
OPTIMUM
ALLOCATION
14&3
...
•1
1•
7•
.17
11
•• 1
51
l4.-943
&.842
3.696
2.187
8.228
4.869
,,
••
7
••
•
....
2
7.357
4.353
7
37
••
SAMPLE WITH T-VALUE .. 1.645
SAMPLE WITH T-VALUE .. 1.960
------_.------------------------------------------SAMPLE ACCLRACY
SAI1PLE ACCURACY
----------------------------T •
T •
sm
TOTAL SAMPLE
STRATA
ACCURACY
OPTItruM
ALLOCATION
--------------------------------------------------SAMf'LE ACCURACY
SAMPLE ACCURACY
----------------------------T •
T'
SAMPLE
SIZE
STRATAtl
STRATA 12
STRATAn
STRATA 14
STRATA IS
WITli
.6 •
TOTAl.
SAMPLE ACCURACY: T • ST AtlJARO ERROR OF MEAN
7. OF TOTAL POPULATION MEAN
SAMPLE SIZE
T-VALUE:;; 1.960
ERR
OF
MEAN
7. OF
'OP
MEAN
7
6.,
14
•
10.5
19.2
33.2
66.6
9.565
9.431
9.693
9.499
9.959
45
7.'
4.353
••
ST1J ERR
SAMPLE
SIZE
OF
7. OF
'OP
MEAN
MEAN
,•
,•
•
....
11.6
20.9
66.3
144.4
17.895
18.263
19.3&7
16.998
21.514
14.9
8.842
11
SAMPLE
SIZE
..
37
'T' ERR
OF
MEAN
X OF
pop
MEAH
79
17.4
33.5
4.957
4.964
4.962
4.999
4.995
241
'.7
2.187
41
41
3.2
•••
•••
SAMPLE
SIZE
'TO
ERR
OF
MEAN
7. OF
'OP
"EAN
7
7.'
17
"
12.5
21.6
35.3
72.3
ll.397
ll.237
10.689
10.te3
10.768
51
8.'
4.869
••
EVAWATION BASED ON lUMBER OF METERS FROI'1 ORIGINAL STRATIFICATION VARIABLE ANALYSIS (WEIGHTEO)
SAMPLE WITH T-VALUE
=
1.645
--------------------------------------------------SAMPLE ACCURACY
SAMPLE ACCURACY
----------------------------T •
T
sm
SAMPLE
SIZE
STR.&.TA
STRATA
STRATA
STRATA
STRATA
II
12
_3
14
15
TOTAL SAMPLE
7
••
•
14
••
ERR
OF
"EAN
pop
MEAN
33.2
66.6
9.565
9.431
9.693
9.499
9.959
7.4
4.353
6.'
10.S
19.1
•ERR
ST.
7. OF
SAMPLE
SIZE
,••
,•
11
OF
7. Of
POP
MEAN
MEAN
11.6
lO.9
38.4
66.3
144.4
17.695
18.863
19.387
18.998
21.514
14.9
8.842
SAMPLE WITH T-VALUE
= 1.960
------------------------------------------------------------------------------T'
T'
SAMPLE ACCURACY
'TO
SAMPLE
SIZE
37
ERR
X OF
OF
MEAN
•••
•••
POP
MEAN
79
17-4
33.5
4.957
4.964
4.982
4.999
".995
'41
'.7
2.1&7
41
43
41
5.5
SAMPLE ACCURACY
STD ERR
OF
SAMPLE
SIZE
"EAN
7
8
1. OF
POP
MEAN
17
35.'
72.3
lL397
1I.l37
10.8M
10.121
10.768
Sl
8.'
4.869
7.'
12.5
21.6
•
10
EVALUATION WITH MINIMUM 30 METERSISTRATA (WEIGHTED)
SAHPLE WITH T-VALUE = 1.645
--------------------------------------------------SA.MPLE ACCURACY
SAMPLE
----------------------------T •
T •
ACC~ACY
SAMPLE
SIZE
STRATA
STRA.TA
STRATA
STRATA.
