Understanding the Latent Variable

Constructs versus Measures
 Typically researchers are concerned with constructs, rather than items or scales.
However, responses to items on a well-defined scale are typically the best way to
assess variability in an underlying construct.
 Measurement scales are proxies of variables that we cannot directly observe. By
assessing the relationship between measures we infer, indirectly, the relationships
between constructs.

Construct Maps
 A construct map can help you to conceptualize your construct. It is a visual
representation that depicts an ordering of respondents, as well as an ordering of
item responses. The following figure represents a generic construct map:
Increasing levels of the Construct “X”
Respondents
Respondents with High “X”
Item Responses
Item response indicates highest level of “X”
Item response indicates higher level of “X”
Respondents with Mid-range “X”
Item response indicates lower level of “X”
Respondents with Low “X”
Item response indicates lowest level of “X”
Decreasing Levels of the Construct “X”
 Creating a concept map at the initial stage of scale development helps the
“measurer” to focus on the essential features of what is to be measured.
 It illustrates both the way in which an individual shows more or less of the
construct. This may be expressed as “higher to lower”, “agree to disagree”,
“weaker to stronger” or “more often to less often”. The particular wording
depends on the context.
 The important idea is that there is a qualitative ordering of the levels inherent in
the construct and underlying that there is a continuum that goes from more to less.
 A good way to approach the development of a construct map is to:
1. Think of the extremes of the continuum, such as “expert” to “novice” to
“expert” or “loves” to “hates”, in the context of an attitude towards
something.
2. Develop descriptions of the extremes to make them more concrete.
3. Develop intermediate stages or levels between the two extremes.
 When creating a construct map it is important to be clear about whether the
construct is defined in terms of the respondents or the item responses. Although
both are eventually needed, in a specific context it often makes sense to start with
one rather than the other. For example, if there is a developmental theory of how
individuals increase on the construct or how people arrange themselves between
the extremes then the respondent side should be developed first. On the other
hand, if the construct is mainly defined by a set of item responses, then the items
should be ordered first.

Latent Variables
 The underlying construct that a scale is intended to reflect is often called the
latent variable. Latent refers to the fact that it is not directly observable and
variable refers to the fact that it is not constant, that some aspects of the construct
(i.e. strength, magnitude) vary with regard to time, place, people, etc.
 Although we cannot observe the latent variable directly, we presume it takes on a
specific value under specific conditions.
 A measurement scale designed to measure a latent variable is intended to measure
the actual magnitude at the time and place of measurement for each person
measured, known as a person's true score.

Latent Variables as the Cause of Item Values
 The latent variable is considered to be the cause of item scores. In other words,
the value of a person's true score causes an item to take on a particular value.
Each item on a scale should give an indication of the strength of the latent
variable.
 This causal relationship implies certain empirical relationships. A set of items
that are designed such that scores are caused by the same latent should be related
to each other.

Path Diagrams
 Path diagrams are a visual method for depicting causal relationships among
variables.
 This causal relationship implies certain empirical relationships. A set of items
that are designed such that scores are caused by the same latent should be related
to each other.
 A straight arrow drawn from one variable label to another indicates that the two
variables are causally related and the direction of causality is indicated by the
arrow, such that the arrow is pointing to the variable that is the cause of the other
variable.
Benefits
Colleagues
Administration
Job Satisfaction
Error (e)
…
Opportunities
 The absence of an arrow indicates that two variables are unrelated. Furthermore,
a path diagram typically depicts error, or the residual, which represents variation
not accounted for by other causes that are depicted in the model. In the case of
the above example, the residual error represents variation in job satisfaction that is
not captured by the particular subscale included in the measure.
 Path diagrams can help us to see how scale items are related to a latent variable
and to understand how certain relationships among items imply certain
relationships among items and the latent variable.
 Typically paths in a path diagram include numbers that represent standardized
path coefficients. These numbers represent the strength of the causal relationship
between the variables joined by arrows, as in the following example.
 When path diagrams only have one common origin, job satisfaction in this case,
the correlation between any of the variables that are caused by the underlying
construct is equal to the product of the path coefficients. Therefore, the
correlation between benefits and opportunities is equivalent to (0.4) (0.5) or 0.2.
 This known relationship allows us to go "backwards" and estimate the path
coefficients using the correlation among variables. Consider the following path
diagram that corresponds to a hypothetical scale.
Depression
Item 1
e1
Item 2
e2
Item 3
e3
Item 4
e4
Item 5
e5
In this case, it is assumed that each item score is caused by the latent trait of
depression and that the error associated with each item score is unique. If we had
data which represented actual responses to these items then we could easily
calculate the correlation between item scores. Given that information, we could
then estimate the path coefficients. For example, if item 1 and 2 were correlated
at 0.36 then we know that the product of the path coefficients leading from
depression to item 1 and item 2 would have to be 0.36. If we assume that the two
path coefficients are equivalent, then they would both have to be 0.6.

