# Parameter estimation in generalised logistic model with application to DIF
detection

Adéla Hladká1,2, Patrícia Martinková1,3, Marek Brabec1
1 Institute of Computer Science of the Czech Academy of Sciences, Prague,
Czech Republic
2 Faculty of Mathematics and Physics, Charles University, Prague, Czech
Republic
3 Faculty of Education, Charles University, Prague, Czech Republic

###### Abstract

This paper proposes innovations to parameter estimation in a generalised
logistic regression model in the context of detecting differential item
functioning in multi-item measurements. The two newly proposed iterative
algorithms are compared with existing methods in a simulation study, and their
use is demonstrated in a real data example. Additionally the study examines
software implementation including specification of initial values for
iterative algorithms, and asymptotic properties with estimation of standard
errors. Overall, the proposed methods gave comparable results to existing ones
and were superior in some scenarios.

##  1 Introduction

Logistic regression <e.g.,¿agresti2010analysis is one of the most popular
tools for describing item functioning in multi-item measurements. This method
might be used in various contexts, including educational measurements,
admission tests, cognitive assessments, and other health-related inventories.
However, its broad applications are not limited to behavioural disciplines. In
the context of psychometrics, logistic regression can be seen as a score-based
counterpart to the 2 Parameter Logistic (PL) Item Response Theory (IRT) model
Birnbaum (1968), since in contrast to this class of generalised linear mixed
effect models, this method uses an observed estimate of the underlying latent
trait.

Furthermore, logistic regression has become a widely used method for
identifying between-group differences on item level when responding to multi-
item measurements Swaminathan Rogers (1990). The phenomenon which is known as
Differential Item Functioning (DIF) indicates whether responses to an item
vary for respondents with the same level of an underlying latent trait but
from different social groups (e.g., defined by gender, age, or socio-economic
status). In this vein, DIF detection is essential for a deeper understanding
of group differences, for assessing effectiveness of various treatments, or
for uncovering potential unfairness in educational tests. Beyond the logistic
regression, various psychometrics and statistical methods have been proposed
for an important task of DIF identification which are still being studied
extensively Schneider . (2021); Schauberger Tutz (2016); Paek Fukuhara (2015);
Suh Bolt (2011).

A natural extension of the logistic regression model to describe item
functioning is a generalised logistic regression model, which may account for
the possibility that an item can be correctly answered or endorsed without the
necessary knowledge or trait. In this case, the model is extended by including
a parameter defining a lower asymptote of the probability curve, which may be
larger than zero. Similarly, the model can consider the possibility that an
item is incorrectly answered or opposed by a respondent with a high level of a
certain trait due to issues such as inattention or lack of time. That is, the
model includes an upper asymptote of the probability curve which may be lower
than one. Analogous to the logistic regression model’s being a counterpart to
the 2PL IRT model, generalised logistic regression models can be seen as
score-based counterparts to 3–4PL IRT models Birnbaum (1968); Barton Lord
(1981).

The estimation in the logistic regression model is a straightforward
procedure, but including additional parameters in this model makes it more
statistically and computationally challenging and demanding. Therefore, this
article examines innovations in the item parameter estimation for the
generalised logistic regression model in the context of DIF detection. This
work proposes novel iterative algorithms and compares the newly proposed
methods to existing ones in a simulation study.

To begin, Section 2 introduces generalised logistic regression models,
examining the estimation techniques. This section provides a detailed
description of two existing methods for parameter estimation (Nonlinear Least
Squares (NLS) and the Maximum Likelihood (ML) method) and their application to
fitting a generalised logistic regression model. Furthermore, this study
proposes a novel implementation of the Expectation-Maximisation (EM) algorithm
and a new approach based on a Parametric Link Function (PLF). Additionally,
this section provides asymptotic properties of the estimates, an estimation of
standard errors, and a software implementation including a specification of
starting values in iterative algorithms. Subsequently, Section 3 describes the
design and results of the simulation study. To illustrate differences and
challenges between the existing and newly proposed methods in practice, this
work provides a real data analysis of anxiety measure in Section 4. Finally,
Section 5 contains the discussion and conclusion remarks.

##  2 Methodology

###  2.1 Generalised logistic model for item functioning

Generalised logistic regression models are extensions of the logistic
regression model which may account for the possibility of guessing or
inattention when answering an item. The simple 4PL model describes functioning
of the item $i$, meaning the probability of endorsing item $i$ by respondent
$p$, by introducing four parameters:

$\displaystyle\pi_{pi}=\mathrm{P}(Y_{pi}=1|X_{p})=c_{i}+(d_{i}-c_{i})\
\frac{\exp(b_{i0}+b_{i1}X_{p})}{1+\exp(b_{i0}+b_{i1}X_{p})},$ (1)

with $X_{p}$ being an observed trait of respondent $p$.

##### Parameter interpretation.

All four parameters have an intuitive interpretation: The parameters $c_{i}$
and $d_{i}$ are the upper and lower asymptotes of the probability sigmoid
function $\pi_{pi}(x)$ since

$\displaystyle\lim_{x\to-\infty}\pi_{pi}(x)=c_{i}\quad\text{and}\quad\lim_{x\to\infty}\pi_{pi}(x)=d_{i},$

where $c_{i}\in[0,1],d_{i}\in[0,1]$ and $c_{i}<d_{i}$ if $b_{i1}>0$ and
$c_{i}>d_{i}$ otherwise. Evidently, with $c_{i}=0$ and $d_{i}=1$, this model
recovers a standard logistic regression for item $i$.

In the context of psychological and health-related assessments, the asymptotes
$c_{i}$ and $d_{i}$ may represent reluctance to admit difficulties due to
social norms. In educational testing, parameter $c_{i}$ can be interpreted as
the probability that the respondents guessed the correct answer without
possessing the necessary knowledge $X_{p}$, also known as a pseudo-guessing
parameter. On the other hand, $1-d_{i}$ can be viewed as the probability that
respondents were inattentive while their knowledge $X_{p}$ was sufficient
Hladká Martinková (2020) or as a lapse-rate Kingdom Prins (2016). Next,
parameter $b_{i0}$ is a shift parameter, a midpoint between two asymptotes
$c_{i}$ and $d_{i}$ related to the difficulty or easiness of item $i$.
Finally, parameter $b_{i1}$ is linked to a slope of the sigmoid curve
$\pi_{pi}(x)$, which is also called a discrimination of the respective item.

##### Adding covariates, group-specific 4PL model.

The simple model (1) can be further extended by incorporating additional
respondents’ characteristics. That is, instead of using a single variable
$X_{p}$ to describe item functioning, a vector of covariates
$\boldsymbol{X}_{p}=(1,X_{p1},\dots,X_{pk})$, $p=1,\dots,n$, is involved,
including the original observed trait and an intercept term. This process
produces extra parameters $\boldsymbol{b}_{i}=(b_{i0},\dots,b_{ik})$. Beyond
this, even asymptotes may depend on respondents’ characteristics
$\boldsymbol{Z}_{p}=(1,Z_{p1},\dots,Z_{pj})$, $p=1,\dots,n$, which are not
necessarily the same as $\boldsymbol{X}_{p}$. This, the general covariate-
specific 4PL model is of form

$\displaystyle\pi_{pi}=\mathrm{P}(Y_{pi}=1|\boldsymbol{X}_{p},\boldsymbol{Z}_{p})=\boldsymbol{Z}_{p}\boldsymbol{c}_{i}+(\boldsymbol{Z}_{p}\boldsymbol{d}_{i}-\boldsymbol{Z}_{p}\boldsymbol{c}_{i})\
\frac{\exp(\boldsymbol{X}_{p}\boldsymbol{b}_{i})}{1+\exp(\boldsymbol{X}_{p}\boldsymbol{b}_{i})},$
(2)

where $\boldsymbol{c}_{i}=(c_{i0},\dots,c_{ij})$ and
$\boldsymbol{d}_{i}=(d_{i0},\dots,d_{ij})$ are related asymptote parameters
for item $i$.

