Use of Correlation to 

Improve Estimates of the 

Mean and Variance

GEOLOGICAL SURVEY PROFESSIONAL PAPER 434-C
Use of Correlation to 

Improve Estimates of the 

Mean and Variance

By MYRON B. FIERING

STATISTICAL STUDIES IN HYDROLOGY

GEOLOGICAL SURVEY PROFESSIONAL PAPER 434-C

An examination of the criteria for 

extending streamftow records 

by correlation

UNITED STATES GOVERNMENT PRINTING OFFICE, WASHINGTON : 1963
UNITED STATES DEPARTMENT OF THE INTERIOR 

STEWART L. UDALL, Secretary

GEOLOGICAL SURVEY 

Thomas B. Nolan, Director

For sale by the Superintendent of Documents, U.S. Government Printing Office

Washington 25, D.C.
CONTENTS

Abstract.- _ --______

Introduction -------

Notation ______-__-

Statistical principles.

Page

Cl

1

1

2

Historical reveiw the two-stream model. 

The three-stream model._______________

Conclusions- _________________________

References. _ __________-------_____-__

ILLUSTRATIONS

Page

C3

5

9

FIGURE 1. Relative information of the mean, three-stream correlation model, ni = 

2. Relative information of the mean, three-stream correlation model, n\ =

Page

C6

6

TABLES

TABLE 1. Range and mesh of arguments for tables of the relative information ratio of the mean___

2. Relative information of the mean, two-stream model______________-_---_-_-_-------_

3. Relative information of the mean, three-stream model, n3 =0-------------------------

4. Relative information of the variance, two-stream model_____________-_-_-_-_-___-_-_

5. Sample of table of relative information of the variance, two-stream model____________

6. Sample of table of relative information of the variance, three-stream model-____________

7. Range and mesh of arguments for tables of the relative information ratio of the variance.

8. Comparison of theoretical and simulated values of the information ratio________________

9. Summary of equations to be used for the several cases considered- -------------------

Page 

C7

7

in

658684-63

STATISTICAL STUDIES IN HYDROLOGY

USE OF CORRELATION TO IMPROVE ESTIMATES OF THE MEAN AND VARIANCE

BY MYRON B. FIERING

ABSTRACT

Statistical techniques are utilized to examine the validity of 

using correlation analysis to augment streamflow data when 

concurrent and additional data are available on a nearby stream. 

The investigation demonstrates that this use of correlation can 

yield significant improvement in estimates of streamflow population parameters, and confirms previous indications that the 

indiscriminate use of poor correlations may produce poorer 

estimates of parameters than could be obtained from the original 

data alone. The mathematical analysis leads to complicated 

equations for judging the relative gain in information, these, 

equations being functions of the length of the original record, 

the length of extension, and the coefficients of correlation. To 

aid in the application of these equations to actual problems, 

tabulated solutions are available.

INTRODUCTION

Frequency curves of various streamflow characteristics are commonly used in planning water-resource 

developments. Parameters of the frequency curve for 

an individual site are usually computed from streamflow data obtained at that site. If the streamflow 

record is short the parameters may be poor estimates of 

the population parameters. Under certain conditions 

the correlation with data obtained at other sites may 

be used to improve the estimates of the two parameters, mean and variance, of the population. The 

use of a least-squares regression line is typical of the 

several methods available for applying this technique 

to hydrologic problems. This method assumes that 

there exists a linear relationship of the form

q = y+0£ (1)

where t\ and £ are concurrent flows on each of two 

streams, Y and X, respectively, and where 7 and ft are 

constants. If additional records of flow from stream 

X are available, it is possible to estimate the corresponding values of flow for stream Y by means of 

equation 1 and to use these values to aid in determining 

estimates of the mean and variance for stream Y.

This paper reviews the previous work that has been 

done on this problem and extends the analysis to the

case of the relationship between streamflow at one 

stream site and the concurrent flows at sites on two 

other streams (the three-stream model).

The analysis indicates that the extension of a short 

record for the purpose of improving the estimate of a 

population parameter is not always desirable. Under 

certain conditions the parameter based on the extended 

record will be less reliable than that based on the 

unextended record. Criteria are derived which define 

the range of useful application of least-squares estimates 

of the type embodied in equation 1 and in the threestream model. These criteria depend on the length of 

original record, the length of the proposed extension, 

and the coefficients of correlation.

The research reported in this paper was done at 

Harvard University under the sponsorship of the U.S. 

Geological Survey. The author acknowledges the 

assistance of Dr. Nicholas C. Matalas and Mr. Walter 

B. Langbein, of the Geological Survey, and Dr. Joan 

R. Rosenblatt, of the National Bureau of Standards, 

for advice, guidance, and encouragement. Mr. Robert 

Gemmell, of Harvard University, checked derivations 

and analyses and Mr. John Burton, of the University 

of New South Wales in Australia, reviewed the manuscript. Finally, the author expresses his sincere appreciation to Professor Harold A. Thomas, Jr., of Harvard 

University, under whose guidance the entire project, of 

which this paper summarizes only apart, was conducted.

NOTATION

The following list includes the majority of the symbols 

which appear in this paper. In addition to this summary, graphic representation of sample sizes that is, 

record lengths is provided by arrays where appropriate, and terms which appear rarely are defined in the 

text as necessary. Where no confusion can exist, subscripts are dropped in the text for convenience.

