LST
Running head: LST
Latent State Trait Theory: An application in sport psychology
Ziegler, M., Ehrlenspiel, F. & Brand, R. (2009). Latent state-trait theory: An application in sport psychology.
Psychology of Sport and Exercise, 10, 344-349.
doi: doi:10.1016/j.psychsport.2008.12.004
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Abstract
Questionnaires are often applied in sports psychology to measure a person’s trait or state.
However, often it is unclear in how far the questionnaire captures differences because of trait or
state influences. The Latent State Trait theory (LST) offers the opportunity to distinguish both
sources. This allows to compute specific relibility values. The present paper gives an
introduction to LST and depicts its basic ideas. Using a real data set with N = 156 athletes
answering a comprehensive inventory - assessing competitive anxiety we go on and examplify
the processes necessary to derive the LST scores. The results confirm the questionnaire’s trait
satuation. Finally, results are discussed in light of practical and theoretical implications.
Keywords: Reliability; Consistency; Specificity; Test theory; Latent Trait State theory; State;
Trait
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Latent State Trait Theory: An Application in Sport Psychology
Psychologists use tests and questionnaires to measure individual differences. Such
individual differences are often perceived as . Traits can be seen as “Enduring characteristics on
which individuals differ.” (Liebert & Liebert, 1998, p. 184). Based on these measurements, they
hope to predict future behavior or performance. Yet, sometimes it is more interesting to measure
states and not traits. States can be regarded as temporary conditions (Liebert & Liebert, 1998, p.
185). Moreover, empirical results have not always confirmed a strong relationship between
measured traits and actual behavior in specific situations. This observation sparked the personsituation debate which has been going on for quite a while now (Bowers, 1973; Epstein, 1997;
Epstein & Obrien, 1985). Deinzer et al. (1995) concluded: “… we always measure persons in
situations, not persons; there is no psychological measurement in the situational vacuum. (p. 7)”
This statement, along with the need to measure states in certain assessment settings, highlights
the importance of situational circumstances in the measurement of individual differences. The
special role of situational circumstances and the importance of states is nothing new in sports
psychology either. Especially here, it is important to know how much variance can be explained
by situational circumstances. Obviously it makes a difference whether an athlete is close to an
important competition or still a long time away from it. On the other hand, sports psychologists
need instruments which are able to assess states and others which measure traits. For example, if
we want to select young athletes based on certain psychological characteristics, we want to
assess individual differences that are consistent across situations and times, and that are not due
to a specific situation. Yet, when treating problems associated with choking under pressure, such
states might be of greater interest.
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Nowadays, it is possible to quantify the amount of variance due to a trait, the situation or
state, their interaction as well as measurement error. Steyer and colleagues introduced the latent
state-trait theory (LST) which provides the theoretical background and statistical techniques
needed (Steyer, Ferring, & Schmitt, 1992; Steyer, Schmitt, & Eid, 1999). The present paper aims
at introducing the basics of LST theory and provides a practical example for conducting the
analyses. Based on this, it will be possible to quantify the amounts of state and trait variance in
assessment tools. Furthermore, research on the functional differences between traits and states
can be facilitated.
Latent state-trait theory: An introduction
The basic mathematical problem which had to be solved was how to separate an item’s
variance into variance due to trait, situation (or state), interaction between trait and situation, and
measurement error. Defining a situation and its properties is a difficult task. Measuring specific
situation attributes is even more challenging. LST theory provides one possible solution for the
problem. Using structural equation modeling, LST theory provides a framework in which it is
possible to distinguish between the different variance sources without defining or measuring
situational aspects. In LST, specific coefficients can be computed containing information on
consistency, occasion specificity, and reliability of an instrument in a given sample and situation.
In the following paragraphs we will outline the basic ideas and concepts before providing a
practical example.
