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ORIGINAL PAPER

Statistical analysis of ratio estimators and their estimators

of variances when the auxiliary variate is measured with error

Christian Salas

Æ

Timothy G. Gregoire

Received: 14 May 2008 / Revised: 2 December 2008 / Accepted: 23 December 2008 / Published online: 3 June 2009

ÓSpringer-Verlag 2009

Abstract Forest inventory relies heavily on sampling

strategies. Ratio estimators use information of an auxiliary

variable (x) to improve the estimation of a parameter of a

target variable (y). We evaluated the effect of measurement

error (ME) in the auxiliary variate on the statistical per-

formance of three ratio estimators of the target parameter

total s

y

. The analyzed estimators are: the ratio-of-means,

mean-of-ratios, and an unbiased ratio estimator. Monte

Carlo simulations were conducted over a population of

more than 14,000 loblolly pine (Pinus taeda L.) trees, using

tree volume (v) and diameter at breast height (d) as the

target and auxiliary variables, respectively. In each simu-

lation three different sample sizes were randomly selected.

Based on the simulations, the effect of different types

(systematic and random) and levels (low to high) of MEs in

xon the bias, variance, and mean square error of three ratio

estimators was assessed. We also assessed the estimators of

the variance of the ratio estimators. The ratio-of-means

estimator had the smallest root mean square error. The

mean-of-ratios estimator was found quite biased (20%).

When the MEs are random, neither the accuracy (i.e. bias)

of any of the ratio estimators is greatly affected by type and

level of ME nor its precision (i.e. variance). Positive sys-

tematic MEs decrease the bias but increase the variance of

all the ratio estimators. Only the variance estimator of the

ratio-of-means estimator is biased, being especially large

for the smallest sample size, and larger for negative MEs,

mainly if they are systematic.

Keywords Sampling Forest inventory

Design-based inference Variance estimators Bias

Introduction

Sampling methods are important for assessing natural

resource abundance. Natural populations in ecology (for-

estry, ﬁsheries, and wildlife) are extremely large; conse-

quently, sampling techniques have to be conducted for

characterizing those populations. Sampling allows us,

based on a very small portion of the population, to extend

the sample results to the population level through the use of

statistical inference. There are three key components to be

deﬁned for any sampling task: sample design, estimator,

and inferential procedure. The sample design elucidates

how to draw the sample, while the estimator is the statistic

that estimates a parameter of interest of the population, and

the inferential procedure determines the reliability of the

estimator. The combination of a particular design and

estimator deﬁnes a sampling strategy in the sense of

Gregoire and Valentine (2008). Here we stay within the

design-based framework of statistical inference (sensu

Gregoire 1998; Gregoire and Valentine 2008), where the

population of interest is regarded as a ﬁxed—not a ran-

dom—quantity, and statistical inference is based on the

distribution of all estimates possible under the given

sampling design.

Communicated by T. Knoke.

This article belongs to the special issue ‘‘Linking Forest Inventory

and Optimisation’’.

C. Salas (&)T. G. Gregoire

School of Forestry and Environmental Studies, Yale University,

360 Prospect Street, New Haven, CT 06511-2104, USA

e-mail: christian.salas@yale.edu

C. Salas

Departamento de Ciencias Forestales,

Universidad de La Frontera, Temuco, Chile

123

Eur J Forest Res (2010) 129:847–861

DOI 10.1007/s10342-009-0277-3

Following a simple random or systematic sampling, the

Horvitz-Thompson (expansion) estimator of the population

mean (l) of total (s) is common. Nevertheless, more

efﬁcient estimators have been developed for the same

sampling designs. Among them are those that use an

auxiliary variable xthat is correlated with the variable of

interest y. As pointed out by Robinson et al. (1999), the

opportunities to integrate auxiliary information into forest

stand inventory are considerable, and the potential beneﬁts

are very attractive. An example of the use of auxiliary

information in a forestry context is two-phase sampling,

where in the ﬁrst phase auxiliary data are obtained for all

sampling units from ‘‘large area measurements’’ tech-

niques such as aerial photographs, remote sensing (e.g.,

Landsat), and laser scanning (e.g., LiDAR) and in the

second phase a portion of them are measured in the ﬁeld.

This has been used in Finnish forest inventory for almost

30 years (Poso et al. 1999), and since 1950’s also in most

states in The United States (Frayer and Furnival 1999). In

most surveys, data are collected on many items beyond the

one variable of primary interest; making the most use of

the additional information collected is an issue of both

practical and theoretical interest (Dryver and Chao 2007).

Examples of estimators that use auxiliary information are

the Grosenbaugh’s (1964) adjusted estimator based on

probability proportional to prediction (3P) sampling,

regression estimators, and ratio estimators. The latter is

particularly interesting because the sampling variability

may be quite smaller than other estimators (Cochran 1977;

Gregoire and Valentine 2008), providing a more reliable

estimate of the population parameter than the comparable

estimate based on the simple arithmetic mean (Sukhatme

and Sukhatme 1970). In general, ratio estimators are the

simplest estimators that incorporate related information

(Mickey 1959).

ME is present in most sampling. The quality of an

estimator is a function of both sampling and nonsam-

pling errors (Scali et al. 2005). Sampling errors arise due

to drawing a probability sample rather than conducting a

census (Stage and Wykoff 1998). Non-sampling errors

are due to data collection and processing procedures. ME

arises when a given measurement differs from the true

value of a variable of interest. MEs depend on the

measuring instruments and the way in which each par-

ticular ﬁeld technician uses these instruments (Cunia

1965; Gertner 1990). ME is also called the ‘‘observa-

tional error’’ or the ‘‘response error’’ (Hansen et al.

1951). Customarily, it is assumed that the data collected

on the units in the sample are the actual values of the

characteristics observed, and that the estimates of the

population values obtained are uniquely subject to errors

solely due to sampling (Sukhatme and Sukhatme 1970).

MEs are unavoidable, yet increasing the sample size is

typically not a viable method for reducing their effects

(Canavan and Hann 2004). An easy way to deal with

MEs is to pretend they do not exist, or if they do,

assume that their effect is negligible (Chandhok 1988).

However, MEs might affect the accuracy (i.e., bias)

and precision (i.e., variability) of some estimators. The

cumulative effect of the various errors on the estimate

is not always negligible, since errors from different

sources may not cancel out one another (Sukhatme and

Sukhatme 1970).