Xf Independent variate value

¥* Dependent variate value (bivariate case) or independent variate value (trivariate case)

Cl
C2 USE OF CORRELATION TO IMPROVE ESTIMATES OF THE MEAN AND VARIANCE

n2

Dependent variate value (trivariate case) 

Estimated value of ( )

Length of record during which measurements are 

available for all variables

Length of record during which measurements are 

available for one variable (bivariate case), or for two 

variables (trivariate case)

Length of record during which measurements are 

available for one variable (trivariate case) 

Population mean of the dependent variable 

Estimate of p utilizing n^ regression estimates and n.j 

measured values

Population variance of the parameter i 

Measured sample mean of ( ) during period j 

Population correlation coefficient between the Xi and 

Yi (bivariate case)

Population correlation coefficient between the measured values of variate i and j (trivariate case) 

Multiple correlation coefficient (trivariate case) 

Population regression coefficient of Y on X (bivariate

b Least-squares estimate of /3

TH Least-squares estimate of pi,-

E( ) Expected value of ( )

7 Relative-information ratio

??, a Population values of dependent variate and mean,

	respectively, in regression equations 

Var( ) Variance of ( ) 

Cov( ) Co variance of ( ) 

MSE( ) Mean-square-error of ( )

STATISTICAL PRINCIPLES

Because the theory of mathematical statistics enters 

prominently into this research, a brief survey of some 

of the more commonly used results is given here. 

1. The mean of a sample of N items, each of which has 

a numerical value Xt} is given by

"N (2)

The mean can be described as a measure of central 

tendency, or average, of the data. The mean of a 

population is generally denoted by ft. 

2. The variance of the sample in (1) above is a measure 

of the degree of dispersion of the individual items 

about their mean value, and is defined by

(3)

The variance of a population is generally denoted 

by a2 . Data which possess a small variance are 

closely grouped about then1 mean, whereas data 

with a large variance are widely spread about 

their mean. Thus, if it is desired to estimate a 

population parameter for example, the mean 

from sample data and if several sets of data are 

available, each possessing a mean, these several 

means form another sample from which it is 

possible to compute the variance of the mean.

The smaller the value of this variance, the more 

reliable is the estimated value of the mean, since 

a small variance implies a small dispersion and 

greater precision. Extensions of this logical 

process will be used throughout this paper. 

Correlation techniques involving the relation between

two variables, say X and Y, require the following

defined terms.

1. The regression coefficient, b, equals

(4)

Xf-N'X*

2. The correlation coefficient r equals

r* 3 _ _!1/2 r* 3 _ -p 

L i * J L i J

(5)

The theory of least squares defines b as the best estimator of j8 in equation (1) and r as a measure of the 

goodness of fit between the variables X and Y. A 

value of r equal to ±1.0 indicates that the variables 

are perfectly correlated; a value of 0.0 indicates that 

there is no relation between the sample values of the 

two variables. Intermediate values indicate the 

closeness of fit between X and F. The correlation 

coefficient of the joint population of X and Fis denoted 

by p, and R is an estimator of p.

To clarify the problem, consider a simplified example 

involving the mean annual flows, X{ and Y{, of two 

nearby streams, X and F respectively. The following 

assumptions are made:

1. A stable hydrologic regime is known to exist, so 

that significant correlation between the X{ and 

Yi can reasonably be expected,

2. The mean annual flows, or some suitable transform 

thereof, are normally distributed,

3. Serial correlation of the mean annual flows is zero,

4. Tii mean annual flows are available on stream F,

5. HI mean annual flows measured concurrently with 

those on stream F, are available on stream X; 

Ti2 additional mean annual flows are also available 

on stream X, and

6. The Xt and Y{ are linearly related by a regression 

equation of the form

MX) (6)

and a fitted trend line is

'Xni) (7) 

The data may be arrayed with regard to time in the
STATISTICAL STUDIES IN HYDROLOGY C3

following form:

Y,, Y2 , (A-I)

The major issue is whether or not the supplementary 

data from stream X can be used to provide a better 

estimate of the population parameters on stream Y, 

whose record is shorter. Since the data or their transforms are assumed normally distributed, the first two 

statistical moments serve to specify completely the 

distribution of the data. Thus we deal here only with 

these two moments, since small-sample correction 

terms involving the higher moments and cumulants 

are neglected.

Since the data of array (-4-1) or their transforms are 

assumed to be normally distributed, it follows that 

the estimates of population moments for this model 

will be normally distributed about their respective 

population values. Thus the magnitude of the variance 

of each such estimate is a measure of the precision 

associated with it, this being a logical extension of the 

method of the simple example given above, wherein 

the variance of the mean was defined as a measure of 

the precision of the estimate of the mean. Well-known 

results from the theory of sampling give the variance 

of the first two statistical moments for a record of 

length HI, taken to include only the measured data. 

If correlation estimates are used to provide nz additional 

values, the result is a combined record consisting of 

HI measured values and n% estimated values. If the 

variance of a parameter computed from this combined, 

or blended, record exceeds that computed from the 

record of size HI alone, it is clear that the combined 

record provides a less precise estimate of the parameter. 

