PSYCH - Psychology
Please see attached - need it in 8 hours. There are 2 parts. APA citation is a must for all parts.
PART 1
Read the attached article and answer the following questions –(MINIMUM 300 WORDS)
• Why are ordinal (or Likert) scales typically the form of data collected when studying
human behavior within the fields of education, management, psychology, and related
social and behavioral sciences?
• Describe the differences found when using a normal-curve (NC) transformation versus
using Likert scales for the psychometric analysis of diversity measures with
entrepreneurial teams.
• In the author's evaluation of three different reliability estimates (standard deviation,
error variance, and unidimensionality), which was found to the preferred method and
which was the least preferred? Why?
• Be sure to use scholarly references to back your assertions and cite!
PART 2
Responses needed for these 2 discussions (minimum 150 words for each discussion)
Discussion 1
Likert scales are very insignificant when it comes to studying human behavior in various fields
like education, behavioral sciences, and psychology for different reasons. The scale is ordinal
primarily because of its association with discrete values like 1,2 and 3, just to mention a few
(Wu & Leung, 2017). The data collected in this case constitute continuous values in a given
interval. It is important to mention that considering the Likert scale, interval value is never
accepted. In the context of studying human behavior is critical to use the ordinal or Likert scale
since the associated data resonate with the nature of humans' engagements. The attitudes and
related opinions are highly critical sources of data that require an ordinal scale to collect. It
follows that the objectivity of acquiring viable and feasibility information based on the discrete
values is very high and significantly possible.
Most of the respondents dealing with the Likert scale are capable of comprehending the
implications of this psychometric scale holistically. Data collected from using the scale is very
pragmatic and assists immensely in addressing views and opinions (Wu & Leung, 2017). The
researchers apply this firm of scale to understand the underlying views and opinions from
associated parties effectively. The Likert scale is a very important aspect of explicitly
understanding the perspectives that human behavior can adopt. It remains critical to affirm
that the scale is essential in acquiring data that is generally relevant. The scale types such as the
Guttman scale, Thurstone scale, and Bogardus scale are essential and efficient in ensuring that
the ultimate goal of collecting discrete data is met. Various questions are provided when using
the Likert scale that serves a major role in promoting authenticity and reliability in various
dimensions.
The differences that exist between using normal-curve (NC) transformation versus using Likert
scales are very vivid and highly pragmatic since they are based on the provided facts. The NC
transformation scale is associated with data that are closer or nearer to the underlying values in
a situation where the normal curve assumption is upheld (Deng et al., 2015). Another key
difference between the nominal curve transformation and the Likert scale is that the associated
transformed data is usually less skewed considering their average value or on average. This is a
very significant component and factor to consider since it can get utilized in situations where
there is a need to measure various diversities. It follows that transformed data is relevant and
essential when it comes to measuring different diversities, thereby giving the most reliable and
effective outcome.
In evaluating the three different reliability estimates that are standard deviation, error variance,
and unidimensionality, the most effective and preferred method is the standard deviation. It Is
important to mention that standard deviation provides an opportunity to engage in accurate
and reliable estimations. The three diversity measures are supposed to embrace effectiveness
in the application when it comes to statistical analysis. In this context, adopting an option that
provides amicable results and operational excellence in various dimensions remains a very
subjective step. Standard deviation is an alternative that is most preferred since it yields a
highly efficient and accurate parameter estimate as oppose to other reliability estimates (Deng
et al., 2015). On the other hand, the unidimensionality reliability estimates are least preferred
following the nature of estimates that are associated with the same. This reliability estimate
compared to the other counterpart options is the least and less reliable.
Discussion 2
An ordinal scale is a scale (of measurement) that uses labels to classify cases into ordered
classes. Ordinal scale tests are often used in social science settings because of how they can
measure behavioral attitudes by asking people to respond to a series of statements topics, in
terms of the extent to which they agree or disagree that would tap into the test taker’s
intellectual and emotional components of attitudes. Ordinal scales tests, such as the Likert
tests, are easy to interpret once the data has been received, and it allows test-takers to provide
opinions based on the statement’s description. Likert scales also offer anonymity on self-
administered questions to reduce social pressure and desirability bias (McLeod, 2019). The
limitation of the tests is the questions in place can have the tendency for the test takers to
present themselves in a generally favorable fashion. An example would be if the test is
measured based on discrimination; test-takers would lie because they do not want to be seen
as bullies or racists.
