NATIONAL CENTER Series 2 

For HEALTH STATISTICS INumber ~1 

PROPERW OF THE -

PUBLICATIONS BRANCH 

EDITORIAL LIBRARY 

VITAL and HEALTH STATISTICS 

DATA EVALUATION AND METHODS RESEARCH 

Pseudoreplication 

Further Evaluation and 

Application of the Balanced 

Half-Sample Technique 

Uses of the method in analysis of data from complex surveys. 

DHEW Publication No. (HSM) 73-1270 

U.S. DEPARTMENT OF HEALTH, EDUCATION, AND WELFARE 

Public Health Service 

Health Services and Mental Health Administration 

Washington, D.C. January 1969 
DHKW Puh]icatinn No. (HSM) 73-1270 

For SAC hy [he Supcrintcndcnt of Documents, U.S. Government Printing office 

Wmhingtnn, D.C. 20402 - Price 35 cents 
NATIONAL CENTER FOR HEALTH STATISTICS 

THEODORE D. WOOLSEY, Director 

PHILIP S. LAWRENCE, SC.D., Associate Director 

OSWALD K. SAGEN, PH.D.,,Assistant Director /or Health Statistics Development

WALT R.SIMMONS,M.A., Assistant Dr’rector for Research and Scientific Development 

ALICE M. WATERHOUSE, M.D.,Medical Consultant 

JAMES E. KELLY, D.D.S.,Dental Advisor 

EDWARD E.MINTY, Executive O//icer 

MARGERY R. CUNNINGHAM, bz/om.ation O//icer 

Public Health service Publication No. 1000-Series 2-No. 31 

Library of Congress Catalag Card Nutnber 68-62232 
PREFACE 

In an earlier publication of the National Center for Health Statistics 

(Series 2, No. 14) the method of balanced half-sample pseudoreplication 

was introduced as a device for estimating precision of estimates which 

come from surveys of modern complex design. The technique was developed by Philip J. McCarthy of Cornell University. The report and 

McCarthy’s work are facets of a series of efforts by the Center to find 

appropriate methods for analyzing data from complex social and health 

surveys, in which the assumptions of standard classical techniques 

rarely are satisfied. 

The present study describes a number of ‘internal evaluation techniques which aid in understanding the khavior of balanced half-sample 

replication as it relates not only to the variance of linear estimators 

but also to precision of ratios. And it extends the method to applications 

of a more analytic character, including adaptations to the sign test and 

to contingency tables. 

Replication and pseudoreplication are concepts that have deep roots 

in statistical theory and that have influenced the thinking of many people. 

Acknowledgment of all who have contributed is not feasible, but the list 

of those who have had special influence on the research presented here 

includes W. Edwards Deming, John Tukey, M.H. Quenouille, Leslie 

Kish, and several members of the staff of the U.S. Bureau of the Census. Professor McCarthy Carried out the research and wrote the report. 

Walt R. Simmons has coordinated Center activities in the area of 

pseudoreplication and, as Project Officer has maintained close contact 

with the present undertaking. 

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CONTENTS


Introduction


Background for EvaluationStudies


Description and Application of Evaluation Techniques


Average of Half-Sample Estimates


Bias of aRatio Esttiator


Estimating the Varianceof F


Estimating the Varianceof A-


Design Effect


Approximate Test of Independence Based on a Set of Balanced Half-Sample


Replicates


Order Statistics ofa Set of Balanced Half-Sample Estimates


References


Detailed Tables


Page 

1 

2 

4 

4 

6 

7 

10 

11 

12 

15 

18 

19 

v 
A REPORT of the National Center fmHealth Statistics (Series2, No. 

14) introduced a pseudoreplicatwn procedure, called balanced halfsample replicatwn, as a basis fir statistical analysis when dealing with 

data derived from complex sample designs and estimation processes. 

It appeared particularly suited fm variance estimation in connection 

with the commonly used multistage design of two primary units per 

stratum, with the individual observations being subjected to various 

kinds of ratio and posts tratipcation adjustments. 

The exact behavior of balanced half-sample replication as a variianceestimating procedure is known only fov a simple linear situation where 

it has no real utility. The fivst portions of this veport review this theory 

and descvibe a number of evaluation techniques which can provide evidence concerning the behavior of balanced half-sample replication for a 

particular set of data. They are illustrated on data derived from the 

Centerts Health Examination Survey (Series 11, No, 1) relating to a 

vaviety of body measurements for a sample of approximately 3,000 U.S. 

adult males and involving estimates of population means, multiple regression coefficients, and multiple correlation coe~cients. These techniques are principally concerned with providing presumptive evidence 

relating to (1) the extent to which the average of a set of balanced halfsample estimates “exhausts” the information in the entire set of 2 L 

half-samples, (2) the existence of bias in the entire-sample estimate, and 

(3) the adequacy of the mviance estimates produced by balmced hulfsample replication. Applications to the Health Examination Survey data 

seem to indicate that balanced half-sample replication is performing 

well as a vaviance estimating procedure. Also, as might be expected.on 

the basis of the sample size, there is no evidence of bias in the estimates. The variables and subpopulations are less extensive than one 

might desire for an exploratory investigation, W this limitation was 

imposed by the fact that these habuhtions were produced as a byproduct 

of another study. 

The last two sections of the report suggest several applications of balanced Mf-sample replication to problems of a more analytic chavacter. 

Thws an investigation of its use to test the hypothesis of independence 

in a contingency table is described, and the relationship of balanced 

half-sample replication to the sign testis explored. 
PSEUDOREPLICATION 

FURTHER EVALUATION AND APPLICATIONS OF THE 

BALANCED HALF-SAMPLE TECHNIQUE 

Philip J. McCarthy, New York State Sckool of Industrial and Labor Relations, Cornell University 

INTRODUCTION 

The extensive literature on theory of sample 

surveys well documents the fact that commonly 

used complex sample designs and estimation 

procedures lead to approximate and extremely 

complicated expressions for variance estimation. 

Furthermore, the survey data often do not satisfy 

the conditions required for the application of even 

elementary statistical techniques of analysis. In 

the absence of simple methods for handling these 

problems, the ordinary researcher will ignore the 

complexities of design and merely treat the data 

as though they had been obtained by simple random 

sampling, the more sophisticated researcher will 

be frustrated in not being able to account for all 

the effects of design in his analysis. These points 

have been discussed in some detail in a publication of the National Center for Health Statistics 

(Series 2, No. 14). 

Durbin (1967) has developed methods for 

designing multistage surveys in such a manner 

that error computations will be relatively simple, 

where the emphasis is on the estimation of population means, percentages, or totals. However, 

it is unlikely that these methods can ever be extended to handle more complicated problems of 

statistical analysis, such as the use of multiple 

regression techniques. Another possible avenue 

of approach to these problems is through the use 

of independent replications of the sample design, 

variously referred to as interpenetrating samples, 

duplicated samples, or random groups. Deming, 

for example, has been a consistent advocate of 

replicated sampling. He first wrote of it as the 

Tukey plan (1950), and his recent book (1960) 

presents references and descriptions of the applications of replicated sampling” to many different 

situations and contains a wide variety of ingenious 

devices that he has developed for solving particular problems. 

A major disadvantage of replicated sampling 

arises from the difficulty of obtaining a sufficient 

number of independent replicates to provide 

adequate sampling stability for estimating variances and for other purposes. Thus the ,commonly 

used design of two primary units per stratum 

(frequently obtained by collapsing strata from 

each of which only a single unit has been drawn) 

gives only two independent replicates. To overcome this problem of having only one degree of 

freedom available for variance estimation, the 

U.S. Bureau of the Census originated a pseudoreplication scheme called half -sample replication. The scheme was adapted and modified by the 

staff of the National Center for Health Statistics 

(NCHS) and has heen used in their Health Examination Survey reliability measurements. A brief 

description of this approach is given in a report 

of the U.S. Bureau of the Census (Technical Paper 

No. 7, 1963, p. 57), and a reference to the Census 

method of half-sample replication was made by 

1

Kish (1957, p. 164). The basic characteristics of 

this procedure have been described by Gurney 

(1962) and in the NCHS publication (Series 2, No. 

14). 

A refinement of this pseudoreplication technique, called balimced hulf-sample replication, 

that increases the precision of variance estimates 

was also introduced in NCHS (Series 2, No. 14) 

and was applied to the analysis of data derived 

from the Health Examination Survey (NCHS, 

Series 11, No. 1). The present report carries 

further the evaluation of the balanced half-sample 

technique, with illustrative data drawn from the 

Health Examination Survey. In addition, several 

applications to problems of a more analytic 

character are suggested. ~hus an investigation 

of the use of balanced half-sample replication 

to test the hypothesis of independence in a 

contingency table is described, and the relationship of balanced half-sample replication to the 

sign test is investigated. 

BACKGROUND FOR EVALUATION 

STUDIES 

In simplest terms, the idea of half-sample 

replication is as follows. Consider a stratified 

sampling procedure where two independent selections are made from within each of L strata. 

Denote the strata weights by W~, h = 1, 2, . . ., L, 

and the observations by Yh,, h = 1, 2, . . . , L and 

i= 1,2. Under these circumstances, a halfsample replicate is obtained by choosing one of 

Yll and Ylz, one of Y21 and YZZ~.. ., and one 

of yLl and YL2. A half-sample estimate of the 

population mean is Y~, = ; W~yh,, where i is 

either 1 or 2 for each h. There are 2L 

possible half samples, If the estimate made from 

the entire sample is denoted by Yst = ~ W~(Y~l + 

Yh2)/2 , then it can be demonstrated that if 

k half samples are independently selected from 

the entire set of 2L possible half samples, with 

means denoted by ~h~,l~ ~h=,z ~. - .) ~hs,k$ 

then l?[~$l(yh,, - yst)~k] is equal to the ordinary varianc~ estimate of Y,t, where the expectation is taken over the entire set of 2L half 

samples. As is explained in NCHS (Series 2, No. 

14), a balanced set of half samples is obtained by 

appropriately choosing the half samples so as to 

eliminate a between-strata contribution which 

influences the stability of this estimate of variance. 

The basic characteristics of balanced halfsample replication, as outlined in the preceding 

paragraph, have been investigated in relation to 

a linear situation where the method obviously 

has no real utility, at least as far as ordinary 

variance estimation is concerned. Under these 

circumstances the method merely reproduces 

results that can be obtained by direct and simple 

methods of analysis. If, however, more complicated methods of sampling and estimation are 

employed, then direct methods of analysis may 

not be available or may require a prohibitive 

amount of computation in comparison with halfsample replication. For example, suppose that a 

multiple regression coefficient is estimated from 

a complex sampling and estimation operation. No 

known theory leads to an estimate of its sampling 

variability. Nevertheless, one may compute a 

value of the regression coefficient from each 

balanced half sample and use the comparisons 

among the computed regression coefficients to 

estimate the variability of the overall coefficient. 

