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Volume 17, Number 3, January 2012 ISSN 1531-7714
Estimating ordinal reliability for Likert-type and ordinal item response
data: A conceptual, empirical, and practical guide
Anne M. Gadermann, Harvard Medical School
Martin Guhn & Bruno D. Zumbo, University of British Columbia
This paper provides a conceptual, empirical, and practical guide for estimating ordinal reliability coefficients
for ordinal item response data (also referred to as Likert, Likert-type, ordered categorical, or rating scale item
responses). Conventionally, reliability coefficients, such as Cronbach’s alpha, are calculated using a Pearson
correlation matrix. Ordinal reliability coefficients, such as ordinal alpha, use the polychoric correlation matrix
(Zumbo, Gadermann, & Zeisser, 2007). This paper presents (i) the theoretical-psychometric rationale for
using an ordinal version of coefficient alpha for ordinal data; (ii) a summary of findings from a simulation
study indicating that ordinal alpha more accurately estimates reliability than Cronbach’s alpha when data
come from items with few response options and/or show skewness; (iii) an empirical example from real data;
and (iv) the procedure for calculating polychoric correlation matrices and ordinal alpha in the freely available
software program R. We use ordinal alpha as a case study, but also provide the syntax for alternative
reliability coefficients (such as beta or omega).
Reliability is an important source of evidence when
establishing the validity of the inferences one makes
based on scores from tests and measures (e.g., Kane,
2006; Zumbo, 2007). Throughout, we use the term ‘test’
to refer to any type of quantitative, multi-item
measurement, such as tests, scales, and surveys in the
social sciences. There are, of course, multiple definitions
and types of reliability (e.g., internal consistency, retest,
inter-rater), and multiple ways of obtaining reliability
coefficients or indices (e.g., via different estimation
methods and by using correlation or covariance
matrices). Given the importance and the complexities of
the concept(s) of reliability, the field has witnessed
recurring debates on the interpretations and purposes of
different types of reliability, on the advantages and
disadvantages of different reliability indices, and on the
methods for obtaining them (e.g., Bentler, 2009; Cortina,
1993; Revelle, & Zinbarg, 2009; Schmitt, 1996; Sijtsma,
2009).
A topic that has attracted particular attention in the
psychometric literature is Cronbach’s alpha (Cronbach,
1951), which remains the most widely and frequently
used reliability index (Sijtsma, 2009). Some of this
attention has been motivated by the fact that Cronbach’s
alpha has repeatedly been misinterpreted and misused
(cf. Cortina, 1993; Schmitt, 1996; Sijtsma, 2009)—as
noted by Cronbach himself (2004). In addition, some of
the attention has centered on the question of whether
there are some alternative reliability coefficients, such as
omega, that may be more appropriate under certain
circumstances (Revelle & Zinbarg, 2009; Zinbarg,
Revelle, Yovel, & Li, 2005).
Some of the debates on reliability indices and on
Cronbach’s alpha have been fairly technical, including
Cronbach’s original paper (1951; see also, for example,
Bentler, 2009; Green & Yang, 2009a, 2009b; Lord &
Novick, 1968; Sijtsma, 2009). The implications of those
technical debates are, however, not just of interest to a
technical audience, but are critically relevant to
practitioners and researchers in the social sciences in
general. In fact, using Cronbach’s alpha—or any other
reliability coefficient—under circumstances that violate
its assumptions and/or prerequisites might lead to
substantively deflated reliability estimates (e.g., Gelin,
Beasley, & Zumbo, 2003; Maydeu-Olivares, Coffman, &
Hartmann, 2007; Osburn, 2000). A substantively deflated
Practical Assessment, Research & Evaluation, Vol 17, No 3 Page 2
Gadermann, Guhn & Zumbo, Estimating Ordinal Reliability
estimation of a test’s reliability, in turn, might potentially
entail some misinformed inferences, such as discarding a
test due to its seemingly low reliability.
Purpose of the paper
In this paper, we provide a tutorial for when and
how to calculate ordinal reliability coefficients—rather
than non-ordinal coefficients, such as Cronbach’s
alpha—for the very common scenario that one’s data
come from measurements based on ordinal response
scales (e.g., rating scales or Likert-type response formats).
We focus on ordinal alpha, which was introduced by
Zumbo, Gadermann, and Zeisser (2007), and which was
shown to estimate reliability more accurately than
Cronbach’s alpha for binary
1
and ordinal response scales.
We focus on alpha because it is the most widely used
reliability coefficient, and because it is useful to use a
familiar scenario as a concrete example. We note,
however, that our discussion applies to other reliability
coefficients as well. In other words, the rationale for
using an ordinal version of a reliability coefficient is not
restricted to alpha, but is equally valid for other reliability
coefficients, such as McDonald’s omega or Revelle’s beta
(please see Zinbarg et al. (2005) for a description of the
omega and beta reliability coefficients).