As a special case of the covariate-specific 4PL model (2), an additional
single binary covariate $G_{p}$ might be considered. This grouping variable
describes a respondent’s membership to a social group ($G_{p}=0$ for the
reference group and $G_{p}=1$ for the focal group). In other words, this
special case assumes $\boldsymbol{X}_{p}=(1,X_{p},G_{p})$ and
$\boldsymbol{Z}_{p}=(1,G_{p})$, which reduces the covariate-specific 4PL model
(2) to a group-specific form:

$\displaystyle\begin{split}\pi_{pi}=\ &\mathrm{P}(Y_{pi}=\
1|X_{p},G_{p})=c_{i}+c_{i\text{DIF}}G_{p}\\\
&+(d_{i}-d_{i\text{DIF}}G_{p}-c_{i}-c_{i\text{DIF}}G_{p})\
\frac{\exp(b_{i0}+b_{i1}X_{p}+b_{i2}G_{p}+b_{i3}G_{p}X_{p})}{1+\exp(b_{i0}+b_{i1}X_{p}+b_{i2}G_{p}+b_{i3}G_{p}X_{p})}.\end{split}$
(3)

The group-specific 4PL model (3) can be used for testing between-group
differences on the item level with a DIF analysis Hladká Martinková (2020).

In this model, $X_{p}$ is an observed variable describing the measured trait
of the respondent, such as anxiety, fatigue, quality of life, or math ability,
here called the matching criterion. In the context of the logistic regression
method for DIF detection, the total test score is typically used as the
matching criterion Swaminathan Rogers (1990). Other options for the matching
criterion include a pre-test score, a score on another test measuring the same
construct, or an estimate of the latent trait provided by an IRT model.

###  2.2 Estimation of item parameters

Numerous algorithms are available to estimate item parameters in the
covariate-specific 4PL model (2). First, this section describes two methods,
which may be directly implemented in the existing software: The NLS method and
the ML method. This study discusses the asymptotic properties of the
estimates. Next, the study introduces two newly proposed iterative algorithms,
which might improve implementation of the computationally demanding ML method:
The EM algorithm inspired by the work of dinse2011algorithm and an iterative
algorithm based on PLF.

####  2.2.1 Nonlinear least squares

The parameter estimates of the covariate-specific 4PL model (2) can be
determined using the NLS method Dennis . (1981), which is based on
minimisation of the Residual Sum of Squares (RSS) of item $i$ with respect to
item parameters $(\boldsymbol{b}_{i},\boldsymbol{c}_{i},\boldsymbol{d}_{i})$:

$\displaystyle\text{RSS}_{i}(\boldsymbol{b}_{i},\boldsymbol{c}_{i},\boldsymbol{d}_{i})=\sum_{p=1}^{n}\left[Y_{pi}-\pi_{pi}\right]^{2}=\sum_{p=1}^{n}\left[Y_{pi}-\boldsymbol{Z}_{p}\boldsymbol{c}_{i}-(\boldsymbol{Z}_{p}\boldsymbol{d}_{i}-\boldsymbol{Z}_{p}\boldsymbol{c}_{i})\frac{\exp(\boldsymbol{X}_{p}\boldsymbol{b}_{i})}{1+\exp(\boldsymbol{X}_{p}\boldsymbol{b}_{i})}\right]^{2},$
(4)

where $n$ is a number of respondents. Since the criterion function
$\text{RSS}_{i}(\boldsymbol{b}_{i},\boldsymbol{c}_{i},\boldsymbol{d}_{i})$ is
continuously differentiable with respect to item parameters
$(\boldsymbol{b}_{i},\boldsymbol{c}_{i},\boldsymbol{d}_{i})$, the minimiser
can be obtained when the gradient is zero. Thus, the minimisation process
involves a calculation of the first partial derivatives with respect to item
parameters $(\boldsymbol{b}_{i},\boldsymbol{c}_{i},\boldsymbol{d}_{i})$ and
finding a solution of relevant nonlinear estimating equations <e.g., ¿[Chapter
5]van2000asymptotic. Since $\boldsymbol{Z}_{p}\boldsymbol{c}_{i}$ and
$\boldsymbol{Z}_{p}\boldsymbol{d}_{i}$ asymptotes represent probabilities, it
is necessary to ensure that these expressions are kept in the interval of
$[0,1]$ which is accomplished using numerical approaches.

The asymptotic properties of the NLS estimator, such as consistency and
asymptotic distribution, can be derived under the classical set of regularity
conditions <e.g.,¿[Theorems 5.41 and 5.42; see also Appendix
A.1]van2000asymptotic. This study proposes a sandwich estimator (A1) which can
be used as a natural estimate of the asymptotic variance of the NLS estimate.

####  2.2.2 Maximum likelihood

The second option for estimating item parameters in the covariate-specific 4PL
model (2) is the ML method <e.g.,¿ren2019algorithm. Using a notation
$\phi_{pi}=\frac{\exp(\boldsymbol{X}_{p}\boldsymbol{b}_{i})}{1+\exp(\boldsymbol{X}_{p}\boldsymbol{b}_{i})}$,
the corresponding likelihood function for item $i$ has the following form:

$\displaystyle
L_{i}(\boldsymbol{b}_{i},\boldsymbol{c}_{i},\boldsymbol{d}_{i})=\prod_{p=1}^{n}$
$\displaystyle\left[\boldsymbol{Z}_{p}\boldsymbol{c}_{i}+(\boldsymbol{Z}_{p}\boldsymbol{d}_{i}-\boldsymbol{Z}_{p}\boldsymbol{c}_{i})\phi_{pi}\right]^{Y_{pi}}\left[1-\boldsymbol{Z}_{p}\boldsymbol{c}_{i}-(\boldsymbol{Z}_{p}\boldsymbol{d}_{i}-\boldsymbol{Z}_{p}\boldsymbol{c}_{i})\phi_{pi}\right]^{1-Y_{pi}},$

and the log-likelihood function is then given by

$\displaystyle
l_{i}(\boldsymbol{b}_{i},\boldsymbol{c}_{i},\boldsymbol{d}_{i})=\sum_{p=1}^{n}$
$\displaystyle\left\\{Y_{pi}\log(\boldsymbol{Z}_{p}\boldsymbol{c}_{i}+(\boldsymbol{Z}_{p}\boldsymbol{d}_{i}-\boldsymbol{Z}_{p}\boldsymbol{c}_{i})\phi_{pi})\right.$
$\displaystyle\ \left.+\
(1-Y_{pi})\log(1-\boldsymbol{Z}_{p}\boldsymbol{c}_{i}-(\boldsymbol{Z}_{p}\boldsymbol{d}_{i}-\boldsymbol{Z}_{p}\boldsymbol{c}_{i})\phi_{pi})\right\\}.$

The parameter estimates are obtained by a maximisation of the log-likelihood
function. Thus this study proceeds similarly to the logistic regression model,
except for a larger dimension of the parametric space. To find the maximiser
of the log-likelihood function
$l_{i}(\boldsymbol{b}_{i},\boldsymbol{c}_{i},\boldsymbol{d}_{i})$, the first
partial derivatives are set to zero and these so-called likelihood equations
must be solved. However, the solution of a system of the nonlinear equations
cannot be derived algebraically and needs to be numerically estimated using a
suitable iterative process.