The effect on inference of ME has not been widely

studied in a design-based inference framework. The effect

of MEs when ﬁtting parametric models, e.g. regression

analysis, has been widely studied not only in the statistical

literature (e.g. Fuller 1987; Myers 1990; Bay and Stefanski

2000) but also in the forestry literature (e.g. Gertner 1988,

1990; Kangas 1996,1998; Stage and Wykoff 1998; Kangas

and Kangas 1999; Canavan and Hann 2004; Hordo et al.

2008). In sampling, MEs have been mostly studied in a

model-based inference setting or using a mathematical

model for the errors of measurement or observational

errors (e.g., Cochran 1977, p. 37; Sukhatme and Sukhatme

1970, p. 390).

The effect of ME in the auxiliary variate on the per-

formance of the ratio estimator only recently has been

studied. Although some studies have assessed the perfor-

mance of ratio estimators in sampling without MEs (Tin

1965;Ek1971; Hutchison 1971; Royall and Cumberland

1981), only a recent theoretical study conducted by Greg-

oire and Salas (2009) has examined the performance of

ratio estimators under MEs. They assessed the effects of

having systematic and random MEs in the auxiliary variate

(x) used in three ratio estimators. They provided mathe-

matical expressions both to determine how the bias of the

ratio estimators change due to systematic ME in xand to

compute the variance of the ratio estimators with ME in x.

In order to assess the effect of random ME of the ratio

estimators these authors conducted simulations over a

population of 501 Eucalyptus nitens leaves. Gregoire and

Salas (2009) neither assess the effect of MEs on the esti-

mates of the variance of the estimators nor of using a larger

population. For practical purposes the estimates of the

variance estimators are crucial for computing conﬁdence

intervals of parameters. Furthermore, higher variability in y

and/or xmight enhance the performance of one ratio esti-

mator over the others. In the present study, our objective is

to assess the statistical performance of three ratio estima-

tors under various forms and magnitudes of ME in the

auxiliary variate in a design-based inference framework,

and of the estimates of the variance of those ratio estima-

tors, using a large tree population.

848 Eur J Forest Res (2010) 129:847–861

123

Materials and methods

Population

Our population data consists of N=14,387 loblolly pine

(Pinus taeda L.) trees collected in southern USA. The data

were provided by the U.S. Forest Service. For each tree of

this population, the following variables were measured:

crown class, diameter at breast height (d), total height (h),

and total volume (v), which was computed based on mul-

tiple measurement points along the standing stem. Trees

from all crown classes are represented, except open-grown

trees. The same data set was used by Gregoire and

Williams (1992) and Magnussen (2001) in a volume

equations and a 3P sampling study, respectively. For these

data volume has the highest variability and skewness,

followed by basal area, diameter, and height (Table 1).

Volume and diameter have a linear correlation coefﬁ-

cient of r=0.91 (Fig. 1b). In the context of our study we

prefer to use d, instead of basal area (g), because it is the

variable that is directly measured in the ﬁeld (i.e., gis only a

function of d, and therefore fully depends on it) and a better

understanding of the ME effect can be achieved using it

instead of g. Although the relationship between vand dis

not linear for the entire range of the data, it is linear across

most of the range. Therefore, we chose diameter to be the

auxiliary variable for the ratio estimators.

Description of estimators

We consider the following estimators of sy¼PN

k¼1yk

based on data from a simple random sample without

replacement:

‘‘ratio-of-means’’ ! b

sy1¼b

Rsx¼y

xsx;ð1Þ

where sx¼PN

k¼1xk:In (1), b

Ris an estimator of R¼

sy=sx¼ly=lx;sy¼PN

k¼1yk;as in Gregoire and Valentine

(2008), and yand xare the sample means for the yand x

variables, respectively. Notice that l

y

and l

x

are the

population average of yand x, and are computed

as ly¼sy=Nand lx¼sx=N;respectively.

Also,

‘‘mean-of-ratios’’ ! b

sy2¼rsx;ð2Þ

where ris the average ratio of r

k

=y

k

/x

k

of those units in

the sample. The population average ratio is denoted by l

r

,

and computed as lr¼1

NPN

k¼1rk:

Also,

‘‘unbiased ratio estimator’’ !

b

sy3¼b

sy2þN1

N

n

n1b

syprb

sxp

;ð3Þ

where b

sypand b

sxpare the Horvitz-Thompson (HT)

estimators (Horvitz and Thompson 1952)ofs

y

and s

x

,

respectively, as follows

b

syp¼Ny;b

sxp¼Nx:ð4Þ

The estimator in (1) is the usual ratio-of-means

estimator of s

y

. The estimator b

sy2in (2) is sometimes

called the mean-of-ratios estimator. It is well known that

the ratio-of-means and the mean-of-ratios are biased

estimators of s

y

. The estimator b

sy3in (3) is the unbiased

ratio-type estimator introduced by Hartley and Ross (1954)

and further developed by Goodman and Hartley (1958).

The usual approximation to the bias of b

sy1may be

deduced from (6.34) in Cochran (1977)as

Bb

sy1:sy

¼1

n1

N

c2

xqcxcy

sy;ð5Þ

where c

x

and c

y

are the coefﬁcients of variation (expressed

in relative units) of xand y, respectively, and qis the

correlation coefﬁcient between yand xin the population.

The bias of b

sy2is exactly

Bb

sy2:sy

¼X

N

k¼1

rklxxk

ðÞ:ð6Þ

The bias of b

sy3is zero (Hartley and Ross 1954).

The usual approximation of the variance of b

sy1

under simple random sampling without replacement (i.e.

Table 1 Descriptive

parameters of the loblolly pine

(Pinus taeda L.) trees

population for different

variables (N=14,387)

Diameter in cm, height in m,

basal area in cm

2

, and volume in

m

3

Parameter Variable

Diameter (d) Height (h) Basal area (g) Volume (v)

Minimum 12.7 4.3 126.7 0.01

Maximum 87.4 43.9 5,996.2 7.79

Mean (l) 28.2 19.9 736.6 0.62

Variance (r

2

) 140.3 41.6 424,706.0 0.56

Total (s) 406,330.7 285,916.7 10,598,091.1 8,932.42

Coefﬁcient of variation (c) (in %) 41.9 32.4 88.5 120.4

Coefﬁcient of skewness 0.9 0.3 2.1 2.5

Kurtosis 0.8 -0.4 6.8 9.7

Eur J Forest Res (2010) 129:847–861 849

123

SRSwoR) is given as (6.16) in Gregoire and Valentine

(2008)as

Vb

sy1

¼N21

n1

N

r2

rm;ð7Þ

where r2

rm ¼1

N1PN

k¼1ykRxk

ðÞ

2:

The variance of b

sy2is

Vb

sy2

¼1

n1

N

s2

xr2

r;ð8Þ

where r2

r¼1

N1PN

k¼1rklr

ðÞ

2;as shown in Goodman

and Hartley (1958), Eq. (8).