The main purpose of this paper is to (i) provide
researchers and practitioners with the psychometric and
conceptual rationale for when, why, and how to use
ordinal reliability coefficients, (ii) present an empirical
example that illustrates the degree to which Cronbach’s
alpha and its ordinal equivalent can differ, and (iii)
present step-by-step practical instructions and an
example for how to compute ordinal alpha and other
reliability coefficients, such as ordinal beta and ordinal
omega, in the freely available software package R
(http://www.R-project.org).
Coefficient alpha for ordinal data:
psychometric rationale
Ordinal alpha is conceptually equivalent to
Cronbach’s alpha. The critical difference between the
two is that ordinal alpha is based on the polychoric
correlation matrix, described in detail below, rather than
the Pearson covariance matrix, and thus more accurately
estimates alpha for measurements involving ordinal data.
1
A special case of coefficient alpha is KR-20, which is
computed from binary data.
In general, the computation of coefficient alpha
involves the matrix of correlations or covariances among
all items of a scale. For Cronbach’s alpha, the Pearson
covariance matrix is routinely used; for example, as a
default in statistical software programs, such as SPSS and
SAS. An important assumption for the use of Pearson
covariances is that data are continuous, and if this
assumption is violated, the Pearson covariance matrix
can be substantively distorted (e.g., Flora & Curran,
2004). In social science measurement, it is very common
to use the Likert-type item response format. (For
example, respondents are asked to indicate their level of
agreement with an item by choosing one of a given
number of ordered response categories, e.g., with five
categories ranging from ‘strongly agree’ to ‘strongly
disagree’.) The data arising from such items are not
continuous, but ordinal; however, they are often treated
as if they were continuous; that is, they are treated “as if
the data had been measured on an interval scale with
desired distributional properties” (p. 443, Olsson, 1979).
It has been shown that the Pearson correlation
coefficient severely underestimates the true relationship
between two continuous variables when the two
variables manifest themselves in a skewed distribution of
observed responses. A tetrachoric/polychoric
correlation, on the other hand, more accurately estimates
the relationship of the underlying variables (Carroll,
1961). Accordingly, for ordinal data, the method of
choice is to use the polychoric correlation matrix. Based
on this argument, Zumbo et al. (2007) introduced a
coefficient alpha for ordinal data—ordinal alpha—that is
derived from the polychoric correlation matrix.
It needs to be noted that ordinal alpha, in line with
the longstanding psychometric tradition of interpreting
ordinal responses as manifestations of an underlying
variable, is focusing on the reliability of the unobserved
continuous variables underlying the observed item
responses. Using a polychoric matrix for calculating
alpha thus represents an underlying variable approach to
covariance modeling of ordinal data. That is, when using
a polychoric correlation matrix, an item’s observed
responses are considered manifestations of respondents
exceeding a certain number of thresholds on that item’s
underlying continuum. Conceptually, the idea is to
estimate the thresholds and model the observed cross-
classification of response categories via the underlying
continuous item response variables. Formally, the
observed ordinal response for item j with C response
categories, where the response option c = 0, 1, 2, …, C-1,
is defined by the underlying variable y* such that
Practical Assessment, Research & Evaluation, Vol 17, No 3 Page 3
Gadermann, Guhn & Zumbo, Estimating Ordinal Reliability
cy
j
=
if
1
*+
<<
cjc
y
ττ
,
where
1
,+cc
τ
τ
are the thresholds on the underlying
continuum, which are typically spaced at non-equal
intervals and satisfy the constraint
∞=<<<<=∞−
−cc
τ
τ
τ
τ
110
L. The underlying
distribution does not necessarily have to be normally
distributed, although it is commonly assumed so due to
its well understood nature and beneficial mathematical
properties (cf. Liu, Wu, & Zumbo, 2010).
In summary, ordinal reliability coefficients may
differ from their non-ordinal counterparts because of
their scaling assumptions. The non-ordinal coefficients
focus on the reliability of the observed scores by treating
the observed item responses as if they were continuous,
whereas the ordinal coefficients focus on the reliability of
the unobserved continuous variables underlying the
observed item responses. In this way, the ordinal
coefficients are nonparametric reliability coefficients in a
nonlinear classical test theory sense (Lewis, 2007).
Review of findings from a simulation study
Zumbo et al. (2007) present findings from a
simulation study that compared ordinal alpha and
Cronbach’s alpha for all combinations of the following
conditions:
(i) The theoretical reliability of a test was simulated
so that it was .4, .6, .8, or .9. As Zumbo et al.
(2007) note, the theoretical reliability was
determined in reference to the underlying
continuum of variation.
(ii) The number of response options of the items was
set to 2, 3, 4, 5, 6, or 7.
(iii) The amount of skewness of the data was set to 0
or –2.