Using van der Vaart’s 1998 Theorems 5.41 and 5.42, consistency and asymptotic
normality can be shown for the ML estimator, see Appendix A.2. Additionally,
the estimate of the asymptotic variance of the item parameters is an inverse
of the observed information matrix (A2).

####  2.2.3 EM algorithm

The ML method may be computationally demanding and iterative algorithms might
help in those situations. Inspired by the work of dinse2011algorithm, this
study adopts a version of the EM algorithm Dempster . (1977) for parameter
estimation in the covariate-specific 4PL model (2).

Next, the original problem can be reformulated using latent variables which
describe hypothetical responses status of test-takers Dinse (2011). In this
study’s setting, the work considers four mutually exclusive latent variables
($W_{pi1}$, $W_{pi2}$, $W_{pi3}$, $W_{pi4}$), where variable $W_{pij}=1$
indicates that respondent $p$ belongs in the category $j=1,\dots,4$ for an
item $i$, whereas $W_{pij}=0$ indicates that respondent does not belong in
this category.

In the context of educational, psychological, health-related, or other types
of multi-item measurement, the four categories can be interpreted as follows:
Categories 1 and 2 indicate whether a respondent who responded correctly to
item $i$ or endorsed it (i.e., $Y_{pi}=1$) was determined to do so
($W_{pi1}=1$, e.g., the respondent guessed correct answer while their
knowledge or ability was insufficient) or not ($W_{pi2}=1$, e.g., had a
sufficient knowledge or ability to answer correctly and did not guessed). On
the other hand, Categories 3 and 4 indicate whether the respondent who did not
respond correctly or did not endorse the item (i.e., $Y_{pi}=0$) was prone to
do so ($W_{pi3}=1$, e.g., did not have sufficient knowledge or ability) or not
($W_{pi4}=1$, e.g., incorrectly answered due to another reason such as
inattention or lack of time). Thus, the observed indicator $Y_{pi}$ and its
complement $1-Y_{pi}$ could be rewritten as $Y_{pi}=W_{pi1}+W_{pi2}$ and
$1-Y_{pi}=W_{pi3}+W_{pi4}$ (Figure 1).

ptObserved answer$Y_{pi}\in\left\\{0,1\right\\}$GuessingCategory
1$W_{pi1}=1$Category 2$W_{pi2}=1$InattentionCategory 3$W_{pi3}=1$Category
4$W_{pi4}=1$ptCorrect $Y_{pi}=1$ ptIncorrect $Y_{pi}=0$Yes NoNo Yes Figure 1:
Latent variables for EM algorithm

Let $\boldsymbol{Z}_{p}\boldsymbol{c}_{i}$ be the regressor-based probability
that the respondent was determined to respond to item $i$ correctly or endorse
it (Category 1), and let $\boldsymbol{Z}_{p}\boldsymbol{d}_{i}$ be the
regressor-based probability of the respondent not prone to respond correctly
or endorse item $i$ (Categories 1–3). Then
$\boldsymbol{Z}_{p}\boldsymbol{d}_{i}-\boldsymbol{Z}_{p}\boldsymbol{c}_{i}$
gives the regressor-based probability that the respondent was not determined
but prone to (Categories 2 and 3). Further, we denote $\phi_{pi}$ and
$1-\phi_{pi}$ – the probabilities to answer given item correctly (Category 2)
and incorrectly (Category 3), respectively, depending on the regressors
$\boldsymbol{X}_{p}$. Finally, the probability that the respondent did not
respond correctly and was not prone to do so is given by
$1-(\boldsymbol{Z}_{p}\boldsymbol{d}_{i}-\boldsymbol{Z}_{p}\boldsymbol{c}_{i})-\boldsymbol{Z}_{p}\boldsymbol{c}_{i}=1-\boldsymbol{Z}_{p}\boldsymbol{d}_{i}$
(Category 4). In summary, the expected values of the latent variables are then
given by the following terms

$\displaystyle\boldsymbol{Z}_{p}\boldsymbol{c}_{i},\ \
(\boldsymbol{Z}_{p}\boldsymbol{d}_{i}-\boldsymbol{Z}_{p}\boldsymbol{c}_{i})\phi_{pi},\
\
(\boldsymbol{Z}_{p}\boldsymbol{d}_{i}-\boldsymbol{Z}_{p}\boldsymbol{c}_{i})(1-\phi_{pi}),\
\ 1-\boldsymbol{Z}_{p}\boldsymbol{d}_{i},$

and the probability of correct response or endorsement is given by

$\displaystyle\mathrm{P}(Y_{pi}=1|\boldsymbol{X}_{p})=$ $\displaystyle\
\mathrm{P}(W_{pi1}+W_{pi2}=1|\boldsymbol{X}_{p})=\mathrm{P}(W_{pi1}=1|\boldsymbol{X}_{p})+\mathrm{P}(W_{pi2}=1|\boldsymbol{X}_{p})$
$\displaystyle=$ $\displaystyle\
\boldsymbol{Z}_{p}\boldsymbol{c}_{i}+(\boldsymbol{Z}_{p}\boldsymbol{d}_{i}-\boldsymbol{Z}_{p}\boldsymbol{c}_{i})\phi_{pi},$

which under the logistic model
$\phi_{pi}=\frac{\exp(\boldsymbol{X}_{p}\boldsymbol{b}_{i})}{1+\exp(\boldsymbol{X}_{p}\boldsymbol{b}_{i})}$
produces the covariate-specific 4PL model (2).

Using the setting of the latent variables, the corresponding log-likelihood
function for item $i$ takes the following form:

$\displaystyle l_{i}^{\text{EM}}=$
$\displaystyle\sum_{p=1}^{n}\left[W_{pi2}\log\left(\phi_{pi}\right)+W_{pi3}\log\left(1-\phi_{pi}\right)\right]$
$\displaystyle+\sum_{p=1}^{n}\left[W_{pi1}\log\left(\boldsymbol{Z}_{p}\boldsymbol{c}_{i}\right)+W_{pi4}\log\left(1-\boldsymbol{Z}_{p}\boldsymbol{d}_{i}\right)+\
\left(W_{pi2}+W_{pi3}\right)\log\left(\boldsymbol{Z}_{p}\boldsymbol{d}_{i}-\boldsymbol{Z}_{p}\boldsymbol{c}_{i}\right)\right]$
$\displaystyle=$ $\displaystyle\ l_{i1}^{\text{EM}}+l_{i2}^{\text{EM}}.$

The log-likelihood function $l_{i1}^{\text{EM}}$ includes only parameters
$\boldsymbol{b}_{i}$ and regressors $\boldsymbol{X}_{p}$, whereas the log-
likelihood function $l_{i2}^{\text{EM}}$ incorporates only parameters related
to the asymptotes of the sigmoid function and includes only regressors
$\boldsymbol{Z}_{p}$. Notably, the log-likelihood function
$l_{i1}^{\text{EM}}$ has a form of the log-likelihood function for the
logistic regression. However, in contrast to the logistic regression model, in
this setting it does not necessary hold that $W_{pi2}+W_{pi3}=1$ since the
correct answer could be guessed or the respondent could be inattentive,
producing $W_{pi2}+W_{pi3}=0$. The log-likelihood function
$l_{i2}^{\text{EM}}$ takes the form of the log-likelihood for multinomial data
with one trial and with the regressor-based probabilities
$\boldsymbol{Z}_{p}\boldsymbol{c}_{i}$,
$\boldsymbol{Z}_{p}\boldsymbol{d}_{i}-\boldsymbol{Z}_{p}\boldsymbol{c}_{i}$,
and $1-\boldsymbol{Z}_{p}\boldsymbol{d}_{i}$.