Goodman and Hartley (1958) also derive a variance

approximation for b

sy3in their Eq. (6), which is

Vb

sy3

¼1

n1

N

s2

yc2

yþc2

x2Cðx;yÞ

lxly

"#

;ð9Þ

where C(x,y) is the covariance between yand x. Finally,

we computed the mean square error (MSE) of each esti-

mator as the sum of its bias square plus its variance, and for

interpretation the square root of the MSE (or RMSE), was

used (Table 2).

Measurement error processes

As mentioned by Rice (1988), a distinction is usually made

between random and systematic ME. Random MEs vary

among units of the population. On the other hand, sys-

tematic MEs, have the same effect on every measurement.

Following Gregoire and Salas (2009) we used 25% of l

x

,

which given our population data is equal to 7 cm in

diameter, as the maximum ME to be tested.

Systematic measurement error in x We suppose that x

k

cannot be measured without a systematic error in mea-

surement denoted by d

k

s

. The magnitude of d

k

s

may be due

to a miscalibrated instrument used in the measurement

process. The measurement of x

k

contaminated with sys-

tematic ME is denoted by

x

k¼xkþds

k;ð10Þ

and likewise s

x¼PN

k¼1x

k:That is to say, for each level of

systematic ME, d

k

s

=d

s

, then a constant level of ME was

added to d

k

, the dfor the kth element of the population.

Thus, (1) computed with ME is b

sy1¼y

xs

x;likewise, (2)

becomes b

sy2¼rs

x;and (3) is calculated similarly. We use

a range of values of d

s

, from -7 to 7 in evenly spaced

increments in order to have a total of 11 classes (5 with

positive MEs, 5 with negative MEs, and 0 ME).

Random measurement error in x Suppose that the error in

the measurement of xis random, rather than systematic,

such that the value that is measured is not x

k

but

x

k¼xkþdr

k;ð11Þ

which implies s

x¼PN

k¼1x

k:In (11), d

k

r

varies among the

x

k

,k=1,..., N. We assume that, on average, the magnitude

of d

k

r

is close to zero, yet in any particular sample of n

elements, its average is not identically zero, viz.,

dr

x¼1

nX

n

k¼1

dr

k6¼ 0:ð12Þ

Let the variance of d

k

r

be denoted by r

d

2

. In summary, when

we are considering systematic MEs, we have E[d

s

]=d

s

and V[d

s

]=0, and E[d

r

]=0 and V[d

r

]=r

d

2

, when con-

sidering random MEs.

We examined three probability density functions (pdf)

to characterize the distribution of the random errors. We

used a uniform, Gaussian, and beta pdf as a way to

mimic uniformly, symmetrically, and asymmetrically

distributed random MEs. We scaled the random MEs in

such a way that the maximum (and minimum) d

k

would

be close to the maximum and minimum systematic error

also tested.

0.25 2.25 4.25 6.25

Volume (m3)

Percent of the total

0

10

20

30

40

50

60

0 20406080

0

2

4

6

8

Diameter at breast height (cm)

Volume (m3)

(a) (b)

Fig. 1 Histogram of volume (a)

and scatterplot between volume

and diameter at breast height (b)

for 14,387 loblolly pine trees

850 Eur J Forest Res (2010) 129:847–861

123

Uniform

Let dr

kf7U½1;1;where Uis a random number

from a uniform distribution, and fis some fraction of the

maximum ME to be tested, and 7 is the maximum ME in x

to be tested. We use a range of values of f, from 0 to 1 in

increments of 0.1, establishing 11 different levels (the same

number of levels used for systematic MEs) of random

uniformly distributed MEs.

Normal

Let dr

kfrd; and Nð0;1Þ;r

d

=0.02r

x

, and fis a

fraction of the random ME to be tested. We use a range of

values of f, from 0 to 1 in increments of 0.1, establishing 11

different levels of random normally distributed MEs.

Beta

We wished to examine the performance of the estimators

under skew ME too. We used the Beta distribution. Spe-

ciﬁcally, we let d

k

r

*b[a,b]9f97 / max(b[a,b]), where a

and bare parameters of the distribution, bis a random

number from a Beta distribution, fis some fraction of the

maximum ME to be tested, 7 is the maximum ME in xto be

tested, and max(b[a,b]) is the maximum random number

from a Beta distribution (obtained when setting the random

number seed). We used a range of values of f, from 0 to 1

in increments of 1, establishing 11 different levels of ran-

dom beta distributed MEs. We ﬁxed the parameters of the

Beta pdf to be a=2 and b=10, positive skewed (right-

skewed) shape distribution.

Monte Carlo simulation study

Statistical properties of estimators can be assessed using

computational re-sampling techniques. We can approximate

expected values of estimators by computing the arithmetic

average for a large number of simulated samples, and also

approximate the distribution of the estimator for these

several samples (i.e., empirical sampling distribution). We

conducted simulations (each simulation corresponds to an

independent random sample) for each combination of ME

type and level with samples of sizes (n) 7, 15, and 37. These

sample sizes correspond to sampling intensities of 0.05%,

0.10%, and 0.25%, respectively. We conducted 100,000

simulations, and all the analysis were programmed using the

free statistical software R (R Development Core Team

2007). The number of simulations was chosen based on a

prior analysis for this population in order to make the

sampling error of the simulation itself negligibly small. A

similar analysis to determine or justify the number of sim-

ulation has been conducted by Gregoire and Schabenberger

(1999).

Based on the simulations, we computed the empirical

estimates of the bias (B), standard error (SE), and root

mean square error (RMSE) of each estimator studied. The

bias of an estimator relates to the accuracy of it, while the

variance of an estimator relates to the precision of it. An

estimator should be judged for both accuracy and precision,

hence the use of the RMSE of an estimator is more suitable

since takes it into account both features. All these statistics

were expressed in percentage terms, after dividing them

by s

y

.

Assessing variance estimators of the ratio estimators

We also examine the behavior of the estimators of variance

for the ratio estimators. The precision of the ratio estima-

tors is judged through an approximate expression for its

variance. Therefore it is important to examine the accuracy

of this approximation (Raj 1964). Therefore, we compute

the empirical bias of the estimates of variance for the ratio

estimators. The following variance estimators were used.