Lovler and Miller (2020) describe Normal Curves (NC) as a theoretical distribution that exists in
our imagination as perfect and symmetrical, with the scores concentrated on the middle than
the tails. The psychometric analysis article describes those 117 individuals who took the
diversity measurement tests, which is about 13 items to measure diversity. Researchers used 4
waves to distribute the tests, only to use the first wave tests because there were not many
changes to the individuals’ answers from the other wave distribution. Researchers found that
the sample skewness and the distribution curve provided a smaller critical value (Deng, et al.,
2014). The Likert scale provided more marginal and well-fitted data by the one-factor model,
which is easier to interpret to measure human characteristics.
The article describes that the reliability estimates following the transformed data tend to be
smaller than the Likert data (Deng, et al., 2014). Based on the descriptions provided, since NC’s
tend to be on a hypothetical side and provide “perfect” scores, using the Likert scale can be
realistic and provides reliability scores. The test takers can manipulate the Likert tests, but it
provides a more measurement ability to convert or transform test scores in a meaningful unit
(Lovler and Miller, 2020).
See discussions, stats, and author profiles for this publication at:
https://www.researchgate.net/publication/273586003
Psychometric Properties of
Measures of Team Diversity
With Likert Data
Article in Educational and Psychological Measurement · May 2014
DOI: 10.1177/0013164414541275
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Measurement
Educational and Psychological
http://epm.sagepub.com/content/early/2014/07/03/0013164414541275
The online version of this article can be found at:
DOI: 10.1177/0013164414541275
published online 4 July 2014Educational and Psychological Measurement
Lifang Deng, George A. Marcoulides and Ke-Hai Yuan
Psychometric Properties of Measures of Team Diversity With Likert Data
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DOI: 10.1177/0013164414541275
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Psychometric Properties of
Measures of Team Diversity
With Likert Data
Lifang Deng1, George A. Marcoulides2, and
Ke-Hai Yuan3
Abstract
Certain diversity among team members is beneficial to the growth of an organization.
Multiple measures have been proposed to quantify diversity, although little is known
about their psychometric properties. This article proposes several methods to evalu-
ate the unidimensionality and reliability of three measures of diversity. To approxi-
mate the interval scale required by the measures of diversity, a transformation on the
Likert-item scores is proposed. Ridge maximum likelihood is used to deal with the
issue of small sample size, and methods for evaluating the significance of the difference
of two reliability estimates with correlated samples are also developed. Results with a
real data set on entrepreneurial teams indicate that different measures of diversity
may correspond to significantly different estimates of reliability. Results also indicate
that diversity measures obtained with the transformed data tend to be more unidi-
mensional than their counterparts obtained from Likert data. However, diversity mea-
sures obtained from Likert data tend to yield greater reliability estimates. Among the
three examined measures of diversity, the standard deviation is found to yield greater
and more efficient reliability estimates than the others and is thus recommended.
Keywords
unidimensionality, reliability, normal-curve transformation, ridge structural equation
modeling
1
Beihang University, Beijing, China
2
University of California, Santa Barbara, CA, USA
3University of Notre Dame, Notre Dame, IN, USA
Corresponding Author:
Ke-Hai Yuan, University of Notre Dame, 123a Haggar Hall, Notre Dame, IN 46556, USA.
Email: [email protected]
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Introduction
The compositional diversity of team members within an organization has been
shown to affect the performance and growth of the organization (Harrison & Klein,
2007; Van Knippenberg, De Dreu, & Homan, 2004). Among various kinds of diver-
sity (e.g., demographic, informational, experiential, or personality attributes), not all
have been determined to be beneficial to the growth of an organization. Some
researchers have indicated that it is merely the differences between team members in
terms of their skill level, knowledge, and perspectives that are needed to foster crea-
tivity and innovation (e.g., Guzzo & Shea, 1992). However, the findings in the extant
literature have not been consistent (Jackson, Joshi, & Erhardt, 2003; Stewart, 2006;
Van Knippenberg et al., 2004; Webber & Donahue, 2001) and indicate that our
understanding of diversity and its role is still relatively limited. To facilitate a better
understanding of diversity, Harrison and Klein (2007) classified diversity into three
distinctive types: separation, variety, and disparity. Separation is for the difference
in position or opinion among team members; variety is used to describe diversity in
expertise, knowledge, or experience; and disparity refers to inequality in status or
resources held among team members. Such a classification allows researchers to
identify different roles of different types of diversity.