Troublesome tIdomain of study” problems> as 

described by Hartley (1959) or Kish (1965), may 

also be handled in the same way. The simplicity 

of the approach is most appealing in that one has 

only to apply the appropriate estimation procedure 

to each half sample, followed by a simple variance computation on the separate half-sample 

estimates. 

Since balanced half-sample replication does 

permit the “easy” com~utation of variance estimates which intuitively seem to mirror most of 

the complexities in survey sampling, estimation, 

and analysis, one would like to argue that it can 

j!correctlYl! account for all of these features. This 

is clearly not the case. For example: 

1. If the y~l represent values of a variable 

associated with individual population elements, if 

the two elements per stratum are selected with 

equal probabilities and without replacement, and 

if the sampling fractions are the same in all 

strata, then a finite population correction can 

easily be inserted at the” end of the variance 

2 
estimation process. The effect of differing 

sampling fractions in the separate strata could 

be taken into account by working with W\’s 

where W; is equal to W~ multiplied by the square 

root of the appropriate finite population correction, but even this simple change effectively 

divorces the estimation procedure from the 

variance estimation process. The situation becomes more involved if the two units are selected 

with unequal probabilities and without replacement. 

2, If the Yhi represent estimates of a cluster 

characteristic, as in two-stage sampling, then 

one must also worry about correctly accounting 

for the within-cluster contribution to the total 

variance. As is evident in Cochran (1963, ch. 10), 

the problem is not substantial if the first-stage 

units are selected with equal probabilities and with 

replacement and the first- stage sampling fractions are small, or if the first-stage units are 

selected with unequal probabilities and with replacement. The situation in which two units are 

selected with unequal probabilities and without 

replacement is the variance estimating problem 

to which Durbin (1967) addressed himself. 

3. Most estimation problems are not linear 

but involve the use of ratios in one form or 

another. Under these circumstances, it will not 

even be necessarily true that the estimate made 

from the entire sample will be equal to the 

average of the balanced half-sample estimates 

and thus new dimensions of. complexity will be 

involved. 

As implied by the foregoing examples, the 

exact characteristics of estimates of variance 

obtained from balanced half -sample replication 

are, for the most part, unknown. These characteristics may be investigated in a variety of 

ways, including the following: (a) exact analytic, 

in which one assumes the functional form of a 

distribution (or a joint distribution) and obtains 

straightforward, although possibly not simple, 

answers; (b) approximate analytic, in which one 

uses Taylor series approximations, obtains 

bounds by analytic methods, and the like; (c) empirical studies, in which one employs data obtained 

from surveys to investigate the behavior of statistical procedures, frequently using such an 

approach to check on results obtained in (b); and 

(d) Monte Carlo sampling from synthetic populations, again usually to check on results obtained 

in (b). 

In the area of present concern, there appears 

to be little hope of obtaining exact analytic solutions to problems. In particular, exact solutions 

require quite restrictive models as evidenced by 

a portion of the work of Durbin (1959) and the 

work of Rao and Webster (1965) on the Quenouille 

version of a ratio estimate. Their developments 

start with the assumption that Yi = a + B x ~ + 

u,, ~(uljxl) = 0, V(uil xi) = nt5 where 13is a 

constant for each n such that n~ is bounded, and 

the variates X,/n have the gamma distribution 

with parameter h. Such investigations are extremely valuable in providing clues concerning 

the behavior of a particular method, but it is not 

possible to devise specific models for the many 

different types of data studied by large survey 

organizations. 

Taylor series approximations have also been 

used extensively to investigate the properties 

of a wide variety of ratio estimators, both with 

and without models. One of the more recent 

accounts and summaries of this type of approach 

is that of Tin (1965). In general, results are obtained on bias and variance estimation which one 

wishes to check against data obtained either 

from actual studies or from Monte Carlo experiments, particularly when the samples are sin-all 

or moderate in size. Thus Kish, Namboodiri, and 

Pillai (1962) use actual survey data while Tin 

(1965) and Lauh and Williams (1963) use Monte 

Carlo techniques. 

In this report, the general view is adopted 

that .the behavior of estimates and of variance 

estimates should be investigated, if possible, on 

the basis of actual survey results. This enables 

one to consider a wide variety of types of data 

that are of current interest and avoids the 

problem of having to construct “representative” 

synthetic populations and having the results 

refer only to these populations. Even the construction of appropriate synthetic populations, 

to which one could apply complex sample designs and estimation procedures, would k a 

difficult undertaking. 
DESCRIPTION AND APPLICATION 

OF EVALUATION TECHNIQUES 

This section of the report will describe a 

number of internal evaluation techniques which 

can provide evidence concerning the behavior 

of balanced half-sample replication. These techniques will be illustrated on data derived from 

the Health Examination Survey (NCHS, Series 11, 

No. 1, and the appendix to NCHS, Series 2, No. 

14) relating to a variety of body measurements 

for a sample of approximately 3,000 U.S. adult 

males. (The total sample also contained approximately 4,000 U.S. adult females.) Two different 

types of estimates will be considered: (1) Estimates of population means and (2) Estimates of 

multiple regression coefficients and multiple 

correlation coefficients. The original tabulations 

upon which these results are based were provided by the Survey Research Center of the 

University of Michigan as a portion of the work 

performed under a contract ketween the Survey 

Research Center and the National Center for 

Health Statistics. The variables and subpopulations are less extensive than one might desire 

for an exploratory investigation, but this limitation was imposed by the fact that these tabulations were produced as a byproduct of another 

study. 

The Health Examination Survey is an example 

of the combination of a highly complex sample 

design and estimation procedure. In brief summary, the roughly 1,900 primary sampling units 

(PSU’S) of the Current Population Survey were 

grouped into 42 strata, some of which contained 

only a single PSU that was then included in the 

sample with certainty. From each of the strata 

containing more than one PSU, a single PSU was 

selected and then subsampled in accordance with 

customary practice to obtain a sample of about 

160 persons (75 males). The estimation procedure 

included four principal operations: (1) Inflation 

by the reciprocal of the probability of selection, 

(2) a first-stage ratio adjustment to 1960 population for eight geographic and population concentration classes, (3) an adjustment for nonresponse 

carried out in 294 age-sex- PSU cells, and (4) a 

poststratification by 12 age-sex cells. These 

observations were forced into a “two observations from each of 27 strata” format by pairing 

“adjacent” noncertainty strata and by defining 

“similar” subsamples from within the certainty 

Psu’s. 

Average of Half-Sample Estimates 

As mentioned earlier, the formal consideration of balanced half samples has been restricted 

to the. ordinary linear situation. One can move 

away from the linear case in a variety of ways, 

but only the ordinary combined ratio estimate will 

be discussed here. The points raised, however, 

apply to any nonlinear situation and will be 

illustrated in Health Examination Survey data. 

In the case of stratified simple random sampling, 

with two units selected from within each of L 

strata, there are a totality of 2L half samples, 

from each of which one may compute a combined 

ratio estimate, say r,. For the linear case, it 

has been shown that analyses carried out on a 

set of balanced half samples give the same results as would be obtained by carrying out the 

same half- sample analyses on the entire set of 

2L elements. We shall now investigate the manner in which this carries over to the case of 

half-sample ratio estimates. 

In the linear case, denote the means of a 

balanced set of half samples by Y~?,l,Yhs,z, . . . , 

Yh~,k> and the mean of the entree sample by 

Y=t. Denote the means of the complementary 

set of half samples by Y&~, y;~ *, . . . P y~,,k * 

where Y;*O, ~ is the mean &f the’half sample obtained using the L elements left out of the ith 

original half sample. If k is greater than L— 

so that each observation appears in half of the 

samples— certain relationships hold among these 

means. In particular, we have that (Y~$, + y#I ,1/2 

= ,~1 Y~S,~/k= & Y;~,i/k = Y.t . In ‘a sen~e, 

we can say that the set of k balanced half samples “exhausts” the information in the totality of 

2L half samples, as far as estimation is concerned. The question can then be raised of whether 

the half -sample combined ratio estimators 

rl, r2, . .. Jr~j separately from or in combination with their complementary estimators 

r;,’ r2, . . ..r~. also “exhaust” the totality of 

4 
all 2L half-sample ratio estimators. Notice that 

we are~here asking about the relationship between ,~lr,/k, ,~lri/k, and the average of the 

ri’s for all 2L half samples and notabout t~e 

relationship between these quantities and” R, 

the combined ratio estimator for the entire 

sample. The latter is a separate question which 

will be discussed shortly. 

It appears that this problem can be attacked 

empirically by the following argument. Consider 

by way of illustration, the three-strata example 

discussed in the NCHS report (Series 2, No. 14, 

p.	 17), where the observations are denoted by 

(x~1, y~l), h= 1, 2, 3, i = 1,2, and WI = W2 = 

W3 = 1/3. Translated into combined ratio estimate 

terms, for a set of four balanced half samples, 

this is 

Yll + Y21 + Y31 Y12 + Y22 + Y32 

‘1= 4 = 

’11 + ’21 + ’31 %2 + ’22 + ’32 

Yll + Y22 + Y32 Y12 + Y21 + Y31 

‘2= r; = 

X1l + X2* + x32 ’12 + ’21 + ’31 

r - Y12 + Y22 + Y31 Y~~ + Y21 + Y32 r; = 

%2 + X22 + X31 %1 + ’21 + ’32 

Y12 + Y21 + Y32 Y~~+ Y22 + Y31 

r4= r~ = 

XII + X22 + X31 ’12 + ’21 + ’32 

In the r~ set, note that each observation 

(Xhl> y~l) occurs in two of the four samples. 

Furthermore, if we look at any two of the three 

strata, say strata h and k, then each of the four 

possible pairs of observations 

(Xhl, Yhl)and (Xkl, Ykl) 

(x ~1, y~l) and (x~2, yk2) 

‘xh2 ! Yh2) and (xkl , ykl) 

‘xh2 J yh2) and (xk2, Yk2) 

appears in one of the four samples. It is this 

“balancing of pairs of observations” which characterizes any set of balanced half samples and, 

in the linear case, makes them so suitable as 

the basis for variance-estimation. The set of 

r,’s is not, however, balanced on triplets of observations. Thus the triplet ( x~z~YIJJ (X22) Y22)) 

(X32, y32 ), - obtained by taking one observation 

from each of The three strata, does not appear in 

the ri set. Similar observations hold for the r; 

set. The combined set of ri’s and r: ‘S in this 

case does account for all possible half- sample 

ratio estimates and is therefore balanced on 

triplets. If, in a particular example involving three 

strata, we found that F and F, the averages of 

the r,’s and r; ‘s, respectively, were in close 

agreement, this would indicate that the effect 

of the balancing of triplets was negligible. Of 

course, the average of F and F’ is in this instance the actual value that one wishes to estimate. Even for the artificial and extreme example 

given in NCHS report (Series 2, No. 14, p. 27), 

E (7) = 1.1985 and E(P) = 1.1968, thus indicating that the effect of triplet balancing is extremely small. 