However, if the variance is less than that computed 

from the original record alone, the corn-elation technique 

has provided a more precise estimate and should be 

utilized. For any particular parameter, with known 

values of ni, n% and p (the population correlation coefficient), it is a simple matter to define the relativeinformation ratio, /, as the ratio of the variance of that 

parameter estimated from the original record to that 

estimated from the combined record. When I exceeds 

unity, it implies that the variance of the estimate of a 

moment made from the original record alone is larger 

than that of the estimate made from the combined 

record, and therefore the more precise estimate is 

computed from the combined data. On the other hand, 

if 7 is less than unity, the implication is that the original 

estimate is more precise than that computed from the 

blended data and that correlation analysis actually 

introduces additional variance, or loss of precision, 

into the estimate. Correlation should, therefore, not 

be used to augment the original data when / <1.0.

HISTORICAL REVIEW THE TWO-STREAM MODEL

Statistical literature contains several references to the 

use of regression estimates and, more generally, the use 

of correlation to improve estimates of parameters.

Wilkes (1952) assumes the following array of data 

to be a large sample from a bivariate normal population, and derives maximum likelihood estimators of 

the population means and variances:

(A-II)

If n^ni and ns/ni remain constant as HI approaches infinity,

^=JLru-K>* > 8X

and similarly

-F)-j , =77 J » .

(l+8)F eF (8b)

wherein

and

"X, ~Y are sample means during the period m,

X, Y are sample means during the periods w2 and ns, respectively.

The variance of the estimator of the mean of X is 

given by

r__ i /i -9\1_ a

(9)

and similarly for a^ . Additional results are given for 

the estimators of the population variances.

For independence between X and Y, that is p=0, 

equation 9 reduces to ax2l(n\-{-n^), which is equal to 

the variance of the mean for an original record of length 

Wi+n2 . In other words, Wilks' results indicate that 

the absence of correlation does not increase the variance 

of the estimated mean, and therefore that the use of 

correlation is always justifiable since it cannot decrease 

the precision of the estimate but can, at worst, effect 

no change. It will be shown that this conclusion is 

erroneous for small samples, in which the variance of 

the regression coefficient must be considered.

Matthai (1951), Edgett (1955), Nicholson (1957), 

and Lord (1956), have given solutions for different 

arrays of data, and their results are based on the assumption of the asymptotic behavior, as % becomes 

large, of the ratio of ntfni, where nt is the length of
C4 USE OF CORRELATION TO IMPROVE ESTIMATES OF THE MEAN AND VARIANCE

additional record on a nearby stream. Lord's results 

are typical of the rest. The data may be arrayed as 

follows:

Y2, Y HI

(A-III)

Maximum likelihood estimators are given for the eight 

parameters nx, ny, /z*, <rx2 , <ry2 , a-z2, pxy, pxz. Since there is no 

overlap, pyz cannot be estimated from the sample. Of 

particular interest is a matrix of the sampling variances 

and covariances of the maximum likelihood estimators 

of the population means, from which it is deduced that 

the reciprocal of the relative-information ratio is:

a 2 (when estimating /*,)

_l_ Pxv2 (when estimating 

It can further be shown that 7~: for the variance is: 

1 2 r z4 (when estimating <r z2

B|/4 (when estimating a,,2)

(10)

(ID

Note that again these values of 7"1 cannot exceed 

unity, and therefore 7 cannot be less than 1.0. The 

result of this lower boundary on 7 is to indicate that 

the correlation technique cannot decrease the precision 

of the estimates of the mean and variance, no matter 

how poor the correlation may be. This conclusion, 

drawn from maximum-likelihood (large-sample) theory, 

is in agreement with the result of Wilks above, and is 

incorrect for small samples. The appropriate values 

for computing the relative information ratio will be 

given below.

Langbein and Hardison (1955) propose a semigraphical method for providing estimates of additional 

data. Their analysis indicates that when correlation 

is poor, the error in these estimates may increase the 

sampling error of the mean.

Defining

e relative reduction in the variance in the mean =(/-!)//,

they derive the following approximate formula for 

useful extension when estimating the mean:

(12) n\

With known values of the record lengths (% and n2 ) it 

is possible to compute the required correlation as a 

function of the relative reduction in variance. Langbein (written communication) indicates that equation

12 applies only when 7ii^>2, say of the order of 10 or 

more, and may be derived from the more general form

(1 P2 )nin2

From equation 13 it may be deduced that e^ 

equivalently, 7^1.0) when

(13)

(or,

p = 

The Langbein-Hardison equations are approximate 

relations, but are significant in that they represent the 

first solution which suggests tha,t correlation may not 

always be useful for improving estimates of the mean. 

One minor inconsistency arises from equation 12 when 

e=Q, or when 7 is unity. For this case, (ni-\-n^)ln\ 

is always equal to 1.0, which implies that n2 is identically zero. This singularity is avoided by use of 

equation 13.

Professor H. A. Thomas, Jr. (written communication), arrived at a formulation which reduces to 

results found earlier and independently by Cochran 

(1953). For the array of data given in array A-l above, 

Professor Thomas derived the relative-information 

ratio, 7, for an estimate of the mean based on a combined sample of n\ original measurements and n2 

regression estimates. For those combinations of n\ } 

n2 and pxv which yield a value of 7 less than unity, 

correlation introduces a retrogression of information 

and should not be utilized. The pertinent results are:

"" Tn.-Fn,) (14a)

and

(14c)

Equations 14 are exact solutions, and do not depend 

upon any approximations assuming that the data conform to assumptions (1) through (6) given above. 