STRATA
11
12
13
14
15
TOTAL SAMPLE
!l
!,.
,.
30
30
3.
3.
3.
IS.
STO ER"
OF
NEAtf
3 .•
5.4
9.'
17.1
45.7
•••
X Of"
'OP
"EAN
SAMPLE
SIZE
4.621
4.870
5.006
4.905
6.803
3.
30
3.
3.
3.
2.354
15.
'TO ERR
OF
"E,,"
3 ..0
5.4
r.
SAMPLE WITH T-VALUE
SAMPLE ACCURACY
MEAN
SAMf'LE
SIZE
17.1
45.7
4.621
4.870
5.006
4.905
6.803
37
41
43
41
79
4 ••
2.354
241
•••
595
SAMPLE ACCURACY
---------------
Of
POP
= 1.960
--------------------------------------------------T'
.TO ERR
OF
MEA~
•••
r.
--------------T'
STD ERR
Of
POP
MEA~
SAMf'LE
SIZE
OF
MEM
'.6
r.
OF
POP
""AN
3.
3.
17.4
33.5
4.957
4.964
4.982
4.999
4.995
3D
3D
3D
54.4
5.505
5.803
5.964
5.844
8.106
'.7
2.167
15.
'.7
2.604
5.5
'.9
6.4
ll.8
20.4
nrr ._COMMERCIAL
SUBSTRATIFICATION VARIABLE: ANN_~WH
SUBPOPULATION: INT ._COJ1MERCIAL
SAMPLE SIZE
SAI1PlE SIZE
IoIITH
WIlH
T-VALUE
POP S10
Z("O)
:1.(95)
459537.000
681517.000
• 100
.050
4025 140437.000
2526 256701.01)0
1573 552468.000
1064 1027734.01)0
648 2073734.000
82115.000
146655.000
753941.000
520079.000
1073912.000
.100
.100
.100
.100
.100
.050
.050
• 050
.050
.050
POPULATIOt-l
---------tJt.ISTRATIFIED
STRATA
STRATA
STRATA
STRATA
STRATA
#1
12
13
14
15
9836
POP MEAN
STRATA
ACCURACY
.
.,
,*
STAHDARD ERROR OF MEAN
OF TOTAL POPULATION MEAt-I
SAMPLE WITH T-VALUE
= 1.645
SAMPLE
SIZE
STRATA
STRATA
STRATA
STRATA
STRATA
#1
12
13
14
#5
TOTAL SAMPLE
'"
MEAN
sm
,. OF
POP
~EAN
504
70
73
14007.1
25572.2
55244.4
102255.4
20b763.2
9.974
'7.962
10.000
9.950
9.971
629
11.601.4
3.613
93
"
EVALUATION BASEO ON
ERR
FOP
MEAN
MEAN
STRATA
STRATA
STRATA
STRATA
STRATA
#1
12
13
14
#5
TOTAL SAMPLE
7
6
OF
'.R
"
70
73
MEAN
14
51055.1
85293.9
43e488.6
302475.6
47Z.139.9
36.354
33.227
79.349
29.431
2Z.766
4'
77827.7
16.936
••
••,.
502
1573
394
413
4•
23
"
Z43
114
143
."
12.
3408
644
16601.449
3.613
45566.820
9.916
5895.426
1.283
22947.754
4.994
= 1. 960
--------------------------------------------------SAMPLE ACCURACY
SAMPLE ACCURACY
--------------T _
--------------T_
SAMPLE
SIZE
52.
STO
OF
;I, OF
'"
POP
MEAH
MEAN
S10 ERR
OF
SAMPLE
SIZE
••,.
MEAH
;I, OF
POP
~EAN
4.994
30101.4
6228'7.9
179012.2
178390.3
32:8046.7
25.706
<:4.<:66
32.402
17.358
15.819
= 1.645
.02
,,.
