Statistical analysis of ratio estimators and their estimators of variances when the auxiliary variate is measured with error

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 performance of three ratio estimators of the target parameter total τy
. 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 simulation 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 x on 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 systematic 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.

Full text

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
sy2þN1
N
n
n1b
syprb
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
n1
N

c2
xqcxcy

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
rklxxk
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
n1
N

r2
rm;ð7Þ
where r2
rm ¼1
N1PN
k¼1ykRxk
ðÞ
2:
The variance of b
sy2is
Vb
sy2

¼1
n1
N

s2
xr2
r;ð8Þ
where r2
r¼1
N1PN
k¼1rklr
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
n1
N

s2
yc2
yþc2
x2Cðx;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
xs
x;likewise, (2)
becomes b
sy2¼rs
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
kf7U½1;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
kfrd; 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
n1
N

s2
rm;ð13Þ
where s
rm
2
is an estimator of r
rm
2
of (7):
s2
rm ¼1
n1X
n
k¼1
ykb
Rxk

2;ð14Þ
and b
R¼y=x:
For b
sy2;we used the unbiased estimator of Vb
sy2

namely
b
vb
sy2

¼1
n1
N

s2
xs2
r;ð15Þ
where s
r
2
is the estimator of r
r
2
of (8):
s2
r¼1
n1X
n
k¼1
rkrðÞ
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
n2

þðn1Þs2
rs2
xþðn3Þc2þ12
n

ðn1Þk22
n2n2
n1
n1
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
ðn1Þðn2Þðn3Þ
ðnþ1ÞS22 2ðnþ1Þ
nS21S01 þS12 S10
ðÞ

ðn1Þ
nðS20S02 þ2S2
11Þ
þ2
nðS20S2
01 þS02S2
10 þ4S11S10 S01Þ6
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
nðn1ÞnX
n
k¼1
ykX
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
n1

X
n
i¼1ðxkxÞðrkrÞ;ð21Þ
*c0. From Goodman and Hartley (1958, Eq. 32)
c0¼1
n1

X
n
k¼1ðxkxÞðrkrÞ2:ð22Þ
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