Solving equation 14c for the case of 7=1.0, it is found 

that the critical, or cut-off value of p2, is

For the range of values which HI generally assumes in 

hydrologic problems, this expression gives values of p 

in fair agreement with those obtained from the earlier 

Langbein and Hardison (1955) approximate solution for 

which p2 =2/iii.

Correlation is also used for improving estimates of 

the variance of a population. Dr. J. R. Rosenblatt
STATISTICAL STUDIES IN HYDROLOGY C5

(written communication, 1959) gives the following results based on the Thomas model.

(15a) 3)

-] (15b)

in which

(m-3)

i 3

and

"MSE (Var (F)) (15c)

The Rosenblatt equations comprise an exact solution, 

and are the first clear statement of the fact that correlation, which can decrease the precision of estimates of 

the variance, should be used only when / exceeds unity. 

Previous solutions for example, equations 11 yield 

information ratios which exceed unity for all nonzero 

values of the correlation coefficient. For typical values 

of HI and n2 , I exceeds 1.0 when p is of the order of 0.8, 

according to the Rosenblatt solution. Thus, conditions 

suitable for improving estimates of the variance are 

much more restrictive than those for improving estimates of the mean.

THE THREE-STREAM MODEL

Previous work has considered only the correlation 

between flow characteristics at two sites (the twostream model). The following development defines the 

conditions under which estimates of the mean and 

variance at one site can be improved by correlation with 

flow characteristics at two other sites. The threestream model is used.

Improving estimates of the mean. The following data, 

or some suitable transform of data, constitute an arrayed sample from a trivariate normal population in 

which serial correlation is zero.

 *!) -*2j * njj -f ni

Z1} Z2, . . .Zni (A-IV)

It is required to examine the effect on the estimate of 

the population mean of Z of using the correlation 

between Z and Y and X. A linear trivariate regression 

model is predicated to represent the data:

(16)

wherein bx and bv are the partial regression coefficients 

computed from the n\ data on three streams by means 

of well-known formulae from the theory of least 

squares. The theory gives:

bx = [Var (F)-Cov (X.Z)-Cov (X,Y) Cov (F,Z)]-A 

6,,=[Var (X) - Cov (Y, Z) -Cov (X, Y) - Cov (X, Z)]-*-A

and

A- Var (X) Var (Y) -[Cov (X, Y)]2 .

An unbiased estimate of the combined mean, M»1+»2 » ^s 

given by

and it is necessary to calculate the variance of the mean, 

<rM 2. The result may be written, after considerable

simplification, as

Var v \ g' Bl+B2 )=_ -

-*!) +2)1 n ^ J (18)

where N=ni+n2 , the total length of record, and R 

is the total correlation coefficient of Z on X and F, 

defined by

Prom equation 18 it may be deduced that /, the 

relative information with respect to the mean, is given

-^^Var (A) n\

or

t

~ N\ m

2)-2\n-i

-4 /J

(19)

This result may be generalized to a model represented 

by a multiple regression on ^-independent variables, 

each of equal length N n\-\-n2- For this case, the 

variance of the mean of the combined sample may be 

expressed as

r- / I» 7D9 I \ "H

(20)

P 2) J

The families of curves in figures 1 and 2 represent the 

relative information of the mean, I, for typical values 

of n2 and R2, with % set equal to 6 and 10, respectively.

Equation 19 can be solved for R2 and evaluated when 

/ is set equal to unity. This gives

m 2

as the critical values of R2, at which it is equally advantageous to correlate or not for estimating the mean.
C6 USE OF CORRELATION TO IMPROVE ESTIMATES OF THE MEAN AND VARIANCE

FIGURE 1. ;Relative information with respect to the mean obtained by using the 

three-stream correlation model, »i=6.

FIGURE 2. Relative information with respect to the mean obtained by using the 

three-stream correlation model, wi=10.

This is an exact solution, and the similarity to the 

solutions for the two-stream case is apparent.

The above analysis may be extended to the case in 

which each of the three streams has a record of different 

length. The following array of data obtains from a 

trivariate normal population, in which serial correlation 

is assumed to be zero:

* It * n,» JL n1

z\,... zB (A-V)

In hydrologic practice, the effect of the use of correlation would probably be investigated in two stages, 

with the equations for the two- and three-stream models 

being applied separately to the applicable lengths of 

record. Analytical solutions are available for testing 

the use of a combined regression model in which Z is 

estimated by means of a trivariate equation for the 

range %-fl to ni-{-n2) and estimated by means of a 

bivariate equation for the range 7i1 +7i2+ 1 to rii -}-n2 -{-n3, 

but owing to the formidable statistical problems encountered they are approximate solutions only. However, for the sake of completeness, two solutions are 

presented here. It must be emphasized that these 

solutions are not exact, and that the first of the two 

leads to overestimation of the relative information by 

a factor of (1/3) (p?x2). The solutions are:

>

«i-4

\ ns T J^NL N

(22)

Equation 21, derived by the author, is based on the 

single simplifying assumption that the partial regression 

coefficients, bx and bv, are each uncorrelated with the 

bivariate regression coefficient, 6, which defines the 

dependence of Z on X (as in Equation 7). Equation 

22, suggested by Langbein (written correspondence), 

has somewhat less theoretical justification but provides 

exact solutions for the case of R2=pgx2=l.Q, whereas 

equation 21 does not. Neither formulation is recommended.

