Introduction to Structural Equation Modelling Using SPSS and AMOS

279 Pages·2008·16.15 MB·English

by  Niels J. Blunch

Introduction to Structural Equation
Modelling Using SPSS and AMOS
1 Introduction
Contributors: Niels J. Blunch
Print Pub. Date: 2008
Online Pub. Date:
Print ISBN: 9781412945578
Online ISBN: 9781446249345
DOI: 10.4135/9781446249345
Print pages: 3-27
This PDF has been generated from SAGE Research Methods. Please note that the
pagination of the online version will vary from the pagination of the print book.
Introduction to Structural Equation
Modelling Using SPSS and AMOS
2 Classical Test Theory
Contributors: Niels J. Blunch
Print Pub. Date: 2008
Online Pub. Date:
Print ISBN: 9781412945578
Online ISBN: 9781446249345
DOI: 10.4135/9781446249345
Print pages: 27-47
This PDF has been generated from SAGE Research Methods. Please note that the
pagination of the online version will vary from the pagination of the print book.
University of Queensland
Copyright ©2013 SAGE Research Methods
10.4135/9781446249345.n2
[p. 27 ↓ ]
2 Classical Test Theory
The concepts in your model will usually be rather diffuse (attitude, skill, preference,
democracy)—i.e. concepts for which no generally agreed measuring instruments exist.
In such situations, therefore, you have to make your own measuring instruments—be
they questions in a questionnaire or some sort of test.
The first requirement for such an instrument is that if you repeat the measurement
under identical conditions, then you will end up with nearly the same result: Your
instrument must be reliable.
Another requirement—which seems just as obvious—is that the instrument shall
measure exactly what it is intended to measure and nothing else: The instrument must
be valid.
Reliability and validity are the two main standards on which we evaluate a measuring
instrument.
I will take as my starting point the classical version of the two concepts and show
how the use of SEM can illustrate the weaknesses of the classical ways of measuring
reliability and validity and perhaps lead to more useful ways of judging these two central
concepts.
In Example 2 in the previous chapter you met examples of summated scales. You will
also learn how to construct such scales and how special computer programs can be
used to that end.
Page 2 of 29 Introduction to Structural Equation Modelling Using
SPSS and AMOS: 2 Classical Test Theory
University of Queensland
Copyright ©2013 SAGE Research Methods
1. Reliability
We evaluate a measuring instrument by its reliability. The reliability of an instrument
is its ability to give nearly identical results in repeated measurements under identical
conditions; in other words reliability is about reproducibility.
Let me take the simple measurement model in Figure 2.1 as our point of departure.
[p. 28 ↓ ]
Figure 2.1 A simple measurement model
I can now write
with the conditions
The expectation of (1a) is
and the variance
While you are able only to observe variation in X, it is of course the latent variable F and
its variance—often referred to as the true variance—that is the center of our interest. I
will now define the reliability coefficient #
XX
Page 3 of 29 Introduction to Structural Equation Modelling Using
SPSS and AMOS: 2 Classical Test Theory
University of Queensland
Copyright ©2013 SAGE Research Methods
as the proportion of measured variance that can be traced back to F:
It is obvious that (1) is an analog to the simple regression model, and that #
XX
is the squared correlation coefficient in this model. Unlike the traditional regression
model, however, the independent variable F is un-observable.
When—as in this case—a latent variable has only one indicator, # is only a scale factor
the value of which is arbitrary. Usually # is then taken to have the value 1.00, in which
case (3a) can be written:
Variances, Covariances and Reliability
Analysis of quantitative empirical data is basically analysis of variation and co-variation.
By how much do the attributes vary among observations, and to what extent do they co-
vary?
Suppose you are interested in measuring the structural correlation #
F
1
F
2
between the two latent variables F
1
Page 4 of 29 Introduction to Structural Equation Modelling Using
SPSS and AMOS: 2 Classical Test Theory
University of Queensland
Copyright ©2013 SAGE Research Methods
and F
2
in Figure 2.2. However, being able to only observe [p. 29 ↓ ] the empirical correlation r
XY
between the manifest variables X and Y, you will underestimate the structural co-
variation.
Figure 2.2 Correlation and reliability
As you can read from the figure
From which you get
