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Coefficient alpha sometimes called cronbach's
alpha or tau-equivalent reliability is one of
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the most commonly used reliability
indices in social science research.
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It may not be the best index for
every scenario but it is commonly
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used and therefore it is important
to understand what it quantifies
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and under which assumptions it
is a good reliability index.
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Coefficient alpha is sometimes referred to as
internal consistency reliability. What that means
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is that is based on or its calculated based on
how consistent the indicators are interrelated. So
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basically it means how highly the indicators are
correlated. And this index is calculated from the
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correlation matrix of the indicators or covariance
matrix depending on which equation you apply.
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Coefficient alpha quantifies what is the
reliability of a scale score calculated as
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a sum of the scale items. So if you
have a scale of five items or five
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measures that are supposed to measure
the same thing and then you take a mean
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or a sum of those five items then alpha
quantifies the reliability of that sum.
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The part about internal consistency
is very important because while alpha
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is an internal consistency measure in
terms that it quantifies how strongly
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the indicators are correlated it doesn't
test where the internal consistency holds.
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So here's an example of six indicators.
We have indicators x1 x2 and x3 that are
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designed to measure one thing. So they're
highly correlated because they measure the
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same thing. x4 x5 and x6 are designed
to measure something else and they are
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highly correlated with one other because
they measure the same thing. But these
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two groups of indicators x1 x2 x3 and x4
x5 x6 are uncorrelated with one another
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because they are designed to measure two
distinct things that are not correlated.
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So this set of six indicators measures
two different dimensions or two different
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things. Yet if we calculate coefficient
alpha using this correlation matrix we
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get the value of 0.7. So the fact that we
got an alpha value that is acceptable for
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some researchers doesn't guarantee that the scale
is a uni-dimensional internally consistent one.
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The important thing here is that
internal consistency is something
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that alpha assumes and it is based on
that assumption. High alpha doesn't
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guarantee that you have an internally
consistent uni-dimensional scale.
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So alpha is a reliability index and it relies
on the classical test theory assumptions.
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Basically it means that your scale that
goes into the Alpha the indicators must
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be uni-dimensional and there is a no other error
there except unreliability which is random noise.
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Then it doesn't provide any test of the
classical test theory assumptions. So
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before you apply alpha at least you have to
check uni-dimensionality and you also can
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check the tau covariance which basically means
whether the indicators are equally reliable.
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The classical test theory states that the measures
score X is a sum of the true score T plus the
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measurement error E. In greek letters the T is tau
and therefore it's tau equivalent. Every indicator
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has the same amount of true score various in
it. That's what the tau equivalence means.
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While alpha is commonly used there are also
common misconceptions that you seen in research.
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The first misconception is that it was developed
by Cronbach. That's not the case. It was developed
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a couple of decades before Cronbach widely sited
paper. It just happens to be the first index in
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his paper where he discussed many different
reliability indices and therefore he got
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the coefficient alpha and it got his name. But
Cronbach himself says that this index probably
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shouldn't carry his name. So that's why we call
it coefficient alpha instead of cronbach's alpha.
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Also alpha does not necessarily equal reliability.
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It is an estimate of reliability under
the assumptions of classical test theory.
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If those assumptions don't hold then
alpha can underestimate reliability.
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Your statistical software when you
calculate alpha also gives you an
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estimate of how much the alpha would
be if you omit one of the indicators.
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For example it would say that if
you have five indicators going
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to the Alpha dropping one of those
indicators could increase the alpha.
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Should you drop the indicator? The answer
to that question is not necessarily because
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dropping an indicator from the... while it
increases the alpha value it can also mean that
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you're just capitalizing on chance factors.
So the actual reliability doesn't increase.
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Remember that alpha is not equal to
reliability. It is an estimate of reliability.
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And we could just have an alpha value that is
slightly overestimated because of random factors.
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Then there are misconception about
cut-offs. So if alpha equals zero
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point seven you're okay if it's zero point
sixteen then your study is unpublishable.
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So that of course doesn't hold true. What
the reliability - what kind of reliability
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you require depends on the context or if you are
measuring something that no one has ever measured
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before then it's perhaps more acceptable
to have an alpha of maybe even less than
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0.7 if you're studying something that others
have studied and have used scales with the
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reliability of 0.85 then 0.7 is not gonna cut
it because there are better scales available.
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Also it is not about yes or no decision. You have
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to explain what the reliability of 70%
or 80% means for your study results.
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So what kind of systematic error
do you expect when your measures
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are unreliable. So it's not about the cutoff.
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And then the final misconception is that alpha
is the best reliability coefficient. People may
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think that it's the best because it's widely
used but sometimes we use a statistic simply
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because they have been used in the past and
if we just used something that's been used in
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the past - it means that we use the oldest thing.
And there are many other reliability indices that
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have been introduced after coefficient alpha
that are more modern and better than alpha.
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Here's a list of some from a paper by
McNeish from psychological methods. He
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starts with alpha and then he goes
and explains omega which is relaxes
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some of the assumptions then he goes on and
explains others that relax the assumptions.
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The important assumptions in alpha were
that the indicators are uni-dimensional. If
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you relax that assumption then you can go with
some of these hierarchical omega coefficients.
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Also another important assumption is that
indicators are both equally reliable. If you relax
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that assumption you can go with Omega coefficient
which is also known as composite reliability.
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Cho's paper presents this nice decision diagram
of which reliability index to choose from. So he
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starts by checking is the scale uni-dimensional
and if the scale is uni-dimensional and he
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measures one thing then you check are the
indicators about equally reliable - do
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they have the same true score. If yes then
you're going to be OK with coefficient alpha.
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If no the second step then you go with
coefficient omega or composite reliability index.
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If on the other hand you don't have
uni-dimensionality then you apply a
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factor model in which case you are do higher omega
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or on a variant of alpha that
doesn't use the factor model.
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So it's a not that you would always use alpha
but instead you have to make a decision based
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on the nature of your data and you have to
- if you use alpha you have to justify the
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uni-dimensionality assumption which
you do with the factor analysis. And
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then you have to justify the Tau equivalence
assumption which basically means that all the
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indicators are equal reliable. Which
you also do with the factor analysis.
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And if those assumptions don't hold
then alpha is not the ideal but you
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have to look at these other coefficients instead.