Performance decline in low-stakes educational assessments: different mixture modeling approaches
List et al. Large-scale Assess Educ
Performance decline in low‑stakes educational assessments: different mixture modeling approaches
Marit K. List 1
Alexander Robitzsch 0 1
Oliver Lüdtke 0 1
Olaf Köller 0 1
Gabriel Nagy 1
0 Centre for International Student Assessment , Munich , Germany
1 Leibniz Institute for Science and Mathematics Education , Kiel , Germany
Background: In low-stakes educational assessments, test takers might show a performance decline (PD) on end-of-test items. PD is a concern in educational assessments, especially when groups of students are to be compared on the proficiency variable because item responses gathered in the groups could be differently affected by PD. In order to account for PD, mixture item response theory (IRT) models have been proposed in the literature. Methods: In this article, multigroup extensions of three existing mixture models that assess PD are compared. The models were applied to the mathematics test in a largescale study targeting school track differences in proficiency. Results: Despite the differences in the specification of PD, all three models showed rather similar item parameter estimates that were, however, different from the estimates given by a standard two parameter IRT model. In addition, all models indicated that the amount of PD differed between tracks, in that school track differences in proficiency were slightly reduced when PD was accounted for. Nevertheless, the models gave different estimates of the proportion of students showing PD, and differed somewhat from each other in the adjustment of proficiency scores for PD. Conclusions: Multigroup mixture models can be used to study how PD interacts with proficiency and other variables to provide a better understanding of the mechanisms behind PD. Differences between the presented models with regard to their assumptions about the relationship between PD and item responses are discussed.
Educational assessments; Mixture IRT models; Performance decline; Group comparisons; Aberrant response behavior
One main purpose of large-scale assessments (LSAs) is to provide policy-makers and
educational institutions with information about students’ proficiency. Having no
personal consequences, educational assessments are low-stakes tests for the test takers
A common observation in educational LSAs is that the probability of a test taker
giving a correct response decreases for items at the end of the test
. On the test
taker’s side, this can be viewed as a performance decline (PD), which can be considered
as a type of aberrant response behavior reflected in unexpectedly high rates of incorrect
or omitted responses for end-of-test items
(Cao and Stokes 2008; Schnipke and Scrams
1997; Suh et al. 2012)
. Whereas, in high-stakes tests, aberrant response behavior on
endof-test items is often attributed to test speededness (Bolt et al. 2002), in low-stakes tests,
it is assumed that aberrant response behavior is related to a decline in test-taking effort
(Wise and Kong 2005)
. In order to explore PD in educational assessments, mixed-effects
models (De Boeck and Wilson 2004) have been proposed to analyze item position effects
by comparing the probabilities of a correct response when the item is presented in
(Debeer and Janssen 2013)
. An alternative strategy proposes using
mixture models with categorical latent variables in order to identify the latent classes of test
takers who differ in their test-taking behavior
(Mislevy and Verhelst 1990; Rost 1990)
Among others, Bolt et al. (2002),
Jin and Wang (2014)
mixture models to separate test takers who show aberrant response behavior that
corresponds to PD from test takers who respond to all items with full effort.
The aim of our study was to examine the utility of three mixture models to handle
PD in low-stakes tests and to explore the differences and similarities in the conclusions
drawn from these models. In this study, we applied the mixture models to investigate
PD in the mathematics test of a German LSA. First, we explored the differences between
the PD of test takers attending different school types. Next, we investigated whether PD
affects the estimation of group differences in proficiency. Then, we compared existing
mixture models with regard to the differences and similarities in their estimation of PD.
Furthermore, we demonstrate how to fit these models using standard software such as
Muthén and Muthén 1998
This article is organized as follows. First, we will provide an overview of the research
on PD in educational assessments. We will then proceed to present three mixture
models for PD that were extended to multigroup settings. After that, we will apply these
models to a low-stakes mathematics test in order to compare their performance and
parameter estimates. Finally, we will present and discuss the differences and similarities
of these models regarding their measurement of PD and estimated group differences in
Performance decline in educational assessments
If PD is present, the probability of providing a correct response does not depend solely
on item parameters and a person’s proficiency. Simulation studies have shown that
parameter estimates of end-of-test items are biased if PD is not taken into account
(Oshima 1994; Suh et al. 2012)
. This is of particular concern when parameter estimates
of items are going to be treated as known in other applications, for example, in adaptive
testing, as part of a calibrated item pool, or for test construction purposes
Lee 2011; van Barneveld 2007)
. In addition, test scores obtained under PD conditions are
likely to be lower than they would be if the test taker had invested full effort throughout
the test. Thus, when PD is ignored, proficiency scores are underestimated—the stronger
the PD effects are and the more test takers experience PD, the stronger the
underestimation of the sample’s average proficiency will be.
Several authors have studied associations between PD and test takers’ characteristics,
such as ethnicity, language skills, or gender. Bolt et al. (2002) analyzed a high-stakes
college placement test for mathematics and found that the number of students showing PD
differed between different ethnic groups.
Yamamoto and Everson (1995)
assessed PD in
a high-stakes reading comprehension test for university students. They also found
differences between the PD of different ethnic groups. Furthermore, they found that
nonprimary-language speakers showed an earlier onset of PD than primary-language speakers.
For a high-stakes reasoning test,
Schnipke and Pashley (1997)
also found that
nonprimary-language speakers were more strongly affected by PD than primary-language
For low-stakes tests, differences in PD have been reported on a country level for the
(Debeer et al. 2014; Hartig and Buchholz 2012; Jin and Wang 2014)
Nagy et al. (2016) also found PD effects to differ between school types in the German
PISA 2012 study. Furthermore, male test takers have been found to show more guessing
behavior and lower levels of test-taking motivation than female test takers
(DeMars et al.
Due to the differential effects found, these results imply that ignoring PD might affect
the estimations of average proficiency levels in a group-specific way. Thus, when
investigating group differences in proficiency, the extent to which the differences found
might be caused by differences in PD instead of true differences in proficiency should be
(DeMars et al. 2013; Denis and Gilbert 2012; Mittelhaëuser et al. 2015)
Mixture models of performance decline
Mixture item response theory
(IRT; e.g., Embretson and Reise 2000)
models can be used
to identify test takers showing PD. We will refer to these models as mixture PD models
in the rest of this article. In general, mixture models assume that the population consists
of subgroups, for which the model parameters differ in particular ways
Verhelst 1990; Rost 1990)
. These subgroups are also referred to as latent classes since
they are not observed. Based on their individual responses, the probabilities for each
test taker of belonging to one of these latent classes are estimated along with the other
model parameters. Mixture PD models consist of two or more latent classes, where one
class represents test-taking behavior that reflects full effort throughout the test (the
nodecline class), and the other classes represent test-taking behavior that reflects PD (the
Of these, two models, the two-class mixture model of Bolt et al. (2002) and the
HYBRID model of
have been applied to several empirical and
simulated data sets
(e.g., Boughton and Yamamoto 2007; Cao and Stokes 2008; Hailey et al.
2012; Mittelhaëuser et al. 2013; Mittelhaëuser et al. 2015; Suh et al. 2012; Wollack et al.
2003; Yamamoto and Everson 1997)
. Bolt et al. (2002) developed their two-class mixture
model to reduce the bias in item parameters that is caused by test speededness in
highstakes tests. The aim of the model is to identify the group of test takers who run out of
time and show PD (i.e., the decline class). PD is reflected in higher difficulty parameters
for end-of-test items. The item parameters of the no-decline class are expected to be
unbiased. Later on, the model was also used to model differences in test-taking
motivation in low-stakes tests in order to separate motivated from unmotivated test-taking
behavior (e.g., Mittelhaëuser et al. 2015).
The HYBRID model of
was initially developed to model changes
in test-taking behavior that occur under conditions of test speededness in high-stakes
tests. The HYBRID model focuses on determining the position in which a test taker
switches from effortful response behavior to aberrant response behavior when running
out of time. The item responses that reflect aberrant response behavior are ignored for
the estimation of proficiency and item parameters; therefore, item parameter and
proficiency estimates are expected to be unbiased. The HYBRID model has also been used to
investigate PD in low-stakes tests (Cao and Stokes 2008).
Jin and Wang (2014)
proposed a multiclass mixture model to account for PD.
HYBRID model, it makes it possible to determine the point
where test takers alter their test-taking behavior. In contrast to the HYBRID model, Jin
and Wang (2014) do not assume that test takers switch to entirely aberrant response
behavior that is unrelated to proficiency. Rather, the authors model PD as a decline in
test performance so that the probability of providing a correct response decreases after
the onset of PD.