Table 2 Concurrence of simulation moments to the exact or approximate moments of ratio estimators

nEstimator Bias (%) SE (%) RMSE (%)

Theoretical Empirical Theoretical Empirical Theoretical Empirical

7b

sy1-4.09 -4.08 31.61 30.23 31.88 30.51

b

sy2-23.05 -23.13 22.63 22.65 32.31 32.37

b

sy30.00 -0.10 35.06 35.03 35.06 35.03

15 b

sy1-1.91 -1.78 21.59 21.15 21.67 21.22

b

sy2-23.05 -22.96 15.46 15.43 27.75 27.67

b

sy30.00 0.12 23.79 23.77 23.79 23.77

37 b

sy1-0.77 -0.76 13.74 13.58 13.76 13.60

bsy2-23.05 -23.07 9.83 9.82 25.06 25.07

b

sy30.00 0.00 15.09 15.04 15.09 15.04

100,000 simulations of each size nwere conducted

Eur J Forest Res (2010) 129:847–861 851

123

For b

sy1;we used (6.31) of Gregoire and Valentine

(2008), as follows

b

vb

sy1

¼N2l2

x

x2

1

n1

N

s2

rm;ð13Þ

where s

rm

2

is an estimator of r

rm

2

of (7):

s2

rm ¼1

n1X

n

k¼1

ykb

Rxk

2;ð14Þ

and b

R¼y=x:

For b

sy2;we used the unbiased estimator of Vb

sy2

namely

b

vb

sy2

¼1

n1

N

s2

xs2

r;ð15Þ

where s

r

2

is the estimator of r

r

2

of (8):

s2

r¼1

n1X

n

k¼1

rkrðÞ

2;ð16Þ

For b

sy3;we used the unbiased estimator presented by

Goodman and Hartley (1958, Eq. 35). For this estimator,

the statistics k

22

,c, and c0(see Appendix for formulas)

must be computed ﬁrst, followed by the variance estimator.

We adjusted

1

the variance estimator of b

sy3presented in

(35) of Goodman and Hartley (1958), and the correction of

Goodman and Hartley (1969), to

b

vb

sy3

¼s2

xs2

r

nþ2sxc0

n2

þðn1Þs2

rs2

xþðn3Þc2þ12

n

ðn1Þk22

n2n2

n1

n1

N

N2;ð17Þ

where s

r

2

and s

x

2

are the sample variance of rand x,

respectively.

We assessed these variance estimators (Eqs. 13,15, and

17) using our simulations described in Section ‘‘Monte

Carlo simulation study’’. The bias in percentage terms of

the estimator of variance was obtained dividing the bias by

the empirical variance of the corresponding ratio estimator.

Results

Without measurement error in x

Both theoretical and empirical results (i.e., bias, SE, and

RMSE) are almost identical, with differences (for most

cases) smaller than 0.1% (Table 2). Theoretical formulas

for bias, SE, and RMSE for the ratio estimators are

presented in Gregoire and Salas (2009). The mean-of-

ratios estimator, b

sy2;had the largest bias (underestima-

tion) for all sample sizes: its magnitude is unacceptably

large, and does not diminish with increasing the sample

size (second row of Fig. 2). The ratio-of-means estimator,

b

sy1had a bias less than that of b

sy2(smaller than -4.1%

for the smallest sample size). Its bias decreases to less

than -0.8% with increasing sample size. b

sy2;although

biased, had the best precision (smaller standard error

values) for all sample sizes. The ratio-of-means estimator

had the smallest RMSE, followed by b

sy3;with small

difference with increasing sample size (Table 2). b

sy2

performs similar to b

sy3when n=7, but the RMSE of

mean-of-ratios does not decrease much with increasing

sample size because its bias is invariant to sample size.

The precision clearly increases when the sample size

increases (Fig. 2).

With measurement error in x

We report the effect of ME on bias by considering the

ratio of bias with ME to bias in the absence of ME, which

we call relative bias henceforth. The extent to which this

ratio is smaller (greater) than unity is a measure of the

relative decrease (increase) in bias due to ME. In an

analogous fashion, we report relative SE and relative

RMSE.

Systematic measurement error in x Positive MEs tend to

decrease the absolute bias of the ratio estimators, espe-

cially for the mean-of-ratios estimator (ﬁrst panel-row

inner plots of Fig. 3). Estimator 3 remained unbiased

under ME in x, as also noticed by Gregoire and Salas

(2009). On the other hand, if we consider the ME effect

in the relative bias (i.e., B*/B), positive MEs tend to

reduce the B*/Bof both the ratio-of-means and the mean-

of-ratios estimator by approximately 10% (ﬁrst panel-row

of Fig. 3).

The mean-of-ratios estimator had better precision than

the other ratio estimators under systematic MEs for all the

sample sizes (second panel-row inner plots of Fig. 3). If we

only consider the effect of ME, it is possible to infer that

positive systematic ME on xdecreases the precision of all

the ratio estimators tested in our study. Conversely, nega-

tive ME produces better precision (second panel-row of

Fig. 3). Gregoire and Salas (2009) found the opposite trend

in a similar study but with a different distribution of both

target and auxiliary variables. The effect of ME in the

precision is reduced with increasing sample size. Overall,

even though large values of MEs were added to x, this

alters the precision of the ratio estimators comparatively

1

The formula given by these authors is for an estimator of the

population mean and assuming inﬁnite populations; but here we are

dealing with population total and ﬁnite populations.

852 Eur J Forest Res (2010) 129:847–861

123

little (\10%) compared to their precision in the absence of

ME.

Root mean squares errors (RMSE%) slightly increase for

positive systematic MEs, except for the mean-of-ratios

estimator. According to this statistic, ratio-of-means per-

forms the best for all conditions, with b

sy3next best (third

panel-row inner plots of Fig. 3). Positive ME increases the

RMSE compared to the RMSE from xwith no ME, for both

ratio-of-means and estimator 3, but decreased for estimator

2 (third panel-row of Fig. 3). b

sy1;even though slightly

biased, performed best among all the ratio estimators tested.

Random measurement error in x The effect of random

ME in xon the ratio estimators is smaller than the effect

of systematic ME. Neither the accuracy nor the precision

of the ratio estimators are very affected by uniform,

Gaussian, and beta (inner plots of Figs. 4,5, and 6,

respectively) distributed MEs. Only a minor change in the

bias of each estimator with and without ME was observed

(Figs. 4,5, and 6for uniform, Gaussian, and beta MEs,

respectively). The effect of random ME is more notable

when these errors are uniformly distributed than when

they are either Gaussian or beta distributed.