A variety of measures have also been proposed to quantify different types of
diversity among individuals within a team. According to Harrison and Klein (2007),
separation should be measured by either the standard deviation or the average of
Euclidean distances, variety should be measured by the so-called Blau’s (1977) index
or entropy (Teachman, 1980), and disparity should be measured by the coefficient of
variation or the ratio of the average of the absolute differences over the mean.
Harrison and Klein (2007) also discussed the type of scales required by each of these
diversity measures and emphasized that measuring separation requires the observed
data on team members to be at the interval scale, whereas measuring disparity
requires the observed data to be at the ratio scale. However, in the study of human
behavior within the fields of education, management, psychology, and related social
and behavioral sciences, it is extremely difficult to obtain data at the ratio or even
interval scales. What are typically obtained are data collected from a survey using
questionnaires that are commonly only ordinal or Likert type. Nevertheless, research-
ers still regularly apply procedures that require interval-scale data to ordinal data.
For example, factor analysis is commonly applied to Likert data for item selection or
scale development (Raykov & Marcoulides, 2011). Although such a practice may
still yield interpretable results, a better method is to factor analyze the polychoric
correlation matrix (Babakus, Ferguson, & Jöreskog, 1987). Given that the observed
values in Likert data used to determine the above-mentioned diversity measures are
somewhat arbitrary, we propose to transform them to avoid the arbitrariness. The
transformation is based on threshold values under the normal curve, parallel to those
used in estimating polychoric correlations (Olsson, 1979). We can call it the normal-
curve (NC) transformation. Although the transformed data are still limited in number
of values, we argue that they are closer to the conditions required by diversity
2 Educational and Psychological Measurement
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measures than the commonly used Likert data. To see the effect of the transforma-
tion, we will study the psychometric properties of several diversity measures when
applied to both Likert and transformed data.
Unidimensionality and reliability are probably the two most basic psychometric
properties one has to consider for any scale or instrument. Unidimensionality implies
that the statistical dependence among the items can be accounted for by a single
underlying latent trait, and reliability informs about the degree to which the observed
individual differences are indicative of true individual differences on the latent
dimension of interest. Measures of diversity, especially those for measuring separa-
tion with Likert data, are also subject to such properties if they aim to properly cap-
ture any underlying trait. In particular, when measurements in a scale are not
unidimensional, the empirical meaning of the scale will be different from the mean-
ing assigned to it, which will create interpretational confounding (e.g., Anderson &
Gerbing, 1982; Bagozzi, 1980; Burt, 1973, 1976). Reliability is equally important
because, for measurements with a low reliability index, the observed values of the
obtained measurements can be mostly due to random errors. Additionally, because
the value of the determined reliability index sets a bound on validity (Allen & Yen,
1979), a high reliability (index) is a necessary condition for high validity (Raykov &
Marcoulides, 2011). We hope that by studying the unidimensionality and reliability
of different measures of diversity, the inconsistent findings obtained to date on the
roles of diversity can be better understood.
The methodological development presented in this article was motivated by the
need to study the psychometric properties of diversity measures based on 13 Likert
items administered to entrepreneurial teams. Because the number of teams plays the
role of sample size, which is not sufficiently large, a method to deal with the issue of
small sample sizes was also needed especially when using factor analysis to evaluate
the unidimensionality of the diversity measures. For such a purpose, we make use of
the ridge maximum likelihood (ML) method originally developed in Yuan and Chan
(2008). This method has been shown to yield more accurate parameter estimates than
the normal-distribution-based maximum likelihood (NML) even for normally distrib-
uted data. We also develop methods for evaluating the significance of the difference
of two reliability estimates with correlated samples. This enables us to determine
whether different measures of diversity correspond to significantly different reliabil-
ity estimates. If different diversity measures yield significantly different reliability
estimates, then it is better to use the one that corresponds to the greatest reliability.
In the next section, the methodological components for studying the unidimen-
sionality and reliability of different measures of diversity are given, including the
formulations of different diversity measures, the NC transformation, ridge ML, and
standard error (SE) for difference of reliability estimates. A real data set with Likert
scale and its analysis are presented in the following section. We conclude with a dis-
cussion and recommendations. It is important to note that our focus is on the psycho-
metric properties (unidimensionality and reliability) of different measures of
Deng et al. 3
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diversity, not on interrater reliability issues (for further details on interrater reliabil-
ity, see Algina, 1978; Schuster & Smith, 2002; Shrout & Fleiss, 1979).