In a recent study, the HYBRID model
and the two-class mixture
model of Bolt et al. (2002) were compared with regard to their capability to reduce the
bias in item parameters that is caused by test speededness (Suh et al. 2012). Both models
were found to be equally capable of estimating the true item parameters and
outperformed a standard IRT model that did not account for test speededness. In addition,
in both cases, the results were not affected by the coding of the items not reached by
the examinees (i.e., missing vs. incorrect). However, the focus of the study of Suh et al.
(2012) was bias in item parameters; the effects of PD on the estimation of proficiency
therefore need further investigation. Furthermore, the model of
Jin and Wang (2014)
not yet been investigated thoroughly, and its performance has not yet been compared
with that of the model of
or the model of Bolt et al. (2002).
A general mixture IRT model of performance decline
In the three aforementioned mixture PD models, several common assumptions are
made. It is assumed that test takers respond to the items in the order in which they
are presented. It is further assumed that PD starts at some point in the test, that is, the
test takers will have responded to at least the first item with full effort before their
testtaking behavior reflects PD. Thus, a switching point
(von Davier and Yamamoto 2007)
exists at which decline onset can be observed, dividing the item response vector of a test
taker into two parts: in the first part, item responses depend solely on the test taker’s
proficiency and on the item characteristics. In the second part, item responses depend
on a latent person variable underlying the aberrant response behavior that reflects
PD. Depending on the definition of PD as specified in the particular model, aberrant
response behavior may or may not be related to proficiency. For both parts, the
relationship between item responses and the latent person variable can be modeled within the
IRT framework. We assume that, before the switching point, a two-parameter logistic
model (2PLM; Birnbaum 1968) holds. However, after the switching point, we assume
that item and person parameters might be different. Furthermore, we assume that some
test takers show full effort throughout the test. Thus, the sample consists of a mixture of
test takers with and without PD.
Based on these assumptions, a general mixture PD model can be described, which
contains the aforementioned models as special cases. In the following, we describe
such a model for the multigroup case, although the framework can be easily adapted
for single group situations. According to the 2PLM, the probability of person p
providing a correct response to item i depends on the person’s proficiency (θp) and the item’s
slope and intercept parameter (αi,βi). The person’s individual switching point (δp) equals
the last item position before decline onset. Thus, the switching point can take values
δp = 1, . . . , I, where I is the total number of items in the test. If δp = i, PD will start after
item i. If δp = 1, PD will start after the first item. If δp = I, PD will not occur at all in the
Let ηpi be the logit of the probability of correct response of person p to item i:
ηpi = logit P Xpi = 1|θp, δp . For the general mixture PD model, the conditional item
response probabilities are defined as
αi · θp + βi,
α˜ig · θ˜p + β˜ ig ,
if δp ≤ i,
if δp > i,
where θp denotes the proficiency variable, αi is the slope parameter or item
discrimination and βi is the intercept parameter1, both before the switching point, that is, for δp ≤ i.
Note that the item parameters applied to responses before the switching point contain
no group index g, so that we assume them to be invariant across groups, although this
assumption can be relaxed.
The parameters α˜ig and β˜ ig denote the slope and intercept parameter in group g after
the switching point, that is, for δp > i, respectively, and θ˜p denotes the person variable
underlying aberrant response behavior. When a test taker shows PD, the proficiency
variable θp can change to θ˜p, which would no longer be interpreted as proficiency.
Equation 1 is a mixture 2PLM where the latent classes capture the magnitude of the
individual PD. If δp = I for all test takers in the sample, Eq. 1 is reduced to ηpi = αi · θp + βi,
which is the standard 2PLM.
The joint distribution of θ and δ is defined based on the assumption of how proficiency
and switching point are related. Typically, θ is considered to be a normally distributed
continuous variable. By definition, δ is considered to be a discrete variable. One way of
specifying the joint distribution P θ, δ|G = g is to consider the conditional distribution
of θ with respect to δ in group g (i.e., δ|G = g), that is, P(θ,δ|G = g) = P(θ|δ, G = g) · P(
δ|G = g). Within each class and each group, a normal distribution of θ is assumed, that
is, P θ|δ = i, G = g = N μig , σi2g . Typically, in the case of many classes, restrictions
are imposed on the means μig and standard deviations σig
(e.g., Yamamoto and Everson
The dependencies between θ and δ as assessed by multigroup mixture PD models
could be of substantive interest in real applications. For example, research suggests that
1 Note that the item difficulty bi is the negative of the ratio of intercept to slope parameter
(Hambleton et al. 1991)
is, bi = − βi/αi.
students with lower proficiency estimates are more likely to show aberrant response
behavior, as reflected in PD
(Bolt et al. 2002; De Boeck et al. 2011; Wise and Kong 2005)
mixture PD models make it possible to examine such hypotheses. In addition,
researchers might expect that the strengths of the relationship between θ and δ differ between
groups (e.g., groups assessed in low-stakes vs. high-stakes conditions). Again, the
multigroup setup of mixture PD models allows such hypotheses to be explicitly tested.
In addition, the multigroup mixture PD models provide estimates for the proportions
of decline classes in group g, πig = P(δ = i|G = g) for i < I, including the proportions
of test takers not affected by PD, πIg = P(δ = I|G = g), thereby allowing researchers to
examine whether the proportion of test takers not affected by PD differs between groups,
and to investigate group differences in the prevalence of earlier and later onsets of PD.
However, the assessment of latent class probabilities, πig, might not succeed in mixture
PD models that contain many latent classes without imposing additional constraints. In
order to solve this problem,
Cao and Stokes (2008)
proposed a cumulative probability
function of switching points, which was also employed by
Jin and Wang (2014)
mixture PD model. In the multigroup case, the function recurs on the probability of the
no-decline class in group g, πIg, and the probabilities of the decline classes, πig, for i < I.
Cao and Stokes (2008)
assume that the cumulative probability function of δp depends on
a shape parameter ω > 0, so that
P δp ≤ i|G = g = (I − 1)ωg , for i < I .
iωg − (i − 1)ωg
(I − 1)ωg
· 1 − πIg , for i < I .
The shape parameter ωg defines the form of the cumulative probability curve in group
g: if ωg = 1, the probability increases linearly, if ωg > 1, the increase is convex, and if
ωg < 1, the increase is concave. With Eq. 2, the curve of πig is described as
In Eq. 3, only two parameters are estimated per group, the proportion of the
nodecline class πIg and the shape parameter ωg; this reduces the number of parameters for
long tests with many switching points. Because of its parsimony, we used Eq. 3 in our
study to model the probability distributions for the decline classes in mixture PD models
with many classes. However, other functions could be used as well.
The cumulative probability function of δp (Eq. 2) used in our application allows
researchers to explicitly test hypotheses about group differences in the proportion of
examinees affected by PD via the estimates of πIg (Eq. 3). In addition, the function allows
for a straightforward comparison of the ωg-parameters that indicate whether the onset
of PD occurs later (higher values of ω) or earlier in the test (lower values of ω). However,
as the group-specific proportions of PD onset points πig is a rather complex function
of ωg and πIg (Eq. 3), we suggest using graphical aids for comparing the distribution of
onset points across groups.
Finally, it should be noted that the multigroup mixture PD models make it possible to
assess not only the group-specific distributions of the onset points of PD but also the
magnitude of PD in each group. However, the magnitude of PD depends on the
specific restrictions imposed on the general mixture PD model (Eq. 1): in some models, the
magnitude of PD is expressed via the β˜ ig-parameters
(Bolt et al. 2002; Yamamoto 1995)
whereas, in others, the magnitude of PD is quantified by the person variable θ˜p
Multigroup mixture IRT models of performance decline
By placing specific parameter restrictions, the general mixture PD model can be
transformed into multigroup versions of each of the three aforementioned mixture PD models
(Bolt et al. 2002; Jin and Wang 2014; Yamamoto 1995)
. Since PD is modeled differently in
the specific mixture PD models, group comparisons for each mixture PD model involve
different parameters. In the following, we will describe how the general mixture PD
model can be extended to realize the mixture PD models of Bolt et al. (2002),
Jin and Wang (2014)
, allowing for multigroup comparisons.
The two‑class mixture model of Bolt et al. (2002)
Bolt et al. (2002) proposed a mixture PD model with two latent classes (which will
further be referred to as the 2PDM): while test takers in the no-decline class do not show
PD, test takers in the decline class show lower performance on end-of-test items.