1573
413
7017.6
Ha'9.'
0.'
51354.4103573.7
.4.997
4.998
0.0
4.997
4.995
243
114
143
SAMPLE WITH T-VALUE
--------------T_
STD ERR
i! OF
POP
...
,.
I'
.44
"""""
MEAN
3380
22947.8
3408
5895.4
1.283
129
45566.8
9.916
OF t1ETERS FROM ORIGINAL STRATIFICATION VARIABLE ANALYSIS (WEIGHTEOJ
23
29
---------------
SAMPLE
SIZE
OPTIt'f1Jt1
AlLOCATION
13.898
12.845
17.159
'7.289
8.488
I.
--------------------------------------------------SAI1PlE ACCURACY
SAI1PlE ACCURACY
T -
= 1.960
19517.5
32912.1
94796.1
95471.3
176017.8
14
,.
SAMPLE WITH T-VALUE
'TO
----------
.04
,. OF
OF
SAMPLE
SIZE
STRATA
ACCURACY
OPTII1IJt1
AlLOCATION
SAMPLE WITH T-VALUE
--------------------------------------------------SAtlPLE ACCURACY
SAMPLE ACClmACY
--------------T _
T _
--------------STO
OF
1.645
93
TOTAL
SAMPLE ACCURACY: T
=
T -VALUE
SAMPLE
SIZE
,
,,,
;I, Of
POP
OF
MEAN
MEAH
= 1.960
--------------------------------------------------SAI1PLE ACCURACY
SAMPLE ACCURACY
--------------T _
--------------T_
i! OF
STD ERR
SAMPLE
SIZE
68.011
66.454
158.738
58.863
49.184
"43
3
95515.4
170567.8
876977.6
604951.3
1019938.7
37
11
157965.4
34.375
POP
OF
MEAN
MEAN
79
26459.3
44891.2
225350.8
159196.4
236516.1
18.841
17.488
40.790
15.490
11.420
241
39251.6
8.542
41
STD ERR
SAMPLE
SIZE
OF
M''''
i! OF
POP
MEAN
17
6t1831.6
101626.• 7
49tS74.7
322348.3
510505.3
43.316
39.590
89.159
31.365
2'1.616
51
87051.5
18.9:+3
7
•,
10
EVALUATION WITH MINIMUM 30 METERS/STRATA (WEIGHTED)
SAMPLE WITH T-VALUE
= 1.645
SAMPLE WITH T-VAlUE
~--------------------------------------------------
SAMPLE ACCURACY
SAMPLE
SIZE
STRATA
Sn!ATA
STRATA
STRATA
STRATA
.1
#2
#3
#4
tiS
TOTAL SAMPLE
T 'TO 'RR
OF
M'AN
,
I! Of
POP
MEAN
..
SAMPLE ACCURACY
--------------T_
--------------SAMPLE
SIZE
'TO
OF
MEAN
MEAN
STD ERR
SAMPLE
SIZE
30
30
30
30
30
24662.0
44045.6
226434.6
156197.7
322532.9
17.561
17.158
40.986.
15.1"8
15.553
30
30
30
30
30
24662.0
44045.6
226434.6
156197.7
322532.9
17.561
17.158
40.tlB6.
15.198
15.553
37
41
.3
41
150
42163.7
9.175
150
42163.7
9.175
596
SAMPLE ACCURACY
--------------T_
Y. OF
POP
= 1.960
-------~------~------------------------------------
SAMPLE ACCURACY
OF
MEAN
--------------n
Y. OF
POP
MEAN
79
26459.3
44891.2
225350.8
159196.4
236.816.1
18.841
17.488
40.790
15.4<;0
11.420
241
39251.6
8.542:
S10 ERR
SAMPLE
SIZE
30
30
OF
MEAN
7. OF
FOP
MEAN
48.834
30
30
29384.5
52479.8
269794.3
186107.9
384294.5
150
50237.5
10.932
"
20.924
20.444
18.109
18.532
POPULATlON:
INT "_COJ1t1ERCIAL
DETERIORATION OF STRATA
-
5USSTRATIFICATION VARIABLE; HAX. KII
SL'BPOPULATlOJ.!: INT._COMMERCIAL
100;': -
5
• STRATA '"
,
IT * s.E.1
STRATA
mETERS fOR:
.-..-..-..-..-
------------
~Oi::
95i::
90;':
~5i::
~Oi::
~5%
90?