To facilitate solution for the relative information, the 

equations representing I for the two exact solutions for 

the mean are evaluated by means of programs written 

by the author for the UNIVAC I computer at the 

Harvard University Computation Laboratory. Equation 14c for the Thomas bivariate model, and equation 

19 for the trivariate model are tabulated for many 

combinations of the several values of % and correlation 

coefficients. To use the tables one enters with known 

values of the nt and the correlation coefficient, and 

reads the corresponding value of 7 directly. Complete 

sets of tables, on deposit at the Widener Library and 

Gordon McKay Library, at Harvard University and 

the U.S. Geological Survey, require some 600 pages and 

are not reproduced here. The range and mesh, or 

spacing between successive entries, of each argument 

in the complete tables is as follows:
STATISTICAL STUDIES EST HYDROLOGY 07

TABLE 1. Range and mesh of arguments for tables of the 

information ratio of the mean

Model

3-stream_____ ___ _ _ _ _____

3-stream_ _______ _ _ __ _

3-stream__ _ _ __ _____ _ _

Argument

»i

n2

Pxv 

Ui

n2 R*

Range

4-30 

0-20 

0. 05-1. 00 

6-20 

2-20 

0. 0-1. 0

relative

Mesh

2 

2 

0.05 

2 

2 

0.05

Condensed results for the two-stream model are given 

in table 2, from which the general behavior of / may be 

deduced. In this table, / is given as a function of 

HI, HZ, and pxy. A condensed version of the complete 

table for the three-stream model with ns=Q is given in 

table 3. 

Estimating the Variance. A solution is presented 

for the three-stream model in which the two independent records have equal lengths. As before, the 

data or their transforms are assumed normally and 

independently distributed, and may be arrayed as in 

array (A-IV) above:

TABLE 3. Relative information of the mean, three-stream model, 

n3=0

raa

2-.__._.

6-.. _ .

10- 

14...... 

18.- 

Ri

0.0 

.2 

.4 

.6 

.8 

1.0 

.0 

.2 

.4 

.6 

.8 

1.0 

.0 

.2 

.4 

.6 

.8 

1.0 

.0 

.2 

.4 

.6 

.8 

1.0 

.0 

.2 

.4 

.6 

.8 

1.0

raj

6

0.800 

.870 

.952 

1.053 

1.176 

1.333 

.667 

.769 

.909 

1.111 

1.429 

2.000 

.615 

.727 

.889 

1.143 

1.600 

2.667 

.588 

.704 

.877 

1.163 

1.724 

3.333 

.571 

.690 

.870 

1.176 

1.818 

4.000

8

0.909 

.962 

1.020 

1.087 

1.163 

1.250 

.824 

.921 

1.045 

1.207 

1.429 

1.750 

.783 

.900 

1.059 

1.286 

1.636 

2.250 

.759 

.887 

1.068 

1.341 

1.803 

2.750 

.743 

.878 

1.074 

1.383 

1.940 

3.250

10

0.947 

.989 

1.034 

1.084 

1.139 

1.200 

.889 

.976 

1.081 

1.212 

1.379 

1.600 

.857 

.968 

1.111 

1.304 

1.579 

2.000 

.837 

.963 

1.132 

1.374 

1.748 

2.400 

.824 

.959 

1.148 

1.429 

1.892 

2.800

12

0.966 

1.000 

1.037 

1.077 

1.120 

1.667 

.923 

1.000 

1.091 

1.200 

1.333 

1.500 

.898 

1.000 

1.128 

1.294 

1.517 

1.833 

.881 

1.000 

1.156 

1.368 

1.677 

2.167 

.870 

1.000 

1.176 

1.429 

1.818 

2.500

14

0.976 

1.005 

1.036 

1.070 

1.105 

1.143 

.943 

1.012 

1.092 

1.185 

1.295 

1.429 

.923 

1.017 

1.132 

1.277 

1.463 

1.714 

.909 

1.020 

1.163 

1.351 

1.613 

2.000 

.899 

1.023 

1.187 

1.413 

1.747 

2.286

16

0.982 

1.007 

1.034 

1.063 

1.093 

1.125 

.957 

1.019 

1.089 

1.170 

1.264 

1.375 

.940 

1.026 

1.130 

1.258 

1.418 

1.625 

.928 

1.032 

1.163 

1.331 

1.557 

1.875 

.919 

1.037 

1.189 

1.393 

1.683 

2.125

18

0.986 

1.009 

1.032 

1.057 

1.084 

1.111 

.966 

1.022 

1.085 

1.157 

1.239 

1.333 

.951 

1.032 

1.126 

1.241 

1.380 

1.556 

.941 

1.039 

1.159 

1.311 

1.509 

1.778 

.933 

1.045 

1.186 

1.373 

1.628 

2.000

20

0.989 

1.009 

1.030 

1.053 

1.076 

1.110 

.972 

1.024 

1.081 

1.145 

1.218 

1.300 

.960 

1.034 

1.121 

1.224 

1.348 

1.500 

.951 

1.043 

1.154 

1.293 

1.469 

1.700 

.944 

1.050 

1.182 

1.352 

1.580 

1.900

Z,,__._ZB1 (A-VI)

It is desired to estimate the variance of stream Z, 

a? using correlation techniques. As above, a linear 

trivariate regression model is postulated, and may be 

expressed as

Zf=ZB1 + bx (X, XB1)+btf (Y YB1 ) (23) 

TABLE 2. Relative information of the mean, two-stream model

MS

2 . 

