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Language: en
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Factor analysis results can
be used for calculation of
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reliability indices similar to coefficient alpha.
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The most commonly used one is the
composite reliability index also
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known as coefficient Omega and the
average variance extracted or AVE.
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The AVE is not that useful of an index
and it has some problems but it's
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useful to understand what it quantifies
because it's still fairly commonly used.
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Let's take an example. The Yli-Renko's
article has both composite reliability
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here and they explain that composite
reliability is similar to coefficient
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alpha and they interpret these
results as reliability indices.
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So the reliability index here quantifies that
if we take a sum of the indicators then the
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reliability of the sum is 0.7 and this sum is
0.75. So it's interpreted the exact same way
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as coefficient alpha. Then they have the
average variances extracted statistics.
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This is often interpreted as the average
reliability of an individual indicator
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in the scale. So it doesn't really
measure the reliability of the scale
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as such but instead it measures the average
reliability of each individual indicator.
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So the scale reliability increases the
reliability of the some of the indicators
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increases as a function of the number of
the indicators and composite reliability
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they extend that into account where
average variants extract it does not.
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And then they claimed that it indicates internal
consistency which these indices do not. So they
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measured reliability assuming internal consistency
but they do not test internal consistency.
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The assumptions are basically
the same as with alpha mean.
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The composite reliability index and average
variance extracted index are present with
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this kind of equations. So the composite
reliability is basically the variance due
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to the factor divided by the total variance of
the composite. So the total variance of the sum
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divided by the variation due to the factor.
And then the average variance extracted is
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approximately the average of the estimated
indicator reliability of the scale. So it's
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an approximately - it's not exactly because
there's some nuances there how it's calculated.
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Let's do an example. So let's check
what kind of reliability indices we
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get from our exploratory factor analysis results.
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So we have he first indicator - first
factor here. We can use the equation
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from the previous slide to calculate indicator
reliabilities. The reliability is the square
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of factor loading. That's an estimate of
reliability of an indicator. We divide by 3
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because we have three indicators and the average
variance extracted for this factor would be 81%.
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Then composite reliability -which is
more useful - is calculator with this
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kind of equation. So we calculate how
much variance the factor explains what
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is the total variance of the data
of the composite and it's a 93%. So
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we can do this calculations pretty easily
by hand based on factor analysis results.
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So of these the composite reliability is useful
because it tells us something that is not obvious
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from the factor analysis results. So it tells
us what is the reliability of the sum. It
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takes into account the individual indicator
reliabilities and the number of indicators.
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Doing this kind of calculation by looking at
the factor loadings would be a bit difficult.
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The average variance extracted on the
other hand doesn't really tell us much
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beyond the factor analysis results. If we
want to know what are the reliabilities of
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each indicator - is there bad indicator
somewhere here we are much better off by
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looking at the factor loadings than looking
at the other variance extracting statistics.
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So I recommend that you apply the
composite reliability and it's nowadays
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more recommended than alpha because it has
less assumptions but I don't think that the
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average variance extract statistics is
that useful even if it's a widely used.
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So what's the difference between composite
reliability and coefficient alpha? The idea
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of an alpha is that it tells us what is the
reliability of the sum of the scale items.
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The composite reliability tells the exact same
information but instead of taking the indicator
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correlations it uses factor analysis
results as input for its calculation.
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The difference is that you make less assumptions
by doing so. So the composite reliability index
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is also called conterigent index because it
doesn't make the Tau equivalence assumption
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that every indicator is equally reliable.
So it allows the indicators to vary in
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how reliable they are and it is therefore
more general index than chronbach's alpha.
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You will need to do a factor analysis anyway
to asses uni-dimensionality to use alpha and
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therefore you have the input for the composite
reliability index regardless. So using the
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composite reliability as a matter of routine is
a better idea than then calculating the alpha
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which makes the more constraining assumption
that all indicators are equally reliable.
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So there is a -there's a counter-argument to
what I just explained. So this argument for
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the superiority of composite reliability over
a coefficient alpha has been made in a number
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of papers for example Cho and Macintosh
make that kind of argument but there's
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also the counter-argument that Peterson
study has found that alpha and composite
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reliability tend to be taken about the
same value. So they went through number
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of studies that reported both of these
two indices and concluded that they give
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you almost the same results. So why bother
choosing because it doesn't really matter.
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Well that can be challenged by stating that if
the composite reliability and chronbach's alpha
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provide different values that is evidence against
the assumptions of alpha and because composite
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reliability is more general and then in that kind
of scenarios alpha probably wouldn't be reported.
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So it's possible that these results
where Petersons are just a selection
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effect. You are better off at using
composite reliability because under
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the alphas assumption it produces the same
result if alphas assumptions don't hold then
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composite reliability can still be applied
and it provides you a less biased estimates.
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One key weakness in composite reliability
is that and alpha is a that they both rely
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on uni-dimensionality assumption. If your
scale has multiple dimensions - for example
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there are cross loadings or you cannot justify
the unit-dimensionaly assumption for some
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other reason - then there are other indices
that are better than composite reliability.
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Also the fact that you calculate composite
reliability doesn't provide you any evidence
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of dimensionality that you have to
check from your factor analysis.