Basic idea
LST theory is based on the idea that a person’s score in any given test is at least partially
determined by situational circumstances. As mentioned above, it makes a difference whether an
athlete is actually in the middle of a competition or still a few weeks away from one. Let us
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assume we want to measure the concerns an athlete generally has regarding failure in a
competition using four items with a 4-point rating scale. If we asked the athlete six weeks before
the year’s most important competition, he/she might still be quite relaxed and involved in
training. The scores on the four items might be 1, 2, 2, and 1. However, if we asked the same
four questions on the day of the competition, the athlete may currently be worried and the scores
might then be 2, 2, 3, and 4. Based on classical test theory, we would assume that the scores at
both times are a compound of the athlete’s true score (true concerns as a trait) and measurement
error. The basic formula is Xpi = Tpi + Epi. The score X of a person p in a test i is a combination
of the person’s true score, Tpi, and measurement error, Epi. LST theory extends this idea. It
assumes that the scores are results of random experiments. The athlete was drawn from a
population of athletes and the measurement times were drawn from a population of t possible
situations. Obviously, the scores change between the measurement times because of the changed
situation. The athlete’s trait cannot be responsible for this change. Neither should the
measurement error be able to fully explain the change. However, the specific situational
circumstances influencing the athlete’s state can be used as an explanation. It is therefore
necessary to extend the basic formula and add a situation-specific index t: Xpit = Tpit + Epit. The
true score variable Tpit is considered as a Latent State Variable in LST theory. However, this is
just a technical term and does not indicate that we are actually measuring a state. Rather than
that, it should be understood as a latent trait biased with situation and the interaction between
situation and trait (Deinzer et al., 1995, p. 5). This definition indicates that the latent state
variable Tpit can be understood as a combination of individual differences in the latent trait ξpit
and variance due to the situation and the situation by trait interaction, called the latent state
residual ζpit. This is the amount of variance in the latent state variable which is not explained by
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differences in the latent trait. Hence, it must contain systematic differences due to the situation
(state) and the situation by trait interaction. All measurement error influences are comprised in
the error term Epit. The basic formula for a test score now reads: Xpit = ξpit + ζpit + Epit. Based on
this reasoning, it is possible to separate the different variance sources. In order to determine the
latent state residual variance, latent trait variance, and measurement error variance, at least two
measurements are necessary. Figure 1 illustrates the variance decomposition. The observed
variable Xit can be divided into latent state variance Tit and error variance Eit. The latent state
variance can then be separated into latent trait ξit and latent state residual ζit variance.
Based on this variance decomposition, the LST coefficients can now be introduced. The
equations are:
Reliability can be operationalized as the amount of true score variance in the measured
variance (Novick, 1966). In this sense, true score variance only includes systematic differences
between persons. All unsystematic differences are defined as error. Thus, the reliability (REL) of
the test score is the amount of observed total variance as explained by latent trait and latent state
residual variance. The total variance represents the observed variance. Systematic true score
differences can be found in the latent trait variable as well as the latent state residual. These
variance sources contain all the systematic variance, either due to trait, situation or their
interaction.
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The consistency (CON) of the score equals the amount of total observed variance as
explained by the latent trait. In other words, the consistency quantifies the amount of trait
specific variance and thus, variance which is present independent of situational circumstances.
The larger the consistency is, the higher the trait impact.
Finally, occasion specificity represents the amount of total observed variance due to the
latent state residual and, hence, due to the situation and the situation by trait interaction. In other
words, occasion specificity shows how much an instrument captures situation-specific variance
in a given sample and situation. From these considerations, it can be derived that the reliability is
the sum of consistency and specificity.
However, with just one single time of measurement, it would not be possible to split the
latent state variance. The problem is solved by using at least two times of measurement. Now it
is possible to define a Multi-State-Single-Trait model. In this model it is assumed that one and
the same trait is measured in more than one situation. In our example, we measure the athlete’s
concerns twice with a gap of six weeks in between. The model looks like that depicted in Figure
2.
As can be seen from the model, we have four items, X1 to X4, each of which was asked at
two measurement points, as indicated by the second index. Because we are measuring the same
trait at both times, the latent state variables only need an index indicating the measurement point.
The same holds true for the latent state residuals. The latent trait variable now explains
systematic differences in the latent state variables at each time caused by individual trait
differences. Consequently, it explains trait variance which is inherent in both situations. The
latent state residuals now contain variance, which cannot be explained exclusively by the trait
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acting at both times. This again can only be variance due to the situation and the situation by trait
interaction.
It can also be seen that several assumptions were made. First of all, it is assumed that the
latent state variable explains each item equally well. The same assumption is made for the
relationships between latent trait and latent state variables. These assumptions are necessary to
compute reliability, consistency, and occasion specificity.
The assumptions also mean that the items are supposed to be parallel measures of the
same trait. Equal assumptions are made for other reliability estimates as well and are therefore
not an LST-specific aspect. Significantly, however, these assumptions are tested in LST. If the
items do not fulfill the requirements, a bad fit will be obtained in the structural equation analyses.
Thus, the LST framework actually offers a further advantage compared with other reliability
estimation techniques where the assumptions are usually not tested.