6-- 

10 

14... 

18 

0.2

.4 

.6 

.8 

1.0 

.2

.4 

.6 

.8 

1.0

.2

.4 

.6 

.8 

1.0 

.2

.4 

.6 

.8 

1.0 

.2

.4 

.6 

.8 

1.0

4

1.000

1.000 

1.000 

1.000 

1.000 

.644

.710 

.856 

1.201 

2.500 

.603

.673 

.833 

1.250 

3.500

.582

.654 

.821 

1.278 

4.500 

.570

.642 

.813 

1.297 

5.500

6

0.934

.970 

1.038 

1.149 

1.333

.877

.943 

1.079 

1.351 

2.000 

.851

.930 

1.100 

1.481 

2.666 

.836

.922 

1.114 

1.572 

3.333 

.826

.917 

1.123 

1.639 

4.000

8

0.970

.998 

1.048 

1.128 

1.250 

.938

.996 

1.110 

1.321 

1.750 

.922

.995 

1.148 

1.461 

2.250 

.911

.994 

1.173 

1.566 

2.750 

.904

.994 

1.191 

1.648 

3.250

n

10

0.984

1.006 

1.046 

1.108 

1.200 

.964

1.015 

1.112 

1.283 

1.600 

.953

1.020 

1.155 

1.417 

2.000 

.946

1.023 

1.185 

1.522 

2.400 

.941

1.026 

1.208 

1.608 

2.800

i

12

0.990

1.009 

1.043 

1.093 

1.166

.978

1.022 

1.106 

1.250 

1.500 

.970

1.031 

1.151 

1.375 

1.833 

.965

1.037 

1.184 

1.477 

2.166 

.961

1.041 

1.209 

1.562 

2.500

14

0.994

1.010 

1.039 

1.082 

1.142 

.986

1.025 

1.099 

1.222 

1.428 

.980

1.036 

1.143 

1.338 

1.714 

.976

1.043 

1.177 

1.436 

2.000 

.974

1.049 

1.204 

1.518 

2.285

16

0.996

1.010 

1.035 

1.073 

1.125 

.990

1.026 

1.092 

1.200 

1.375 

.987

1.038 

1.135 

1.308 

1.625 

.984

1.046 

1.169 

1.400 

1.875 

.982

1.053 

1.196 

1.479 

2.125

18

0.997

1.010 

1.032 

1.065 

1.111

.994

1.026 

1.086 

1.182 

1.333 

.991

1.038 

1.127 

1.282 

1.555 

.989

1.047 

1.161 

1.368 

1.777 

.988

1.054 

1.188 

1.445 

2.000

If the values of Z«1+i to Z«1+B2 are computed using 

this model, and the variance of Z estimated without 

regard to the fact that these values are not measured

but estimated variates, the following expressions are 

obtained:

(24)

whereupon

MSE (^)=

j

4n2 (n2 + 3) 

ni 4

4n2 (2n2 +l) 

n\ 4

8n2 (n2 +2)

1 (m-4)(m-6) 

and the relative information may be written

(1-E2 ) 2 }

i 1

(26)

(27)

As in the case of the mean, the relative information 

of the variance is tabulated by means of UNIVAC I 

evaluation of equation 15b for the Rosenblatt twostream solution and of equation 27 for the three-stream 

solution. Table 4 gives a concise summary of the relative information of the variance for the two-stream 

model. Table 5 is a sample of the complete tables 

which are available in the Harvard University libraries. 

Entering the tables with values of n1} n2 , and correlation coefficients, the relative information is read directly.

Table 6 is used in the same manner, except that 

72,1, n-2, and R2 are required to read the relative information in the 3-stream case.
C8 USE OF CORRELATION TO IMPROVE ESTIMATES OF THE MEAN AND VARIANCE

The range and mesh of each argument is tabulated 

in table 7.