Therefore, in the decline class, items appear to be more difficult than in the no-decline class,
resulting in lower item intercept parameter estimates after the switching point. In their
original formulation of the 2PDM, Bolt et al. (2002) suggested specifying the switching
point i0 in advance
(see also De Boeck et al. 2011; Wollack et al. 2003)
, so that it refers to
an item position up to which responses are expected to not be affected by PD. When put
into the multigroup context, this specification implies that, in each group, the switching
point is a dichotomous variable that can take two values, that is, δp = i0 for all test takers
showing PD, and δp = I for all test takers not showing PD. Accordingly, in each group,
there are two latent classes with proportions πi0g = P δ = i0|G = g for the decline class
and πIg = P(δ = I|G = g) for the no-decline class.
The model was first proposed as a mixture Rasch model with equal item
discrimination parameters for all items. Recent applications have also considered extensions based
on a 2PLM
(Cao and Stokes 2008; Suh et al. 2012)
, which we also used in our study. For
each group, we restricted the item discriminations to be equal in both latent classes (i.e.,
α˜ig = αi; see Cao and Stokes 2008). After the switching point, item intercept parameters
were allowed to change but item responses to still depend solely on the item parameters
and proficiency. Thus, the person variable underlying the aberrant response behavior
would still be regarded as proficiency, hence θ˜p = θp. Within each group, item intercept
parameters were constrained to be lower in the decline class (i.e., β˜ ig ≤ βi for each g).
Equation 1 can be altered to realize the multigroup version of the 2PDM with
αi · θp + βi,
αi · θp + β˜ ig ,
if δp ≤ i0,
if δp > i0.
The mean proficiency is allowed to vary across groups and classes, while the
standard deviation is held equal across latent classes, that is, θ ∼ N (μig , σg2). We assumed that
groups could differ in the intercept parameters after the switching point, β˜ ig (see Eq. 4).
In addition, we did not impose any constraints on the latent class proportions, πi0g and
Note that, in the common specification of the 2PDM, the (arbitrarily) specified
switching point i0 does not identify the item position in which PD first occurs. Rather,
researchers are required to compare the estimates of βi and β˜ ig and to identify the
position from which these estimates show meaningful deviations from each other. An
alternative is to identify the point at which PD first occurs by comparing the model-data-fit
of the 2PDM, specified with different values of i0, and selecting the switching point that
provides the best fit to the data at hand. This procedure has the advantage that many
βi-parameters that would otherwise be specified to be class- and group-specific (i.e.,
βig-parameters) are estimated on the basis of a larger number of item responses. The
procedure can be extended to identify group-specific switching points, but the merits
of such an approach seem to be limited. When a common switching point is assumed,
researchers can inspect the results for group differences in the onset of PD by comparing
the estimates of β˜ ig with the corresponding estimates of βi. Results showing negligible
discrepancies between β˜ ig- and βi-parameters after the switching point i0 indicate a later
onset point of PD in this group.
Hence, the multigroup 2PDM makes it possible to (1) assess the relationship between
proficiency and PD on the basis of the latent class means μig, (2) estimate the proportion
of examinees not affected by PD in each group (πIg), and (3) quantify the strengths of PD
in each group by examining the discrepancies between β˜ ig- and βi-parameters. In
addition, the latter parameters also make it possible to (4) inspect the results for group
differences in the onset points of PD.
The HYBRID model of Yamamoto (1995)
In his HYBRID model (further referred to as the HYBRID),
that, after the switching point, item responses no longer reflect test takers’ proficiency.
Instead, the probability of a test taker providing a correct response to items after PD
onset is independent of proficiency; rather, it corresponds to an item-specific response
threshold. While the model was originally proposed to model random guessing in
highstakes tests under speededness conditions, it can also be applied to other types of
aberrant response behavior reflecting PD
(e.g., Suh et al. 2012)
PD onset can occur in all item positions except the first one, hence, there are multiple
decline classes, one for each item position, so that δp = 1, . . . , I in each group. Within
each group, the response thresholds can be specified to be either item-specific or equal
for all items. The latter assumption is reasonable in cases where all items share a similar
response format, such as multiple-choice items with the same number of options.
Since item responses after the switching point do not depend on proficiency, the slope
parameter after the switching point is set to zero in each group (α˜ig = 0, for each g).
Since the slope parameters after the switching point are set to zero, the item responses
do not depend on the person variable, θ˜p, meaning that the person variable underlying
aberrant response behavior is not defined. To keep the model identified, we set θ˜p = θp.
Note that θ˜p can be fixed to any other value, as it does not impact on ηpi after the
switching point. The intercept parameter after the switching point is constrained to a common
response threshold within each group, which we restricted to be the same for all items in
all decline classes, that is, β˜ ig = β˜ g for all i.
Thus, within each group, the probability of a correct response after PD onset is the
same for all test takers and items regardless of proficiency (or any other person
variable underlying aberrant response behavior) and regardless of the location of PD onset.
Based on these specifications, the multigroup HYBRID can be derived by altering Eq. 1
αi · θp + βi,
0 · θp + β˜ g ,
The distribution of decline class probabilities within each group was assumed to follow
the function described in Eqs. 2 and 3. We assumed that the mean of θ in each PD class
assessed in each group is a function of δ, while the standard deviation of θ is the same for
all latent classes but differs between groups. More specifically, we modeled the mean of
the conditional θ distributions across latent classes as a linear function
with the mean of the no-decline class, μIg, as the intercept and a
groupspecific slope parameter ρg:
μig = μIg + ρg · (I − i).
The value of ρg in Eq. 5 shows how θ and δ are related: if ρg is negative, average
proficiency is lower, the earlier the PD onset. If ρg is positive, average proficiency would be
lower for later PD onsets.
Taken together, the multigroup HYBRID makes it possible to assess the
group-specific (1) relationships between proficiency and PD via the parameters ρg, (2) proportions
of examinees not affected by PD by means of the latent class proportions πIg, and (3)
strengths of PD effects that are governed by the response thresholds β˜ g. Compared to
the multigroup 2PDM, the HYBRID model presented here allows for (4) a more
finelygrained assessment of the group-specific onset points of PD by inspecting the
groupspecific cumulative probability functions of δp (Eq. 2) and their parameters ωg.
The multiclass mixture performance decline model of Jin and Wang (2014)
Jin and Wang (2014)
proposed another multiclass mixture PD model (further referred
to as the MPDM). Similar to the HYBRID, PD onset can occur after any item position
throughout the test. In the MPDM, it is assumed that, after the switching point, θp is
reduced according to a decrement function which depends on δp. Hence, when specified
in a multigroup context, in the MPDM, PD is modeled via a group-specific change in the
θ-variable instead of via changes in the intercept parameters as is the case in the 2PDM
and the HYBRID model. More specifically, in the multigroup MPDM, it is assumed that,
within each group, the value of the decrement function is the same for all test takers who
have the same switching point. It is further assumed that, in all classes, the decrement is
smaller when PD onset occurs later in the test.
Jin and Wang (2014)
assume that, after PD onset, the item parameters are the same as
before the switching point (α˜i = αi and β˜ i = βi), but that the person variable underlying
the aberrant response behavior, θ˜p, corresponds to the difference between proficiency
and the decrement for the respective switching point, that is, θ˜p = θp − κg · I − δp .
The multigroup MPDM can be formalized by altering Eq. 1 to
αi · θp + βi,
αi · [θp − κg · (I − δp)] + βi,
As shown in Eq. 7, the decrement in θ occurring after PD onset is a linear decreasing
function across switching points with a group-specific decrement parameter κg > 0 as
the slope so that, for the last decline class, the decrement is κg.2 Note that
Jin and Wang
modeled θ and δ to be independent of one another. However, we suggest applying
Eq. 6 in the same manner as described for the HYBRID for two reasons. First, in this
way, it is possible to empirically investigate whether the independence assumption that
Jin and Wang (2014)
propose holds. Second, we wanted to base our comparisons of the
three mixture PD models on similar assumptions regarding proficiency and PD
throughout all models. In our multigroup version of the MPDM, the proportions of decline
classes were defined in the same way as for the HYBRID, with group-specific shape
parameters ωg (Eqs. 2, 3).