Using this model it is not only possible to determine the LST coefficients, it is also
possible to test whether there actually is state variance or method variance. Method variance
refers to systematic influences due to the measurement technique used (Podsakoff, MacKenzie,
Lee, & Podsakoff, 2003). In our example, method variance would be present because the same
items were used twice.
We will now run you through an analysis using the ideas presented here and showing you
some of the possibilities this technique offers.
Latent state-trait theory: A practical example
Our exemplary analyses are based on a data set collected as part of the development of a
German comprehensive inventory - assessing competitive anxiety, the WAI1 (Brand, Graf, &
Ehrlenspiel, 2008). It follows an interactional perspective, distinguishing state from trait
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competitive anxiety (Spielberger, 1966). Consequently, the inventory consists of two
questionnaires, the WAI-Trait and the WAI-State. The inventory also takes a multi-dimensional
view (Liebert & Morris, 1967; Martens, Burton, Vealey, Bump, & Smith, 1990), as it
distinguishes between somatic anxiety (“emotionality”) and cognitive anxiety (“worry”).
Somatic anxiety reflects the perception about the physiological changes occurring, such as a
pounding heart. The cognitive dimension reflects the worry cognitions and negative expectations
of success and subsequent evaluation. The WAI-Trait is loosely based on the Sport Anxiety
Scale (SAS) by Smith, Smoll, and Schutz (1990) and its revised versions (e.g., Dunn, Dunn,
Wilson, & Syrotuik, 2000), whereas the WAI-State can be seen to be in line with the
Competitive State Anxiety Inventory – 2 (Martens et al., 1990) and its revisions (Cox, Martens,
& Russell, 2003). The data used in this example were originally collected in a study in which the
stability of the constructs and sensitivity of the two instruments were assessed simultaneously
(see Ehrlenspiel, Brand, & Graf, submitted, for details), while only data for the WAI-T will be
reported in this example. After briefly describing methods of the original study, we will describe
the different steps in the LST analyses and explain the results.
Methods
Sample and procedure. N=156 competitive athletes (49 female, age between 14 and 36
years, M =22.76, SD = 4.60) filled out the inventory immediately after awakening in the morning
on two days before a competition. The first time of measurement was 4 days prior to
competition, while the second measurement time was on the day of the competition.
Instrument. The WAI-T consists of three subscales, “somatic anxiety”, “worry”, and
“concentration disruption”, each subscale consisting of four items. Athletes have to rate the
intensity with which they are usually experiencing different symptoms before a competition on a
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four-point rating scale ranging from “not at all” to “very much”. Items are, for example, “Before
competitions … I feel nervous” (somatic), “… I’m concerned about reaching my goal” (worry),
“… I’m disrupted by the audience” (concentration disruption). Only the subscale “worry” was
used in this practical example. Considering their brevity, the WAI-T subscales show acceptable
internal consistency (Cronbach’s α between .72 (concentration disruption) and .78 (worry)) and
retest-reliability across 5 weeks (between rtt=.67 (concentration disruption) and rtt=.84
(somatic)). Evidence for construct validity (through confirmatory factor analysis) as well as for
criterion validity (WAI-T predicted competitive state anxiety immediately before a competition)
has been found.
In the following paragraphs, we will describe the different steps in the analyses and
explain the results.
Models
The basic model for our analyses was conducted with a model similar to that seen in
Figure 2. The questionnaire used here also consisted of four items and participants were asked at
two different times. However, we added correlated errors between the same items at time one
and two. These correlated errors represent any possible method variance.
All in all, three models will be tested and compared. In model 1 (no correlated errors), the
correlated errors are all fixed at zero. Model 2 (no latent state residuals) allows for correlated
errors, but the latent state residual variances are fixed at zero. Finally, in model 3 (full model),
correlated errors as well as latent state residual variances are estimated. If model 1 fits better than
model 3, we can conclude that there is no method variance. If model 2 fits better than model 3,
the situation would not have any impact on the measurements.
After these analyses, we will compute reliability, consistency and occasion specificity.
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Statistical analyses
Confirmatory factor analyses (maximum likelihood) were conducted using AMOS 16.0.
One important assumption for such an analysis is a multivariate normal distribution. We tested
this assumption with the Mardia Test. The assumption of multivariate normality was violated
(multivariate kurtosis = 13.52, c.r. = 6.68) and Bollen-Stine bootstrap corrections for the p-value
of the χ² test were conducted with N = 1000 samples.
