WMOD00440

• Title: HIPLEX-1: Statistical Evaluation
• Date: 4/4/1984
• Weather Modification - Doc Type: Report

514

JOURNAL OF CLIMATE AND APPLIED METEOROLOGY

VOLUME 23

analysis space which is congruent to the data space
being investigated (Mielke and Berry, 1983; Mielke et
al., 1982). Then the MRPP statistic used in this study
is given by

g (ni)
t5=L N ~i,

,~1

where ni ;;;. 2 is the number of objects in group Si
(i = 1, . . . , g),

g
N= Lni,
i~1

( )-1
ni
~i = 2

L A/,Jtfii(WI)tfii(WJ)
I<J

is the average between-object distance for all objects
within group Si (i = 1, . . . , g), L is the sum over all
I<J
I and J such that 1 .,;; 1< J.,;; N, and tfii(WI) is 1 if WI
belongs to Si and 0 otherwise (i = 1, . . . , g; I = 1,
. . . , N). The underlying permutation distribution of
15 (the null hypothesis) assigns equal probabilities to
the

g
M = N!(II ni!)-l
i~l

possible allocations of the Nobjects to the g groups.
Since small values of 15 imply a concentration of re-
. sponse measurements within at least some of the g
groups, the null hypothesis is rejected when the ob-
served value of 15 is small. The exact P-value (i.e., the
probability under the null hypothesis of a value of 15
being as or more extreme than the observed value of
15) is the proportion of all M values of 15 which are
equal to or less than the observed value of 15. Berry
(1982) has developed an efficient algorithm for cal-
culating exact P-values; it is practical for values of M
up to 20 000. EffiCient and accurate moment approx-
, imation procedures exist for calculating P-values for
large values of M (Mielke, 1979; Mielke et ai" 1976,
1981a, 1982). It should be noted that when a single
response variable is analyzed, then MRPP is a uni-
variate analysis procedure. However, the term MRPP
will designate both univariate and multivariate analyses
in this paper.

3. Preliminary sample size estimates

Because the power characteristics of permutation
tests, such as the MRPP, are highly dependent upon
both the actual distribution and the hypothesized al-
ternative in question,' it is important that reasonable
approximations be obtainable for both these entities.
To this end, a simulation program was designed to
yield power-characteristic results and sample size es-
timates forHIPLEX-l. The data base for this simu-
lation consisted of measurements on CIC5, TFPI, and
PIC8 collected on 35 clouds (11 of which were seeded)

as part of the calibration seeding trials during the sum-
mer of 1978 and prior to the actual experiment. Ap-
pendix E of the design document (Bureau of Recla-
mation, 1979) provides details of the data set. From
this data base, four sets of random samples (N = 50,
100, 150, and 200) were drawn with replacement.
For analyses involving more than a single response
variate (i.e., joint effects), the investigator must insure
that the response measurements are commensurate,
i.e., the sample ranges of the different response variates
must be equalized. Let the kth response measurement
for a seeded case be represented by XkjS (k = 1, . . . ,
rand) = 1, . . . , ns); similarly, let the kth response
measurement for a nonseeded case be represented by
XkjNS (k = 1, . . . , rand) = 1, . . ., nNS)' Seeded and
nonseeded variates of the kth response measurement
are respectively denoted by

. YkjS = bkNS(akS + CkSXkjS)} ,

YkjNS = bkNS(XkjNS)

where bkNS is a commensuration adjustment, akS is. a
location adjustment, and CkS is a scale adjustment for
the kth response measurement. In this way, the input
for the MRPP simulation program was generated; val-
ues for akS, bkNs, and CkS for CIC5, TFPI, and PIC8
are given in Table 1. Response variables CIC5, TFPI,
and PIC8 were considered most important at the time.
The definitions of 1) all primary and secondary re-
sponse variables and 2) the final test statistics are given
in Table 2. Values of bkNS were based on the observed
ranges. Since prior information on akS and CkS did not
exist, these initial values were based on a concensus
judgment by the HIPLEX-l investigators.
Given the specified adjustments of Table 1, the re-
sults of the MRPP computer simulation, using the
four sets of random samples (N = 50, 100, 150, and
200) with ns = nNS = N12, indicated that 50-150 test
cases of a given class of cloud (25~ 7 5 each of seeded
and nonseeded clouds in a given class) would be re-
quired to reject the null hypothesis with a = 0.10,
where a is the probability of a type-I statistical error.
Unfortunately, because 1) the summer of 1980 was
extremely dry in eastern Montana, 2) HIPLEX-l was
pre-empted by the Cooperative Convective Precipi-
tation Experiment (CCOPE) in the summer of 1981,
and 3) the funding for HIPLEX-l was suddenly ter-

TABLE I. Specified changes associated with the location adjustment
OkS, the commensuration adjustment bkNS, and the scale adjustment
CkS of the three response variates CIC5, TFPI, and PICS,

. Adjustment
Response
variate OkS bkNS CkS
CIC5 5.0 1.0 1.0
TFPI 0.0 2.4 0.6
PICS 0.2 16,0 1.0