Hence, it can be summarized that the multigroup MPDM shares some similarities
with the HYBRID: it makes it possible to examine the group-specific (1) relationships
between proficiency and PD (ρg-parameters), (2) proportions of examinees showing no
PD (proportions πIg), and (3) distributions of PD onsets δp (Eq. 2). However, the MPDM
differs from the HYBRID because it assumes (4) that the magnitude of PD depends on
δp, although the size of PD could be group-specific, and (5) that the responses affected
by PD still depend on proficiency.
The present investigation
The aim of the present study was to examine the extent to which the multigroup mixture
PD models make it possible to assess PD, and whether the conclusions about the
presence and group differences in PD that can be drawn from these models differ. In
addition, we examined whether accounting for group differences in PD affected the results
of the group comparisons of students’ proficiencies, and whether the mixture models
envisaged differed in their estimates of group differences in proficiencies. Finally, we
investigated whether the mixture PD models provided item parameters estimates that
differed from those given by the multigroup 2PLM
(Suh et al. 2012)
. To this end, we
drew on a LSA of German students who worked on a mathematics test. We fitted the
multigroup versions of the 2PDM, the HYBRID, the MPDM, and a standard multigroup
2PLM. We chose school track as a grouping variable, since school track comparisons are
at the core of many large-scale educational programs, and school track has been found
to be strongly related to PD, as reflected in item position effects
(Nagy et al. 2016)
In Germany, after attending primary school, children are assigned to different school
tracks based on their school achievement. There is one academic school track (higher
secondary school, Gymnasium) and there are several non-academic school tracks,
including comprehensive (Gesamtschule), intermediate (Realschule), and lower
secondary schools (Hauptschule), though the number of non-academic school tracks can vary
(Pietsch and Stubbe 2007)
. Achievement differences between students at academic and
Jin and Wang (2014)
also discuss quadratic functions for the decrement function, in their empirical
application, they found that a linear function was sufficient.
non-academic school tracks are known to be large
(e.g., Prenzel et al. 2013)
Furthermore, students at the non-academic tracks appear to take participation in LSAs less
seriously (Baumert and Demmrich 2001), possibly giving rise to stronger PD. As such, group
differences in PD are likely, and group comparisons of students’ proficiencies could be
affected by group differences in PD (
Mittelhaëuser et al. 2015
; Nagy et al. 2016).
The analyses reported in this article examined both substantive and methodological
research questions. From a substantive point of view, we applied the multigroup mixture
PD models in order to examine (Q1) the existence of PD in item responses and track
differences therein. Regarding the differences between tracks, we examined (Q2) the
proportions of students not affected by PD, (Q3) the distributions of onset points of PD,
(Q4) the magnitude of PD effects in each group, and (Q5) the relationships between PD
behavior and students’ proficiencies.
We expected to obtain the following results: regarding Q1, since the assessment
considered was a low-stakes test, we assumed that our data would show some degree of PD
and, thus, we expected all mixture PD models to provide a better fit to the data than the
2PLM. Regarding Q2–Q4, in line with research on school track differences regarding
item position effects
(Nagy et al. 2016)
and test-taking motivation
, we expected the amount of PD to be larger for students attending the
nonacademic tracks. More specifically, we expected the proportion of students not affected
by PD to be higher in the academic track (Q2). We did not feel able to derive detailed
expectations about group differences in the onset point of PD (Q3), or in the
magnitude of PD effects (Q4) because, to the best to our knowledge, such issues have not been
investigated previously. We therefore treated Q3 and Q4 as open research questions to
be examined in our application.
So far, only few studies have dealt with the relationship between PD and proficiency
(Q5). However, as research on response times indicates that low test-taking effort is
associated with low proficiency
(Wise and DeMars 2005; Wise et al. 2009)
, it seemed
reasonable to assume that less proficient students would be more likely to show PD. We
therefore expected that, within each school track, average proficiency would be lower,
the earlier PD occurs.
Our second set of research questions targeted methodological issues. Here, we were
interested in (Q6) whether accounting for PD affects the estimates of item parameters,
(Q7) whether the multigroup mixture PD models differ in the conclusions that can be
drawn from them about the prevalence of PD, and (Q8) the impact that PD has on the
results of group comparisons of proficiency.
When PD occurs, the intercept parameters of end-of-test items are likely to be
underestimated, whereas their slope parameters are likely to be overestimated when a
standard 2PLM is employed
(Bolt et al. 2002; Oshima 1994; Suh et al. 2012)
. Thus, regarding
Q6, we expected the estimates of the intercept parameters of end-of-test items in the
no-decline class to be higher for the mixture PD models than for the 2PLM, whereas we
expected to obtain the opposite result for the item discriminations. We expected this
result to hold for all mixture PD models, as Suh et al. (2012) found that most mixture
PD models showed quite similar behavior in this respect. Regarding Q7, we did not feel
able to derive detailed expectations because the results provided by the models depend
on their representation of PD. Therefore, we treated the question of whether the models
presented here converge to similar conclusions as an open research question.
Regarding Q8, we expected that not accounting for PD would lead to an
underestimation of the proficiency for the decline classes. Thus, we expected to obtain higher
estimates of average proficiency when mixture PD models were employed (Q5a). Since
PD was expected to be higher at the non-academic school tracks, we further expected
the underestimation of proficiency to be more pronounced in the non-academic than
in the academic group. As a consequence, accounting for PD was expected to result in
smaller group differences in proficiency between both school tracks. However, the
question of whether all models lead to a similar reduction in group differences in proficiency
Sample and test design
We considered the mathematics achievement test of the first measurement point of a
German large-scale longitudinal educational study, “Aspects of learning background and
learning development”, which was conducted in the federal state of Hamburg
für Schule und Berufsbildung 2011)
. Participation in the assessment was mandatory. The
test had no individual consequences for the test takers and, hence, it can be regarded
as a low-stakes test. The sample consisted of N = 12,182 students in the fifth grade at
different school tracks, an academic track (n = 5333 students) and two non-academic
The mathematics assessment was presented as a paper-and-pencil test with a fixed
item order, consisting of 30 multiple-choice items, scored as correct or incorrect.
Missing item responses caused by omitted and not-reached items were also coded as
incorrect. A distinction between incorrect and missing responses was not possible because the
original item responses were not made available by the primary investigator. Although
this means that PD effects were also influenced by the number of not-reached items,
this issue does not appear to be of importance in mixture PD models. Suh et al. (2012)
found the 2PDM and HYBRID model to perform equally well regardless of whether
notreached items were coded as incorrect or missing. Furthermore,
Jin and Wang (2014)
explicitly recommended coding not-reached items as incorrect in their MPDM.
We applied the three mixture models (2PDM, HYBRID, MPDM) in the multigroup
extensions presented and a multigroup 2PLM to the mathematics achievement test data.
The grouping variable was school track and we considered two groups—students
attending the academic school track, and students attending a non-academic school track. The
models were compared by means of the Bayesian information criterion
and Akaike’s information criterion
(AIC; Akaike 1987)
. Group differences in PD
were investigated by comparing the group-specific parameters for each model, and
Wald tests were used to test for significant parameter differences between groups. Note
that some parameters could not be compared across models; therefore, we investigated
whether the patterns found in each model led to similar conclusions regarding PD.
Model specification and parameter estimation
For model identification purposes, the mean and standard deviation of the
proficiency variable in the no-decline class at the academic track were constrained to 0 and
1, respectively. The item position of PD onset i0 for the 2PDM was defined by means
of model fit comparisons: the model was repeatedly estimated with switching points
δi0 = 10, . . . , 23, and the switching point was chosen where the BIC value was lowest.
Based on the BIC, the model with a PD onset i0 = 18 showed the best fit to the data and
was then used for the further analyses.
The models were estimated by means of marginal maximum likelihood estimation
using an expectation–maximization algorithm and using numerical integration with 15
integration points in Mplus 7.4 (
Muthén and Muthén 1998
–2012). One problem with
maximum likelihood estimation for mixture IRT models is that the solution can
converge to a local rather than the global maximum
(Finch and French 2012)
usage of multiple random starting values is recommended to ensure replication of the
best likelihood value
(Lubke and Muthén 2005)
. The Mplus code for the estimated
models is given in the additional file to this article.
We first present the goodness-of-fit statistics for the four models (Q1) and show how
accounting for PD impacted item parameter estimation (Q6). Next, we present the
comparisons between students at the academic track [academic group (aca)] and students at
the non-academic school tracks [non-academic group (nac)] across models; these results
are divided into three subsections. In the first subsection, we examine school-type
differences in PD model parameters (Q2–Q4). The next subsection discusses our findings on
the relationship between PD and proficiency (Q5) and, in the last subsection, we present
the results concerning the impact of PD on proficiency estimates (Q5a, Q8). While
presenting the results, we also highlight the similarities and differences between the results
provided by the different mixture PD models (Q7).