In general, a set of k half samples, where 

k is a multiple of four, will be balanced on the 

number of strata corresponding to the highest 

power of two that is an exact divisor of k. Thus 

if a set of eight balanced half samples is used 

for five strata, then the r, and r; sets will each 

be balanced on singles, pairs, and triplets of 

observations while the combined set will also 

be balanced on quadruplets. Under these circumstances, if F and 7’ agree, then one can argue 

that quadruplet balancing is unnecessary, although this can be achieved by combining the two 

sets. 

When k is not exactly equal to two raised to 

an integer power, and the Plackett- Burman 

(1943-46) method of constructing an orthogonal 

matrix is used, the balancing situation is not 

as clear cut. For example, the case of k=12 

has been completely investigated. Thus the r i 

and r; sets are both balanced for pairs of observations since 12 is divisible by four. Furthermore, 

the combined ri and r: sets, which contain 24 

half samples, are balanced on triplets of observations. On the other hand, f~r the case of 

k = 24 which is given in NCHS report (Series 

2, No. 14, p. 19), the r, and r: sets are each 

5 

3 
balanced on triplets of observations, since 24 

is divisible by eight, but the combined set of 48 

half samples is not balanced on quadruplets of 

observations even though 48 is divisible by 16. 

Nevertheless, a small investigation has indicated 

that the- combined sets will be b&ter balanced on 

quadruplets than will either set separately. 

In ordinary practice one would perform all 

computations on a single set of k balanced half 

samples and would not be concerned with the 

complementary set. The two sets can, however, 

be compared if there is any doubt concerning 

the adequacy of a single set. This comparison 

has been made, for methodological purposes, 

with data derived from the Health Examination 

Survey which, as noted earlier, was based on 

approximately 3,000 adult males. The analyses 

were performed on a set of 28 balanced half 

samples, the sample being treated as if two 

independent selections had been made within 

each of 27 strata, together with the complementary 

set of 28 half samples, Two different types of 

estimates were considered: 

1. Ratio estimates of population means were 

made for 15 physical body characteristics (e.g., 

chest girth, waist girth, and knee height). 

2. Each of eight body characteristics were 

separately regressed on the independenrvariables 

of age, height, and weight. Estimates were made 

of the partial regression coefficients and of the 

multiple correlation coefficient. 

In each case— 15 means, 24 partial regression 

coefficients, and eight multiple correlation coefficients—the average of the estimates made 

from the original set of 28 balanced half samples 

can be compared with the average of the estimates made from the complementary set. These 

comparisons are given in tables 1 and 2. In only 

seven of the 47 comparisons do the two means 

differ by more than .03 percent of their average 

and these instances could well have arisen from 

rounding error. Hence we conclude that for this 

situation half -sample estimates, based on a set of 

28 half samples, effectively give the same relTen groups of four columns were selected randomly from 

the population of *1C4 possible groups of four columns. Perfect balance was achieved by combining the r, and r; sete 

in five of the 10. Improved balance was achieved in four of 

the 10; and in only one of the 10 did no improvement occur. 

suits as would be obtained by working with the 

entire set of 227 half samples. 

Bias of a Ratio Estimator 

The estimate that is customarily used for 

the type of sample design to which the present 

discussion refers is the combined ratio estimator which, when hAsed on the total sample, 

will be designated by R. This estimator may be 

biased, especially for small samples, and this 

topic has been extensively researched in a wide 

variety of ways, e.g., Kish, Namboodiri, and 

Pillai (1962) and Tin (1965). When balanced half 

samples are used for variance estimation, it is 

possible to obtain, as a byproduct, an empirical 

check on the existence and magnitude of bias. 

The quantities 7, F!, and their average, denoted by 7*, are of exactly the same form as 

1$, but are based on half samples. Since the 

bias of a ratio estimator decreases with increasing sample size, we ca~n expect F* to be 

subject to greater bias than R. If, for a given 

set of data, 7, T’, F* and ~ are essentially the 

same, this is presumptive evidence that the differential bias is close to zero and it therefore 

follows that the absolute bias of either estimator 

is essentially zero. For the artificial example 

used in NCHS report (Series 2, No. 14, p. 27), 

this is clearly not :he- case since E (n = 1.1985, 

13( F’) = 1.1968, E(R) = 1.1666, and R = 1.1548. 

It is true that the bias in ~ may be small even 

though there is differential bias between F and 

k. 

On the basis of previously cited empirical 

investigations and on the basis of the size of the 

sample upon which the illustrative data of this 

paper are based, one would expect little, if any, 

evidence of bias. This expectation is borne out 

by the summary results presented in table A, which 

is based on the individual values given in column 

7 of table 1 and column 8 of table 2. As can be 

observed from the table, the differential bias of 

the 15 ratio estimates of population means is 

negligible, although one does observe that the 

sign of the difference is negative in 12 of the 15 

cases, possibly indicating some small residual 

bias in the half-sample estimators. The results 

for the estimates of partial regression coefficients and multiple correlation coefficients lead 

to the same general conclusion, although the 

6 
-----

-----

Table A. Differential bias of F* and l?, 

as a fract$on of the estimated standard 

error of R 1 

(P - ii)/s (k) 

-.30 to -.25-----

;:;: : -.20-----

-.15-----

-.15 to -.lo 

-.10 to -.05-----

-.05 to .oo------

.00 to +.05------

+.05 to+.lo 

I-.1O to +.15-----

Estimates 

of partial 

regression 

Ratio coeffiestimates cients and 

of popula- multiple 

tion means correlation 

coefficients 

1 

1 

; 

1: 

5 

7 

1 

-.0?: 

-.011 

Isee the text for a description of the 

data upon which these estimates of differential bias are used. Individual values from which these distributions are 

derived are given in column 7 of table 1 

and column 8 of table 2. 

values of the differential bias (relative to the 

standard error) tendto be somewhat larger than 

in the former case. The three largest (in absolute value) of these arise from the independent 

variable, height, but no specific reason for this 

could be found. In general, one must conclude 

that, for this type of data and for this size of 

sample, there is no need to be concerned about 

bias in either the half-sample estimators or, 

more particularly, in the entire-sample estimator. 

If the possibility of “serious” bias ap~ars 

to exist for any or all of 7, T’, 7“ and R, as 

evidenced by large differences amongthesequantities in a specific situation, one can consider 

the alternative of moving to a Quenouille-type 

(or Jackknife-type) transformation. There is some 

evidence, primarily based on Taylor series 

investigations of the ordinary ratio estimate, 

that such transformations may reduce bias without having deleterious effects on the varianceof 

the estimate. (See, forexample, Brilltiger(1964), 

Durbin(1959), NCHS (Series2,No. 14), Quenouille 

(1956), Rao (1965), and Tin (1965). There have 

been no theoretical or empirical investigations 

of complex design and estimation situations 

similar to the one being discussed in this paper. 

Estimating the Variance ofT 

Under ordinary circumstances, onehas aset 

ofk IMancedh alfs amplesand the accompanying 

half- sample ratio estimates, rl , r2, . . ., rk. 

Carrying over the argument out~ned earlier for 

the linear case, the variance of R(ratioestitnate 

computed f:om the entire sample) would be estimated as i>l(ri–~J2/k. To keep the argument 

in simplest terms, we shall first restrict attention to the prob~em of estimating the variance 

of 7 inste~d of R and hence use the varia:ce 

estimate ~~l(ri –F)2/k, here denoted by V5H~ 

(F).If the data for the complementary set of 

half samples are also processed, as is the case for 

the illustrative example being used in this paper, 

then a second estimate of the same form ca: k 

computed from r!, r;, . . ., r;, namely V~Hs 

@).In the linear case, these two estimates are 

identical. 

‘Another estimate of variance with which to 

compare V5H~m and VSHS(F’)may be obtained 

by the following argument. Viewing a particular 

half sample and its complement as two independent samples, the variance of a half-sample ratio 

can be estimated with a single degree of freedom as 

= (lA ) (r, - r/)2 

The average of ri and r;, say ~“, makes use 

of all the sample information and has estimated 

variance 

(~4)(ri – r~)2 

7 
---------------------

---------------------

---------------------

---------------------

---------------------

---------------------

---------------------

-------- -------- ------

-------- -------- -------- ---- ------------------ --------- -- --------

� 

If this estimate of variance is obtained from a 

single half sample and its complement, then it is 

the simplest form of a replicated estimate of 

variance and has often been suggested as a crude 

first approximation to such variances. In the 

linear case, the value of ~“ would be equal to 

the whole-sample estimator for any i and hence 

we shall regard ( % ) (r, - r; )2 as an approximation to the variance of the whole-sample estimator, whether it be computed as F, F‘, or as 

F* . 

If the foregoing estimate of variance could 

be computed for each of the 2‘- 1 complementary 

pairs of half samples in the entire set of 2L half 

samples, then the averagq of these estimates, 

each of which has the correct expected value, 

would appear to provide an “excellent” estimate 

of the variance F*. Since this is impossible, we 

shall substitute its computation over a set of k 

Table B. Difference between ~BH~(F), o 

balanced half samples, together of course with 

the complementary set. Thus 

ocBH~ (7*)= (lA) i~l(ri – r/)2/k . 

It seems likely, on the basis of the argument 

previously set forth on the balancing of singles, 

pairs, and triplets, and of computations carried 

out on simple examples, that these estimates are 

extremely close to those that would be obtained 

by working with the entire set of 2L half samples. 

Although no way has been found to put the 

foregoing considerations on a formal basis, it 

would appear that V~BH~(~*) if a “better” estimate of variance than either VBH~(7) or ~BH~(F’). 

In practice, of course, one would have available only one of the latter two estimates. In the 

+CBH5(F*), as a fraction 

Estimates of partial estimates coefficients 

.-44=$


Ratio 

Fractional difference 

-.08 to -.07 

-.07 to -.06 

-.06 to -.05 

-.05 to -.04 

-.04 to -.03 

~::; : -.02 

-.01 

-.O1 to .00----------------------

,00 to .01----------------------

.01 to .02----------------------

.02 to .03----------------------

.03 to .04----------------------

.04 to .05----------------------

.05 to .06----------------------

.06 to .07----------------------

J& : .;;----------------------

regression of population means and multiple correlation 

coefficients 

g“p 

1 

2 

2 

4 

2 

Total 15 32 32 

Mean -.0:? +. 019 +. 010 +0 001 

Median --.----- . ------- . . . -.018 +. 017 +* 012 -.001 
present investigation, all three estimates have 

been computed for the 15 ratio estimates of 

population means and for the previously described regression coefficients and multiple correlation coefficients. Agreement of these three 

variance estimators will be taken as presumptive 

evidence that the balanced half- sample variance 

estimation procedure is appropriate for a given 

set of data. A summary of the results of this 

comparison is given in table B, while the actual 

values on which these distributions are based are 

presented in tables 3 and 4. It will be observed 

that the distributions of fractional differences 

show the three estimates to be in relatively close 

agreement. Several further observations concerning these distributions are: 

1. For any given estimate ~c~H~(T*) tends 

to be between I$~H~m and #~H~@). 