Assessing estimators of variance of the ratio estimators

Let us explain our results for the following three scenarios:

When xdoes not contain ME, the variance estimator of the

ratio-of-means estimator (i.e., b

vb

sy1

) is highly biased for

the smallest sample size (Table 3). Although b

vb

sy2

and

b

vb

sy3

do not achieve a zero bias, the largest value is only

0.77%. With systematic MEs, both b

vb

s

y2

hi

and b

vb

s

y3

hi

retain their unbiasedness, while b

vb

s

y1

hi

is more biased

(underestimation) for negative MEs than positive MEs.

With random MEs (Table 4), both b

vb

s

y2

hiand

b

vb

s

y3

hi

exhibit small bias. For b

vb

s

y1

hi

;the magnitude of its

0 10000 20000

0

10

20

30

40

τ

^y (Total volume, m3)

Percent of total

Est.1, n=7

0 10000 20000

0

10

20

30

40

τ

^y (Total volume, m3)

Percent of total

Est.1, n=15

0 10000 20000

0

10

20

30

40

τ

^y (Total volume, m3)

Percent of total

Est.1, n=37

0 10000 20000

0

10

20

30

40

τ

^y (Total volume, m3)

Percent of total

Est.2, n=7

0 10000 20000

0

10

20

30

40

τ

^y (Total volume, m3)

Percent of total

Est.2, n=15

0 10000 20000

0

10

20

30

40

τ

^y (Total volume, m3)

Percent of total

Est.2, n=37

0 10000 20000

0

10

20

30

40

τ

^y (Total volume, m3)

Percent of total

Est.3, n=7

0 10000 20000

0

10

20

30

40

τ

^y (Total volume, m3)

Percent of total

Est.3, n=15

0 10000 20000

0

10

20

30

40

τ

^y (Total volume, m3)

Percent of total

Est.3, n=37

Fig. 2 Empirical distribution of

three different ratio estimators

and three sample sizes (100,000

simulations were conducted) in

predicting total volume of

loblolly pine. The dashed

vertical line represents the value

of the target parameter s

y

, Est.1

is b

sy1, Est.2 is b

sy2, and Est.3 is

b

sy3

Eur J Forest Res (2010) 129:847–861 853

123

underestimation decreases slightly as rd=rxincreases.

There are only slight differences in the estimators of the

variance among the types of the distribution of the random

ME, being the bias greater for the Gaussian, then the Beta,

and ﬁnally the uniform.

Discussion

Our results show important differences in accuracy and

precision for all the estimators evaluated when xis mea-

sured without error. While Gregoire and Salas (2009)

1.05

−0.2 −0.1 0.0 0.1 0.2

0.80

0.85

0.90

0.95

1.00

1.10

Ratio of bias with/without ME

(a) (b) (c)

(a) (b) (c)

(a) (b) (c)

(%)

n=7 n=15 n=37

−25

−20

−15

−10

−5

0

−0.2 −0.1 0.0 0.1 0.2

0.80

0.85

0.90

0.95

1.00

1.05

1.10

(%)

−25

−20

−15

−10

−5

0

−0.2 −0.1 0.0 0.1 0.2

0.80

0.85

0.90

0.95

1.00

1.05

1.10

(%)

−25

−20

−15

−10

−5

0

−0.2 −0.1 0.0 0.1 0.2

0.80

0.85

0.90

0.95

1.00

1.05

1.10

Ratio of SE with/without ME

(%)

5

15

25

35

−0.2 −0.1 0.0 0.1 0.2

0.80

0.85

0.90

0.95

1.00

1.05

1.10

(%)

5

15

25

35

−0.2 −0.1 0.0 0.1 0.2

0.80

0.85

0.90

0.95

1.00

1.05

1.10

(%)

5

15

25

35

−0.2 −0.1 0.0 0.1 0.2

0.80

0.85

0.90

0.95

1.00

1.05

1.10

Ratio of RMSE with/without ME

(%)

5

15

25

35

−0.2 −0.1 0.0 0.1 0.2

0.80

0.85

0.90

0.95

1.00

1.05

1.10

(%)

5

15

25

35

−0.2 −0.1 0.0 0.1 0.2

0.80

0.85

0.90

0.95

1.00

1.05

1.10

(%)

5

15

25

35

μδ / μxμδ / μxμδ / μx

Fig. 3 Bias (ﬁrst panel-row), standard error (second panel-row), and

root mean square error (third panel-row)ofb

s

y1(solid line), b

s

y2(dot-

dash line), and b

s

y3(dashed line) relative to that of b

sy1;b

sy2;and b

sy3;

respectively, with systematic measurement error in the auxiliary

variate, and having samples of 7 (a), 15 (b), and 37 (c) trees. The

inner plots represent bias (ﬁrst panel-row), standard error (second

panel-row), and root mean square error (third panel-row) expressed as

a percentage of s

y

. The quotient ld=lxrepresents the relative level of

measurement of error with respect to the population mean of the

auxiliary variate x. The horizontal axis of the inner plots span the

same range as the axis of the larger plots

854 Eur J Forest Res (2010) 129:847–861

123

found that the mean-of-ratios estimator had a bias less than

2%, with the loblolly pine data b

sy2was very biased. We

believe this is due to the fact that in our study the target

variate has much greater variability, with a coefﬁcient of

variation of 120.4% versus 29.7% for the leaf area popu-

lation studied by Gregoire and Salas (2009). Both studies

reafﬁrm the recommendation of Ek (1971), who advocated

against the use of the mean-of-ratios estimator because of

its sometimes severe bias. On the other hand, the mean-of-

ratios estimator always had better precision than the other

two estimators. Even though b

sy1is slightly biased it per-

formed better than the unbiased b

sy3in terms of RMSE.