Model fit and item parameter estimates across models
The model fit indices (AIC, BIC) for the different mixture PD models as well as for the
2PLM are presented in Table 1. The MPDM had the best fit to the data, but all
mixture PD models were better than the 2PLM. Furthermore, the fit indices of the HYBRID
appeared to be closer to the MPDM than to the 2PDM.
2PLM 63 − 217,085 434,296 434,763
2PDM 91 − 214,224 428,630 429,304
HYBRID 71 − 213,978 428,098 428,624
MPDM 71 − 213,964 428,070 428,596
LL: log likelihood; AIC (BIC): Akaike’s (Bayesian) Information Criterion; 2PLM: multigroup 2PLM; 2PDM: multigroup two‑ class
performance decline model; HYBRID: multigroup HYBRID model; MPDM: multigroup multiclass performance decline model
Item parameter estimates for the no‑decline class
Accounting for PD is expected to reduce bias in the parameter estimates (intercepts
and slope) of items affected by PD
(Suh et al. 2012; Wollack et al. 2003)
. Since the true
item parameters were unknown, we were not able to evaluate whether any of the
mixture PD models were able to do this. However, by comparing the item parameters of the
no-decline class across models, we were able to investigate whether the estimates of the
mixture PD models differed from the 2PLM as well as from one another.
In Fig. 1 the estimated item parameters of the no-decline class (intercept, slope) for
all models are depicted. For the intercept parameters, we found differences between the
2PLM and the mixture PD models for the end-of-test items. All of the mixture PD
models showed higher estimates for the intercept parameters of the end-of-test items than
the 2PLM, which means that items appeared to be less difficult than in the 2PLM. The
mixture PD models also differed from one another: while the HYBRID and the MPDM
appeared to have rather similar parameter estimates, the intercept estimates of the
2PDM were somewhat lower and closer to the 2PLM.
The slope parameter estimates were quite similar across all of the mixture PD models.
Slope estimates differed between the 2PLM and the mixture PD models, not only for
the end-of-test items but also for the items at the beginning of the test: while, early in
the test, the estimates for the decline models were higher than those for the 2PLM, they
were lower at the end of the test. These results indicate that, in the 2PLM, end-of-test
Fig. 1 Estimates of item parameters across models. For the mixture PD models, the displayed item
parameters are those of the no-decline class. 2PLM: multi-group 2PL model; 2PDM: multi-group two-class
performance decline model; HYBRID: multi-group HYBRID model; MPDM: multi-group multi-class performance
items appeared to be more discriminating, a result also reported by Suh et al. (2012).
However, in our application, items positioned early in the test appeared to be less
discriminating than in the mixture PD models.
Group differences in performance decline
The main results of all of the group comparisons regarding PD as well as proficiency are
displayed in Table 2.
Proportions of no‑decline classes and distribution of decline class proportions
The proportions of the no-decline class within each group (i.e. πIg) are displayed in
Table 2, in the section entitled Distribution of latent classes. The size of πIg indicates the
number of students who did not show PD during the test. The smaller the πIg, the smaller
the number of students who did not show PD was and, thus, the higher the proportion
of students showing PD in the sample was.
Note that parameters cannot be compared across models. See main text for more information
nac: non‑academic group; aca: academic group; 2PDM: multigroup two ‑ class performance decline mode; HYBRID:
multigroup HYBRID model; MPDM: multigroup multiclass performance decline model. All parameter estimates are group‑
a See Fig. 3. b Parameters are fixed
Consistently, in all mixture PD models, πIg was significantly lower in the
non-academic group (Wald’s χ2-test statistics in Table 2), indicating that more students at the
non-academic tracks showed PD. However, the no-decline class proportions varied
across models. In the 2PDM, the proportions of the no-decline classes were highest:
πI,aca = 0.91 versus πI,nac = 0.81 (see Table 2, section Proportion of no-decline class). As
a consequence, the total number of students showing PD was smallest for the 2PDM. In
contrast, the proportions of the no-decline class were smallest for the MPDM: here, the
no-decline class proportion in the academic group was πI,aca = 0.68, which means that
one-third of the students showed PD. In the non-academic group, the no-decline class
proportion was significantly lower, with πI,nac = 0.47. Thus, one half of the students in
the non-academic group showed PD.
While, for the 2PDM, there was only one decline class for each group, for the HYBRID
and the MPDM, there were multiple decline classes. As displayed in Eq. 2, the
distribution of decline classes was defined by the no-decline class proportion, πIg, and the
shape parameter, ωg: the higher ωg, the steeper the increase in decline class proportions
across item positions; yet, the higher the πIg, the smaller the cumulated probability of
the decline classes and, hence, the smaller the total number of students showing PD.
The cumulated class probabilities across item positions for both models are displayed in
Fig. 2. The values for ωg are displayed in Table 2 (Shape parameter).
For the HYBRID, ωg was significantly higher in the academic group, ωaca = 7.33 versus
ωnac = 4.78. Thus, the increase in decline class proportions toward the end of the test
was steeper in the academic group (Fig. 2). Similar results were obtained for the MPDM:
likewise, ωg was significantly higher in the academic group (ωaca = 10.27 vs. ωnac = 6.47,
Table 2), which means that the increase in class probabilities in the academic group was
steeper than in the non-academic group (Fig. 2).
Thus, the HYBRID and the MPDM provided a qualitatively similar distribution of PD
classes and group differences within those classes. In both models, the no-decline class
was larger in the academic track (πI,aca > πI,nac) and the majority of academic track
students showing PD experienced PD onset later in the test (ωaca > ωnac). However, in
the MPDM, class proportions πI,g were smaller and shape parameters ωg were higher
in both groups, leading to a steeper increase in cumulated class probabilities for the last
five item positions (i.e., for switching points δ ≥ 24; Fig. 2). However, in earlier positions,
0 10 20 30
Switching point (δ)
Fig. 2 Cumulated decline class probabilities across item positions for multi-class mixture PD models. HYBRID:
multi-group HYBRID model; MPDM: multi-group multi-class performance decline model
the cumulated class probabilities for both models appeared to be almost identical (see
Fig. 2). Thus, differences between the decline class proportions of the two models were
mainly found for PD onset in the last five items.
Magnitude of performance decline
Group differences in the magnitude of PD were evaluated separately for each mixture
PD model according to the respective model parameters assessing the magnitude of
PD. In the 2PDM, the group differences found in the item intercept parameters after
the switching point (i.e., β˜ ig, Eq. 4) were regarded as a measure of differential PD effects,
while the differences in item intercept parameters between classes affected by PD and
no-decline classes (i.e., βi vs. β˜ ig, Eq. 4) were regarded as a measure of the magnitude of
PD. In Fig. 3, the intercept parameters for the no-decline class as well as for the decline
classes for both groups are displayed. The intercept parameter estimates were lower in
the decline classes and, thus, the probability of giving a correct response to items after
the switching point was lower for students showing PD. Moreover, the intercept
parameters decreased across item positions, so that items appeared to become gradually more
difficult toward the end of the test in both groups. Furthermore, the results displayed
in Fig. 3 do not provide a sound indication of group differences in the onset point of
PD as the estimates of the β˜ ig-parameters decreased immediately after the onset point
in both groups. However, the intercept parameters after the switching point were
significantly lower in the non-academic group (Wald’s χ2(12) = 25.24, p = 0.01), which
means that the average size of the PD appeared to be larger in the non-academic school
tracks. However, compared to the large difference in intercept parameters between the
no-decline and the decline classes, the group differences appeared rather small.
In the HYBRID, group differences in the magnitude of PD were reflected in
differences in the response threshold β˜ g (Eq. 5). The response thresholds were β˜ nac = −3.76
and β˜ aca = −3.48 for the non-academic and academic group, respectively, which did not
differ significantly from one another (see Table 2, section Magnitude of PD). After the
switching point, the probability of a correct response being provided was 0.02 in the
nonacademic group and of 0.03 in the academic group of obtaining a correct response after
the switching point.
In the MPDM, the probability of obtaining a correct response was not constant after
the switching point. Rather, in each decline class, this probability decreased, depending
decline class, non−academic
decline class, academic
Fig. 3 Intercept parameters of no-decline and decline classes (multigroup two-class performance decline
model). Item parameters of the no-decline classes are invariant between groups
on the PD onset point, with group-specific decrement parameters κg (Eq. 7). However, κg
did not differ significantly between groups (κnac = 0.80 vs. κaca = 0.91, see Table 2,
section Magnitude of PD), indicating that the magnitude of PD depended only on the PD
onset point, and not on the school track the students attended.