2. For the ratio estimates of population 

means (physical body characteristics of U.S. 

adult males), the estimates #~H&F J tend to be 

on the same side of ~~~~~(F*). In other words, 

the values of #~~~(~) are correlated with one 

another. This effect is not as pronounced for the 

estimates of partial regression coefficients and 

multiple correlation coefficients. 

3, The effects of (1) and (2) show up in the 

two distributions on the left of table B, one 

distribution being located somewhat below zero 

and the other somewhat above zero. 

4. If the comparison were made on the basis 

of estimated standard errors, the distributions 

would show less dispersion. This follows immediately from the fact that (/a - /b )/~b = (~a/ 

~b)- 1 is always smaller in absolute value than 

iS (a - b)/b = (a/b) -1. 

The foregoing results can be presented in a 

slightly different manner by introducing the concept of correlated variables. Walsh (1947) summarizes the following results. If x ~, X2,. . ., Xn 

are observations where E(X, ) = ~, V(Xi ) = U*, 

and E(x1 - p) (x,-p)= PU2, then V(Y) =-$ 

[1+ (n - l)p] and EIZ(XI - Y)*/(n - l)]= u2(l- p). 

Furthermore, if the x, ‘S are normal, the quantity 

(X – K)[(l-p)/ (l+(n- l)p)]7* 

[Z(x, -Y)2/n (n - 1)] ~ 

has a Student’s distribution with (n – 1) degrees 

of freedom. For present purposes, we replace 

the X,’s by the balanced half-sample ratio estimators ‘1? ‘z) . . .) ‘k. The r,’s do have a 

common variance, and they are correlated since 

the half samples within a balanced set do have 

elements in common. For the moment, the complementary set is ignored. 

The ordinary practice for balanced halfsample replication has been to take z (ri - k )*/k 

as an estimate of the variance of *. As has been 

done earlier in ~his discussion, let us consider 

F rather than R and see under what circumstances we can treat 

T-Pr 

as having a Student’s t distribution with ( k -1 ) 

degrees of freedom, at least as far as variance 

considerations are concerned. 

If the aforementioned equality is to hold, then 

it is necessary that 

1+ (k-l)p 

(1 - p) =( JC-1) 

which means that 

k-2 

‘= 2(k-1) . 

This condition can be shown to hold, through 

the use of a common-element argument set forth 

in NCHS (Series 2, No. 14, pp. 20-21), if 

1. We are dealing with a linear situation in 

which W1=W2=. ..= WL,and S;=.$ =. . .=S~. 

2. The number of strata, L, is one less than 

a multiple of four and the set of k = L + 1 balanced half samples is obtained by deleting the 

column whose sign entries are all the same from 

a k x k orthogonal matrix. 

In the present instance, and in more complicated 

ones, these conditions cannot be expected to hold 

exactly. The strata weights and variances will 

not be equal; the half samples may not have 

exactly the same number of elements in common, 

which was required for the “common element” 

argument leading to a theoretical value of P; and 

9 
a ratio estimator may behave in a different 

fashion from a linear estimator. Nevertheless, 

one would expect these results to hold in some 

average sense, especially when there is a reasonable number of strata. It is possible to investigate this matter empirically. 

Suppose we consider a set of balanced half 

samples and their complements, together with 

the half -sample ratio estimators rl , r2, ..., 

r~ and r;, r; , . . ., r:. Then take 

2oc8H~(7*) = (1/2) ,~1 (r, - r;) 2/k 

as an estimate of u2, the variance of a single 

half-sample estimator, Furthermore, take 

$ (r, - F)2 ,$1(r/ -F’)2 i=l and 

k-1 k-1 

as estimators of u2 ( 1- P), The correlation o 

can then be estimated as 

! (ri - 7)2/(k - 1) 

2-1- ‘=1 

(Y2) ,~1 (r, - r~)~k 

or as 

d=~ _ ,i, +~’-l, 

(1/2)$1(r, - r{)2/k 

and these values, or their average, can be 

checked against the theoretical value of 

(k - 2)/2(k - 1). 

The foregoing estimates of P have been computed for the illustrative data used in this paper. 

Their distributions are shown in table C and the 

individual values are given in tables 3 and 4. The 

left portion of table C presents the results for 

ratio estimates of mean body characteristics of 

U.S. adult males, while the right portion gives the 

same data for regression coefficients and multiple correlation coefficients. The theoretical 

value of P with which these can be compared is 

(k- 2)/2( k- 1) = 26/54 = .4815. The general 

agreement of the empirical results with the 

theoretical value again leads to the conclusion 

that, for the illustrative data being used in this 

paper, one can take ~Z1(ri- n2/k as an approxi­ . mation to 

,&(rl-F)2 . [1+ (k- l)P] 

k(k-1) (1 -p) 

in estimating the variance of 7. The general observations made concerning the relationship 

among distrilmtions in table B also appl here 

.:ince the $‘s are simple functions of J’BHS(7)’ 

VBHS (F1), and $ CBHS(FbEstimating the Variance of k 

Thus- far the discussion of variance estimation has focused on 7, F’, and F*, rather than 

R, for the simple reason that we are actually 

dealing with half-sample ratio estimators. The 

ordinary procedure in either half-sample or 

balanced half-~ampl: replication has ken to use 

the quantity ,~l(ri~- R)2/k as an estimator of 

the variance ;f R. This es~imate of variance is 

larger than the estimate Zl(ri - F )~k by the 

amount (F - A )2. Since we have already observed that 7 and F 1do not differ by an appreciable amount from ~ for the illustrative data of 

this paper, it makes little difference which estimate is used. In general, it would seem that 

,~1 (ri - * )?k would overestimate even the 

mean square error of R since it contains a 

contribution for the differential bias of 7 and 

R, and this differential bias may well be larger 

than the absolute bias in ~. 

It is possible to obtain ~ formal relationship connecting V(r) and v(R). Consider the expression E(F - ~ )2. It follows that 

E(T-A)2== V(F)+ VA -2 CO”(7, 4) 

+[E(T) - A]* 

- V(F)+ V(fi) -2 p_@ra V(79 

+ [Em- E(A)]*. 

10

-------- -------- ---- ------------------ -----------

---------------------------

A 

Table C. Estimated value of the intraclass correlation coefficient for a set of 28 

balanced half-sample estimators 

Estimates of partial 

Ratio estimates regression coefficients 

of population means md multiple correlation 

Estimated value of p coefficients

.43 to 

.44 to 

.45 to 

.46 to 

.47 to 

.48 to 

.49 to 

.50 to 

.51 to 

-.52 to 

.44-----------------------


.45-----------------------

.46-----------------------

.47-----------------------

.48-----------------------

.49-----------------------

.50-----------------------

.51-----------------------

.52-----------------------

.53-----------------------

1 

2 2


3 2


5


8 

.48% .48?:


.4890 a .4838 

V(F) and v(A) on the basis of a single sample. 

The bounds obtained will not bepartic;larly good 

if the neglected term [E(F) - E (J%]* is at all 

appreciable in relation to E(T - A) 2, andthishas 

Total ---.----


Mean


Median


Regrouping ofterms leadsto 

V(F)+ v(A)- 2P7,# V!Xm 

= E(F - R) 2- [E(F) - E(fi)]2 . 

Now we observe that P7A<I. Actually, it appears that this correla~~on will bevery closeto 

one under almost any circumstances, and this 

conjecture has been confirmed by direct computation in several ’’small,” synthetic examples. 

Thus 

< E(F - i%’ - [Em - E(#)]2 

or 

[6- @2 <E(F-~)2. 

Since, as shown earlier, we can estim:te 

V(F) from asingle sample, an$shce (F-”R)2 

is an approximation for E(F-R)2 this lastexpression provides a crude way of comparing 

been the case with.various synthetic examples 

that have been investigated. On the other hand, 

this may provide quite reasonable bounds in 

practical situations where theneglected termmay 

be expected to be small. It should be observed 

that the development of the final expression 

includes sampling withoutreplacementfromfinite 

populations. Furthermorei equality will hold 

throughout if (r, + r~)/2=R for each i, as it 

does in the linear case, and we then have theresuit that V(F) = V(l?). For the illustrativedataof 

this paper, as summarized in tableA, allindications :rethat we are obtaining ’’good” estimates 

of V(R) aswellasof V(7). 

Design Effect 

As a final point, we note that column 2in 

table 3 and column 3 in :able 4 contain estimates of variance, labeled V~AN(.), computed as 

if the observations had arisen from simple ran11

dom sampling. Since the estimate ficBH~(.) takes 

into account most of the survey features (stratification, clustering, po:tstratificati:n estimation, 

etc.), the ratio of v~BH~L) to VRAN(.) is an 

estimate of the design effect. These ratios are 

given in column 8 of table 3 and column 9 in 

table 4. In table 3, which refers to estimates of 

mean body characteristics based on approximately 

3,000 adult males, these ratios range from 1.158 

to 5.729 with an average value of 3.223. In table 

4, which refers to estimates of regression coefficients and multiple correlation coefficients, 

the ratios range from .570 to 3.675 with an 

average value of 1.807. The fact that design 

effect is smaller for the regression coefficients 

than for the mean body characteristics is consistent with other studies. 

APPROXIMATE TEST OF 

INDEPENDENCE BASED ON A SET OF 

~BALANCED HALF-SAMPLE REPLICATES 

The statistical analysis of data that have 

been produced by the use of a complex sample 

design and then processed through an involved 

estimation procedure poses many difficult problems. Although it appears that the application of 

some form of pseudoreplication will provide 

a reasonable solution to the problem of estimating the sampling variability of a statistic, 

it is not immediately apparent that one can use 

these same techniques to solve certain other 

commonly occurring problems of statistical 

analysis. This section gives a summary of the 

results of an investigation carried out by Chapman (1966) concerning the use of balanced halfsample replications to test the hypothesis of 

independence in a contingency table. 

The contingency-table test of independence 

is ordinarily phrased in the following manner. 

Observations are classified jointly on each of two 

qualitative variables, the categories of variable 

one being denoted by i, i = 1, 2, . . .,r, and 

those of variable two by j, j = 1, 2,. . .,c. If 

P, i denotes the probability y of an observation 

falling in the ith category of variable one and 

in the jth category of variable two, then the 

hypothesis of independence states that I+j = p ~. 

p-j where pi refers to the marginal distribution 

of the first variable and p.~ refers to the marginal distribution of the second variable. Assuming that a “large” sample of n independent 

observations is available, the statistic 

(nij– ni.nj )

2
 n.


X2=	 ; ‘ i=l j21 ni. 

n.j 

n 

is approximately distributed as Chi- square with 

(r – 1) (c – 1) degrees of freedom when the 

hypothesis is true, and the hypothesis is rejected for “large” values of X2. Data derived 

from a complex survey operation certainly do 

not conform to this model, particularly in view 

of the dependencies that are introduced by the 

use of stratification and clustering techniques. 