The EM algorithm estimates item parameters in two steps – expectation and
maximisation. These two steps are repeated until the convergence criterion is
met, such as until the change in log-likelihood is lower than a predefined
value.

##### Expectation.

At the E-step, conditionally on the item responses $Y_{pi}$ and the current
parameter estimate
$(\widehat{\boldsymbol{b}}_{i},\widehat{\boldsymbol{c}}_{i},\widehat{\boldsymbol{d}}_{i})$,
the estimates of latent variables are calculated as their expected values:

$\displaystyle\begin{aligned} \widehat{W}_{pi1}=&\
\frac{\boldsymbol{Z}_{p}\widehat{\boldsymbol{c}}_{i}Y_{pi}}{\boldsymbol{Z}_{p}\widehat{\boldsymbol{c}}_{i}+(\boldsymbol{Z}_{p}\widehat{\boldsymbol{d}}_{i}-\boldsymbol{Z}_{p}\widehat{\boldsymbol{c}}_{i})\widehat{\phi}_{pi}},\
\ \ &\widehat{W}_{pi2}=&\ Y_{pi}-\widehat{W}_{pi1},\\\ \widehat{W}_{pi4}=&\
\frac{\left(1-\boldsymbol{Z}_{p}\widehat{\boldsymbol{d}}_{i}\right)\left(1-Y_{pi}\right)}{1-\widehat{\boldsymbol{Z}_{p}\boldsymbol{c}}_{i}-(\boldsymbol{Z}_{p}\widehat{\boldsymbol{d}}_{i}-\boldsymbol{Z}_{p}\widehat{\boldsymbol{c}}_{i})\widehat{\phi}_{pi}},\
\ \ &\widehat{W}_{pi3}=&\ 1-Y_{pi}-\widehat{W}_{pi4}.\end{aligned}$ (5)

##### Maximisation.

At the M-step, conditionally on the current estimates of the latent variables
$\widehat{W}_{pi2}$ and $\widehat{W}_{pi3}$, the estimates of parameters
$\boldsymbol{b}_{i}$ maximise the log-likelihood function
$l_{i1}^{\text{EM}}$. The estimates $\widehat{\boldsymbol{c}}_{i}$ and
$\widehat{\boldsymbol{d}}_{i}$ are given by a maximisation of the log-
likelihood function $l_{i2}^{\text{EM}}$ conditionally on current estimates of
the latent variables $\widehat{W}_{pi1}$, $\widehat{W}_{pi2}$,
$\widehat{W}_{pi3}$, and $\widehat{W}_{pi4}$.

The EM algorithm is designed to gain the ML estimates of the item parameters,
so estimates have the same asymptotic properties as described above.

####  2.2.4 Parametric link function

In this study’s setting, the covariate-specific 4PL model (2) can be viewed as
a generalised linear model with a known PLF

$\displaystyle
g(\mu_{pi};\boldsymbol{c}_{i},\boldsymbol{d}_{i})=\log\left(\frac{\mu_{pi}-\boldsymbol{Z}_{p}\boldsymbol{c}_{i}}{\boldsymbol{Z}_{p}\boldsymbol{d}_{i}-\mu_{pi}}\right),$
(6)

where the parameters $\boldsymbol{c}_{i}$ and $\boldsymbol{d}_{i}$ are unknown
and may depend on regressors $\boldsymbol{Z}_{p}$. Subsequently, the mean
function is determined by $\mu_{pi}=\pi_{pi}$ as given by (2) with a linear
predictor $\boldsymbol{X}_{p}\boldsymbol{b}_{i}$.

Keeping this setting in mind, this study proposes a new two-stage algorithm to
estimate item parameters using the PLF (6), which involves repeating two steps
until the convergence criterion is fulfilled.

##### Step one.

First, conditionally on current estimates $\widehat{\boldsymbol{c}}_{i}$ and
$\widehat{\boldsymbol{d}}_{i}$ of the PLF, the estimates of parameters
$\boldsymbol{b}_{i}$ maximise the following log-likelihood function:

$\displaystyle
l_{i1}^{\text{PL}}(\boldsymbol{b}_{i}|\widehat{\boldsymbol{c}}_{i},\widehat{\boldsymbol{d}}_{i})=\sum_{p=1}^{n}$
$\displaystyle\left\\{Y_{pi}\log(\boldsymbol{Z}_{p}\widehat{\boldsymbol{c}}_{i}+(\boldsymbol{Z}_{p}\widehat{\boldsymbol{d}}_{i}-\boldsymbol{Z}_{p}\widehat{\boldsymbol{c}}_{i})\phi_{pi})\right.$
$\displaystyle\ \ \left.+\
(1-Y_{pi})\log(1-\boldsymbol{Z}_{p}\widehat{\boldsymbol{c}}_{i}-(\boldsymbol{Z}_{p}\widehat{\boldsymbol{d}}_{i}-\boldsymbol{Z}_{p}\widehat{\boldsymbol{c}}_{i})\phi_{pi})\right\\}.$

The log-likelihood function
$l_{i1}^{\text{PL}}(\boldsymbol{b}_{i}|\widehat{\boldsymbol{c}}_{i},\widehat{\boldsymbol{d}}_{i})$
has a similar form to the log-likelihood function
$l_{i}(\boldsymbol{b}_{i},\boldsymbol{c}_{i},\boldsymbol{d}_{i})$ using the ML
method. However, the parameters $\boldsymbol{c}_{i}$ and $\boldsymbol{d}_{i}$
are here replaced by their current estimates, $\widehat{\boldsymbol{c}}_{i}$
and $\widehat{\boldsymbol{d}}_{i}$.

##### Step two.

Next, estimates $\widehat{\boldsymbol{c}}_{i}$ and
$\widehat{\boldsymbol{d}}_{i}$ of the PLF (6) are calculated conditionally on
the current estimates $\widehat{\boldsymbol{b}}_{i}$ as the arguments of the
maxima of the following log-likelihood function

$\displaystyle
l_{i2}^{\text{PL}}(\boldsymbol{c}_{i},\boldsymbol{d}_{i}|\widehat{\boldsymbol{b}}_{i})=\sum_{p=1}^{n}$
$\displaystyle\left\\{Y_{pi}\log(\boldsymbol{Z}_{p}\boldsymbol{c}_{i}+(\boldsymbol{Z}_{p}\boldsymbol{d}_{i}-\boldsymbol{Z}_{p}\boldsymbol{c}_{i})\widehat{\phi}_{pi})\right.$
$\displaystyle\left.\ \ +\
(1-Y_{pi})\log(1-\boldsymbol{Z}_{p}\boldsymbol{c}_{i}-(\boldsymbol{Z}_{p}\boldsymbol{d}_{i}-\boldsymbol{Z}_{p}\boldsymbol{c}_{i})\widehat{\phi}_{pi})\right\\}.$

Again, the parameters $\boldsymbol{b}_{i}$ are replaced by their estimates
$\widehat{\boldsymbol{b}}_{i}$, and $\phi_{pi}$ is thus replaced by
$\widehat{\phi}_{pi}$.

In summary, the division into the two sets of parameters makes the algorithm
based on PLF easy to implement in the R software and can take an advantage of
its existing functions. Because the algorithm is designed to produce the ML
estimates, their asymptotic properties are the same as described above.