Hence, it can be summarized that, of the mixture PD models, the HYBRID and the
MPDM provided no indication that the magnitude of PD differs between school tracks.
In contrast, the 2PDM revealed statistically significant group differences in the
magnitude of PD, with stronger declines in non-academic track students. However, from a
substantive point of view, the difference was rather small.
Relationship between performance decline and proficiency
In the 2PDM for both groups, the mean of θ appeared to be somewhat higher in the
decline class. In the non-academic group, the mean proficiency was μθ,I,nac = −1.02 in
the no-decline class and μθ,i0,nac = −0.95 in the decline class, but the difference was not
statistically significant at the p ≤ 0.05 level (Table 2, section No-decline class: mean). In
the academic group, the mean and standard deviation in the no-decline class were fixed
at 0 and 1, respectively. The mean proficiency in the decline class was μθ,i0,aca = 0.03
which did not differ significantly from the no-decline class (Table 2).
For the HYBRID and the MPDM, the mean θ followed a linear function over δ with
group-specific slopes ρg (Eq. 6). For the HYBRID, in both groups, ρg was negative. This
indicates that the mean of θ was lower in the decline classes than in the no-decline class
and that it further decreased when PD onset occurred in earlier item positions.
Furthermore, ρg was significantly lower in the academic group (ρaca = −0.05 vs. ρnac = − 0.01,
Table 2, section Decline classes), indicating a stronger negative association between
θ and PD in the academic group. More specifically, the value of the ρaca-parameter of
− 0.05 means that students who had a PD onset at item 25 (i.e., 5 items before the end
of the test) were expected to score 0.25 units lower on the proficiency variable. As the
standard deviation of θ was fixed to one in the no-decline class in the academic track,
this result directly reflects a standardized effect size.
The results for the MPDM were quite similar to those for the HYBRID. In both groups,
ρg was also negative and, likewise, the slope in the academic group was significantly
lower (ρaca = − 0.03 vs. ρnac = − 0.01, Table 2). Thus, the average proficiency was lower
in the decline classes and the decrease in proficiency when PD onset occurred earlier in
the test was steeper in the academic group.
Hence, the HYBRID and the MPDM converged in their conclusion about the
relationships between students’ proficiencies and their PD onset points, as both models indicated
that the earlier the students’ PD onset occurred, the lower their proficiency was, and, in
both models, this relationship was more exaggerated in academic track students. In
contrast, the 2PDM provided no indication of a relationship between proficiency and PD.
Impact of accounting for performance decline on proficiency estimation
Comparisons of individual proficiency scores
In order to illustrate how accounting for PD affects the proficiency distribution, we
compared the expected a posteriori (EAP) scores for each model between groups and PD
classifications [i.e., students not showing PD (no-PD class) vs. students showing PD (PD
class)].3 In order to be able to compare score estimates, we transformed the EAP scores
obtained by the PD models to the metric of the 2PLM by means of linear equating
In Fig. 4, the equated EAP scores for each mixture PD model are plotted against the
EAP scores estimated by the 2PLM, displayed separately for school tracks and PD
classification. Across all mixture PD models and across both groups, for the no-PD classes,
the EAP score estimates of the mixture PD models appeared to be almost identical to
those of the 2PLM. However, for students belonging to the PD classes, the EAP scores
appeared to be higher when estimated by one of the mixture PD models than those
estimated by the 2PLM model. Assuming that scores estimated by the mixture PD
models are more accurate reflections of proficiency, the EAP scores estimated by the 2PLM
appeared to underestimate the proficiency of test takers showing PD. Looking at the
score estimates for the PD classes across models, the correspondence between the EAP
scores of the 2PLM and the 2PDM was very high because, in the 2PDM, only one decline
class was estimated for each group. For both the HYBRID and the MPDM, the difference
in EAP scores between the mixture PD model and the 2PLM depended on the
switching point and, thus, the correspondence between scores was lower (Fig. 4). Interestingly,
the EAP scores derived from the HYBRID and the MPDM showed very high agreement,
although they were based on models that differed in the specification of PD.
Group differences in proficiency
We also investigated whether accounting for PD would affect the estimation of group
differences in proficiency. In order to compare the magnitude of group differences as
estimated by the four models, we calculated effect sizes between the group means of
proficiency. For the mixture PD models, the group means were calculated from the
marginal distributions of θ across latent classes.5 The group means and standard deviations
of θ and the effect sizes (d) for all models are displayed in Table 3.
Across all models, the average proficiency in the non-academic group was lower than
in the academic group, as expected. In all of the mixture PD models, the effect sizes of
the group difference appeared to be very similar to one another (about d = 1.09) and
smaller than for the 2PLM (d = 1.17; Table 3). Thus, groups differed to a smaller degree
when mixture PD models were employed, regardless of the specification of PD.
However, in the present application, proficiency differences between groups were about one
standard deviation for all models, which means that the differences in the results
provided by the 2PLM and the mixture PD models are rather small when considered on a
3 For PD classification purposes, the probability of the no-decline class was dummy coded so that students were
classified as belonging to the no-PD class if their probability of belonging to the no-decline class was larger than 0.5.
Otherwise, they were classified as belonging to the combined PD class.
4 In order to obtain equating constants, the subsample of students classified to the no-PD class across all mixture PD
models (n = 8,003) was used. Equating parameters were then applied to all students.
5 The resulting distribution is a normal mixture distribution with a group-specific mean μg = iI=1 πig · μig and a
standard deviation as σg = iI=1 πig · μi2g + σg2 − μ2g with i = 1, . . . , I.
a 2PLM versus 2PDM
no−PD class PD class
EAP score (2PLM)
b 2PLM versus HYBRID
no−PD class PD class
EAP score (2PLM)
c 2PLM versus MPDM
no−PD class PD class
EAP score (2PLM)
Fig. 4 EAP scores estimated by the 2PLM versus mixture PD models by group and PD classification. x-axis:
EAP scores estimated by the multigroup 2PLM; y-axis: equated EAP scores estimated by the multigroup
twoclass performance decline model (2PDM, a), by the multigroup HYBRID model (b), and by the multigroup
multiclass performance decline model (MPDM, c)
Summary and discussion
The aim of the present article was to compare the potential of three mixture PD
models, the 2PDM
(Bolt et al. 2002)
, the HYBRID
, and the MPDM (Jin
and Wang 2014), for assessing PD in proficiency tests administered in LSAs. The models
were extended to accommodate multiple groups, thereby making it possible to examine
and test group differences in PD, and to adjust group differences in proficiencies for PD.
In order to examine the similarities and discrepancies in the results and conclusions
provided by the multigroup mixture PD models, the models were applied to a
mathematics tests administered in a German LSA. The results indicate that the models provided
similar conclusions about key aspects of PD, but differed with respect to some results.
However, the main results gathered by the mixture PD models were in line with results
obtained in other settings
(e.g., Nagy et al. 2016)
The mixture PD models consistently indicated that the mathematics test was affected
by PD, as all of the mixture PD models fitted the data better than a standard 2PLM.
Thus, it seems reasonable to assume that, for a subsample of test takers, the responses
obtained for end-of-test items reflected PD. In addition, in all of the mixture PD
models, the decline class proportions were higher in the non-academic group, but the
estimates of the magnitude of PD, such as the intercept parameters of items affected by PD
(2PDM), the response thresholds (HYBRID), and the decrement parameters (MPDM)
appeared to be similar between groups. Thus, all models indicated that groups differed
mainly in the proportions of students showing PD and not in the magnitude of PD.
Furthermore, in all models, group differences in the proportions of students showing PD
affected the results of the group differences in proficiency. Interestingly, all mixture PD
models adjusted the group differences provided by the 2PLM to a similar extent.
However, the reductions in the effect sizes were not large when considered on a relative scale;
this was due to the very strong track differences in proficiency. Nevertheless, in other
settings, where group differences in proficiencies are smaller, the adjustments provided
by the mixture PD models might lead to qualitatively different conclusions.