Furthermore, the effects of commonly used 

estimation procedures would add to these complications. 

One might attempt to find a solution to this 

problem in the following manner. If large samples 

are used, the distribution of the sample estimates of (rc - 1) of the pijfs, which will be 

denoted by j,j, can be approximated by a nondegenerate, normal, multivariate distribution. 

The exponent (ignoring the - 1/2) in this distribution is a quadratic form in the 4, j’s and the 

coefficients of this form are the elements in the 

inverse of the variance- covariance matrix of the 

:i, ‘s. The variance- covariance matrix of the 

Pij ‘s can be estimated by a pseudoreplication 

method, such as balanced half samples, and these 

estimates would mirror the effects of the design 

and estimation. It should be observed, however, 

that the volume of computation would be large 

since even for r = 3 and c = 3 there would be 

eight variances and 28 covariances. It would then 

be necessary to invert this matrix and evaluate 

the quadratic form to take the hypothesis Pi, = 

Pi p.j into account. The resulting statistic would 

be treated as a Chi-square variable with some 

appropriate number of degrees of freedom. This 

approach has not been pursued since we shall now 

describe a “simpler” alternative. 

The basic idea upon which the proposed test 

procedure rests is the following, Suppose that an 

estimate #lj is obtained from a sample and that 

a product estimate, ~~ # j, is obtained from an . . 

12 
1

2

3

4

5

6

7

1

2

3

4

5

6

7

8

------

--

--

independent sample. Form the difference of these 

two estimates ($1, - ~,. j! j ), and consider the 

sign of the difference. Define the variable Sij to 

be one when the sign is plus and zero when it is 

minus. If the hypothesis of independence does 

hold, then this is the difference of two independent 

estimates of the parameter, Pi ~, and one would 

intuitively expect the sign of this difference to be 

plus or minus with equal probability. Actually, 

even with two independent random samples, each 

of size n, it can be shown that this result does 

not generally hold. In particular, 

1. If n is “small,” then there may be an 

appreciable probability that the difference is 

exactly equal to zero. 

2. The distributions o~ ~ij and ~; j: j will 

not be identical and therefore the distribution of 

the difference may not have a median at zero. 

For example, it can be demonstrated that 

v(f),,)= (l/n)p,,(l - pi, ) 

V(fy,$:,)= (l/n) p,, (1 - p,]) 

- [(n -l)\n2] pij(l - p,c)(l-poj). 

On the other hand, if we can assume that both 

Al, and fl; A:, are approximately normally 

distributed, ”then their difference will also be 

approximately normally distributed and the mean 

of this distribution will be ze~o when the hypothesis of independence is true. Hereafter, we shall 

make this assumption of approximate normality. 

Now suppose that it were possible to replicate this situation k times and thus obtain k 

independent observations on the variable S,j. 

The sum of the k S,, ‘s, X,, , is a binomial vari­ ,1 II 

abIe with 

E(X,j) =(kt2) 

V(xi,) = ( V4M 

when the null hypothesis is true and, if k is 

“large,” the quantity 

is distributed approximately as Chi- square with 

one degree of freedom. 

In general it would be impossible to consider using this form of analysis since it requires 2k independent replicates, each of which 

is large enough to ensure that jij and ~; 3:, 

are approximately normally distributed. Furthermore, it is also desirable that k be sufficiently 

large so that the test will have adequate power. 

However, if one is using a sample design for 

which a balanced half-sample replication method 

of estimating variances is appropriate, it appears 

that these circumstances can be realized in an 

approximate manner. Consider, as an example, 

a situation in which there are two independent 

selections from within each of seven strata and 

that one is therefore using a set of eight balanced half samples. Denoting the first element 

selected in each stratum by a + and the second 

by a -, the following table shows one set of 

eight balanced half samples and their complements: 

Half Strata Comple- Strata 

samples 1234567 ments 1234567 

+++++-!-+- -

--E-i--+- + -i---!-. + 

+- ..+-+-. -++ i-+ 

-.++-.+- ++ --+-+-

-F ++---- -i-++”+­

-+-- +-+ ?-- ++-+--

+---.++ --P i-++-­

s --i--+-F- ++ -+ --+ 

It will be observed that a half sample and its 

complement have no elements in common, that a 

half sample and any other half sample in the 

same set have three elements in common, and 

that a half sample and any half sample from the 

complementary set, other than the complement 

itself, have four elements in common. In general, 

if there are L strata and one can use a set of 

(L+l)=k balanced half samples, where k 

is a multiple of four, these numbers of common 

elements will be zero, (k/2) - 1, and (k/2), 

respectively. 

13 
Now consider the following simplified version of the stated problem: 

1. Jjj is the estimate of Plj made from 

half- sample one and #j, ~fi.\ is the corresponding product estimate obtained from the 

complementary half sample. These are independent estimators. 

2. Jl,j is the estimate of pi] made from 

half sample two and .#~. z~~j is the corresponding product estimate obtained from the 

complementary half sample. 

3. Assume that ~J$ll and ~~~j are ordinary 

linear estimates based on L strata of equal weight 

and that the within- strata estimates have common 

variance S;. It then follows, using the argument 

set forth in NCHS (Series 2, No. 14, p. 21), that 

Cov (12,1, #,,) ={[(~-da}{v’’’’}s: 