###  2.3 Implementation and software

For all analyses, software R, version 4.1 R Core Team (2022) was used. The NLS
method was implemented using the base nls() function and the "port" algorithm
Gay (). The sandwich estimator (A1) of the asymptotic covariance matrix was
computed using the calculus package Guidotti (2022). The ML estimation was
performed with the base optim() function and the "L-BFGS-B" algorithm Byrd .
(1995). The EM algorithm implements directly (5) in the expectation step using
the base glm() function and the multinom() function from the nnet package
Venables Ripley (2002) in the maximisation step. Next, the step one of the
newly proposed algorithm based on PLF is implemented with the base glm()
function with the modified logit link, which includes asymptote parameters.
The estimation of the asymptote parameters in the step two is conducted using
the base optim() function. The maximum number of iterations was set to 2,000
for all four methods, and the convergence criterion was set to $10^{-6}$ when
possible.

##### Initial values.

Starting values for item parameters were calculated as follows: The
respondents were divided into three groups based upon tertiles of the matching
criterion $X_{p}$. Next, the asymptote parameters were estimated: $c$ was
computed as an empirical probability for those whose matching criterion was
smaller than its average value in the first group defined by tertiles. The
asymptote $d$ was calculated as an empirical probability of those whose
matching criterion was greater than its average value in the last group
defined by tertiles. The slope parameter $b_{1}$ was estimated as a difference
between mean empirical probabilities of the last and the first group
multiplied by 4. This difference is sometimes called upper-lower index.
Finally, the intercept $b_{0}$ was calculated as follows: First, a centre
point between the asymptotes was computed, and then we looked for the level of
the matching criterion which would have corresponded to this empirical
probability. Additionally, smoothing and corrections for variability of the
matching criterion were applied.

##  3 Simulation study

A simulation study was performed to compare various procedures to estimate
parameters in the generalised logistic regression model, including the NLS,
the ML method, the EM algorithm, and the newly proposed algorithm based on
PLF. Two models were considered – the simple 4PL model (1) and the group-
specific 4PL model (3).

###  3.1 Simulation design

##### Data generation.

To generate data with the simple 4PL model (1), the following parameters were
used: $b_{0}=0$, $b_{1}=1.5$, $c=0.25$, and $d=0.9$. In the case of the group-
specific 4PL model (3), additionally $b_{2}=-1$, $b_{3}=0.5$,
$c_{\text{DIF}}=-0.15$, and $d_{\text{DIF}}=0.1$ were considered. Next, the
matching criterion $X_{p}$ was generated from the standard normal distribution
for all respondents. Binary responses were generated from the Bernoulli
distribution with the calculated probabilities based upon the chosen 4PL
model, true parameters, and the matching criterion variable. The sample size
was set to $n=500;1,000;2,500$; and $5,000$, i.e., $250$; $500$; $1,250$; and
$2,500$ per group in the case of the group-specific 4PL model (3). Each
scenario was replicated $1,000$ times.

##### Simulation evaluation.

To compare estimation methods, we first computed mean and median numbers of
iteration runs and the convergence status of the methods, meaning the
percentage of converged simulation runs; the percentage of runs which crashed
(caused error when fitting, e.g., due to singularities); and the percentage of
those which reached the maximum number of iterations without convergence.
Next, we selected only those simulation runs for which all four estimation
methods converged successfully and computed the mean parameter estimates,
together with parametric confidence intervals. When confidence intervals for
asymptote parameters exceeded their boundaries of 0 or 1, confidence intervals
were truncated at the boundary value.

###  3.2 Simulation results

##### Convergence status.

All four methods had low percentages of iterations that crashed for all sample
sizes in the simple 4PL model (1), but the rate was mildly increased in the
group-specific 4PL model (3) for the NLS method (4.3%) and for the algorithm
based on PLF (3.6%) when $n=500$. With the increasing sample size, convergence
issues disappeared. The EM algorithm struggled to converge in a predefined
number of iterations, especially for small sample sizes in both models.
Additionally, the method based on PLF reached the maximum limit of 2,000
iterations only in a small percentage of simulation runs when smaller sample
sizes were considered in the group-specific 4PL model (Table 1).

##### Number of iterations.

Furthermore, the methods differed in a number of iterations needed until the
estimation process successfully ended. The EM algorithm yielded the largest
mean and median numbers of iterations, which were somehow overestimated by
simulation runs which did not finish without convergence (i.e., the maximum
limit of 2,000 iterations was reached). The fewest iterations were needed for
the NLS method. As expected, all the methods required fewer simulation runs
when the simple 4PL model (1) was considered than in the group-specific 4PL
model (3). Beyond this, the number of iterations was decreasing with the
increasing sample size in both models for all the methods except the EM
algorithm, where the number of iterations was not monotone (Table 1).

In the group-based model (3) with a sample size of $n=500$, some of the
estimation procedures produced non-meaningful estimates of parameters
$b_{0}$–$b_{3}$ (absolute value over 100) despite successful convergence.
Those 11 simulations affected mean values significantly, so they were removed
from a computation of the mean estimates and their confidence intervals for
all four estimation methods. In those 11 simulations, non-meaningful estimates
were received twice for the NLS, three times for the ML, eight times for the
EM algorithm, and once for the PLF-based method.

Table 1: Convergence status and a number of iterations for the four estimation
methods

Simple 4PL model (1) Group-specific 4PL model (3) Method Convergence status
[%] Number of iterations Convergence status [%] Number of iterations Converged
Crashed DNF Mean Median Converged Crashed DNF Mean Median $n=500$ NLS 99.60
0.40 0.00 10.11 9.00 95.70 4.30 0.00 14.36 12.00 MLE 99.60 0.40 0.00 22.99
22.00 99.90 0.10 0.00 90.63 80.00 EM 84.70 0.00 15.30 684.83 361.00 89.30 0.10
10.60 627.13 354.00 PLF 99.80 0.20 0.00 43.73 20.00 95.90 3.60 0.50 131.48
35.50 $n=1,000$ NLS 100.00 0.00 0.00 7.98 7.00 99.60 0.40 0.00 11.07 10.00 MLE
100.00 0.00 0.00 21.40 21.00 100.00 0.00 0.00 80.25 75.00 EM 83.70 0.00 16.30
744.01 437.00 88.20 0.00 11.80 769.27 542.00 PLF 100.00 0.00 0.00 31.13 18.00
99.30 0.60 0.10 76.85 27.00 $n=2,500$ NLS 99.90 0.10 0.00 5.90 6.00 99.90 0.10
0.00 7.58 7.00 MLE 99.90 0.10 0.00 19.98 19.00 100.00 0.00 0.00 73.01 72.00 EM
92.60 0.20 7.20 634.03 475.50 90.90 0.00 9.10 695.99 499.00 PLF 100.00 0.00
0.00 17.77 16.00 100.00 0.00 0.00 38.08 19.50 $n=5,000$ NLS 99.90 0.10 0.00
5.07 5.00 99.80 0.20 0.00 6.02 6.00 MLE 100.00 0.00 0.00 19.21 19.00 100.00
0.00 0.00 69.81 69.00 EM 95.60 0.00 4.40 588.82 477.50 92.60 0.00 7.40 808.35
647.50 PLF 100.00 0.00 0.00 15.40 15.00 100.00 0.00 0.00 26.07 16.00
---

x
Note. DNF = did not finish, NLS = nonlinear least squares, MLE = maximum
likelihood estimation, EM = expectation-maximisation algorithm, PLF =
algorithm based on parametric link function.