As the denominator is generally smaller than one you underestimate the structural
correlation, the size of underestimation depending on the reliability of the two
measurements. It is therefore quite possible to overlook an existing correlation because
of unreliable measuring instruments.
In order to develop satisfactory measuring instruments we must find a way to estimate
the reliability of such instruments. This, however, necessitates that we have at least two
measurements, as shown in Figure 2.3.
From (3a) the reliability coefficients of the two measurements X
Page 5 of 29 Introduction to Structural Equation Modelling Using
SPSS and AMOS: 2 Classical Test Theory
University of Queensland
Copyright ©2013 SAGE Research Methods
1
and X
2
are
, respectively. However, as F is non-observable you cannot use (3a) for computation.
From Figure 2.3 it is evident that X
1
and X
2
—both being influenced by F—must correlate, and the more they correlate with F, the
stronger their mutual correlation. It should therefore be possible to judge the reliabilities
of the two measurements based on the empirical correlation between X
1
and X
2
.
We can now write:
Figure 2.3 A measurement model with two indicators
Page 6 of 29 Introduction to Structural Equation Modelling Using
SPSS and AMOS: 2 Classical Test Theory
University of Queensland
Copyright ©2013 SAGE Research Methods
[p. 30 ↓ ] with the usual regression assumptions:
to which we add one further assumption:
This leaves us with four possibilities:
If the measurements are strictly parallel or tau-equivalent, the values of # are arbitrary
and are usually set to 1.00—a tradition I will follow.
In classical test theory the latent variables F are not formally taken into account, which
means that the coefficients # are not known.
Page 7 of 29 Introduction to Structural Equation Modelling Using
SPSS and AMOS: 2 Classical Test Theory
University of Queensland
Copyright ©2013 SAGE Research Methods
Consequently, classical theory builds on strictly parallel (and to a lesser extent tau-
equivalent) measurements, where values of # can be set to 1.00. Therefore, unless
otherwise stated, parallel in the following means strictly parallel.
Parallel Tests
Setting # = 1, it follows from (6) and (7) that if X
1
and X
2
are parallel measurements, then
It should be possible to use parallel measurements to evaluate the reliability. I will
therefore calculate their correlation:
[p. 31 ↓ ] or—read from right to left:
In other words: Parallel measurements have the same reliability, which is equal to their
correlation.
Page 8 of 29 Introduction to Structural Equation Modelling Using
SPSS and AMOS: 2 Classical Test Theory
University of Queensland
Copyright ©2013 SAGE Research Methods
If you are able to create parallel measurements, you can then estimate their reliability.
In the following paragraphs, I will show three different ways of doing so.
‘Test-Retest’
The obvious way to create parallel measurements is to repeat the measurement using
the same instrument on the same respondents with a shorter or longer interval, and
then estimate the reliability as the correlation coefficient between the two. The problem
with this method is that it is difficult to say whether we are measuring reliability or
changes in the latent variable: If the value of the latent variable has changed between
the two points in time the correlation could be small, even if the measurements are
reliable.
Besides, the correlation could be large if the respondent could remember her answer to
the first measurement, when the second is taken.
‘Alternative Form Method’
In order to rule out the possibility that the respondents at the second measurement
remember their answer to the first—which will contradict assumption (6c)—it is
necessary to construct another measuring instrument, which is parallel to the first, and
use that in the second round.
If e.g. you intend to measure the skills of children in arithmetic by administering a
test comprising various arithmetical problems, you can let them have another test
with different problems at a later time. Of course the two tests have to be of the same
difficulty in order to obtain parallelism. One way to secure the same difficulty in both
tests is to split the pool of problems between the two tests by randomization.
If you are interested in measuring attitudes or other more diffuse concepts it is difficult
enough to construct one measuring instrument—and to construct two parallel ones can
be a formidable task.
Page 9 of 29 Introduction to Structural Equation Modelling Using
SPSS and AMOS: 2 Classical Test Theory