- (L-1) S2 

2L2 w “ 

4. Assume that the product estimates lpi.A‘ 

~~~i bshave in exactly the same manner as the 

A,, ‘s. As a result, we have 

and 

- {fL+I)/2}{1/L2}S~ 

(L+ 1) ~2 

2L2 w “ 

5. Using the foregoing results and assumptions, we find that 

Var(#,j - #~. #j) = Var(2$lj - 2?;. 24)= + s; 

and that the correlation between the two differences is equal to - l/L. For any reasonable number of strata this correlation will be very close 

to zero. Since the various estimates are assumed 

to be approximately normally distributed, it follows that the k differences (#1j - jj. ~~j), as 

computed from a set of balanced half samples, can 

be treated as independent observations. Some 

evidence concerning the reasonableness of this 

type of argument is provided by the data on the 

intraclass correlation coefficient given in table 

C.	 It was shown that the estimated values agreed 

quite well with the theoretical value predicted on 

the basis of the “common element” approach. 

The application of the foregoing procedure to 

a set of k balanced half samples leads to an observed value of Xlj for each of the rc cells in the 

contingency table, where 

E(XI1) = k/2 

i=l,2, . . ..r 

{ ,“=1,2, . . ..C \ 

V(x,j) = (y4)k 

when the hypothesis of independence is true. We 

would, of course, like to combine these rc values 

into one overall test. However, such a combination 

cannot be carried out in a straightforward manner since the + j are not independent of one 

another. Chapman (1966) presents an extensive 

analysis of this problem which culminates in the 

suggestion of an approximate test statistic that 

appears to have reasonable properties and which 

is simple to use in practice. The basic steps 

in the development of the statistic are: 

1. The rc variables X,j are assumed to 

have a nondegenerate, normal, multivariate distribution. Although there are constraints on the 

X,j‘s, namely that they cannot all be simultaneously equal to zero or one, these constraints 

are not of a form that reduces the rank of the 

quadratic form. 

2. Approximations to the covariance and the 

correlation of any two Xi j‘s are obtained in the 

following manner: 

a.	 If two normal variables, with zero 

means and unit variances, y and 

Y have a normal bivariate dist;;bution with correlation p(q, ~), 

14 
then the correlation p (Si ,Sj ) between 4. Combining the preceding results, we obtheir signs-i. e., Si=l if ~>0, tain the test statistic 

and S~= O if ~ <0, and similarly for 

~ —is given by i i - X)2 1=1 j=l(x,, 

T=


P(SI,S,J = -# arc sin p(ylj~). (74)k [1 – # arc sin (- J-- ~ )] rc -

This result is given in Cram& which, when the hypothesis of independence is 

(1946, p. 290). true, has approximately a X2 distribution with 

(rc – 1) degrees of freedom. The hypothesis b, Assuming that #1] and j: ~~ are of independence is rejected for “significantly” based on random samples,’ each of large values of T. Since arc sin x = x when x
 size n, from a population in which is small, this can usually be simplified to


% = (1/rc), p,. = (l/r), p., = (l/c),


it can be shown that the average


correlation between (~ij - j3~ j \ )


and (fi~t - fi~. fi:t 1 where either &


both of i 4 h and j 4 t are true, is (~4)k [1 + (2/r) (l~rc- 1))],


equal to - l/(rc – 1).


c.	 Thus we have the result that 5. The use of the independent Chi- square 

with one degree of freedom 

k2 

P(x,,
,Xht)= -# arc sin [- l/(rc - 1)] rc(X - ~)

T’= 

where either or both of i ~ h and 2 1 (1\4)k [1+ (rc -1) ~arc sin(- —)] j * t hold. Chapman (1966) inves- rc–1 

tigates some aspects of this approxi- is not recommended since ~(~) will tend to be mation procedure. It appears to be close to k/2 whether the hypothesis is true or 

a reasonable one except under ex- not. Under these circumstances, the inclusion 

treme circumstances, e.g., where of T‘ in the test statistic will weaken the power one or more of the Pll are ex- of the test. tremely small and other PI, are extremely large. ORDER STATISTICS OF A SET OF 

3. Walsh (1947) proves that if one has n BALANCED HALF-SAMPLE ESTIMATES 

normal variables, Xl, X2, . . ., Xn, each with 

mean P and variance u2 and with a common cor- A discussion of the relationship between 

relation coefficient. P then balanced half-sample replication and the sign 

test is presented in NCHS (Series 2, No. 14, pp. 

T- ~ 

U*(1- p) ,~1(xl - X)2 20- 23). The problem can briefly Ix stated as 

follows. Suppose that each half-sample replicahas a X2distribution with (n - 1) degrees of tion provides an estimate, say d~.., , of a popu­ ... .. freedom and lation difference, and that one wishes to test the 

. hypothesis that the dh= ,‘s were drawn from 

T’= 

n(X– p)’ a population whose med;;n is equal to zero. If 

u2[l+(n-l)pJ the dh~, ‘s were independent of one another, 

then one’ solution to this problem would be obhas an independent X2distribution with one de- tained by a straightforward application of the 

gree of freedom. ordinary sign test. The half- sample estimates 

15 
are not, however, independent of one another, as 

is observed in the earlier sections of this paper, 

and the question can be raised of whether it 

is possible to devise some suitable modification 

of the sign test. 

As soon as one introduces dependence among 

the members of a set of observations, it is no 

longer possible in general to devise nonparametric or distribution-free procedures. If, however, it is reasonable to assume that the ‘s 

‘hs, I 

have a ,normal multivariate distribution, then existing tables can, under certain circumstances, 

be applied to test the stated hypothesis. Thus, 

Gupta (1963, p. 817) provides a table which gives 

the “probability y that N standard normal random 

variables with common correlation p are simultaneously less than or equal to H.” In particular, 

if H is taken to be zero, the appropriate probabilities form the basis for obtaining the levels 

of significance for an “extreme” sign test, i.e., 

all signs plus or all signs minus. 

For any reasonably large number of balanced half- sample estimates, the Gupta tables 

are not especially helpful. To obtain meaningful 

levels of significance, one must be able to move 

in from the extremes on the distribution of the 

number of plus signs. Since such tables of the 

normal multivariate distribution do not presently 

exist, we shall here present a procedure for 

approximating these probabilities. The degree 

of approximation appears to be quite acceptable 

for ordinary applications of the sign test. 

Consider n independent and normally distributed variables Xl, X2, . . ., Xn with E (Xl) = O 

and E (X;) = 1. If the collection {Xl~ is ordered 

so that X(l) 2 X(2) 2 X(n), then certain moments 

of the X(’) have been extensively tabulated. In 

particular, Teichroew (1956) has tabulated E(X(’)) 

and E(X(i) X(l)) for n = 1 (1) 20 and Harter 

(1960) has tabulated E (X(’)) for n = 2 (1) 100, 

plus values for a number of additional n < 400. 

Furthermore, Owen and Steck (1962) have shown 

that the moments of the order statistics of a 

sample from the equicorrelated, multivariate, 

normal distribution can be readily obtained from 

the corresponding moments for n independent 

variables. Thus, if 21, 22,. . ., Zn are jointly 

distributed random variables with 

E(Z,) = O, E(Z,2) =1, and E(Zi Zj) = p for i ~j, 

E(Z(’)) = (1 – p)l/2E(x(i)) 

and 

E[(Z(’))– E(z(i))]2 = p + (1‘p) E[X(’)– E(X(’))]2. 

Our present concern is with the probability 

that U out of the n values 21,22,. . ., Zn are 

positive, where U takes on the values 0,1,2,... . 

For example, for the first three values, it is clear 

that 

Pr(U= O)= Pr(Z (1) < _ ~, 

Pr(U=O or l)= Pr(Z (*) <0, _ 

Pr(U = O or 1 or 2) =Pr(Z(3) < O). 

Under ordinary circumstances the calculation 

of the probabilities on the right side of these 

expressions would be extremely difficult since 

they must be obtained from the distributions 

of 2(1) 2(2) ,. ... and this is the extreme-value 

problem’ that has been considered in some detail 

by Gumbel (1958). In the present problem, however, moderate values of p, i.e., in the neighborhood of 1/2, appear to ensure tiiat the distributions of the Z(’) can be approximated reasonably 

well by a normal distribution. Some third and 

fourth moments of the extreme order statistic, 

for various values of P, are exhibited by Owen and 

Steck (1962), and these values confirm this 

observation. Owen and Steck note that this behavior is to be expected as p increases since 

z(i) has a unit normal distribution when p = 1. 

As an illustration of this approach consider 

the case where n=20 and p= .5o. From 

Teichroew’s table 

E@) = ~ &j74~ 

Ax(’))= .27568, “


therefore 

E(Z(l)) = 1.32051 

U2(Z(1))= .63784. 

16 
---------

-------- -- -------- -- ------- --- ----------

‘lable D. Probability that u or fewer, out of n normal correlated observations, will 

be greater than their common mean (less than their common mean) 

Number I 16 20 24 48 

observat~ons “ b--l-= .40 

I

.50 .40 

I

.50 

u I I 

0--------- - .0361 .0603 .0272 .0492 .0235 .0433 

1(.0335) ~(.0588) 1(.0251) 1(.0476) 1(.0197) 1(.0440) 

1 - .0806 .1178 .0607 .0956 .0487 .0809 

2 .1346 .1746 .1015 � 1430 .0808 .1204 

3 

4 

5 I I 

lThe values in parentheses are exact and are from Gupta’s tables. 

.40 .50 

+ 

.0096 I .0227 

. 019; , 041i 

.0321 .0615 

.0467 .0818 

.0628 . ;:;;: 

.0805 

Assuming a normal distribution for Z(l) , we 

obtain 

Pr(Z(l) SO)= .0492. 

The correct value for this probability, fromGupta’s tables, is ,0476, This appears to be quite 

acceptable accuracy for ordinary purposes. Furthermore, the approximation should be better 

for Z(2), Z(3), . . . since the distributions of 

these variables cannot be as skewed as is the 

distribution of Z ‘1). For n > ’20,it was also 

necessary to approximate U*(Z(’)). This was 

done byconverting J!?(X(’)), fromHarter’s table, 

to acumulative percentage and using theasymptotic variance formula for the percentage point 

of a distribution (see Wilks, 1962, p. 273). 

The values of these probabilities for n= 16, 

20, 24, and48, and for P=.40and .50 have been 

computed and are presented in table D. Where 

possible, the exact values from Gupta’s table 

are given for purposes of comparison. Not only 

do these values lead toasigntesc, but they also 

lead to confidence intervals for the population 

mean (or median). Thus with 16 half-sample 

estimates, and with a P of .50, the largest and 

smallest estimates provide a 1 - (2 x .06) = 88 

percent confidence interval; similarly, the second 

smallest and second largest estimates provide 

a 1- (2 x .1178)= 76 percent confidence interval. 

In order to apply the foregoing theory it is, 

of course, necessary to have a value of p with 

which to enter the tables. The data given in table 

C on the intraclass correlation coefficient suggest that a reasonable approach is to assume that 

P has the value suggested by the “common element” argument. That is, if a set of k balanced 

half samples is used for (k – 1) strata, p would 

be assigned the value (k – 2)/2 (k - 1). 

Finally, we observe that if one is willing to 

use the set of complementary half samples, then 

still another approach to this problem is suggested by some of the earlier arguments of this 

report. Suppose one wishes to comJ are the means 

of two variables X and Y. Let Xi be the estimate of ~ made from the ith mAember of a set of 

k balanced half samples and Yi’ be the estimate 

of ~ made from the complementary half sample. 

The development on p. 15 suggests that the corr.#atio~n between any two of the k differences 

(Xi – Y;) will be very small and that it would not 

be unreasonable to treat them as k independent 

observations. Thus the ordinary sign-test tables 

could be applied to the signs of the k differences. 