Assumptions
1. The residual error associated with individual items varies randomly and has a
mean of zero when aggregated among a large number of people. Thus, the mean
of an item, obtained from a large number of respondents, tends to be unaffected
by error.
2. Residual errors are NOT correlated with another. Thus, the only routes in a path
diagram which links items are those that pass through the latent variable.
3. Residual errors are NOT correlated with the true score of the latent variable. Note
that the first path diagram illustrated was NOT related to a measurement scale, per
se, but rather to the overall relationship among variables.

Parallel “Tests” or Items : The Strictest Model
 The rationale underlying the model of parallel “tests” or items is that each item of
a scale is as good of a measure of the latent trait as any other item on the scale.
Furthermore, this model assumes that the amount of error present in each item is
also identical. In the path diagram illustrating the five-item measure of
depression this is equivalent to saying e1 = e2 = e3 = e4 = e5. In this case, items
are said to be strictly parallel.
 These additional assumptions imply that the correlation between each item and
the true score is identical. This can only happen if the variances for each item are
identical.
 This model makes the strong assumption that the amount of influence from the
latent variable is the same for each item and the amount from other sources (i.e.
residual error) is also the same for each item. This in turn implies that the
correlations among all items are identical. It further implies that all of the item
level means and variances will be equivalent.
 Although this model allows us to reach many interesting conclusions, it is
obviously a somewhat stringent model with assumptions that typically will not be
met with real data.

Alternative Models
 Fortunately, not all the above assumptions need be met in order to make
inferences about the relationship of true scores and observed scores.
 Essentially tau-equivalent models relax the assumption of equal error variance
between items, which allows the standardized values of the paths from a latent
variable to items to be non-equivalent, although it is still assumed that the
unstandardized values from a latent variable to items are identical. Conceptually,
this implies that items are still parallel with respect to how much they are
influenced by the latent variable; however the residual errors need not be
identical. Because errors may vary among items, so may means and variances.
This model allows us to reach many of the same conclusions as we obtain
assuming the strictly parallel model.
 Congeneric models relax the assumptions even more, but allowing each item to be
influenced by the latent variable to a different degree. In other words, this model
merely assumes that all item scores are caused by a common latent variable (i.e.
each item reflects the true score to some degree). Neither the standardized nor
unstandardized paths from the latent variable to the items have to be equivalent.
Nor do the paths from the errors to the items need to be the same. However, the
more strongly each item correlates with the true score, the more reliable the scale
will be.
 General factor models are even less stringent. These models allow multiple latent
variables to underlie a given set of items. Structural equation modeling often
incorporates a factor analytic model into their measurement model.
 The congeneric model is a special case of the factor model, in which only one
factor underlies responses. Likewise, a tau-equivalent model is a special case of a
congeneric model, for which the relationships between items and the latent trait
are assumed to be equal and the strictly parallel model is a special case of an
essentially tau-equivalent model.
 Item response theory is another measurement approach that can be used. This
approach assumes that each individual item has its own characteristic sensitivity
to the latent variable.