1.02

0.00 0.10 0.20 0.30

0.95

0.96

0.97

0.98

0.99

1.00

1.01

Ratio of bias with/without ME

(%)

n=7 n=15 n=37

−25

−20

−15

−10

−5

0

0.00 0.10 0.20 0.30

0.95

0.96

0.97

0.98

0.99

1.00

1.01

1.02

(%)

−25

−20

−15

−10

−5

0

0.00 0.10 0.20 0.30

0.95

0.96

0.97

0.98

0.99

1.00

1.01

1.02

(%)

−25

−20

−15

−10

−5

0

0.00 0.10 0.20 0.30

0.95

0.96

0.97

0.98

0.99

1.00

1.01

1.02

Ratio of SE with/without ME

(%)

5

15

25

35

0.00 0.10 0.20 0.30

0.95

0.96

0.97

0.98

0.99

1.00

1.01

1.02

(%)

5

15

25

35

0.00 0.10 0.20 0.30

0.95

0.96

0.97

0.98

0.99

1.00

1.01

1.02

(%)

5

15

25

35

0.00 0.10 0.20 0.30

0.95

0.96

0.97

0.98

0.99

1.00

1.01

1.02

Ratio of RMSE with/without ME

(%)

5

15

25

35

0.00 0.10 0.20 0.30

0.95

0.96

0.97

0.98

0.99

1.00

1.01

1.02

(%)

5

15

25

35

0.00 0.10 0.20 0.30

0.95

0.96

0.97

0.98

0.99

1.00

1.01

1.02

(%)

5

15

25

35

σδ / σxσδ / σxσδ / σx

(a) (b) (c)

(a) (b) (c)

(a) (b) (c)

Fig. 4 Bias (ﬁrst panel-row), standard error (second panel-row), and

root mean square error (third panel-row)ofb

s

y1(solid line), b

s

y2(dot-

dash line), and b

s

y3(dashed line) relative to that of b

sy1;b

sy2;and b

sy3;

respectively, with Uniform distributed measurement error in the

auxiliary variate, and having samples of 7 (a), 15 (b), and 37 (c) trees.

The inner plots represent bias (ﬁrst panel-row), standard error (second

panel-row), and root mean square error (third panel-row) expressed as

a percentage of s

y

. The quotient rd=rxrepresents the relative level of

variation of the random measurement error with respect to the

variation of the auxiliary variate xin the population. The horizontal

axis of the inner plots span the same range as the axis of the larger

plots

Eur J Forest Res (2010) 129:847–861 855

123

Overall, we did not ﬁnd any important advantage of using

b

sy3(which adds a correction to b

sy2using the HT estima-

tors) over the ratio-of-means estimator. The ratio-of-means

estimator is easier to compute than b

sy3;which might be

important for practitioners.

Systematic MEs had a slight effect on the performance

of the ratio estimators. There is only a slight effect of MEs

on the bias of the ratio-of-means and the mean-of-ratios

estimator (Fig. 3). B*/Bcan also be checked without con-

ducting any simulations using the formulas given by

Gregoire and Salas (2009). On average, adding 7 to xfor

our population is equivalent to a 25% of the average value

of dwhich is a large value of ME in diameter in the ﬁeld.

The SE*/SE values lower than 1 for negative MEs of all the

0.00 0.10 0.20 0.30

0.95

0.96

0.97

0.98

0.99

1.00

1.01

1.02

Ratio of bias with/without ME

(%)

n=7 n=15 n=37

−25

−20

−15

−10

−5

0

0.00 0.10 0.20 0.30

0.95

0.96

0.97

0.98

0.99

1.00

1.01

1.02

(%)

−25

−20

−15

−10

−5

0

0.00 0.10 0.20 0.30

0.95

0.96

0.97

0.98

0.99

1.00

1.01

1.02

(%)

−25

−20

−15

−10

−5

0

0.00 0.10 0.20 0.30

0.95

0.96

0.97

0.98

0.99

1.00

1.01

1.02

Ratio of SE with/without ME

(%)

5

15

25

35

0.00 0.10 0.20 0.30

0.95

0.96

0.97

0.98

0.99

1.00

1.01

1.02

(%)

5

15

25

35

0.00 0.10 0.20 0.30

0.95

0.96

0.97

0.98

0.99

1.00

1.01

1.02

(%)

5

15

25

35

0.00 0.10 0.20 0.30

0.95

0.96

0.97

0.98

0.99

1.00

1.01

1.02

Ratio of RMSE with/without ME

(%)

5

15

25

35

0.00 0.10 0.20 0.30

0.95

0.96

0.97

0.98

0.99

1.00

1.01

1.02

(%)

5

15

25

35

0.00 0.10 0.20 0.30

0.95

0.96

0.97

0.98

0.99

1.00

1.01

1.02

(%)

5

15

25

35

σδ / σxσδ / σxσδ / σx

(a) (b) (c)

(a) (b) (c)

(a) (b) (c)

Fig. 5 Bias (ﬁrst panel-row), standard error (second panel-row), and

root mean square error (third panel-row)ofb

s

y1(solid line), b

s

y2(dot-

dash line), and b

s

y3(dashed line) relative to that of b

sy1;b

sy2;and b

sy3;

respectively, with Gaussian distributed measurement error in the

auxiliary variate, and having samples of 7 (a), 15 (b), and 37 (c) trees.

The inner plots represent bias (ﬁrst panel-row), standard error (second

panel-row), and root mean square error (third panel-row) expressed as

a percentage of s

y

. The quotient rd=rxrepresents the relative level of

variation of the random measurement error with respect to the

variation of the auxiliary variate xin the population. The horizontal

axis of the inner plots span the same range as the axis of the larger

plots

856 Eur J Forest Res (2010) 129:847–861

123

estimators imply that the SE is decreased in comparison to

the SE when xis measured without error. On the other

hand, SE*/SE values greater than 1 for positive MEs imply

an increasing of the SE in comparison to the SE when xin

measured without error. Overall, systematic ME slightly

increases the accuracy but decreases the precision of the

ratio estimators.