17 
REFERENCES


Brillinger, D. R.: The asymptotic behavior of Tukey’s 

general method of setting confidence limits (the Jacknife) 

when applied to maximum likelihood estimates. Review of fntemational Sta.ti8tical Irwtitute 32:202.206, 1964. 

Chapman, D. W.: An Approximate Test of Independence 

Baeed on Replications of a Complex Sample Survey Design. 

M.S. theeis, New York State School of Industrial and Labor 

Relatione, Cornell University, 1966. 

Cochran,. W. G.: Sampling Technique, ed. 2. New York. 

John Wiley & Sons, Inc., 1963. 

Cram&r,H.: Matheraaticat Methods of Statietic8. Princeton. 

Princeton University Press, 1946. 

Deming, W. E.: Some Theory of Sampling. New York. John 

Wiley & Sons, Inc., 1950. 

Deming, W. E.: Sampling De gign in Bu8inee8 Research. 

New York. John Wiley & Sons, Inc.,’ 1960. 

Durbin, J.: A note on the application of Quenouille’s 

method of bias- ‘reduction to the estimation of ratios. Biometr<ka 46:447-480, 1959. 

Durbin, J.: Design of multi-stage surveys for the estimation of sampling errors. App?ied Statiatica 16:152.164, 1967. 

Gumbel, E. J.: Statistic o~ Eztreme8. New York. Columbia University Press, 1958. 

Gupta, S. S.: Probability integrals of multivariate normal 

and multivariate t. Ann.math.Stati8t. 34: 792-R28. 1963. 

Gurney, M.: The Variance of the Replication Method for 

Estimating Variances for the CPS “SampleDesign. Unpublished 

memorandum, U.”S.Bureau of the Census, 1962. 

Harter, H. L.: f!%pected Value8 of Normal Order S.tatiatica. 

Aeronautical Research Lab. Technical Report 60.292. WrightPatterson Air Force Base, Ohio, 1960. 

Hartley, H. O.: Ana2ytic Studies of Survey Data. Instituto 

di Statistic, Rome, Volume in onora di Corrado Ghri. Ames, 

Iowa. “Statistical Laboratory, Iowa State University of ‘Science 

and Technology. Heprint Series 63. 1959. 

Kleh, L.: Confidence intervals for cluetered samples. 

Americun Sociological Reaiew 22(2): 154165, Apr. 1957. 

Kish, L.: Survey Sampling. New York. John Wiley & Sons, 

Inc., 1965. 

Kish, L., Nsmboodiri., N. K., and Pillai, P. K.: The ratio 

bias in surveys. J.Am.Stotist.A. 56(300): 863-876, Dec. 1962. 

Lauh, E. and Williams, W. H.: Some small sample results 

For the variance of a ratio. Proceeding of the Social Stati8-

tiC8 Sec$ion of the American Statistical A880ciation, 1963. 

PP. 273-283. 

National Center for Health Statistics: Cycle 1 of the Health 

Examination Survev. samole and resconee, United States, 

1960-62. Vital a~’ fiea~th Statiatic81 PHS Pub. No. 1000. 

Series 11-No. 1. Public Health Service. Washington. U.S. 

Government Printing Office, Apr. 1964. 

National Center for Health Statistics: Replication, an ap. 

preach to the analysie of data from complex surveys. Vital 

and ffealth Statistic8. PHS Pub. No. 1000-Series 2-No. 14. 

Public Health Service. Washington. U.S. Government Printing 

office, Apr. 1966. 

Owen, D. B., and Steck, G. P.: Moments of order statistics 

from the equicorfelated multivariate normal distribution. 

Ann.math.Statiit. 33:1286-1291, 1962. 

Placke tt, R. L.: and Burman, J. P.: The design of optimum 

multifactorial experiments. Biometr%%z33:305-325, 1943-46. 

QuenouiHe, M. H.: Notes on bias in estimation. Biometrika 

43:353-360, 1956. 

Rae, J. N. K.: A note on estimation of ratios by Que. 

nouille’s method. Biometrika 52:647-649, 1965. 

Rae, J. N. K.: and Webstar, J. T.: On Two Methods of 

Bias Reduction in the Estimation of Ratios. Unpublished paper, 1965. 

Teichroew, D.: Tables of expected valuee of oder statistics for eamples of size twenty and less from the normal distribution. Ann.math.Statiat. 27:410-426, 1956. 

Tin, M.: Comparison of some ratio estimators. J.Am.Stati8t.A. 60(309): 294307, 1965. 

U.S. Bureau of the Censue: The Current Population SW. 

vey, A Report on Methodology. Technical Paper No. 7. Washington. U.”S.Government Printing Office, 1963. 

Walsh, J. E.: Concerning the effect of intraclaee correlation on certain significance teste. Ann.math.Statist. 18(1): 

88-96, 1947. 

Wilks, S. ‘S.: Mathematical Statistics. New York. John 

Wiley & Ekms, Inc., 1962. 

000 

18 
-----------------------------------------------

----------

--------------------------

---------------------------------------------------------------

DETAILED TABLES 

Page


Table 1.	 Estimates of mean body characteristics for population of U.S. adult males based 

on Health Examination Survey data 20 

2.	 Estimstesof regression coefficients and the multiple correlation coefficient for


population of U.S. adult males based on Health Examination Survey data 21


3.	 Estimated sampling variability of mean body characteristics for population of


U.S. adult males basedon Health Examination Survey data,and estimated intraclass


correlation coefficients among pseudoreplicate samples 22


4.	 Estimated sampling variability of regression coefficients and the multiple correlation coefficient for population of U.S. adult males based on Health Examination Survey data,and estimated intraclass correlation coefficients among pseudoreplicate samples 23 

19

1 -------- ------------- --------

2------------------

3------------------ ---

4-----------------------------

5-----------------------------

6----------------------------

7---------------- --

8-------- -----

9---------------------------

--- --- --- --- ---

Table 1. Estimates of mean body characteristics for population of U.S. adult males based ~
 Health Examination Survey data


Variable 4 F ~1 F* (F* _*) 

(1) (2) (3) (4) (5) (6) (7) I 

39.76991 39.76854 39.76836 39.76845 -.00146 -.020 

-----.--..- 30.82899 30.82636 30.82650 30.82643 -.00256 -.025 

-.------ 99.57059 99.56900 99.56929 99.56914 -.00145 -.007 

89.00548 89.00882 89.00932 89.00907 .00359 .016 

1.26982 1.26907 1.26896 1.26902 -.00080 -.026 

1.50193 1.50136 1.50143 1.50140 -.00053 -.028 

----.------ 86.63670 86.63346 86.63346 86.63346 -.00324 -.026 

.------------.-- 90.60693 90.60443 90.60411 90.60427 -.00266 -.022 

-. 54.32630 54.32346 54.32339 54.32342 -.00288 -.028 

10-----------------------------44.02929 44.02836 44.02843 44.02840 -.00089 -.013 

11----------------------------- 14.51358 14.51357 14.51361 14.51359 .00001 .000 

12----------------------------- 59.22556 59.22364 59.22371 59.22368 -.00188 -.022 

13----------------------------- 49.45498 49.45439 49.45421 49.45430 -.00068 -.007 

14-----------------------------35.62331 

15-----------------------------42.22215 42.22332 42.22350 42.22341 .00126 .012 

16-----------------------------24.26486 24.26357 24.26364 24.26360 -.00126 -.012 

column 1 Variable 1. Biacromial diameter 9. Knee height


2. Right arm girth 10. Popliteal height 

3. Chest girth 11. Thigh clearance height


4. Waist girth 12. Buttock-knee length


5. Right arm skinfold 13. Buttock-popliteal length


6. Infrascapular skinfold 14. Seat breadth


7. Sitting height (normal) 15. Elbw-elbw breadth


8. Sitting height (erect) 16. Elbow rest height


column 2	 his the combined ratio estimate based on the entire sample, including adjustments for


nonresponse and poststratification.


column 3	 The estimate ~ is obtained for each member of a set of 28 balanced half samples. These 

estimates are denoted by r,,i-1, 2,. . .,28,and Fis their average. 

column 4	 If r,is the eatilnatefrom a half sample, then r; is the corresponding estimate made from


the complementary half sample. F is the average of these 28 complementary estimates.


column 5 F*ia obtained by averaging columns 3 and 4.


Column 6 column 5-column 2.


column 7 The difference G“-1$) expressed aa a fraction of the eatimsted standard error of


k. #(&is computed as (~4),~1(r1
 – r~)2/28.


The actual valuea of the estimate of variance used in the computations are g ven in column 7 of table 3. In the text discussion, these estimstea are denoted by #CBHS(F*) “ 

20

--------

--------

--------

--------

--------

--------

--------

--------

--------

--------

--------

--------

--------

--------

--------

--------

Table 2. Estimates of regressioncoefficients and the multiple correlationcoefficientfor populationof 

U.S. adult males based on Health ExaminationSurvey data 

Dependent Independent —0 -* variable variable b b (7)/F­v(p)

(1) (2) (5) (6) (8)


1


2


3


4


5


6


8


Age-----------
 Height


Weight


Multiple R----


Age-----------


Height


Weight


Multiple R----


Age-----------


Height


Weight


Multiple R----


Age-----------
 Height


Weight


Multiple R----


Age-----------


Height


Weight


Multiple R----


Age-----------


Height


Weight


Multiple R----


Age-----------
 Height


.Weight


Multiple R----


Age-----------


Height


Weight


Multiple R----


-.02551 -.02555


.05260 .05228


,02999 ,02999


.55033 .55024


-.02724 -.02733


-.12620 -.12689


.11140 .11142


.88439 .88481


+ 

.03889 .03885


-.20126 -.20248


.28900 .28908


.90431 .90470


.20804 .20781


-.38187 -.38291


.38092 .38117


.91499 .91529


-.002’/9 -.00275


-.02281 -.02283


,01766 .01764


.59132 .59274


, 

.00037 .00010


.34698 .34455


.02407 .02423


.73569 .73549


-.01787 -.01824


.37403 .37154


.01639 .01654


.78113 .78183


-.02555 -.02555 -.00004 -.016


.05227 .05228 -.00032 -.056


.02998 .02998 -.00001 -.011


.55028 .55026 -.00007 -.006


-.02734 -.02734 -.00010 -.038 

-.12688 -.12688 -.00068 -.127 

.11142 .11142 .00002 .016 

.88479 .88480 .00041 .103 

.03884 .03884 -.00005 -.007 

-.20247 -.20248 -.00122 -.075 

.28908 .28908 .00008 .026 

.90468 .90469 .00038 .092 

.20779 .20780 -.00024 -.028 

-.38293 -.38292 -.00105 -.068 

.38119 .38118 .00026 .075 

.91529 .91529 .00030 .069 

-.00276 -.00276 .00003 .037


-.02284 -.00284 -.00003 -.011


.01763 .01764 -.00002 -.037


.59282 .59278 .00146 .065


.00024 .00024 .00001 .011 

-.03650 -.03650 -.00009 -.042 

.02311 .02311 -.00002 -.035 

.77312 .77312 .00005 .015 

.00009 .00010 -.00027 -.084 

.34455 .34455 -.00243 -.232 

.02422 .02422 .00015 .058 

.73544 .73546 -.00023 -.019 

-.01824 -.01824 -.00037


.37152 .37153 -.00250


.01653 .01654 .00015


.78180 .78182 .00069


column 1


column 2 

column 3 

Column 4 

column 5 

cokIml 6 

column 7 

Cohlml 8 

Dependentvariable 1. Biacromialdiameter


2. Right arm girth


Chest girth
 :: Waist girth


5. Right arm skinfold


6. Infrascarxtlar
 skinfold


Sittingheight (normal)
 :: Sittingheight (erect)


Independentvariables as identified. Note that the last entry in each set is the multiple corre­
 lation coefficient.


These are the regressioncoefficients and the multiple correlationcoefficient as estimatedfrom


the entire sample, includingadjustmentsfor nonresponseand postsgratification.


The estimate ~ is obtained for each gember of a set of 28 balancedhalf samples. These esti~tes
 are denoted by b,, i- 1,2,...,28,and b is their average.


If b, is the estimatefrom a half sample,then b;is the correspondingestimatemade from the comple­


mentary half sample. Etis the averageOf these 28 complementaryesti~tes.


Average of columns 4 and 5.


COhnnn 6-column3.


The difference c~’-~) expressedas a fraction of the estimatedstandarderror of ~. ~(j)is cOm­


puted as (1/4)lfl(bl- b;)2/28.


7%&clyl values of the estimateof varianceused in the come tatiQnsare given in COIUmn 8 Of


. In the text discussion,these estimatesare denotedby ?csHs[b*)
 .


21

------- ----

------------

-----------

----------- --- ---

Table 3. Estimatedsamplingvariability of mean body characteristics for population of U.S. adult males


based on Health ExaminationSurvey data, and estimatedintraclass correlationcoefficientsamong pseudo-


replicatesamples


Intraclasscorrelations


Variable $RA@) 05H@) ;BH5m ;5H5(7) (7)/(2) 

; P 

(1) (2) (3) (4) (6) (8) (9) (lo) (11) 

1 .00143 .00551 .00551 .00564 .00564 .00556 3.888 

2----------- .00342 .01034 .01033 .01086 .01085 .01057 3.,090 

At ;*