Methodology
This section first introduces the three diversity measures that will be used in the anal-
ysis of the real data. Then, the NC transformation is described. Ridge ML for factor
analysis is reviewed next. Formulas for SE of the difference of two reliability esti-
mates are developed at the end of this section. These measures and techniques will
be used to analyze the real data in the subsequent section.
Diversity Measures
Let xijk be the score of person k on item j within team i, k = 1, 2, . . . , ni;
j = 1, 2, . . . , p; i = 1, 2, . . . , N: Three measures of diversity derived from xijk will be
studied. These are the average of absolute distances (aad) among team members,
aadij =
2
ni(ni � 1)
Xni�1
k1 = 1
Xni
k2 = k1 + 1
jxijk1 � xijk2j; ð1Þ
the average of absolute deviations from the mean (aadm) of team members,
aadmij =
1
ni � 1
Xni
k = 1
jxijk � �xijj, ð2Þ
where �xij =
Pni
k = 1 xijk=ni; and the standard deviation (sd) among team members,
sdij = ½
1
ni � 1
Xni
k = 1
(xijk � �xij)2�1=2: ð3Þ
Two measures for separation were recommended by Harrison and Klein (2007).
One is the standard deviation in which the denominator is ni instead of ni � 1:
Another is the square root of the average of the squared Euclidean distances
(xijk1 � xijk2 )
2, in which k1 = k2 is not distinguished from k1 6¼ k2: According to
Biemann and Kearney (2010), these measures may contain substantial bias due to
including terms that are obviously 0 or without correcting for the loss of degrees of
freedom. The diversity measure in (1) only includes the absolute distances for differ-
ent team members, and degrees of freedom loss are accounted for in (2) and (3).
Parallel to the average Euclidean distance (aed) in Harrison and Klein (2007) or
Biemann and Kearney (2010), we define aedij as
aedij =
2
ni(ni � 1)
Xni�1
k1 = 1
Xni
k2 = k1 + 1
(xijk1 � xijk2 )
2
" #1=2
:
4 Educational and Psychological Measurement
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Because aedij is proportional to sdij (see Hays, 1981) and any results of reliability
and unidimensionality analysis of aedij would be identical to those of sdij, we do not
separately examine aedij in this article.
As we were not able to locate any references in the literature in which the aadmij
in (2) or an index that is proportional to aadmij has been proposed to measure diver-
sity, aadmij can be regarded as a new measure. The psychometric properties of the
three measures, aad, aadm, and sd, will be examined through real data analysis in
the following section.
Quantities in the form of the average of absolute distances (e.g., aad) are not pre-
sented as stand-alone measures in either Harrison and Klein (2007) or Biemann and
Kearney (2010). Instead, they are divided by the team mean score for measuring dis-
parity. Another measure for disparity recommended by Harrison and Klein (2007) is
the coefficient of variation. Since these measures require the observed data to possess
the properties of ratio scale, they may not be applicable to Likert data and will not be
studied in this article. Similarly, variety will not be measured through Likert data and
neither do we study Blau’s index or the entropy.
Normal-Curve Transformation
Since all three measures of diversity (aad, aadm, sd) are obtained by arithmetic oper-
ations, they are ideally applicable to data that are of interval scale (Harrison & Klein,
2007). However, as indicated previously, measurements in the social and behavioral
sciences are typically Likert or ordinal scale. To approximate interval scales, we pro-
pose a transformation to Likert data in this subsection.