For all conditions, the number of items (p) was set
to 14. We reanalyzed the data from Zumbo et al.’s (2007)
simulation study, which invoked a paradigm introduced
by Zumbo and Zimmerman (1993), specifying the
underlying continuous scale as the reference for the
theoretical reliability. Figure 1 illustrates the degree of
underestimation of the theoretical reliability by
Cronbach’s alpha and ordinal alpha, for those different
conditions, respectively. In Figure 1, the degree to which
ordinal alpha as well as Cronbach’s alpha accurately
estimate or underestimate the theoretical reliability of the
underlying variable is illustrated in terms of an attenuation
index, which is calculated by the following formula
(Equation 1):
Percent attenuation = [100 * (alpha – theoretical
reliability) / theoretical reliability)]
In equation 1, alpha denotes either ordinal alpha or
Cronbach’s alpha. When alpha is equal to the theoretical
reliability, the term becomes zero, indicating no
attenuation. The more alpha diverges from the
theoretical reliability, the closer the term gets to (-100),
which would indicate the highest possible degree of
attenuation.
The graphs in Figure 1 indicate that ordinal alpha
provides a suitable estimate of the theoretical reliability,
regardless of (i) the magnitude of the theoretical
reliability, (ii) the number of scale points, and (iii) the
skewness of the scale point distributions. The accuracy
of Cronbach’s alpha, on the other hand, decreases (i) as
the skewness of the scale items increases, (ii) as the
number of response options becomes smaller, and (iii) as
the theoretical reliability of the scale is lower.
The findings from the simulation study thus
corroborate the general psychometric recommendation
to use a polychoric correlation matrix for ordinal data,
and they indicate that ordinal alpha is an unbiased
estimator of the theoretical reliability for ordinal data (at
least for scenarios like or similar to those tested in the
simulation study). If one assumes that the observed item
responses are manifestations of a continuous underlying
item response variable, particular care should be taken in
the interpretation of Cronbach’s alpha, especially when
one has very few item response options and/or highly
skewed observed item responses. In those cases,
Cronbach’s alpha is a substantially attenuated estimate of
the lower bound of the reliability of the underlying item
response variables, whereas ordinal alpha tends to
estimate this reliability more accurately—as the
polychoric correlations correct for the attenuation caused
by the scaling of the items (cf. Carroll, 1961).
Practical Assessment, Research & Evaluation, Vol 17, No 3 Page 4
Gadermann, Guhn & Zumbo, Estimating Ordinal Reliability
Illustration of an example
We now provide an example that illustrates the
potential for a large discrepancy between ordinal alpha
and Cronbach’s alpha, using real data. The data are from
a sample of 43,644 kindergarten children (48% girls; M
age
= 5.7 years), and were collected with the Early
Development Instrument (EDI; Janus & Offord, 2007). The
EDI is a teacher-administered rating scale with 103
ordinal/binary items on children’s developmental status
in Kindergarten (see also Guhn, Janus, Hertzman, 2007;
Guhn, Zumbo, Janus, & Hertzman, 2011). For our
example, we used data from the physical independence
subscale of the EDI. This subscale is composed of 3
binary items, which are scored as 0 (no) and 1 (yes).
From a statistical point of view, the binary case can be
thought of as a special case of ordinal data, and therefore
the same methods apply here. Table 1 shows the means,
standard deviations, skewness, and kurtosis for the 3
items. Table 2 shows the Pearson correlations, Pearson
covariances, and polychoric/tetrachoric correlations for
the three items, the average correlations/covariance, and
the respective alphas. Table 3 shows the factor loadings,
communalities, and uniquenesses for the 3 items, and the
alphas, calculated based on a factor analysis model, and
using the matrix of the (i) Pearson correlations, (ii)
Pearson covariances, or (ii) polychoric correlations
2
,
2
We note that the term polychoric correlation refers to all
correlations based on ordinal variables that measure an
(assumably) continuous underlying variable. In the special case
that this involves two dichotomous/binary variables, the term
tetrachoric correlation is used. Because the tetrachoric correlation
is a special case of the polychoric correlation, calculating a
polychoric correlation for binary variables is, in fact, equivalent
to calculating a tetrachroic correlation; see Uebersax, 2006.
Figure 1: Percent attenuation of Cronbach’s alpha and ordinal alpha
Practical Assessment, Research & Evaluation, Vol 17, No 3 Page 5
Gadermann, Guhn & Zumbo, Estimating Ordinal Reliability
respectively. As can be seen in Table 1, the skewness and
kurtosis of these three items are very high.
Table 1: Means, standard deviations, skewness and
kurtosis of the three items of the physical independence
subscale of the EDI
Item
Would you say this child …
Mean
(SD)
Skew
(SE)
Kurtosis
(SE)
… is independent in
washroom habits most of the
time (item 1)
.98
(.12)
-7.9
(.03)
59.9
(.07)
… shows an established hand
preference (item 2)
.98
(.16)
-6.1
(.03)
34.9
(.07)
… is well coordinated (moves
without running into things)
(item3)
.92
(.28)
-3.0
(.03)
7.2
(.07)
The results in Table 2 illustrate the degree to which
the polychoric/tetrachoric correlations differ from the
Pearson correlations in this case. The differences are all
statistically significant (as tested via a Fisher z-
Transformation; cf., Cohen, Cohen, West, & Aiken,
2003), and also substantial in their magnitude. If one
applies the guidelines provided by Cohen (1988), the
Pearson correlation coefficient for items 1 and 2
(r
Pearson
=.23) is considered a small-medium effect, whereas
the size of the polychoric correlation coefficient for the
same two items (r
Polychoric
=.63) is considered to indicate a
large effect.