The assessment of the global-goodness-of-fit was based on the Standardized Root Mean
Square Residual (SRMR) and the Root Mean Squared Error of Approximation (RMSEA) as
recommended by Hu and Bentler (1999). The authors also give some advice regarding possible
cutoffs for the indices. Thus, the SRMR should be lower or equal to .11 and the RMSEA should
be less than .06 for N > 250 and less than .08 for N < 250. Additionally, we looked at the
Comparative Fit Index (CFI) as advised by Beauducel and Wittmann (2005). According to Hu
and Bentler the CFI should have a value of approximately .95. Marsh, Hau and Wen (2004)
criticized these “golden rules” and pointed out that the recommended cutoffs are very restrictive
and hardly achievable when using personality questionnaires. Nevertheless, we will apply the
cutoffs, keeping in mind that they are very strict when dealing with personality questionnaires.
Model comparisons were conducted with Χ2 difference tests.
Results
Method and situation variance. Table 1 contains the results regarding model fit for all
three models. According to the χ² test, only the full model fitted the data. However, judging by
the fit indices, the model without correlated errors had an acceptable model fit. The model
without latent state residuals was slightly worse.
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The Χ2 difference tests showed that model 3 fitted significantly better than model 1 (ΔΧ2
= 22.53, Δdf = 4, p < .001) and model 2 (ΔΧ2 = 48.49, Δdf = 2, p < .001). Thus, we can conclude
that there is method variance as well as situation variance in our data set. All further results will
therefore focus on model 3.
Only three of the four error correlations reached significance: item 1 (r = .28, p = .002),
item 2 (r = .17, p = .077), item 3 (r = .19, p = .045), and item 4 (r = .24, p = .008). These small
correlations indicate rather little method variance.
The paths from the latent state variable to the items have a value of .74 at time 1 and .76
at time 2. This shows that the combined impact of trait, situation, and trait by situation
interaction on the items was about equally high at both times.
The path from the latent trait to the latent state variable at time 1 was .90 and it was .88 at
time 2. Consequently, trait impact seems to be identical at both times.
After showing how method or situation variance in the data can be determined, we will
now compute the LST coefficients.
LST coefficients. Table 2 displays the size of the variances at both times. All variances
were significant. Because we only measured one trait, its variance is the same at both times.
Error and latent state residual variance are comparable in size. The item variance is the sum of all
three sources.
Using this information we can now compute the LST coefficients with the equations
stated above.
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The results show that each single item contains more trait influence than situation
influence. Thus, the questionnaire used here primarily captures trait variance. The amount of
occasion-specific variance is slightly higher at time 2. However, this difference can be neglected.
The rather low reliabilities should not be surprising because they were computed for the
individual items. The reliabilities for full test length, obtained with the Spearman-Brown
equation, are considerably higher and satisfying. Moreover, a comparison with the Cronbach α
estimation also shows that neglecting situational influences does not lead to reliability
overestimations.
Using this example, we have shown the steps necessary in LST analyses as well as the
possibilities attached to them. In the following paragraphs we will discuss the advantages of this
procedure.
Discussion
Using a LST design allows one to clearly determine the amount of variance due to trait
and situation (including trait by situation interaction). In our practical example, this has provided
further evidence for the WAI-T as being a reliable and valid measure of trait competitive anxiety.
First of all, reliability is fairly high, even if systematic situational effects are considered.
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Secondly, consistency, i.e., the situation-independent trait influence, is considerable, especially if
compared to specificity of situation, which, as a third aspect, is seen to be low. Looking beyond
the instrument, LST has advantages regarding test construction, assessment as well as research.
Test construction and Assessment
When constructing a test, the aim might be the measurement of a trait or a state. Using
LST designs enables the test constructor to differentiate between trait and state variance. In this
way, the psychometric properties of the test can be better quantified. Moreover, as pointed out
above, certain assumptions such as the parallelism of items can be tested directly. The LST
framework can also be extended to a multi-state-multi trait or to a multi-method design (Steyer et
al., 1992; Steyer et al., 1999; Steyer & Schmitt, 1990). Using these methods, the ideas presented
above can be applied to more complex measurement tools.
Measures or estimates of reliability are often used in individual assessment, for example
in order to compute confidence intervals around individual scores before diagnoses. In order to
compute a confidence interval, the standard error of measurement (SEM) has to be calculated.
The SEM is defined as the square root from the difference between one and reliability. Usually,
Cronbach α is used as a reliability estimate. However, as we have just shown, systematic
variance at a specific measurement point does not only contain differences due to the measured
trait, but is also affected by the situation and the situation by trait interaction. The latter two also
cause systematic differences. Consequently, Cronbach α might be an overestimation. Moreover,
when computing the confidence interval, it is mostly the trait that is the aim of measurement.