##### Parameter estimates.

In the simple 4PL model (1), the smallest biases in estimates of parameters
$b_{0}$ and $b_{1}$ were gained by the PLF-based algorithm with the narrowest
confidence intervals when smaller sample sizes were considered ($n=500$ or
$n=1,000$). Additionally, in these scenarios, the NLS method yielded slightly
more biased estimates with wider confidence intervals. The precision of the
estimation improved for both parameters when the sample size increased in all
four methods, whereas differences between estimation procedures narrowed. The
precision of the estimates of the asymptote parameters $c$ and $d$ was similar
for all four methods, while the differences between estimation approaches were
small. The NLS and the EM algorithm provided slightly wider confidence
intervals for a small sample size of $n=500$ (Figure 2, Table A1).

Figure 2: Mean estimated parameters in the simple 4PL model (1) with
confidence intervals with respect to sample size; horizontal lines represent
true values of parameters

In the group-specific 4PL model (3), the PLF-based algorithm yielded the least
biased estimates of parameters $b_{0}$–$b_{3}$, especially for the smaller
sample sizes. On the other hand, the NLS method produced the most biased
estimates with somewhat wider confidence intervals in such scenarios. The ML
method provided less biased estimates than the NLS, but accompanied with wider
confidence intervals, even for a sample size of $n=1,000$. Similar to the
simple 4PL model (1), the differences in the precision of the parameter
estimates were narrowing with the increasing sample size, and all four
estimation approaches gave estimates close to the true values of the item
parameters (Figure 3, Table A2). The estimates of the asymptote parameters
$c$, $c_{\text{DIF}}$, $d$, and $d_{\text{DIF}}$ were similar for all four
methods. The EM algorithm provided slightly less biased mean estimates of the
asymptote parameters, but with slightly wider confidence intervals, especially
for $n=1,000$.

Figure 3: Mean estimated parameters in the group-specific 4PL model (3) with
confidence intervals with respect to sample size; horizontal lines represent
true values of parameters

##  4 Real data example

###  4.1 Data description

This study demonstrated the estimation on a real-data example of the PROMIS
Anxiety scale111http://www.nihpromis.org dataset. The dataset consisted of
responses to 29 Likert-type questions (1 = Never, 2 = Rarely, 3 = Sometimes, 4
= Often, and 5 = Always) from 766 respondents. Additionally, the dataset
included information on the respondents’ age (0 = Younger than 65 and 1 = 65
and older), gender (0 = Male and 1 = Female), and education (0 = Some college
or higher and 1 = High school or lower).

For this work, item responses were dichotomised as follows: 0 = Never (i.e.,
response $=1$ on original scale) or 1 = At least rarely (i.e., response $\geq
2$ on original scale). The overall level of anxiety was calculated as a
standardised sum of non-dichotomized item responses. This work considered the
simple 4PL model (1) and the group-specific 4PL model (3) using all four
estimation methods: NLS, ML, the EM algorithm, and the algorithm based on PLF.
In both models, the computed overall level of anxiety was used as the matching
criterion $X_{p}$. In the group-specific 4PL model (3), respondents’ genders
were included as the grouping variable $G_{p}$. Overall, there were 369 male
participants and 397 female participants.

###  4.2 Analysis design

The same approach used in the simulation study for computing starting values
was used for the analysis of the Anxiety dataset. In the case of convergence
issues, the initial values were re-calculated based on successfully converged
estimates using other methods.

In this study, item parameter estimates were computed and reported with their
confidence intervals. Confidence intervals of the asymptote parameters were
truncated at boundary values when necessary. Next, this work compared the
estimation methods by calculating the differences in fitted item
characteristic curves (i.e., estimated probabilities of endorsing the item) on
the matching criterion for both models. Finally, likelihood ratio tests were
performed to compare the two nested models (simple and group-specific) to
identify the DIF for all items and all four estimation methods. Significance
level of 0.05 was used for all the tests.

###  4.3 Results

##### Simple 4PL model.

The smallest differences between the four estimation methods and fitted item
characteristic curves in the simple 4PL model (1) were observed for item R8
(”I had a racing or pounding heart”; Figure 4(a)). The greatest differences
were observed for item R29 (”I had difficulty calming down”; Figure 4(b)). The
smallest overall differences were found between the EM algorithm and the
algorithm based on PLF, whereas the greatest overall differences were noted
between the NLS and the algorithm based on PLF. Beyond this, similar patterns
appeared in the estimated item parameters (Figure 5, Table
LABEL:tab:anxiety:pars_simple). Although the lower asymptotes were mostly
estimated at 0, the upper asymptotes were often estimated below 1, suggesting
a reluctance of the respondents to admit certain difficulties, such as those
due to social norms.

(a) Anxiety item R8

(b) Anxiety item R29

Figure 4: Estimated item characteristic curves for the simple 4PL model (1)
Figure 5: Item parameter estimates of the Anxiety items for the simple 4PL
model (1).

##### Group-specific 4PL model.

The smallest differences between the four estimation algorithms and the fitted
item characteristic curves in the group-specific 4PL model (3) were once more
observed in item R8 (”I had a racing or pounding heart”; Figure 6(a)). The
greatest differences were noticed in item R24 (”Many situations made me
worry”; Figure 6(b)). The smallest overall differences were found between the
EM algorithm and the algorithm based on PLF. The greatest overall differences
were observed between the NLS and the algorithm based on PLF, analogous to the
simple 4PL model. Additionally, similar patterns were seen in the estimated
item parameters (Figure 7, Table LABEL:tab:anxiety:pars_dif).

(a) Anxiety item R8

(b) Anxiety item R24

Figure 6: Estimated item characteristic curves for the group-specific 4PL
model (3) Figure 7: Item parameter estimates of the Anxiety items for the
group-specific 4PL model (3)

##### DIF detection.

Using the likelihood ratio test, the simple 4PL model (1) was rejected for
item R6 (”I was concerned about my mental health”), item R10 (”I had sudden
feelings of panic”), and item R12 (”I had trouble paying attention”) when
considering at least one estimation method (i.e., these items functioned
differently). While item R6 was identified as a DIF item by all four of the
estimation methods (all $p$-values $<0.004$), items R10 and R12 were only
identified as functioning differently with the NLS ($p$-value = 0.042) and
with the algorithm based on PLF ($p$-value = 0.047), respectively.

In item R6, there were no significant differences between the estimated
asymptotes, and DIF was caused by various intercepts and slopes of the two
genders (Figure 7). Male participants seemed to have a higher probability of
being concerned about their mental health at least rarely (original response
$\geq 2$) than female participants of the same overall anxiety level. This
difference was especially apparent for those with lower levels of anxiety,
whereas the differences between these two genders narrowed as the overall
anxiety level increased (Figure 8).

Figure 8: Estimated item characteristic curves for item R6 of the Anxiety
dataset for the group-specific 4PL model (3)

##  5 Discussion

This work explored novel approaches for estimating item parameters in the
generalised logistic regression model. We described in detail two existing
procedures (NLS and ML) and their applications for fitting the covariate-
specific 4PL model (2). Additionally, the study proposed two iterative
procedures (a procedure using the EM algorithm and a method based on PLF).
With a simulation study, we demonstrated satisfactory precision of the newly
proposed PLF-based procedure even for small sample sizes and when additional
covariates were considered. However, these pleasant properties were not
observed for the NLS and ML methods, which produced either biased estimates or
wide confidence intervals. On the other hand, the EM algorithm performed
satisfactorily, but it sometimes failed to converge in a predefined number of
iterations, so its fitting was inefficient. As the sample size increased,
differences between the estimation methods vanished, and all estimates were
near the true values of the item parameters.