Uniform, Gaussian, and Beta distributed MEs in x

degrade neither the accuracy nor the precision of these

ratio estimators. Gregoire and Salas (2009) found that the

0.00 0.10 0.20 0.30

0.95

0.96

0.97

0.98

0.99

1.00

1.01

1.02

Ratio of bias with/without ME

(%)

n=7 n=15 n=37

−25

−20

−15

−10

−5

0

0.00 0.10 0.20 0.30

0.95

0.96

0.97

0.98

0.99

1.00

1.01

1.02

(%)

−25

−20

−15

−10

−5

0

0.00 0.10 0.20 0.30

0.95

0.96

0.97

0.98

0.99

1.00

1.01

1.02

(%)

−25

−20

−15

−10

−5

0

0.00 0.10 0.20 0.30

0.95

0.96

0.97

0.98

0.99

1.00

1.01

1.02

Ratio of SE with/without ME

(%)

5

15

25

35

0.00 0.10 0.20 0.30

0.95

0.96

0.97

0.98

0.99

1.00

1.01

1.02

(%)

5

15

25

35

0.00 0.10 0.20 0.30

0.95

0.96

0.97

0.98

0.99

1.00

1.01

1.02

(%)

5

15

25

35

0.00 0.10 0.20 0.30

0.95

0.96

0.97

0.98

0.99

1.00

1.01

1.02

Ratio of RMSE with/without ME

(%)

5

15

25

35

0.00 0.10 0.20 0.30

0.95

0.96

0.97

0.98

0.99

1.00

1.01

1.02

(%)

5

15

25

35

0.00 0.10 0.20 0.30

0.95

0.96

0.97

0.98

0.99

1.00

1.01

1.02

(%)

5

15

25

35

σδ / σxσδ / σxσδ / σx

(a) (b) (c)

(a) (b) (c)

(a) (b) (c)

Fig. 6 Bias (ﬁrst panel-row), standard error (second panel-row), and

root mean square error (third panel-row)ofb

s

y1(solid line), b

s

y2(dot-

dash line), and b

s

y3(dashed line) relative to that of b

sy1;b

sy2;and b

sy3;

respectively, with Beta distributed measurement error in the auxiliary

variate, and having samples of 7 (a), 15 (b), and 37 (c) trees. The

inner plots represent bias (ﬁrst panel-row), standard error (second

panel-row), and root mean square error (third panel-row) expressed as

a percentage of s

y

. The quotient rd=rxrepresents the relative level of

variation of the random measurement error with respect to the

variation of the auxiliary variate xin the population. The horizontal

axis of the inner plots span the same range as the axis of the larger

plots

Eur J Forest Res (2010) 129:847–861 857

123

greater the variability of the random error distribution, the

greater the bias, variance, and RMSE of the estimator.

Nevertheless, the difference between B, SE, and RMSE

with and without ME reported by them is smaller than 3%.

There is no difference between a symmetric ME distribu-

tion (i.e. Gaussian and uniform) and a skewed ME distri-

bution (i.e. Beta) on the performance of the ratio

estimators.

Only the variance estimators of the ratio-of-means

estimator is biased. The bias in b

vb

sy1

is especially large

for an extremely small sample size such as n=7.

Cochran (1977) pointed out that b

vb

sy1

is based on large

sample theory. Our results show underestimation of the

variance of the ratio-of-means estimator, conﬁrming

Cochran’s (1977, p. 162) assertion that the large sample

approximation results in underestimation. We have also

found similar results to those reported by Rao (1968)

where the bias in the variance estimators, mainly for ratio-

of-means, are more serious in small samples. Rao (1968)

mention as well that these results are unsatisfactory at

least up to n=12, which is similar to our medium-size

sample (n=15). The usual approximation of the variance

estimator of ratio-of-means would be adequate in large

samples if the data follow a bivariate normal distribution

(Sukhatme and Sukhatme 1970). The highest bias reported

here is larger than the -9% mentioned by Cochran

(1977), who used Sukhatme and Sukhatme’s (1970) the-

oretical results, the estimator of variance of ratio-of-

means, but smaller than the -25% of Koop (1968) for

small population sizes. Neither systematic nor random

MEs affect the unbiasedness of the variance estimators of

b

s

y2and b

s

y3:Only systematic MEs in xaffect the bias of

the variance estimates of the ratio-of-means estimator.

Finally, the bias of b

vb

sy1

raise an interesting point

regarding its effect in statistical inference (e.g., in com-

puting conﬁdence intervals), where further research is

needed. Furthermore, the precision of the variance esti-

mators can be also assessed.

Concluding remarks

The statistical performance of ratio estimators is not very

affected by the presence of either systematic or random ME

in the auxiliary variate. Only some slight effect on bias was

found when having systematic MEs. This resistance of the

ratio estimators to ME revealed in this study conﬁrms the

results of Gregoire and Salas (2009). The ratio-of-means

estimator performs the best in terms of RMSE. The unbi-

ased estimator, b

sy3;does not provide precise enough esti-

mation to perform better than the ratio-of-means estimator.

The mean-of-ratios estimator is highly biased, yet always

was more precise. The unacceptably large bias of b

sy2

observed in this study contrasts with the results of Gregoire

and Salas (2009). We suspect that it is due to characteris-

tics of the population being sampled that we have yet to

identify. Neither systematic nor random ME affect the bias

of the variance estimates of the ratio estimators. For small

sample sizes, the estimator of the variance of b

sy1has

unacceptably large negative bias.