Table 2: Polychoric and Pearson
correlations/covariances, and alphas for the physical
independence subscale items
Items
1,2
Items
1,3
Items
2,3
Averag
e
Pearson correlation
(
r
)
.23 .2
5
.30 .2
6
Covariances
(
cov
)
.004 .00
8
.013 .008
5
Polychoric/tetrachoric
correlation
3
(r)
.63 .6
5
.6
8
.6
5
Item 1 Item 2 Item 3
V
ariances
(
va
r
)
.01
5
.024 .074 .03
8
Standardized alpha α = (k* r
average
)/(1+(k-1) * r
average
) =
(3*.26)/(1+(3-1)*.26) = .51
Cronbach’s (raw) alpha α = (k* cov
average
)/(var+(k-1) * cov
average
)
= (3*.0085)/(.038+(3-1)*.0085) =
.46
Polychoric ordinal
alpha
α = (k* r
average
)/(1+(k-1) * r
average
) =
(3*.65)/(1+(3-1)*.65) = .85
k: number of items in the scale
3
Polychoric covariances are equal to polychoric correlations,
as they are based on standardized variables.
Table 3 shows that, for this example, the factor
loadings (
λ
) obtained from a factor analysis of the
Pearson correlation or covariance matrices are
substantially lower than those obtained from a
polychoric correlation matrix. Correspondingly, these
differences are reflected in the communalities (h
2
) and
uniquenesses (u
2
). In some cases, differences in Pearson-
based loadings versus polychoric correlation-based
loadings might lead to different decisions about which
items to include or not to include in one’s factor
model—for example, if item loadings, respectively, fall
below or above a commonly used convention to only
consider items with factor loadings greater than .40
(Ford, MacCallum, & Tait, 1986).
In addition, ordinal alpha and Pearson covariance-
based (Cronbach’s/raw) alpha are substantially different
(.46 versus .85, respectively), with a percent attenuation
of (-46). (The table provides the factor model-based
formula for the 1-factor model for calculating alpha
4
; cf.
McDonald, 1985). Typically, the psychometric literature
(e.g., Nunnally, 1978) recommends that alpha for a scale
should not be smaller than .70 when used for research
purposes, at least .80 for applied settings, and greater
than .90 or even .95 for high-stake, individual-based
educational, diagnostic, or clinical purposes. In our
example, interpreting the reliability of the physical
independence scale by using ordinal versus Cronbach’s
alpha would make a difference with regard to these
conventional recommendations.
General procedure for computing ordinal
reliability coefficients in R
To calculate ordinal reliability coefficients, one
needs to estimate a polychoric correlation matrix, and
then calculate the reliability coefficient from the
polychoric correlation matrix. In this paper, we show
how to calculate these steps in the statistical software
package R (R Development Core Team, 2011). There are
alternative options for obtaining ordinal reliability
coefficients, but the procedure in R has the following
advantages: (1) Recent advancements and newly installed
applications in R (Fox, 2005, 2006, 2011; Revelle, 2011)
allow one to obtain polychoric correlations, ordinal
reliability coefficients, and corresponding (ordinal) item
statistics in a few simple steps; (2) R can be downloaded
for free, and can be installed for Windows, Unix (Macs),
4
By using this formula, readers who obtain the polychoric
correlation matrix in MPlus or PRELIS/LISREL (via a 1-
factor EFA with categorical data, see syntax in Elosua Oliden
& Zumbo, 2008) may calculate ordinal alpha.
Practical Assessment, Research & Evaluation, Vol 17, No 3 Page 6
Gadermann, Guhn & Zumbo, Estimating Ordinal Reliability
and Linux operating systems; (3) A graphic user interface
developed for R—R Commander (Fox, 2005, 2006)—
allows one to easily import data files in textfile, URL,
clipboard, Minitab, SPSS, or Stata format into R; (4) The
procedures to obtain polychoric correlation matrices in
(proprietary) software programs—such as MPlus, SAS,
Stata, and PRELIS/LISREL—involve more elaborate
syntax (or additional macros)
5
, and/or one would have
to calculate ordinal reliability “by hand”
once the
polychoric correlation matrix is obtained; (5) Some
widely used statistical software packages (e.g., SPSS) do
not produce a polychoric correlation matrix.