Using a real data example for the anxiety measure, we illustrated practical
challenges in estimation procedures, including specification of initial
values. The smallest differences between the estimation procedures were
observed for the EM algorithm and the procedure based on PLF. Additionally the
largest dissimilarities were found for the NLS and the PLF-based method,
supporting the findings of the simulation study, especially when considering a
smaller sample size in the Anxiety dataset.

In recent decades, the topic of the parametric link function has been
extensively discussed in the literature by many authors, including
basu2005estimating, flach2014generalized, and scallan1984fitting. For example,
pregibon1980goodness proposed the ML estimation of the link parameters using a
weighted least squares algorithm. In the same vein, mccullagh1989generalized
adapted this approach and presented an algorithm in which several models with
the fixed link functions were fitted. Furthermore, kaiser1997maximum proposed
a modified scoring algorithm to perform simultaneous ML estimation of all
parameters. scallan1984fitting proposed an iterative two-stage algorithm,
building on the work of richards1961method. In this study’s approach, we
examined generalised logistic regression, accounting for the possibility of
guessing and inattention or lapse rate, whereas these features may depend upon
the respondents’ characteristics.

The crucial part of the estimation process is specifying starting values for
item parameters because these values may significantly impact the speed of the
estimation process and its precision. For instance, initial values which are
far from the true item parameters may lead to situations in which the
estimation algorithm returns only a local extreme or even it does not
converge. In this work we used an approach based on upper-lower index which
resulted in a low rate of convergence issues with the satisfactory estimation
precision. However, other possible naive estimates of discrimination (and
other parameters), such as a correlation between an item score and the total
test score without given item, could be considered.

There were several limitations to this study, and several possible further
directions for study exist. First, the simulation study was limited to two
models – the simple 4PL model (1) and the group-specific 4PL model (3), both
of which included only one or two covariates. The simulation study suggested
requiring a larger sample size with the increasing number of covariates.
Second, the work considered only one set of item parameters, noting that
various values of asymptotes were especially prone to producing computational
challenges. Third, this article described the NLS method as a simple approach,
not accounting for the heteroscedasticity of binary data. For such data, the
Pearson’s residuals might be more appropriate to use. This weighted form
<e.g.,¿ritz2015dose takes the original squares of residuals and divides them
by the variance $\pi_{pi}(1-\pi_{pi})$. Next, the RSS of item $i$ (4) would
take the following form:

$\displaystyle\text{RSS}_{i}(\boldsymbol{\gamma}_{i})=\sum_{p=1}^{n}\frac{\left(Y_{pi}-\pi_{pi}\right)^{2}}{\pi_{pi}\left(1-\pi_{pi}\right)}.$

However, the number of observations on tails of the matching criterion is
typically small and provides only small variability at most. These heavy
weights would require a nearly exact fit for cases with few observations.
Nevertheless, the computation of the NLS estimates demonstrated in this work
was straightforward and efficient, providing sufficient precision. Thus, this
method could be useful in certain cases, such as producing an initial idea
about parameter values and using these estimates as starting values for other
approaches.

Additionally, this study’s real data example explored item functioning in the
multi-item measurement related to anxiety. However, the task of the parameter
estimation in the presented models would also be relevant for several other
situations. The later could include data from educational measurement, where
the lower asymptote may represent item guessing, and the upper asymptote may
represent the lapse rate (item slipping or inattention). Moreover, the
generalised logistic regression model is not limited to multi-item
measurements since the class determined by Equation (2) represents a wide
family of the covariate-specific 4PL models. This model might be used and
further extended in various study fields, including but not limited to
quantitative pharmacology Dinse (2011); applied microbiology Brands . (2020);
modelling patterns of urban electricity usage To . (2012); and plant growth
modelling Zub . (2012). Therefore, estimating parameters and understanding
limitations of used methods are crucial for a wide range of researchers and
practitioners.

To conclude, this study illustrated differences and challenges in fitting
generalised logistic regression models using various estimation techniques.
This work demonstrated the superiority of the novel implementation of the EM
algorithm and the newly proposed method based on PLF over the existing NLS and
ML methods. Thus, improving the estimation algorithms is critical since it
could increase precision while maintaining a user-friendly implementation.

## Acknowledgement

The study was funded by the Czech Science Foundation grant number 21-03658S.

## Supplementary Material

Additional tables and figures, and accompanying R scripts are available at
https://osf.io/bk8a7/.