Additionally, all mixture PD models provided relatively similar estimates of item
parameters that were different from the parameters estimated by the standard 2PLM. As
expected, end-of-test items, that is, those items that were most strongly affected by PD,
were estimated to be less difficult than in the 2PLM, and the slope parameters belonging
to these items were estimated to be lower by the mixture PD models as compared to the
2PLM. These results are in line with the simulation study of Suh et al. (2012). However,
in contrast to Suh et al. (2012), we found that the mixture PD models provided higher
slope parameters for items positioned at the beginning of the test. One reasonable
explanation for this result is that individual differences in PD onset points not only increased
the dependencies between items at the end of the test, but also reduced the relationships
between items at the beginning of test. Hence, when PD was not accounted for, the slope
parameters of the 2PLM were estimated to account for the strong dependencies between
end-of-test items (i.e., higher slopes for items affected by PD) and the weaker
associations between items located at the beginning and the end of the test (i.e., lower slopes
for items not affected by PD). Of course, more research is clearly needed to examine the
plausibility of our explanation.
Despite the similarities, the three mixture PD models differed from each other in some
respects. All of the mixture PD models differed in their estimation of the number of
students showing PD: the decline class proportions were smallest in the 2PDM and
largest in the MPDM. However, the differences between the MPDM and the HYBRID were
rather subtle because the distribution of PD onset points differed only with respect to
the last five items. In addition, the mixture PD models provided different conclusions
about the relationships between PD onset points and proficiency. In the 2PDM, there
were no significant proficiency differences between decline and no-decline classes,
neither for students at the academic nor for students at the non-academic school tracks. For
both the HYBRID and the MPDM, the mean proficiency was lower in the decline classes
and it decreased when PD onset occurred nearer the beginning of the test. Furthermore,
in both cases, this decrease was stronger in the academic group. Similarly, the mixture
PD models differed in the adjustments of proficiency scores relative to the 2PLM. As
we have shown for the EAP scores, for students showing PD, the individual proficiency
scores were estimated to be higher in the mixture PD models than in the 2PLM, but the
EAP scores provided by the 2PDM were closer to those estimated by the 2PLM. Here,
the HYBRID and the MPDM provided EAP scores that were quite similar to each other.
Conclusions and future directions
Taken together, the HYBRID and the MPDM that both consist of many latent classes
performed rather similar in many respects, although they differ in their
representation of PD. Hence, the number of latent classes combined with the assumptions about
their distribution appears to be the main divider between the mixture PD models
envisaged in this article. The mixture PD models are based on different assumptions on how
PD affects test-taking behavior. In the 2PDM, it is assumed that the switching point is
identical for all test takers showing PD within a group. In contrast, in both the HYBRID
and the MPDM, multiple switching points are considered, but the models differ in their
assumptions about PD. In the HYBRID, it is assumed that the probability of a correct
response occurring after the switching point no longer depends on proficiency, whereas,
in the MPDM, it is assumed that responses given after the switching point still depend
on proficiency. This assumption is also embedded in the 2PDM. However, the
question of whether the specification of PD provided by different mixture models are of less
importance than the assumptions about the number of PD classes remains open, as
suggested by the results provided in this article. Therefore, further research on this issue is
In further applications, the suitability of the proposed model restrictions should be
analyzed thoroughly and adapted if necessary. For the HYBRID and the MPDM, we
assumed that there was a linear relationship between proficiency and switching points.
For some applications, this restriction might be too strict, for example, the decrease in
proficiency could be stronger for a switching point in earlier item positions and could
diminish when PD affects only the very last items. Moreover, the distribution of decline
classes could be modeled by functions other than the one proposed in our study. The
issue of model restrictions warrants consideration in subsequent research.
In our study, the HYBRID and the MPDM lead to similar results. Thus, both models
appear to be comparably well suited to explore PD in educational LSAs. However, more
research is needed on the similarities between these two models.
Finally, the variations of PD effects across domains should be explored. Research on
item position effects shows that the strength of the effects varies with respect to the
domain being studied
(Debeer and Janssen 2013; Nagy et al. 2016)
. Comparing PD in
several domains for the same population could also be useful in determining whether
PD can be regarded as an overarching person characteristic or, rather, as a test-specific
phenomenon. Mixture PD models with multiple decline classes provide estimates of
the switching point and its covariation with proficiency, and this allows for fine-grained
analyses of PD. In this sense, mixture PD models can be used to study how PD interacts
with proficiency and other variables to provide a better understanding of the
mechanisms behind PD.
Additional file 1. The Mplus code to analyze the data.
OL and GN provided the initial idea for the study. AR and GN provided the initial idea for the statistical analyses. MKL
conducted major parts of the statistical analyses. GN and AR conducted minor parts of the statistical analyses. MKL and
GN conducted the evaluation of the results. MKL reviewed the literature and wrote the manuscript. AR, OL, OK, and GN
read and revised the manuscript. All authors read and approved the final manuscript.
The authors would like to thank Gráinne Newcombe for editorial assistance with this article.
The authors declare that they have no competing interests.
Availability of data and materials
This paper uses data from the longitudinal study “Aspects of learning background and learning development”. The data
set was generated by the Free and Hanseatic City of Hamburg through the Ministry of Schools and Vocational Training
between 1995 and 2005 and have been provided to the MILES scientific consortium (Methodological Issues in
Longitudinal Educational Studies) for a limited period with the aim of conducting in-depth examinations of scientific questions.
MILES is coordinated by the Leibniz Institute for Science and Mathematics Education.
The Mplus code used to analyze our data is provided in Additional file 1.
There is no funding for this study.
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Akaike , H. ( 1987 ). Factor analysis and AIC . Psychometrika, 52 , 317 - 332 . doi: 10 .1007/BF02294359.
Baumert , J. , & Demmrich , A. ( 2001 ). Test motivation in the assessment of student skills: The effects of incentives on motivation and performance . European Journal of Psychology of Education , 16 , 441 - 462 . doi: 10 .1007/BF03173192.
Behörde für Schule und Berufsbildung . ( 2011 ). LAU-Aspekte der Lernausgangslage und der Lernentwicklung: Klassenstufen 5, 7 und 9 [LAU-Aspects of learning background and learning development: Grades 5, 7 and 9] . Münster: Waxmann.
Birnbaum , A. ( 1968 ). Some latent trait models and their use in inferring an examinee's ability . In F. M. Lord & M. R. Novick (Eds.), Statistical theories of mental test scores (pp. 397 - 549 ). Reading: Addison-Wesley.
Bolt , D. M. , Cohen , A. S. , & Wollack , J. A. ( 2002 ). Item parameter estimation under conditions of test speededness: Application of a mixture Rasch model with ordinal constraints . Journal of Educational Measurement , 39 , 331 - 348 . doi: 10 .1111/j.1745- 3984 . 2002 .tb01146.x.
Boughton , K. A. , & Yamamoto , K. ( 2007 ). A HYBRID model for test speededness . In M. von Davier & C. H. Carstensen (Eds.), Multivariate and mixture distribution rasch models: Extensions and applications (pp. 147 - 156 ). New York: Springer.
Cao , J. , & Stokes , S. L. ( 2008 ). Bayesian IRT guessing models for partial guessing behaviors . Psychometrika , 73 , 209 - 230 . doi: 10 .1007/s11336-007-9045-9.
Davey , T. , & Lee , Y.-H. ( 2011 ). Potential impact of context effects on the scoring and equating of the multistage GRE revised general test (Report No . GREB-08-01) . Princeton, NJ: Educational Testing Service. doi: 10 .1002/j.2333- 8504 . 2011 . tb02262.x.
De Boeck , P. , Cho , S.-J., & Wilson, M. ( 2011 ). Explanatory secondary dimension modeling of latent differential item functioning . Applied Psychological Measurement , 35 , 583 - 603 . doi: 10 .1177/0146621611428446.
De Boeck , P. , & Wilson, M. (Eds.). ( 2004 ). Explanatory item response models: A generalized linear and nonlinear approach . New York: Springer.
Debeer , D. , Buchholz , J. , Hartig , J. , & Janssen , R. ( 2014 ). Student, school, and country differences in sustained testtaking effort in the 2009 PISA reading assessment . Journal of Educational and Behavioral Statistics , 39 , 502 - 523 . doi: 10 .3102/1076998614558485.
Debeer , D. , & Janssen , R. ( 2013 ). Modeling item-position effects within an IRT framework . Journal of Educational Measurement , 50 , 164 - 185 . doi: 10 .1111/jedm.12009.
DeMars , C. E. ( 2000 ). Test stakes and item format interactions . Applied Measurement in Education , 13 , 55 - 77 . doi: 10 .1207/ s15324818ame1301_ 3 .