'lS?
~Oi::
'lS?
IOi::
Si::
10;':
5i::
10i::
5i::
JO?
S?
10i::
5?
7
37
2
7
SRS
SRS
OPT
OPT
OVERALL SRS
OVERAl.l S~S
SFlS-STABlE
SRS-STABLE
OPT-STABLE
OPT-STABLE
POPULATION:
3
•
• • •
5
41
2
43
2
41
2
10
14
7.
3
17
30
30
43
30
30
30
41
30
3.
30
7.
30
30
• •
TOTALS
,.,
45
11
51
262
(AS l:: OF MEAN)
WORST
BEST
STRATA
SlRATA
90.57
95.04
82.10
89.88
90.04
95.00
78.49
88.60
95.38
95.04
95.38
94.49
93.20
95.00
93.20
91.89
1483
30
37
30
3.
41
30
3.
ISO
241
ISO
ISO
tNT ._COMt1ERCIAl
DETERIORATION OF STRATA
SUBSTRATIFIcATION VARIABLE: 'NILKWH
SUBPOPULATIOJ.l: INT._COMMERCIAL
• S-mATA
=
5
•
#METERS FOR:
90i::
'75;':
90i::
1;57.
90i::
95i::
90i::
95i::
90?
95?
.-..-...-.-..-.-
------------
-
10i::
S?
10i::
5i::
10i::
Si::
10i::
5?
lOX
5i::
T It A T A
•
•
•
•
.,
• •
2
7
37
2
7
SRS
SitS
OPT
OPT
OVERALL SRS
OVERALL SRS
SRS-STABlE
SRS-STABLE
OPT-STABLE
OPT-STABLE
POPULATION:
3
2
43
2
41
2
10
30
41
30
43
30
41
3.
30
5
TOTALS
14
7.
3
17
45
241
11
51
262
1483
3•
37
30
30
"30 3."
30
7.
30
30
IS.
241
IS'
150
100;': IT * 5.E.l
(AS ;.: OF MEAN)
BEST
WORST
S-mATA
STRATA
.....
.....
.....
n.23
SO.82
75.38
20.63
59.21
31. 99
10.84
84.8ll
81.89
!>9.01
59.21
59.01
51.17
INT._COMMERCIAL
OVERALL SUMI'IARY TABLE
T
*
S.E. AS X OF MEAN
lBASEO ON ORIGINAL METER ALLOCATIONS)
DESIGN
90;':
95;':
907.
957.
907.
957.
907.
957.
907.
957.
.-...-
......-
10;': SRS
4.353
16.936
'.0
•••
•••
'.0
57. SRS
2.187
8.542
0.0
0.'
0.0
0.0
107. OPT
8.842
34.375
0.0
0.0
0.0
0.0
57. OPT
4.869
18.943
0.0
0.0
0.0
0.0
107. SRS STABLE
2.354
9.175
0.0
0.0
0.0
0.0
57. SRS STABLE
2.187
8.542
0.0
0.0
0.0
0.0
10;': OPT STABLE
2.354
9.175
0.0
0.0
0.0
0.0
57. OPT STABLE
2.804
10.932
0.0
0.0
0.0
0.0
107. OVERALL SRS
9.982
15.072
'.0
0.'
0.0
0.0
57. OVERALL SRS
4.999
7.548
0.0
0.0
0.0
0.0
597