We note that one limitation of using ordinal
reliability coefficients may arise from the procedure with
which the polychoric correlation matrix is calculated:
Depending on the nature of the data as well as on the
estimation method that is employed (i.e., whether
correlations are calculated pair-wise for variables or
simultaneously for the entire matrix), the correlation
matrix may be non-positive definite which is not an issue
when one calculates the Pearson correlation matrix from
the same data, in the case of no missing data (Rigdon,
1997; Wothke, 1993). We note that R simultaneously
5
For example, for SAS, see
http://www.ats.ucla.edu/stat/sas/faq/tetrac.htm, and for
STATA, see
http://www.ats.ucla.edu/stat/stata/faq/tetrac.htm.
estimates polychoric correlations from the entire data
matrix, and we did not encounter this problem with the
data used for this study. However, this limitation may
arise in other software environments (i.e., with other
estimation methods) or with other data. On that note, it
should also be mentioned that the estimation of
polychoric correlation matrices for scales containing a
relatively large number of items may require substantial
time/computer processing power.
In the remainder of the paper, we provide an
example for how to calculate ordinal reliability
coefficients with a data set that is included in the R
software. We also provide instructions for preparing and
importing data from other sources into R, so that
researchers can easily calculate ordinal reliability
coefficients in R for data files that already exist in the
databases of their respective software programs.
R can be downloaded at the R website,
http://www.R-project.org. Appendix A provides a
description for downloading and installing R, and lists a
number of useful online resources. Once R is
downloaded, starting R will open the R menu and
console, and one can install and load so-called R
packages—installing specific packages will allow one to
conduct specialized analyses, such as, in our case,
calculating polychoric correlation matrices, ordinal
reliability coefficients, and ordinal item statistics. The
packages that need to be installed for our purpose are
psych (Revelle, 2011), and GPArotation (Bernaards &
Table 3: Item characteristics, ordinal alpha, Cronbach’s alpha, and discrepancy
Pearson correlation matrix Pearson covariance matrix Polychoric matrix
λ h
2
u
2
λh
2
u
2
λ h
2
u
2
Item 1 .44 .19 .81 .46 .21 .79 .78 .60 .40
Item 2 .53 .29 .71 .49 .24 .76 .81 .66 .34
Item 3 .57 .32 .68 .48 .23 .77 .84 .71 .29
Average .51 .27 .73 .475 .23 .77 .81 .66 .34
Formula for alpha based on a 1-factor model (cf. McDonald, 1985)
α = [ k / (k-1) ] * [ ( k * (λ
average
)
2
– h
2average
) / ( k * (λ
average
)
2
+ u
2average
) ]
Standardized α
Pearson correlations
[ 3/(3-1) ] * [ ( 3*(.51)
2
– .27 ) / ( 3*(.51)
2
+ .73 ) ] = .51
Cronbach’s (raw) α
Pearson covariances
[ 3/(3-1) ] * [ (3*(.475)
2
– .23 ) / (3*(.475)
2
+.77 ) ] = .46
Ordinal α
polychoric correlations
[ 3/(3-1) ] * [ ( 3*(.81)
2
– .66 ) / ( 3*(.81)
2
+ .34 ) ] = .85
Percent attenuation: (100 * (Cronbach’s alpha – Ordinal alpha) / ordinal alpha) = (100 * (.46 - .85)/.85) = - 46
λ: Factor loading; h
2
:
Communality (for a 1-factor model h
2
=
λ
2
); u
2
: Uniqueness (u
2
= 1 - h
2
); k: number of items on the scale
Practical Assessment, Research & Evaluation, Vol 17, No 3 Page 7
Gadermann, Guhn & Zumbo, Estimating Ordinal Reliability
Jennrich, 2005). Once R and the required packages
within R have been installed and loaded, we suggest
closing R, and opening a new session, so that the
following example can be replicated by readers in a
manner that represents a typical new session in R.
In Appendix B, we present the syntax that needs to
be typed (not copied and pasted
6
) into the R console
(which opens when R is started) to run the example.
Please note that (i) the syntax is entered after the greater-
than sign (>) that always appears on the last line of the R
console (i.e., the syntax file), (ii) R is case sensitive, and
(iii) the return key needs to be pressed at the end of each
syntax command to run the command. Please also note
that the #-signs in our example are not part of the
syntax, but simply indicate that we inserted an
explanatory comment. In the syntax, bolded font
indicates necessary steps, and regular font indicates steps
that are optional, but will help to obtain commonly
requested information in the context of calculating
ordinal reliability coefficients.
Importing data into R via the graphic user
interface R Commander
Once the R package Rcmdr has been installed and
loaded, the syntax command library (Rcmdr), entered into
the R console, will open the graphic user interface, R
Commander (see Fox, 2005 for a tutorial on R
Commander). R Commander lets you import data in the
following file formats: url, textfile (e.g., ASCII),
clipboard, SPSS/PASW, Stata, and Minitab. Here, we
briefly delineate the procedure for importing an SPSS
file: In the menu of the graphic user interface R
Commander, which is located on the top, click on ‘Data’,
then on ‘Import data’, and then on ‘from SPSS data
set…’. In the window that opens (‘Import SPSS Data
Set’), highlight the word ‘Dataset’ (in the box ‘Enter
name for data set’), choose and type a name under which
you wish to save the dataset to be imported (e.g., mydata),
and click ‘OK’. This opens your computer’s directory.