TABLE 4. Relative information of the variance, two-stream model

m

2__ 

6- _

10

14____._

18_ 

P

0.2

.4

.5

.8

1.0

.2

.4

.6

.8

1.0

.2

.4

.6

.8

1.0

.2

.4

.6

.8

1.0

.2

.4

.6

.8

1.0

6

0.887

.884

.908

1.013

1.400

.628

.659

.742

.977

2.200

.520

.561

.661

.946

3.000

.466

.509

.616

.925

3.800

.434

.478

.587

.911

4.600

8

1.155

1.098

1.053

1.075

1.285

.955

.940

.961

1.110

1.858

.782

.801

.870

1.105

2.428

.680

.714

.808

1.095

3.000

.617

.658

.765

1.084

3.570

10

1.163

1.112

1.068

1.076

1.222

.969

.963

.989

1.121

1.668

.769

.800

.885

1.118

2.111

.648

.695

.808

1.104

2.556

.573

.627

.753

1.089

3.000

n

12

1.146

1.104

1.066

1.070

1.190

.954

.957

.990

1.114

1.545

.741

.782

.877

1.108

1.909

.610

.666

.790

1.091

2.273

.528

.589

.727

1.071

2.638

i

14

1.129

1.094

1.061

1.063

1.154

.940

.950

.988

1.104

1.461

.720

.767

.869

1.096

1.769

.583

.643

.774

1.074

2.079

.497

.561

.704

1.050

2.386

16

1.115

1.084

1.056

1.057

1.133

.931

.944

.985

1.095

1.400

.809

.845

.923

1.093

1.667

.628

.686

.810

1.073

1.934

.476

.542

.688

1.032

2.200

18

1.103

1.076

1.051

1.052

1.118

.925

.940

.983

1.086

1.354

. 700

!753

.859

1.074

1.589

.554

.618

.755

1.047

1.825

.462

.528

.675

1.016

2.060

20

1.093

1.069

1.047

1.048

1.105

.921

.938

.981

1.079

1.316

.697

.750

.857

1.065

1.526

.547

.612

.750

1.036

1.737

.453

.519

.667

1.003

1.948

TABLE 5. Sample of table of relative information of 

two-stream model

variance,

ni

16......... .

16 ..........

16 

16. . _

16 

16 

16. _ -

16 

16 

16 . 

16 

16 

16 

16 

16 . _ -

16 _ -

16 

16 .

16 

16 

16 

16 

16 . 

16 

16 

16 

16 _ ....

16 

16 

16 ..........

n2

14

14

14

14

14

14

16

16

16

16

16

16

16

16

16

16

16

16

16

16

16

16

16

16

18

18

18

18

18

18

ft

0.70

.75

.80

.85

.90

.95

.10

.15

.20

.25

.30

.35

.40

.45

Kfl

.55

.60

.65

.70

.75

.80

.85

on

.95

.10

.15

.20

.25

.30

.35

I

0 880

QOQ

1.059

1.188

1.358

1

c no

.508

.516

.527

.541

.559

Cot

607

.638

.676

779

.848

.935

1.045

1.190

1 OJM

1.657

.462

.468

.476

.488

.502

.520

n'

16 

16 

16 

16 

16 

16 

16 .

16 

16 

16 

16 

16 

16 

16 

16 

16 .

16 

16 - 

16 

16 .

16 

16 ..

16 

16 

16 

16 

16 

16 

16 

16 

n2

10

1Q

1 C

18

1C

1 Q

1 Q

1 Q

1C

10

10

1C

20

20

9(1

20

20

20

20

20

20

20

20

20

20

20

20

20

20

ft

4fl

AK

.50

.55

.60

.70

.75

QA

QC

on

Q*.

.10

.15

.20

.30

4O

AK

Kf\

KK

.60

.70

.75

.80

QC

90

QS

I

K49

.601

640

.688

Q19

1.032

1 190

1 Af\n

1 718.

436

444

.455

.470

.487

KrtQ

eoe

569

fiftQ

658

7QK

OQ9

1.018

1 1QQ

1.426

The most significant result of the derivation and 

tabulation of / for the variance is the fact that the 

maximum information gain in the three-stream model 

for estimating the variance rarely exceeds, and generally 

is less than, the gains associated with the two-stream 

model. Owing to the formidable analytic and computational difficulties which would be encountered, the 

case of three different lengths is not solved.

Numerical Checking of Results. A check on the derivations is provided by examining a multiple regression 

model predicated on a table of normal random sample 

deviates with mean=5.0 and variance=1.0. A 540- 

item sequence of these deviates was divided into 18

TABLE 6. Relative information of the variance, three variables, for 

, 0.4, 0.6, 0.8, 1.0

2 ... ..______._--.--_-_-

6_.__ _ - -

10_ _.-_-_-_---------

K*

0.2

.4 

. 6 

. 8 

1.0 

.2

. 4 

.6 

.8 

1.0 

.2

.4 

.6 

.8 

1.0

8

0. 891

.921 

.976 

1.090 

1.279 

. 698

.778 

.918 

1. 192 

1.840 

.598

. 700 

.879 

1.250 

2.430

Wl

12

1.045

1.045 

1.051 

1.092 

1. 180 

.954

.985 

1.058 

1.220 

1.545 

.809

.905 

1.045 

1.309 

1.895

16

1.046

1.044 

1.044 

1.077 

1.138 

.970

.998 

1.068 

1. 193 

1.410 

.811

.904 

1.042 

1.261 

1.660

TABLE 7. Range and mesh of arguments for tables of the relative 

information ratio of the variance

Model Argument

HI

n2

P

HI

n2

R2

Range

6-20

2-20

0. 10-0. 9

8-16

2-10

0. 2-1. 0

Mesh

2

2

 0.05

4

4

0.2

sets of 30 items. Ten of the 30 items in each set were 

discarded in favor of estimated values, computed by 

means of a multiple regression that is, 3-stream 

model using assumed population values of the regression 

coefficients. The variance of the mean and the variance 

of the variance are computed from a sample of 18 sets, 

using the combined 20-10 item record, where each such 

record consists of 20 original and 10 estimated deviates. 

These measured values are then compared to theoretical 

values of the variance of both the mean and the variance 

as computed from assumed population parameters. 