With a total of Nt =
PN
i = 1ni individual observations and c categories for a given
item, let q̂l be the proportion of observations
1
for category l: Following the conven-
tion of polychoric correlations (Olsson, 1979), we may assume that, for each
observed xijk , there is an underlying continuous variable zijk ;N (0, 1) such that
xijk = l whenever zijk belongs to the interval (hl�1, hl�, where h0 \h1 \ � � � \hc
are threshold values to be estimated. This implies that the probability for xijk = l is
given by
ql = F(hl ) � F(hl�1), l = 1, 2, . . . , c,
where F(�) is the cumulative distribution function of z;N (0, 1), with h0 = � ‘ and
hc = ‘: Thus, the marginal maximum likelihood estimate of hl is given by
ĥl = F
�1(
Xl
t = 1
q̂t ), l = 1, 2, . . . , c � 1: ð4Þ
Based on this underlying NC assumption, we propose a transformation to the Likert
xijk by
yijk = (ĥl�1 + ĥl )=2 if xijk = l, l = 1, 2, . . . , c: ð5Þ
Deng et al. 5
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Notice that there are only c � 1 finite values of ĥl in (4), and we cannot use ĥ0 = � ‘
or ĥc = ‘ because they will result in yijk = � ‘ when xijk = 1 or yijk = ‘ when xijk = c:
We further propose to use
ĥ0 = F
�1(:5=Nt ) and ĥc = F
�1(1 � :5=Nt ): ð6Þ
The proposed values in (6) are equivalent to assigning a value of .5 to cells with zero
number of observations in the analysis of contingency tables, because we can think
of an extra category xijk = 0 below xijk = 1 and another extra category xijk = c + 1 above
xijk = c, and both had zero number of observations. The proposed values in (6) are
also similar to the so-called continuity correction in applying the central limit theo-
rem to categorical data (Feller, 1945), where a step of .5 is used when jumping from
1 to the next whole number.
Notice that the correction in (6) is for yijk to avoid being �‘ or ‘ whenever
xijk = 1 or c: If the nominal number of categories is c but only c � 1 or fewer number
of categories are observed, we may simply treat the unobserved categories in the
middle as having probability of zero by just applying the correction to the end points
of xijk:
We need to note that the transformed yijk do not possess the property of interval
scales, although they avoid the arbitrary nature of Likert data that assign consecutive
whole numbers to ordered categories. Closely related to polychoric correlation, the
rationale of the transformation in (5) depends heavily on the assumption of a normal
curve underlying the observed frequencies. If the NC assumption holds, the yijk
obtained by the NC transformation determined by equations (4), (5), and (6) is sim-
ply the middle point of the interval zijk belongs, and thus represents the best predic-
tion of the true value of zijk in the sense of smallest absolute mean difference.
Applying each of the three diversity measures, aadij, aadmij, and sdij, to the trans-
formed yijk yields three more measures of diversity. In the next section, their reliabil-
ity and unidimensionality are examined, and the results are contrasted with those
obtained based on Likert data.
Ridge Maximum Likelihood for Factor Analysis With Small Sample Sizes
As indicated in the previous section, the number of teams, N , plays the role of sample
size when evaluating the psychometric properties of the diversity measures aad,
aadm, and sd. Since it can be expensive to have a large N , we use ridge ML for factor
analysis of the diversity measures in (1) to (3) when studying their unidimensionality.
Unless all the ni are sufficiently large, the diversity measures in (1) to (3) cannot be
regarded as normally distributed. As such, we expect ridge ML to work better than
NML when factor analyzing the diversity measures.
Let S be a sample covariance matrix of size p, and we are interested in modeling
S = E(S) by a confirmatory factor model
S(u) = LFL
0
+ C, ð7Þ
6 Educational and Psychological Measurement
by guest on July 5, 2014epm.sagepub.comDownloaded from
http://epm.sagepub.com/
where L is a factor loading matrix, F is a factor correlation matrix, and C is a diago-
nal matrix of measurement errors/uniquenesses. The widely used NML procedure for
covariance structure analysis is to minimize
FML(S, S(u)) = tr½SS�1(u)�� log jSS�1(u)j� p
for parameter estimation. Let a . 0 be a small number and Sa = S + aI, with I
being the identity matrix. The ridge ML developed in Yuan and Chan (2008) is to
estimate ua by minimizing FML(Sa, S(ua)), and let the estimates be denoted by ûa:
The corresponding estimates û for u are obtained by subtracting a from each of
the elements of ûa corresponding to the diagonal elements of C, leaving the other
elements of ûa unchanged. Standard errors of û are obtained by a sandwich-type
covariance matrix, which accounts for the unknown underlying population distri-
bution of the involved diversity measure. As for overall model evaluation, Yuan
and Chan (2008) showed that, unless a = 0, TML = (N � 1)FML(Sa, S(ua)) does
not asymptotically follow the nominal chi-square distribution x2df even if data are
normally distributed. They developed a rescaled statistic TRML and an adjusted sta-
tistic TAML: Parallel to the development for NML in Satorra and Bentler (1994),
TRML asymptotically follows a distribution whose mean equals df , and TAML
asymptotically follows a distribution whose mean and variance equal those of the
approximating distribution. Since the details of ridge ML have already been
described in Yuan and Chan (2008), no further elaboration is given here. Our pur-
pose is to apply ridge ML to evaluate the unidimensionality of each of the three
measures of diversity in (1) to (3) and to determine whether the corresponding
sample covariance matrix can be reasonably fitted by a one-factor model.