Choose the dataset to be imported (e.g., spssdatafile.sav),
and click on ‘Open’. Please note that unless one wishes
to create a new data frame
7
in R, (i) the imported dataset
6
Please note that copying syntax into the R console sometimes
leads to error messages.
7
It is possible to import data sets, and to then create subsets—
so-called data frames (using the syntax command data.frame—
in R, in order to conduct analyses, such as calculating ordinal
alpha, for a subset of the imported items/data (see, e.g.,
http://cran.r-project.org/doc/manuals/R-intro.html)).
should only contain the items of the scale for which the
reliability coefficient(s) is to be computed, (ii) the items
should have ordinal data with consecutive numbers (e.g., 1, 2,
3; or 0, 1, 2, 3, 4)—so if, for example, an ordinal variable
is coded as 10, 20, 30, 40, 50, it should be recoded to 1,
2, 3, 4, 5; otherwise, R will produce erroneous results, (iii)
item labels should be a maximum of 8 characters long,
(iv) the SPSS column ‘Values’ should be set to ‘None’ for
all items (so that all values are displayed as numbers, not
as text/response categories), (v) the SPSS column
‘Missing’ should be set to ‘None’ for all items, (vi)
missing data should be empty cells (in R, cells with
missing data will then appear as NA), not numerical
values, such as 88 or 999, and (vii) the SPSS column
‘Measure’ should be set to ‘Ordinal’. Once the data file is
imported, clicking on ‘View data set’ in the menu will
display the imported data file. As a last step, entering the
syntax command attach (mydata) into the R console will
attach the imported data set (in this case, mydata) to the
current R session. Then, the syntax for calculating
ordinal reliability coefficients, as described above, can be
used for the imported dataset.
Conclusions
We recommend considering ordinal, polychoric
correlation-based versions of reliability coefficients, such
as alpha or omega, when one’s data are binary and/or
ordinal—that is, from Likert-type or mixed items, with 2
to 7 response options. In so doing, one invokes an
underlying continuous variable for each item and the
reliability coefficient is then defined by the covariation
among these underlying variables. In this light, it is useful
to think of the tetrachoric and polychoric strategy as akin
to a data transformation, so that one is quantifying the
reliability of the item response data in this transformed
metric.
This recommendation is in line with general current
thinking in the psychometric literature about using
polychoric correlations for ordinal data (cf., Flora &
Curran, 2004). Also, since the introduction of ordinal
alpha by Zumbo et al. (2007), the use of a polychoric
correlation-based version of alpha for ordinal or binary
data has been applied elsewhere (e.g., Bentler, 2009,
Green & Yang, 2009a). We reiterate that the strategy of
using the polychoric correlation could be applied to any
reliability estimate that can be computed from a
correlation matrix. We provided the R syntax for alpha
and alternative reliability coefficients, such as omega, but
it needs to be noted that one could also compute an
ordinal version of generalizability theory (e.g., G
Practical Assessment, Research & Evaluation, Vol 17, No 3 Page 8
Gadermann, Guhn & Zumbo, Estimating Ordinal Reliability
coefficients) or test-retest reliability by using the
polychoric correlation with the respective equations.
For future research, it will be of particular interest
to better understand the interdependent, interacting
effects that a scale’s number of items, number of item
response options, skewness, kurtosis, and factor
structure have on ordinal alpha and Cronbach’s alpha,
and the discrepancy between them. Our data suggest a
diminishing return model with regard to the number of
items and number of response options, and also indicate
that item skewness is associated with the attenuation of
Cronbach’s alpha. However, the exact nature of the
multivariate relationship between these factors remains
to be determined.
We would like to conclude with a note of caution
and with an endorsement of a unitary, holistic approach
to validation. In a recent review (Cizek, Rosenberg, &
Koons, 2008), it was found that a majority of articles in
the social sciences that report on the ‘validity’ of tests
rely on none (7%), one (29%), or two (33%) sources of
evidence for ‘validity’. Cronbach’s alpha is the most
commonly reported piece of validity evidence for tests
(reported in 76% of the cases). This practice is not in line
with current recommendations provided by the large
scientific and professional associations in the
psychological and educational fields (e.g., American
Educational Research Association, American
Psychological Association, & National Council on
Measurement in Education, 1999). Furthermore, such
practice is not in line with current scholarly thinking in
the areas of reliability analysis, generalizability theory
(Cronbach, Gleser, Nanda, & Rajaratnam, 1972; see also
Brennan, 2001; Shavelson & Webb, 1991; Shavelson,
Webb, & Rowley, 1989), and holistic perspectives on
validity theory (e.g., Lissitz, 2009; Zumbo, 2007). Rather,
a unitary, holistic perspective on validity emphasizes the
importance (i) of uncovering and understanding multiple
sources of measurement variance, and (ii) of validating
the interpretations, meanings, inferences, and social consequences
that are attributed to or based on measurement scores. In
line with this thinking, we recommend using ordinal
reliability coefficients for binary and Likert-type and
mixed response data as one of several sources of
information on a scale’s reliability and validity.