Thus, changes between measurements occurring because of the situation or the situation by trait
interaction are not of interest. Consequently, the confidence interval should be computed with
the amount of systematic trait variance. This is the consistency. Instead of using Cronbach α, the
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consistency coefficient should be used. In our example, this would not make a difference because
the two are almost identical. However, there are other examples where larger discrepancies have
occurred (Deinzer et al., 1995). The neglect of situational influences in the computation of
Cronbach α might yield confidence intervals which are too narrow because of an overestimated
α. This, however, can have positive or negative effects for the person assessed. Assuming the
person is just below a critical cutoff, a too narrow confidence interval might prohibit a correct
diagnosis. The other scenario of a wrong diagnosis can result as well given the person’s score is
just above the cutoff. In both cases, the wrong decision is influenced by the impact of situational
variance. Using reliability estimates obtained from LST designs might yield confidence intervals
which are broader. Importantly, however, these intervals ensure the consideration of the
measurement error attached to the trait alone!
Research
In sports psychology, states play an important role. Athlete’s who are excellent in
training but choke under pressure obviously have difficulties with the situational influences.
Thus, it is necessary to have instruments for measuring states. LST design helps to confirm this.
If an instrument really measures a state, only little variance should be accounted for by a trait.
Consequently, the specificity coefficient should be larger.
Moreover, there might be settings when it is less clear whether state or trait influence an
individual’s feelings and performance. Applying the LST design to research settings will help to
clarify this. If the amount of variability does not change over time, trait differences are more
prominent. However, it is also possible that trait differences lose in importance over time. In that
case, researchers will know that situational circumstances have gained in impact. Such settings
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will then help to tailor interventions which either try to help coping with trait impact or change
situational aspects or perception. Both would help the athlete to perform better.
LST also offers more complex designs in which method effects or multiple traits can be
modeled. With these designs it is possible to estimate trait intercorrelations free of any method or
situational impact. The paper by Steyer, Schmitt, and Eid (1999) gives more advice on how the
LST theory can be applied to different research questions.
Limitations
Obvious limitations are the need for more than one time of measurement and relatively
large samples. The first is an absolute prerequisite for LST analyses. The latter is a necessity for
reasonably good estimations in the structural equation modelling. However, the added value of
this kind of analysis is definitely worth the effort.
The present paper introduces the basics of LST theory and details the analysis steps with
a practical example from sports psychology research. LST theory and designs offer many
advantages to psychologists in general and specifically to sports psychologists. Not only is it
possible to obtain information on the influence of trait and state, but it is also possible to obtain
better reliability estimates for traits as well as guidance in specific research questions.
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Foot Notes
1
The German acronym will be used in order to avoid confusion with other similar instruments; it
translates into Competitive Anxiety Inventory
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Table 1
Model fits
Model
Χ2(df)
1 (no correlated errors)
65.14 (31) .017
.084 (.055 - .113)
.94
.049
2 (no latent state residuals) 91.10 (29) .001
.118 (.091 - .145)
.90
.062
3 (full model)
.061 (.020 - .095)
.97
.041
BS-p RMSEA (90% CI) CFI SRMR
42.61 (27) .10
Notes. Χ2(df) = Chi² value and degrees of freedom; BS-p = Bollen-Stine bootstrap p-value for the
Χ2 test; RMSEA (90% CI) = root mean squared error of approximation with 90% confidence
interval; CFI = comparative fit index; SRMR = standardized root mean square residual.
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Table 2
Variances, LST coefficients, and reliability estimates
Situation
REL
CON
SPE α
1
.285 .067 .278 .630 .55 (.83) .44 (.76) .11
.83
2
.264 .080 .278 .622 .58 (.84) .45 (.76) .13
.84
Notes.
trait variance;
= measurement error variance;
= latent state residual variance;
= latent
= total score variance; REL = reliability; CON = consistency; SPE = occasion
specificity; reliability and consistency for full test length using the Spearman-Brown equation are
printed in parentheses; α = Cronbach α.
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Figure Captions
Figure 1. Variance decomposition.
Eit = measurement error variance of item i at time t; Xit = observed score for item I at time t; Tit
= latent state variable i at time t;ζit = latent state residual; ξit = latent trait.
Figure 2. Multi-State-Single-Trait model.
E11 to E41 = measurement error variances for item 1 to 4 at time 1; X11 to X14 = observed score
for item 1 to 4 at time 1;T1 = latent state variable at time 1; T2 = latent state variable at time 2;
ζ1 = latent state residual at time 1; ζ2 = latent state residual at time 2; ξ = latent trait.
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