DeMars , C. E. , Bashkov , B. M. , & Socha , A. B. ( 2013 ). The role of gender in test-taking motivation under low-stakes conditions . Research & Practice in Assessment , 8 , 69 - 82 . Retrieved from http://www.rpajournal.com/.
Denis , P. L. , & Gilbert , F. ( 2012 ). The effect of time constraints and personality facets on general cognitive ability (GCA) assessment . Personality and Individual Differences , 52 , 541 - 545 . doi: 10 .1016/j.paid. 2011 . 11 .024.
Eklöf , H. ( 2010 ). Skill and will: test-taking motivation and assessment quality . Assessment in Education: Principles, Policy & Practice , 17 , 345 - 356 . doi: 10 .1080/0969594X. 2010 . 516569 .
Embretson , S. E. , & Reise , S. P. ( 2000 ). Item response theory for psychologists . Mahwah: Lawrence Erlbaum.
Finch , W. H. & French , B. F. ( 2012 ). Parameter estimation with mixture item response theory models: A Monte Carlo comparison of Maximum Likelihood and Bayesian methods . Journal of Modern Applied Statistical Methods , 11 , 167 - 178 . Retrieved from http://digitalcommons.wayne.edu/jmasm/vol11/iss1/14.
Finn , B. ( 2015 ). Measuring motivation in low-stakes assessments (Report No . RR-15-19) . Princeton, NJ: Educational Testing Service. doi: 10 .1002/ets2. 12067 .
Hailey , E. , Callahan , C. M. , Azano , A. , & Moon , T. R. ( 2012 ). An evaluation of test speededness in an assessment for thirdgrade gifted students . Journal of Advanced Academics , 23 , 292 - 304 . doi: 10 .1177/1932202X12462575.
Hambleton , R. K. , Swaminathan , H. , & Rogers , H. J. ( 1991 ). Fundamentals of item response theory . Newbury Park: Sage.
Hartig , J. , & Buchholz , J. ( 2012 ). A multilevel item response model for item position effects and individual persistence . Psychological Test and Assessment Modeling , 54 , 418 - 431 .
Jin , K.-Y., & Wang , W.-C. ( 2014 ). Item response theory models for performance decline during testing . Journal of Educational Measurement , 51 , 178 - 200 . doi: 10 .1111/jedm.12041.
Livingston , S. A. ( 2014 ). Equating test scores (without IRT) (2nd ed .). Princeton: Educational Testing Service.
Lubke , G. H. , & Muthén , B. ( 2005 ). Investigating population heterogeneity with factor mixture models . Psychological Methods , 10 , 21 - 39 . doi: 10 .1037/ 1082 - 989X . 10 .1.21.
Mislevy , R. J. , & Verhelst , N. ( 1990 ). Modeling item responses when different subjects employ different solution strategies . Psychometrika , 55 , 195 - 245 . doi: 10 .1007/BF02295283.
Mittelhaëuser , M.-A. , Béguin , A. A. , & Sijtsma , K. ( 2013 ). Modeling differences in test-taking motivation: Exploring the usefulness of the mixture Rasch model and person-fit statistics . In R. E. Millsap , L. A. van der Ark , D. M. Bolt , & C. M. Woods (Eds.), New developments in quantitative psychology (pp. 357 - 370 ). New York: Springer.
Mittelhaëuser , M.-A. , Béguin , A. A. , & Sijtsma , K. ( 2015 ). The effect of differential motivation on IRT linking . Journal of Educational Measurement , 52 , 339 - 358 . doi: 10 .1111/jedm.12080.
Muthén , L. K. , & Muthén , B. O. ( 1998 - 2012 ). Mplus user's guide (7th ed .). Los Angeles, CA: Muthén & Muthén.
Nagy , G. , Lüdtke , O. , & Köller , O. ( 2016 ). Modeling test context effects in longitudinal achievement data: Examining position effects in the longitudinal German PISA 2012 assessment . Psychological Test and Assessment Modeling , 58 , 641 - 670 .
Oshima , T. ( 1994 ). The effect of speededness on parameter estimation in item response theory . Journal of Educational Measurement , 31 , 200 - 219 . doi: 10 .1111/j.1745- 3984 . 1994 .tb00443.x.
Penk , C. , Pöhlmann , C. , & Roppelt , A. ( 2014 ). The role of test-taking motivation for students' performance in low-stakes assessments: An investigation of school-track-specific differences . Large-scale Assessments in Education. doi:10.1186/ s40536-014-0005-4.
Pietsch , M. , & Stubbe , T. C. ( 2007 ). Inequality in the transition from primary to secondary school: School choices and educational disparities in Germany . European Educational Research Journal , 6 , 424 - 445 . doi: 10 .2304/eerj. 2007 . 6 .4.424.
Prenzel , M. , Sälzer , C. , Klieme , E. , & Köller , O . (Eds.). ( 2013 ). PISA 2012 . Fortschritte und Herausforderungen in Deutschland [PISA 2012 . Advances and challenges in Germany]. Münster: Waxmann.
Rost , J. ( 1990 ). Rasch models in latent classes: An integration of two approaches to item analysis . Applied Psychological Measurement , 14 , 271 - 282 . doi: 10 .1177/014662169001400305.
Schnipke , D. L. , & Pashley , P. J. ( 1997 , March). Assessing subgroup differences in item response times . Paper presented at the Annual Meeting of the American Educational Research Association , Chicago, IL.
Schnipke , D. L. , & Scrams , D. J. ( 1997 ). Modeling item response times with a two-state mixture model: A new method of measuring speededness . Journal of Educational Measurement , 34 , 213 - 232 . doi: 10 .1111/j.1745- 3984 . 1997 .tb00516.x.
Schwarz , G. ( 1978 ). Estimating the dimension of a model . The Annals of Statistics , 6 , 461 - 464 .
Suh , Y. , Cho , S.-J., & Wollack , J. A. ( 2012 ). A comparison of item calibration procedure in the presence of test speededness . Journal of Educational Measurement , 49 , 285 - 311 . doi: 10 .1111/j.1745- 3984 . 2012 . 00176 .x.
van Barneveld , C. ( 2007 ). The effect of examinee motivation on test construction within an IRT framework . Applied Psychological Measurement, 31 , 31 - 46 . doi: 10 .1177/0146621606286206.
von Davier , M. , & Yamamoto , K. ( 2007 ). Mixture-distribution and HYBRID Rasch models . In M. von Davier & C. H. Carstensen (Eds.), Multivariate and mixture distribution Rasch models: Extensions and applications (pp. 99 - 115 ). New York: Springer.
Wise , S. L. , & DeMars , C. E. ( 2005 ). Low examinee effort in low-stakes assessment: Problems and potential solutions . Educational Assessment , 10 , 1 - 17 . doi: 10 .1207/s15326977ea1001_ 1 .
Wise , S. L. , & Kong , X. ( 2005 ). Response time effort: A new measure of examinee motivation in computer-based tests . Applied Measurement in Education , 18 , 163 - 183 . doi: 10 .1207/s15324818ame1802_ 2 .
Wise , S. L. , Pastor , D. A. , & Kong , X. J. ( 2009 ). Correlates of rapid-guessing behavior in low-stakes testing: Implications for test development and measurement practice . Applied Measurement in Education , 22 , 185 - 205 . doi: 10 .1080/08957340902754650.
Wollack , J. A. , Cohen , A. S. , & Wells , C. S. ( 2003 ). A method for maintaining scale stability in the presence of test speededness . Journal of Educational Measurement , 40 , 307 - 330 . doi: 10 .1111/j.1745- 3984 . 2003 .tb01149.x.
Wu , M. ( 2010 ). Measurement, sampling, and equating error in large-scale assessments . Educational Measurement: Issues and Practice , 29 ( 4 ), 15 - 27 . doi: 10 .1111/j.1745- 3992 . 2010 . 00190 .x.
Yamamoto , K. ( 1995 ). Estimating the effects of test length and test time on parameter estimation using the HYBRID model (Report No . TR-95-2) . Princeton, NJ: Educational Testing Service.
Yamamoto , K. , & Everson , H. ( 1995 ). Modeling the mixture of IRT and pattern responses by a modified hybrid model (Report No . RR-95-16) . Princeton, NJ: Educational Testing Service.
Yamamoto , K. , & Everson , H. ( 1997 ). Modeling the effects of test length and test time on parameter estimation using the hybrid model . In J. Rost & R. Langeheine (Eds.) , Applications of latent trait and latent class models in the social sciences (pp . 89 - 98 ). New York: Waxmann.