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Appendix A
Downloading and installing R and loading the R packages pscyh and Rcmdr
R can be downloaded from the R website, at http://www.r-project.org/. The website provides background information
on the R project, manuals, a FAQ page, the open access journal The R Journal, and links to multiple additional resources (e.g., R
search; R conferences; related projects; see also Revelle, 2009a, and 2009b). The software can be downloaded by clicking on
CRAN mirror, in the box Getting started. This opens the site CRAN Mirrors, on which users can choose an URL that is close to
one’s (geographic) location. Clicking on the link for your location will open a page containing a text box entitled Download and
Install R. Here, users may choose the “precompiled binary distributions of the base system and contributed packages” for
Linux, Mac OS X, and Windows. Mac users, after clicking on Mac OS X, can download R under Files, by clicking on R-
2.##.#.pkg (latest version). Please note that you can choose between a 32- and a 64-version, and that this choice depends on the
settings—under Applications
Æ
Utilities
Æ
Terminal—of your computer. Please refer to the frequently asked questions section,
under the hyperlink R for Mac OS X FAQ, at http://cran.stat.sfu.ca/bin/macosx/RMacOSX-FAQ.html. Windows users can
download R, by clicking on Windows, and then on base.
Once R is installed, starting R opens the R menu and console, and the R packages and their dependencies that one needs for
specific calculations—in our case, Rcmdr, psych, and GPArotation, and their dependencies—can be installed by clicking on the
menu option Packages & Data, choosing Package Installer, and then clicking on Get List. In the list, highlight the needed package,
and install it by checking the boxes At System Level (or At User Level) and Install dependencies, and by clicking Install Selected. Once
R packages are installed and loaded, they become part of the R environment. However, every time R is started for a new
session, and one wishes to use one of the packages, one needs to type in the syntax library, and specify the name of the package
in parentheses—e.g., library (psych). For each package, users may open and/or download a pdf-format user manual
(www.personality-project.org/R/psych.manual.pdf for the psych package; and http://cran.r-
project.org/web/packages/Rcmdr/Rcmdr.pdf for the R Commander;), and the R help function allows one to open package-
specific information in the R help window (by entering ??psych or ??Rcmdr into the R console, and then clicking on the
respective package name in the list of help topics).
Please note that R is an open source software program environment that develops quickly. Our syntax was developed and
tested for R 2.14.0 (32 bit version) on a computer with a Mac OS X 10.6.8 and on a computer with a Windows 7 operating
system.
Practical Assessment, Research & Evaluation, Vol 17, No 3 Page 11
Gadermann, Guhn & Zumbo, Estimating Ordinal Reliability
Appendix B
Syntax for calculating ordinal alpha and other ordinal reliability coefficients in R
library(psych) # This activates the R package
8
‘psych’ (Revelle, 2011) for the
current session in R.
data(bfi) # This loads the dataset bfi contained in the R package psych.
attach(bfi) # This attaches the dataset bfi to the current session in R.
bfi5items<-data.frame(N1,N2,N3,N4,N5) # This creates a new dataset, labeled bfi5items, containing only five
(ordinal) variables, N1 to N5, of the original 15-item dataset bfi.
describe(bfi5items) # This describes the dataset bfi5items, providing descriptives, such
as n, mean, sd, min, max, range, skew, and kurtosis.
bfi5items # This displays the object/dataset called bfi5items.
polychoric(bfi5items) # This provides the polychoric correlation matrix for the dataset
bfi5items.
cor(bfi5items, y=NULL,
use="complete.obs", method=c("pearson”))
# This calculates the Pearson correlation matrix for the dataset
bfi5items, only taking into account cases with complete data
(“complete.obs”).
cov(bfi5items, y=NULL,
use="complete.obs", method=c("pearson”))
# This calculates the Pearson method covariance matrix for the
dataset bfi5items, only taking into account cases with complete data
(“complete.obs”).
skew(bfi5items) # This provides the skewness for all items in the bfi5items dataset.
kurtosi(bfi5items) # This provides the kurtosis
9
for all items in the bfi5items dataset.
scree(bfi5items) # This provides the scree plots of the eigenvalues for a factor
analysis and a principal component analysis for the dataset
bfi5items.
examplename<-polychoric(bfi5items) # This saves the polychoric correlation matrix, and
corresponding tau values, under the name examplename. You may
choose any name to save the matrix. (Note: R will not produce
any output for this step.)
alpha(examplename$rho) # This provides (raw and standardized) alpha, and corresponding
item statistics, based on the data set or matrix that is specified in
brackets. (The $rho command specifies that only the correlation
matrix is used for the calculation, disregarding the tau values that
are saved in conjunction with the matrix.) In the output of this
calculation, alpha represents ordinal alpha, because it is based on
the polychoric correlation matrix for the bfi5items dataset saved
under the name examplename. One should obtain the following
results as part of the R output: raw_alpha = .84; std.alpha = .84;
average_r = .51. (Please note that raw alpha and standardized alpha
are the same when they are calculated from a correlation matrix.)
alpha(bfi5items) # This provides raw/Cronbach’s and standardized alpha of the
object specified in brackets. In this case, the object is a data
8
As stated in the previous section, a package in R has to be installed once first, before it can be loaded for a current session in R. Please
refer to Appendix A.