---.------ .02265 .03740 .03740 .03943 .03943 .03833 1.692 :----------- .04148 .04623 .04622 .05007 .05006 .04802 1.158 

5----------- .00019 .00096 .00096 .00100 .00100 .00098 5.158 

6----------- .00019 .00039 .00039 .00038 .00038 .00038 2.000 

.00449 .01591 .01590 .01599 .01598 .01592 3.546 L---------- .00444 .01480 .01479 .01528 .01527 .01501 3.380 

9---------- .00282 ,01012 .01011 .01083 .01082 .01045 3.706 

.00241 .00509 ,00509 .00503 .00503 .00505 2.096 :L---------- .00096 .00556 .00556 .00546 .00546 .00550 5.729 

12----------- .00281 .00713 .00713 .00727 .00727 .00718 2.555 

13----------- .00310 .01061 .01061 .01111 .01111 .01084 3.496 

.00243 .01126 .01030 .01075 4.424 K---------- .00707 .01109 .01109 .01156 .01156 .01129 1.596 

16----------- .00292 .01187 .01187 .01184 .01184 .01184 4.054 

column 1 Variable 1. Biacromia1 diameter Knee height

2. Right arm girth J: Poplitealheight


3. Chest girth Thigh clearanceheight


$.	 Waist girth +:: Buttock-kneelength
 . Rizht arm skinfold 13. Buttock-popliteallength


6. In&ascaDular skinfold 14. Seat breadth


7. Sittingkeight (normal) 15; Elbow-elbowbreadth


8. Sittingheight (erect) 16. Elbow rpst height
 . .
 column 2 ~~AN(~)
is computedas if the sample had been drawn as a simple random sampleof approximately3,000


adult males from the total U.S. population. The effectsof clustering,stratification,
 and estima­


tion have been ignored.


column 3	 1$~~(~)is the varianceestimate that is ordinarilycomputed from a setof 28 balancedhalf samples. 

If the individualhalf-sampleestimatesare denotedby rl, i = 1,2, ...,28 then 

flsHs(Z$) = ,;l(r,
– &2/28


column 4 ~B~s(F)is similar to ts~~ (J%,
except that deviationsare taken about the mean of the ri’s,namely F.


;BH~(r)=
~1 (ri- 7)2/28


column 5	 #~Hs(& correspondsto #sHs(fqexcept that it is computed from the complementaryhalf-eample estimates. Denoting these by r:, i = 1, 2, . . ., 28 

fi&& = ,~l(r~-
i$)2/28


COh.nml6


;B~~(r’)=
,~l(r~ - F’)2/28 

column 7 VC6HS,
(P) is an estimateof “variancebased on the comparison of a half-sample estimater,with its


complementary estimater~.This estimate is not availablein the ordinaryapplicationof balanced


half-samplereplicationfor variance estimation,


#c5H~(P) = (1%),~l(rl -r/)2/28 

Column 8	 The ratio of column 7tocoluum 2 estimatesthe effect on samplingvariabilityof sample design and


estimationin comparisonwith simple random sampling.


column 9 # is an estimate of the int~aclasscorrelationamong the rl’. $. It was computedas 

2 flc5H5(7*)-(28/27
 )fis@) ;=


2 ~sH&*)


Column 10 $’is an estimateof the int~aclasscorrelationamong the r~’s. It was computedas 2Vc~Hs@)-(28/27)&(F) # = 

2Vcmls(~’j


Column 11 j“is the average of$ and j’,


22

-----

-----

Table 4. Estimated sampling variability of regression coefficients and the multiple correlation coefficient for popu­
 lation of Us. adult males basedOn Health Examination Survey data,l and eati~ted intraclass cOrrelatiOn coefficients
 among pseudoreplicate samples


Intraclass correlation 

Dependenl 

(5) $s 

SHS(~) ;EHS(Z’) icBH~(r) :8)/(3~ * ;, )* 

(1) (2) (4) (5) (6) (7) (8) (9) (:0) (11) (12)” 

Age-------- .43432 ,59S23 .59807 .66892 .66876 .63165 1.454 .5091 .4510 .4800 

Height----- .26396T .32710T .32608t .33045t .32936T .32603T 1.235 .4814 .4762 .4788 

Independent 

I var iabh variable OSHJA tisH~ 

1 

Weight----- ,15524 ,08954 .08954 .08896 .08895 .08854 .570 .4756 .4791 .4774 

Multiple R- .15748TT .16116TT .16115tt .15374~t .15374tt .15669TT .995 .4667 .4913 .4790 

Age-------- .32395 .65564 .65483 .63545 .63445 .64223 1.982 .4713 .4878 .4796 

Height----- .19689T .31062+ .30586t .28883t .28421t .29365+ 1.491 .4599 .4982 .4790 

2 Weight----- .11579 .17278 .L7274 .15962 .15958 .16537 1.428 .4584 .4997 .4790 

Multiple R- .01538TT .o1514tT .01496tt .01695tT .01679TT .o1575tt 1.024 .5075 .4473 .4774 

Age-------- .17959+ .39902T .39900t .35124t .35122t .373277 2.078 .4458 .5121 .4790 

Height----- .109157T .28912~T .28763’fT .24561TT .24415‘T .26411tt 2.420 .4353 .5207 .4780 3 Weight----- .64194 .93129 .93065 .94957 .94893 .93292 1.453 .4828 .4726 .4777 

Multiple R- ,01076+t .01762Tt .o1747tt .01702tt .o1688+t .o1710tt 1.589 .4703 .4882 .4792 

Age-------- .29386T .71384t .71331’ .73237+ .73174‘ .72068T 2.452 .4868 .4735 .4802 

Height .17860TT 25191+T .25083TT ,23508Tt .23396‘T ,24096tt 1.349 .4603 .4966 .4784 4 

Weight ,10504+ 11596+ .l1534t .129387 .12865t ,12097+ 1.152 .5056 .4486 .4771 

Multiple R- .00859tt 01870tT .o1861tt .o1994tt ,01985‘t ,o1914tt 2.228 .4959 .4623 .4791 

Age-------- .05323 ,09286 .09270 .08917 .08908 ,09038 1.698 .4682 .4890 .4786 

Height----- .03235+ ,05407’f .05407f .05597t .05596f ,05461+ 1.688 .4866 .4687 .4;76 

5 Weight----- .01903 ,04491 .04487 .04726 .04717 ,04576 2.405 .4916 .4655 .4786 

Multiple R- .13705tt ,52915~t .52713tt .49128tt .48903T’f .503657r 3.675 .4573 .4966 .4770 

Age-------- .03276 ,04977 .04973 .05018 .05017 ,04982 1.521 .4824 .4779 .4802 

Height----- .o1991t ,04614t .046067 .04438t .04430t .04503t 2.262 .4696 .4899 .4798 6 Weight----- .01171 ,03260 .03256 .03196 .03192 .03216 2.746 .4751 .4854 .4802 

Multiple R- .05246Tt ,01384TT .01384TT .01416tT .01416tt .o1390tt 2.650 .4837 .4718 .4778 

Age-------- .08967T ,l1505t .11432+ .10333 t .01255T .10774+ 1.202 .4498 .5065 .4782 

Height----- .05450tl ,11689tt ,11099TT .11577~t .10987t+ .lo991tt 2.017 .4764 .4817 .4790 

7 

Weight----- .32052 ,73399 .73143 .71576 .71351 .71920 2.244 .4727 .4856 .4792 

Multiple R- .06320‘1 ,12980TT .12976tT .139337t .13927+t .133757t 2.116 .4970 .4601 .4786 

Age-------- .07538t .13850t .13713t .13500+ .13486t .13500t 1.791 .4733 .4820 .4776 

Height----- .45812t .82611t .76411T ,80067T ,73767T .74612t 1.629 .4690 .4874 .4782 

8 

Weight----- .26943 .48075 .47850 .47475 .47279 .47351 1.757 .4760 .4823 .4792 

Multiple R- .04924T1 .07338t+ .07289tT .07896Tt .07851t7 ,07524TT 1.528 .4977 .4590 .4784 

See footnotes on next page.


23

‘In columns 3-8, the absence of a dagger meana that the entry ia to be multiplied by 10-s,the presence of a single


dagger signifies multiplication by 10-4,and the presence of a double dagger aignifiea multiplication by 10-3.


Column 1 Dependent variable 1. Biacromial diameter
 2. Right arm girth
 Cheat girth


?: Waist girth
 5. Right arm skinfold
 6. Infrascapular akinfold


Sitting height (normal)
 :: Sitting height (erect)


Column 2	 Independent variablea as identified. Note that the laat entry in each set is the multiple c~rr~lation 

coefficient. 

Column 3	 ~~(j) ia computed as if the sample had been drawn aa a simple random sample of approximately 3,000 adult
 malea from the total U.S. population. The effects of clustering, stratification, and estimation have been
 ignored.


Column 4	 ~BH5(~)ia the variance estimate that ia ordinarily computed from a aet of 28 balanced half ssmples. If the individual half-sample estimates are denoted by b,, i= 1,2,. . .,28 then 

i&(ii)= ,;l~ - ;)2/28


Colunm 5 $BH5(G) is similar to OBH~ (fi),excepthat deviationa are taken about the mean of the hi’s, namely ~. 

#BH5(b) +(4 - ~)2/28 

Column 6 $~HS (fi)corresponda toI?BH~(~),
 except that it LS computed from the complementary half-sample estimatea. 

Denoting these by b;, i=l,2, . . .,2g 

$;H@= #l(b; - ;)2/28 

column 7


Column 8	 Vce @)ia an estimate of variance basedon the comparison of a half-sample estimateh with its complementary eat‘!mate b~. This estimate ia not available in the ordinary application of balauced half-sample replication for variance estimation. 

$cBH~(&)= (~4),~l(bl -~;)2/28 

Column 9	 The ratio of column 8 to column 3 eatimatea the effect on sampling variability of sample design and eati­


awtion in comparison with simple random sampling.


Column 10 $ia an estim]te of the intraclass correlation among the bi’s. It waa computed as 

2; _(&)-- (28/27)
;&~(h


)= .
 2 vcBH~(E*)


Column 11 j! is an estimate of the intraclaaAacorrelation among the b~ta. It was computed aa


p= 2 ‘CBHS(6*)-(28/27)
;BH@) 

2 t&H~(i%


Column 12 ~“ ia the average of j and ;B,


24

VITAL AND HEALTH STATISTICS PUBLICATION SERIES 

Originally fiblic Health Sewice tiblication No. 1000 

Sevies 1. Programs and collection procedures. — Reports which describe the general programs of the National 

Center for Health Statistics and its offices and divisions, data collection methods used, definitions, 

and other material necessary for under standing the data. 

Series 2.	 Data evaluation and methods research. — Studies of new statistical methodology including: experimental tests of new survey methods, studies of vital statistics collection methods, new analytical 

techniques, objective evaluations of reliability of collected data, contributions to statistical theory. 

Series 3. Analytical stvdies. -Rep0rts presenting analytical or interpretive studies based on vital and health 

statistics, carrying the analysis further than the expository types of reports in the other series. 

Swies 4. Documents and committee veports. — Final reports of major committees concerned with vital and 

health statistics, and documents such as recommended model vital registration laws and revised 

birth and death certificates. 

Sevies 10.	 Data from the Health Interview Su?wev. —Statistics on illness, accidental injuries, disability, use 

of hospital, medical, dental, and other services, and other health-related topics, based on data 

collected in a continuing national household interview survey. 

Series 11. Data jvom the Health Examination Survey. —Data from direct examination, testing, and measurement of national samples of the civilian, noninstitutional population provide the basis for two types 

of reports: (1) estimates of the medically defined prevalence of specific diseases in the United 

States and the distributions of the population with respect to physical, physiological, and psychological characteristics; and (2) analysis of relationships among the various measurements without 

reference to an explicit finite universe of persons. 

Series 12.	 Data from the Ins titutional Population Surveys. — Statistics relating to the health characteristics of 

persons in institutions, and their medical, nursing, and personal care received, based on national 

samples of establishments providing these services and samples of the residents or patients. 

Series 13.	 Data from the Hospital Discharge Survey. —Statistics relating to discharged patients in short-stay 

hospitals, based on a sample of patient records in a national sample of hospitals. 

Series 14.	 Data on health resources: manpower and facilities. —Statistics on the numbers, geographic distribution, and characteristics of health resources including physicians, dentists, nurses, other health 

occupations, hospitals, nursing homes, and outpatient facilities. 

Sevies 20.	 Data on mortality. — Various statistics on mortality other than as included in regular annual or 

monthly reports —special analyses by cause of death, age, and other demographic variables, also 

geographic and time series analyses. 

Swies 21. Data on natality, marriage, and divorce. —Various statistics on natality, marriage, and divorce 

other than as included in regular annual or monthly reports+special analyses by demographic 

variables, also geographic and time series analyses, studies of fertility. 

Series 22. Data from the National Natality and Mortality Surveys. — Statistics on characteristics of births 

and deaths not available from the vital records, based on sample surveys stemming from these 

records, including such topics as mortality by socioeconomic class, hospital experience in the 

last year of life, medical care during pregnancy, health insurance coverage, etc. 

For a list of titles of reports published in these series, write to: Office of Information 

Nat ional Center for Health Statistics 

Public Health Service, HSMHA 

Rockville, Md. 20852