A trial was made with %=20, 7i2 =10, p2a; =0.9, pzj,=0.9, 

and pxy=Q.8. Theoretical values taken from the tables 

are 7=1.42 for the mean and 1.34 for the variance, 

which compare favorably with measured values of 1.33 

and 1.14.

All the derivations are made using p instead of its 

estimatoi, r. To evaluate the effect of bias in estimating p, a simulation technique was devised. A 540-item 

sequence of normal random sample numbers of mean= 

5.0 and variance=1.0 is used to generate two additional 

records of equal length by means of a linear regression 

model with a random additive component using assumed 

population values of the p's. This random component 

is numerically equal to the product of the standard 

error of estimate and a new normal random sampling
STATISTICAL STUDIES IN HYDROLOGY C9

deviate. These three concurrent sequences are then 

divided into 18 sets of 30 items, from each of which 10 

values of the dependent variate are discarded in favor 

of estimates computed using the least-squares partial 

regression coefficients measured from the 20 values 

remaining in each set. It does not follow that the 

partial regression coefficients computed in this manner 

will produce the same value of dependent variable 

given in the original sequence, owing to the nature of 

the random-number table. As before, the variance of 

the mean and of the variance are computed from the 

sample of 18 sets by consideration of the population 

values which obtain from the original sample of 20 

versus its combined 20-10-item counterpart. The 

results of several runs are summarized in table 8.

TABLE 8. Comparison of theoretical and simulated values of the 

information ratio

pa and pig

0. 9_._. _________..__._.

0. 8______. _____________

0. 7-___-_-_---__.____-

0. 6____________________

I (mean)

Theory

1.42 

1.34 

1. 26 

1.20

Simulated

1.42 

1.47 

1.46 

1.45

7(var)

Theory

1.22 

1.09

. 97 

. 89

Simulated

1.21 

1. 17 

1.09 

1.07

These results are not conclusive. For example, 

although agreement seems to be quite close as to order 

of magnitude, the simulated data for 7 (mean) exhibits 

a trend opposite to that which is predicted. However, 

it must be noted that all the entries in table 8 are based 

on the same sequence of random sampling numbers and 

any bias in the mean or variance of the random deviates 

would tend to distort all the results in the same fashion. 

It should also be noted that the theoretical solution for 

/ (var) is approximate so that perfect agreement is not 

expected. Based on these considerations, these test 

results are accepted as an aid in verifying the analyses. 

An analytical approach to the sampling variance of / 

due to the variance of r is hopelessly complicated.

CONCLUSIONS

1. Much of the statistical literature, of which the 

work of Lord (1956) is typical, does not admit the 

possibility of loss of information on statistical parameters by correlation, since, as can be seen in equations 

10 and 11, efficiencies are always less than unity. 

This implies that at worst the application of correlation 

techniques to augment data will not lead to a retrogression of information about parameters, and always 

implies a gain of information when p is not zero. As 

indicated approximately by Langbein and Hardison 

(1955), and verified subsequently by the exact analyses

of Professor Thomas, Cochran (1953), and Dr. Rosenblatt, the inclusion of the variance of the regression 

coefficient introduces the possibility of dilution of good 

information by estimates based on poorly correlated 

data. This can lead to a loss of hydrologic information.

2. For the case of a three-stream model, in which 

either the data or a suitable transform thereof are 

normally distributed, expressions for the relative 

information of the mean and of the variance are derived. 

Both solutions are exact.

3. The several cases which have been considered, 

and the equations to be used for analysis in each case, 

are summarized below in table 9.

TABLE 9. Summary of equations to be used for the several cases 

considered

Case Equation No.

14c

15b, 15c

19

26, 27

4. Owing to the great complexity of the several 

functions described above, a table of the relative 

information versus appropriate arguments has been 

prepared in each case by a program written for the 

UNIVAC computer. Entering the tables with known 

values of the n, and p or R2, the corresponding / is read 

directly. When / exceeds unity, the variance of the 

moment under consideration is reduced. Conversely, 

it is increased by the correlation technique when the 

tabulated value of / lies below 1.0. In the latter case, 

the technique should not be used to augment records.

5. Results given in this paper apply to hydrologic 

data which may be, or are transformed to be, reasonably 

normally distributed and without serial correlation.

REFERENCES

Cochran, W. G., 1953, Sampling techniques: New York, John 

Wiley & Sons, Inc.

Edgett, G. L., 1955, Multiple regression with missing observations among the independent variables: Statist. Assoc. 

Jour., v. 50, p. 122-131.

Langbein, W. B., and Hardison, C. H., 1955, Extending streamflow data; Am. Soc. Civil Engineers Proc., v. 81, Paper 

No. 826, 13 p.

Lord, F. M., 1956, Estimation of parameters from incomplete 

data, Statist. Assoc: Jour., v. 51, p. 870-876.

Matthai, A., 1951, Estimation of parameters from incomplete 

data with application to design of sample surveys: Sankhya, 

v. 11, p. 145-152.

Nicholson, G. E., Jr., 1957, Estimation of parameters from 

incomplete multivariate samples: Statist. Assoc. Jour., 

v. 52, p. 523-526.

Wilks, S. S., 1952, Moments and distributions of estimates of 

population parameters from fragmentary samples: Annals 

of Math. Statistics, v. 23, p. 163-190.

o