9
Please note that, in R, the command for kurtosis is spelled without the final ‘s’ (i.e.: kurtosi).
Practical Assessment, Research & Evaluation, Vol 17, No 3 Page 12
Gadermann, Guhn & Zumbo, Estimating Ordinal Reliability
matrix (bfi5items), and R therefore calculates raw/Cronbach’s and
standardized alpha, respectively, from the Pearson covariance and
the Pearson correlation matrices of the data set. This step, in
combination with the previous one, will allow one to compare
ordinal alpha with raw/Cronbach’s alpha. One should obtain the
following results as part of the R output: raw_alpha = .81; std.alpha
= .81; average_r = .47.
fa(bfi5items) # This provides the factor loadings (MR1), communalities (h
2
),
and uniquenesses (u
2
) for a 1-factor solution of the bfi5items data
matrix.
fa(examplename$rho) # This provides the factor loadings (MR1), communalities (h
2
),
and uniquenesses (u
2
) for a 1-factor solution of the polychoric
correlation matrix that was saved under the name examplename.
guttman(examplename$rho) # This provides alternative estimates of reliability for the data
matrix that is specified in brackets (i.e., examplename$rho). In the
R output, these estimates are labeled as beta, Guttman bounds
L1, L2, L3 (alpha), L4 (max), L5, L6 (smc), TenBerge bounds
mu0, mu1, mu2, mu3, alpha of the first PC (=principal
component), and the “estimated greatest lower bound based upon
communalities”. Since the specified data matrix is, in this case, a
polychoric correlation matrix, all the reliability estimates represent
ordinal versions. (We note that the guttman syntax command
includes alpha (=L3) as one of the reliability estimates—however,
the alpha syntax command provides additional item
characteristics, such as the item-total correlations, that may be of
interest to the user.)
[Further details and references with regard to the different
reliability coefficients featured in the guttman command can be
found in Revelle, 2011.]
guttman(bfi5items) # Equivalent to the command above, this provides a list of
alternative estimates of reliability for the data matrix specified in
brackets. Since bfi5items is a raw data matrix, the reliability
estimates represent, in this case, Pearson correlation based
reliability estimates.
omega(examplename$rho) # This provides the ordinal versions of the reliability coefficients
omega (hierarchical, asymptotic, and total), because their
calculation is based on the polychoric correlation matrix
‘examplename’.
omega(bfi5items) # This provides omega coefficients for the data matrix bfi5items.
(For details, see Revelle, 2011.)
Bolded font indicates necessary steps, and regular font indicates steps that are optional, but will help to obtain
commonly requested information in the context of calculating ordinal reliability coefficients.
Practical Assessment, Research & Evaluation, Vol 17, No 3 Page 13
Gadermann, Guhn & Zumbo, Estimating Ordinal Reliability
Citation:
Gadermann, Anne M., Guhn, Martin & Bruno D. Zumbo (2012).
Estimating ordinal reliability for Likert-type and
ordinal item response data: A conceptual, empirical, and practical guide.
Practical Assessment, Research & Evaluation,
17(3). Available online: http://pareonline.net/getvn.asp?v=17&n=3
Acknowledgement
We would like to thank Dr. John Fox and Dr. William Revelle, developers of the R software package, for their email
responses, giving advice on calculations and syntax for polychoric correlations in R. Also, Martin Guhn gratefully
acknowledges the financial support provided to him in the form of a postdoctoral research fellowship from the Michael
Smith Foundation for Health Research, British Columbia. Bruno Zumbo wishes to acknowledge support from the Social
Sciences and Humanities Research Council of Canada (SSHRC) and the Canadian Institutes of Health Research (CIHR)
during the preparation of this work.
Authors:
Anne M. Gadermann
Harvard Medical School
180 Longwood Avenue
Boston, MA 02115
AnneGadermann [at] googlemail.com
Martin Guhn
The Human Early Learning Partnership
University of British Columbia
Suite 440, 2206 East Mall
Vancouver, BC, V6T 1Z3, Canada
Martin.Guhn [at] ubc.ca
Bruno D. Zumbo, Corresponding Author
University of British Columbia
Scarfe Building, 2125 Main Mall
Vancouver, B.C. CANADA V6T 1Z
Bruno.Zumbo [at] ubc.ca