Table 3 Bias of the estimator of the variance, as a percentage of the

Monte Carlo variance, for the ratio estimators under several levels of

the quotient between the population mean of the systematic

measurement error (l

d

)inxand the population mean of x(l

x

), and

different sample sizes

nEstimator l

d

/l

x

-0.25 -0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2 0.25

7c

var b

s

y1

hi -24.59 -22.98 -21.55 -20.31 -19.22 -18.25 -17.40 -16.61 -15.91 -15.30 -14.70

c

var b

s

y2

hi -0.26 -0.19 -0.26 -0.26 -0.25 -0.23 -0.23 -0.21 -0.22 -0.21 -0.18

c

var b

s

y3

hi 0.64 0.60 0.61 0.57 0.52 0.53 0.52 0.48 0.50 0.48 0.44

15 c

var b

s

y1

hi -11.93 -11.16 -10.36 -9.80 -9.22 -8.73 -8.26 -7.87 -7.51 -7.15 -6.84

c

var b

s

y2

hi 0.46 0.53 0.54 0.41 0.43 0.53 0.51 0.43 0.44 0.46 0.49

c

var b

s

y3

hi 0.43 0.42 0.42 0.47 0.41 0.48 0.47 0.50 0.43 0.46 0.44

37 c

var b

s

y1

hi -4.37 -3.94 -3.73 -3.50 -3.18 -2.99 -2.89 -2.69 -2.49 -2.40 -2.26

c

var b

s

y2

hi 0.47 0.40 0.33 0.44 0.43 0.39 0.39 0.46 0.45 0.39 0.49

cvar bs

y3

hi 0.82 0.72 0.69 0.75 0.71 0.77 0.71 0.71 0.65 0.70 0.74

100,000 simulations of each size nwere conducted

858 Eur J Forest Res (2010) 129:847–861

123

Table 4 Bias of the estimator of the variance of the ratio estimators,

as a percentage of the Monte Carlo variance, of ratio estimators under

several levels of the quotient between the standard deviation of three

different random distributed measurement error (r

d

)inxand the

standard deviation of x(r

x

), and different sample sizes

Distribution nEstimator r

d

/r

x

0 0.03 0.06 0.09 0.12 0.15 0.18 0.21 0.24 0.27 0.3

Uniform 7 c

var b

s

y1

hi-18.25 -18.30 -18.28 -18.26 -18.23 -18.19 -18.15 -18.04 -17.99 -17.92 -17.84

c

var b

s

y2

hi -0.23 -0.24 -0.27 -0.24 -0.22 -0.23 -0.26 -0.30 -0.36 -0.34 -0.33

c

var b

s

y3

hi 0.53 0.49 0.52 0.50 0.49 0.48 0.48 0.49 0.45 0.48 0.45

15 c

var b

s

y1

hi -8.73 -8.75 -8.71 -8.70 -8.63 -8.59 -8.57 -8.57 -8.52 -8.49 -8.41

c

var b

s

y2

hi 0.53 0.48 0.49 0.44 0.46 0.42 0.44 0.41 0.45 0.44 0.51

c

var b

s

y3

hi 0.48 0.41 0.44 0.48 0.43 0.40 0.46 0.43 0.42 0.41 0.41

37 c

var b

s

y1

hi -2.99 -3.07 -3.09 -3.05 -3.09 -3.08 -3.14 -3.15 -3.10 -3.13 -3.11

c

var b

s

y2

hi 0.39 0.47 0.41 0.42 0.30 0.24 0.25 0.33 0.29 0.31 0.22

cvar bs

y3

hi 0.77 0.66 0.69 0.73 0.64 0.69 0.61 0.54 0.61 0.55 0.50

Gaussian 7 c

var b

s

y1

hi-18.25 -18.30 -18.28 -18.25 -18.26 -18.25 -18.22 -18.23 -18.23 -18.22 -18.19

cvar bs

y2

hi -0.23 -0.22 -0.19 -0.24 -0.18 -0.19 -0.18 -0.25 -0.21 -0.24 -0.25

c

var b

s

y3

hi 0.53 0.52 0.52 0.52 0.53 0.53 0.54 0.49 0.50 0.51 0.52

15 c

var b

s

y1

hi -8.73 -8.72 -8.70 -8.75 -8.70 -8.72 -8.73 -8.72 -8.69 -8.65 -8.69

c

var b

s

y2

hi 0.53 0.42 0.45 0.50 0.43 0.52 0.49 0.47 0.48 0.37 0.42

c

var b

s

y3

hi 0.48 0.48 0.48 0.48 0.40 0.40 0.41 0.42 0.43 0.44 0.46

37 c

var b

s

y1

hi -2.99 -2.99 -2.97 -2.94 -3.04 -2.98 -3.04 -2.95 -2.99 -3.01 -3.02

c

var b

s

y2

hi 0.39 0.40 0.43 0.48 0.54 0.42 0.51 0.42 0.55 0.50 0.46

c

var b

s

y3

hi 0.77 0.77 0.77 0.77 0.77 0.77 0.78 0.79 0.79 0.80 0.82

Beta 7 c

var b

s

y1

hi-18.25 -18.24 -18.22 -18.19 -18.20 -18.14 -18.13 -18.10 -18.07 -18.02 -17.97

c

var b

s

y2

hi -0.23 -0.17 -0.19 -0.20 -0.20 -0.27 -0.24 -0.20 -0.23 -0.26 -0.27

c

var b

s

y3

hi 0.53 0.54 0.49 0.51 0.52 0.54 0.50 0.52 0.48 0.51 0.48

15 c

var b

s

y1

hi -8.73 -8.69 -8.65 -8.68 -8.61 -8.61 -8.61 -8.59 -8.57 -8.53 -8.56

c

var b

s

y2

hi 0.53 0.55 0.45 0.49 0.55 0.48 0.43 0.52 0.50 0.48 0.48

c

var b

s

y3

hi 0.48 0.46 0.45 0.44 0.43 0.43 0.42 0.50 0.50 0.50 0.50

37 c

var b

s

y1

hi -2.99 -3.02 -3.03 -3.02 -3.01 -2.99 -2.95 -2.91 -2.99 -2.93 -2.85

c

var b

s

y2

hi 0.39 0.33 0.50 0.47 0.45 0.44 0.45 0.47 0.49 0.53 0.38

c

var b

s

y3

hi 0.77 0.71 0.79 0.73 0.67 0.75 0.70 0.78 0.73 0.68 0.77

100,000 simulations of each size nwere conducted

Eur J Forest Res (2010) 129:847–861 859

123

Acknowledgments We gratefully acknowledge Roy C. Beltz, U.S.

Forest Service, Forestry Sciences Lab, Starkville, Mississippi for

providing the population data used in our study.

Appendix: Expressions needed for computing

the estimator of the variance of b

sy3

We used the unbiased estimator presented by Goodman

and Hartley (1958, Eq. 35). This estimator requires the

computation of k

22

,c,c0. These statistics are computed as

follows,

*k

22

k22 ¼n

ðn1Þðn2Þðn3Þ

ðnþ1ÞS22 2ðnþ1Þ

nS21S01 þS12 S10

ðÞ

ðn1Þ

nðS20S02 þ2S2

11Þ

þ2

nðS20S2

01 þS02S2

10 þ4S11S10 S01Þ6

n2ðS2

10S2

01Þ;

ð18Þ

where

Stj ¼X

n

k¼1

xt

krj

k;ð19Þ

for example S22 ¼Pn

k¼1x2

kr2

k¼Pn

k¼1x2

kðy2

k=x2

kÞ¼Pn

k¼1

y2

k:

Note that our expression for k

22

(Eq. 18) has some

algebraic manipulations compared that one gave by

Goodman and Hartley (1958, Eq. 30), and also considering

the corrections made by Goodman and Hartley (1969).

*c. From Goodman and Hartley (1958, Eq. 36, part a)

c¼1

nðn1ÞnX

n

k¼1

ykX

n

k¼1

xkX

n

k¼1

rk

!"#

;ð20Þ

which is the sample covariance between xand ras at the

bottom of page 497 of Goodman and Hartley (1958), as

follows

c¼1

n1

X

n

i¼1ðxkxÞðrkrÞ;ð21Þ

*c0. From Goodman and Hartley (1958, Eq. 32)

c0¼1

n1

X

n

k¼1ðxkxÞðrkrÞ2:ð22Þ

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