Following the recommendation of Yuan and Chan (2008), a = p=N is used in
applying the ridge ML.
In order to fully justify applying a factor analysis to each of the diversity measures,
we do not need to assume that each of aadij, aadmij, or sdij is identically distributed
across i = 1, 2, . . . , N: The development in Lee and Shi (1998) implies that the vector
di = (aadi1, aadi2, . . . , aadip)
0 does not need to have the same population covariance as
i varies. Since for both reliability and unidimensionality the analysis is based on the
sample covariance matrix S of the corresponding diversity measures with the assump-
tion E(S) = S, our study of the psychometric properties of di is for the population rep-
resented by the sample di, i = 1, 2, . . . , N: We will further discuss this point in the
concluding section.
Standard Error for Difference of Two Reliability Estimates With Correlated
Samples
Among the many available estimates of reliability for equally weighted composite
scores, coefficient alpha is most widely used in practice even though it can over- or
underestimate the population reliability (Raykov, 1997). Another popular estimate is
coefficient omega defined through the factor loadings and error variances by fitting
Deng et al. 7
by guest on July 5, 2014epm.sagepub.comDownloaded from
http://epm.sagepub.com/
the sample covariance matrix to a one-factor model (McDonald, 1999). Both are
applicable when evaluating the reliability of the different diversity measures. Our
interest is whether different diversity measures will yield significantly different relia-
bility estimates. Thus, we need to have an estimate of the SE of the difference of two
estimates of alpha or omega. When the two estimates are independent, the variance
of the difference of the two estimates is simply the summation of the variances of
the two estimates of alpha or omega. However, with respect to the three diversity
measures, the variance or SE of the difference of two estimates of alpha or omega
depends on their correlation. Since the SE for the difference of two reliability esti-
mates with correlated samples will facilitate comparison of reliabilities in other con-
texts, and the literature to date does not contain such a development, we provide
more details for obtaining consistent SEs of the difference of two estimates of alpha
and omega, respectively. We also present the necessary notation and formulas for
calculating the SEs. The complete details leading to the calculation formulas are
given in Appendices A and B.
Let S = (sjk ) be a sample covariance matrix of size p, and s = vech(S) be the vector
by stacking the elements in the lower-triangular part of S: Then, with p� = p(p + 1)=2,
s is a vector of p�31, and the sample coefficient alpha is given by
â = g(s) =
p
p � 1
(1 �
Xp
j = 1
sjj=
Xp
j = 1
Xp
k = 1
sjk ) =
p
p � 1
(1 �
a0s
b0s
),
where a is a p�31 vector whose elements are 1 corresponding to sjj and 0 elsewhere;
and b is also a p�31 vector whose elements are 1 corresponding to sjj and 2 corre-
sponding to sjk when j 6¼ k: For example, at p = 3, s = (s11, s21, s31, s22, s32, s33)0,
a = (1, 0, 0, 1, 0, 1)0, and b = (1, 2, 2, 1, 2, 1)0: We need to have the Jacobian matrix or
the matrix of derivatives of g(s) with respect to the elements of s, and it is given by
_g(s) = �
p
p � 1
½
1
b0s
a �
a0s
(b0s)
2
b�:
With s1 = vech(S1) and s2 = vech(S2) from two correlated samples, standard error for
â2 � â1 = g(s2) � g(s1) also involves the variance-covariance matrices of s1 and s2:
Denote these by G11 = Var(
ffiffiffi
n
p
s1), G22 = Var(
ffiffiffi
n
p
s2), and G12 = Cov(
ffiffiffi
n
p
s1,
ffiffiffi
n
p
s2),
where n = N � 1: These are consistently estimated by their sample counterparts, with
details given in Appendix A. With the introduced notation, the result given in
Appendix B implies that
ffiffiffi
n
p
½(â2 � a2) � (â1 � a1)� is asymptotically normally dis-
tributed with mean zero and variance consistently estimated by
t̂
2
a = _g
0(s1)Ĝ11 _g(s1) + _g
0(s2)Ĝ22 _g(s2) � 2 _g0(s1)Ĝ12 _g(s2): ð8Þ
It follows from (8) that the SE of (â2 � â1) is consistently estimated by t̂a=
ffiffiffi
n
p
,
which will be used in the next …
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