## References

  * Agresti (2010) agresti2010analysisAgresti, A.  2010\.  Analysis of ordinal categorical data Analysis of ordinal categorical data (Second ).  John Wiley & Sons.  10.1002/9780470594001
  * Barton Lord (1981) barton1981upperBarton, MA. Lord, FM.  1981\.  An upper asymptote for the three-parameter logistic item-response model An upper asymptote for the three-parameter logistic item-response model.  ETS Research Report Series198111–8.  10.1007/s13398-014-0173-7.2
  * Basu Rathouz (2005) basu2005estimatingBasu, A. Rathouz, PJ.  2005\.  Estimating marginal and incremental effects on health outcomes using flexible link and variance function models Estimating marginal and incremental effects on health outcomes using flexible link and variance function models.  Biostatistics6193–109.  10.1093/biostatistics/kxh020
  * Birnbaum (1968) birnbaum1968statisticalBirnbaum, A.  1968\.  Some latent trait models and their use in inferring an examinee’s ability Some latent trait models and their use in inferring an examinee’s ability.  FM. Lord MR. Novick (), Statistical theories of mental test scores Statistical theories of mental test scores ( 397–479).  Addison-Wesley, Reading, MA.
  * Brands . (2020) brands2020methodBrands, B., Schulze Struchtrup, S., Stamminger, R. Bockmühl, DP.  2020\.  A method to evaluate factors influencing the microbial reduction in domestic dishwashers A method to evaluate factors influencing the microbial reduction in domestic dishwashers.  Journal of Applied Microbiology12851324–1338.  10.1111/jam.14564
  * Byrd . (1995) byrd1995limitedByrd, RH., Lu, P., Nocedal, J. Zhu, C.  1995\.  A limited memory algorithm for bound constrained optimization A limited memory algorithm for bound constrained optimization.  SIAM Journal on Scientific Computing1651190–1208.  10.1137/0916069
  * Dempster . (1977) dempster1977maximumDempster, AP., Laird, NM. Rubin, DB.  1977\.  Maximum likelihood from incomplete data via the EM algorithm Maximum likelihood from incomplete data via the EM algorithm.  Journal of the Royal Statistical Society: Series B (Methodological)3911–22.  10.1111/j.2517-6161.1977.tb01600.x
  * Dennis . (1981) dennis1981adaptiveDennis, JEJ., Gay, DM. Welsch, RE.  1981\.  An Adaptive Nonlinear Least-Squares Algorithm An adaptive nonlinear least-squares algorithm.  Transactions on Mathematical Software73348–368.  10.1145/355958.355965
  * Dinse (2011) dinse2011algorithmDinse, GE.  2011\.  An EM algorithm for fitting a four-parameter logistic model to binary dose-response data An EM algorithm for fitting a four-parameter logistic model to binary dose-response data.  Journal of Agricultural, Biological, and Environmental Statistics162221–232.  10.1007/s13253-010-0045-3
  * Flach (2014) flach2014generalizedFlach, N.  2014\.  Generalized linear models with parametric link families in R Generalized linear models with parametric link families in R .  MünchenTechnische Universität München, Department of Mathematics.
  * Gay () portGay, D.  .  Port library documentation. Port library documentation.  http://www.netlib.org/port/.  Accessed: 2022-12-13
  * Guidotti (2022) guidotti2022calculusGuidotti, E.  2022\.  calculus: High-Dimensional Numerical and Symbolic Calculus in R calculus: High-dimensional numerical and symbolic calculus in R.  Journal of Statistical Software10451–37.  10.18637/jss.v104.i05
  * Hladká Martinková (2020) hladka2020difnlrHladká, A. Martinková, P.  2020\.  difNLR: Generalized logistic regression models for DIF and DDF detection difNLR: Generalized logistic regression models for DIF and DDF detection.  The R Journal121300–323.  10.32614/RJ-2020-014
  * Hogg . (2018) hogg2005introductionHogg, RV., McKean, J. Craig, AT.  2018\.  Introduction to mathematical statistics Introduction to mathematical statistics (Eighth ).  Pearson Education.
  * Kaiser (1997) kaiser1997maximumKaiser, MS.  1997\.  Maximum likelihood estimation of link function parameters Maximum likelihood estimation of link function parameters.  Computational Statistics & Data Analysis24179–87.  10.1016/S0167-9473(96)00055-2
  * Kingdom Prins (2016) kingdom2016psychophysicsKingdom, FA. Prins, N.  2016\.  Psychophysics: A practical introduction Psychophysics: A practical introduction (Second ).  Academic Press.  10.1016/C2012-0-01278-1
  * McCullagh Nelder (1989) mccullagh1989generalizedMcCullagh, P. Nelder, JA.  1989\.  Generalized linear models Generalized linear models (Second ).  Chapman & Hall.
  * Paek Fukuhara (2015) peak2015investigationPaek, I. Fukuhara, H.  2015\.  An investigation of DIF mechanisms in the context of differential testlet effects An investigation of DIF mechanisms in the context of differential testlet effects.  British Journal of Mathematical and Statistical Psychology681142–157.  10.1111/bmsp.12039
  * Pregibon (1980) pregibon1980goodnessPregibon, D.  1980\.  Goodness of link tests for generalized linear models Goodness of link tests for generalized linear models.  Journal of the Royal Statistical Society: Series C (Applied Statistics)29115–24.  10.2307/2346405
  * R Core Team (2022) r2022R Core Team.  2022\.  R: A Language and Environment for Statistical Computing R: A language and environment for statistical computing [].  Vienna, Austria.  https://www.R-project.org/
  * Ren Xia (2019) ren2019algorithmRen, X. Xia, J.  2019\.  An algorithm for computing profile likelihood based pointwise confidence intervals for nonlinear dose-response models An algorithm for computing profile likelihood based pointwise confidence intervals for nonlinear dose-response models.  PLOS ONE141e0210953.  10.1371/journal.pone.0210953
  * Richards (1961) richards1961methodRichards, FS.  1961\.  A method of maximum-likelihood estimation A method of maximum-likelihood estimation.  Journal of the Royal Statistical Society: Series B (Methodological)232469–475.  10.1111/j.2517-6161.1961.tb00430.x
  * Ritz . (2015) ritz2015doseRitz, C., Baty, F., Streibig, JC. Gerhard, D.  2015\.  Dose-response analysis using R Dose-response analysis using R.  PLOS ONE1012e0146021.  10.1371/journal.pone.0146021
  * Scallan . (1984) scallan1984fittingScallan, A., Gilchrist, R. Green, M.  1984\.  Fitting parametric link functions in generalised linear models Fitting parametric link functions in generalised linear models.  Computational Statistics & Data Analysis2137–49.  10.1016/0167-9473(84)90031-8
  * Schauberger Tutz (2016) schauberger2016detectionSchauberger, G. Tutz, G.  2016\.  Detection of differential item functioning in Rasch models by boosting techniques Detection of differential item functioning in Rasch models by boosting techniques.  British Journal of Mathematical and Statistical Psychology69180–103.  10.1111/bmsp.12060
  * Schneider . (2021) schneider2021rSchneider, L., Strobl, C., Zeileis, A. Debelak, R.  2021\.  An R toolbox for score-based measurement invariance tests in IRT models An R toolbox for score-based measurement invariance tests in IRT models.  Behavior Research Methods1–13.  10.3758/s13428-021-01689-0
  * Suh Bolt (2011) suh2011nestedSuh, Y. Bolt, DM.  2011\.  A nested logit approach for investigating distractors as causes of differential item functioning A nested logit approach for investigating distractors as causes of differential item functioning.  Journal of Educational Measurement482188–205.  10.1111/j.1745-3984.2011.00139.x
  * Swaminathan Rogers (1990) swaminathan1990detectingSwaminathan, H. Rogers, HJ.  1990\.  Detecting differential item functioning using logistic regression procedures Detecting differential item functioning using logistic regression procedures.  Journal of Educational Measurement274361–370.  10.1111/j.1745-3984.1990.tb00754.x
  * To . (2012) to2012growthTo, W., Lai, T., Lo, W., Lam, K. Chung, W.  2012\.  The growth pattern and fuel life cycle analysis of the electricity consumption of Hong Kong The growth pattern and fuel life cycle analysis of the electricity consumption of Hong Kong.  Environmental Pollution1651–10.  10.1016/j.envpol.2012.02.007
  * van der Vaart (1998) van2000asymptoticvan der Vaart, AW.  1998\.  Asymptotic statistics Asymptotic statistics.  Cambridge University Press.  10.1017/CBO9780511802256
  * Venables Ripley (2002) venables2002modernVenables, WN. Ripley, BD.  2002\.  Modern applied statistics with S Modern applied statistics with S (Fourth ).  New YorkSpringer.  10.1007/978-0-387-21706-2
  * Zub . (2012) zub2012lateZub, H., Rambaud, C., Béthencourt, L. Brancourt-Hulmel, M.  2012\.  Late emergence and rapid growth maximize the plant development of Miscanthus clones Late emergence and rapid growth maximize the plant development of Miscanthus clones.  BioEnergy Research54841–854.  10.1007/s12155-012-9194-2

##  Appendix A Asymptotics

###  A.1 Nonlinear least squares

Asymptotic properties of the NLS estimator, such as consistency and asymptotic
distribution, can be derived under the classical set of regularity conditions
<e.g.,¿[Theorems 5.41 and 5.42]van2000asymptotic. Next, one must reformulate
these conditions for the covariate-specific 4PL model (2) considering
$\boldsymbol{\gamma}_{iX}=(\boldsymbol{b}_{i},\boldsymbol{c}_{i},\boldsymbol{d}_{i})$
is a vector of true parameters:

  1. [R0]

A vector of true parameters $\boldsymbol{\gamma}_{iX}$ satisfies

$\operatorname{\textsf{E}}\left(\boldsymbol{\psi}_{i}(Y_{pi},\boldsymbol{X}_{p};\boldsymbol{\gamma}_{iX})\right)=\operatorname{\textsf{E}}\left(-2(Y_{pi}-\pi_{pi})\frac{\partial\pi_{pi}}{\partial\boldsymbol{\gamma}_{iX}}\right)=\boldsymbol{0}.$

  2. [R1]

The true parameter $\boldsymbol{\gamma}_{iX}$ is an interior point of the
parameter space.

  3. [R2]

The function $\boldsymbol{\psi}_{i}(y,\boldsymbol{x};\boldsymbol{\gamma}_{i})$
is twice continuously differentiable with respect to $\boldsymbol{\gamma}_{i}$
for every $(y,\boldsymbol{x})$.

  4. [R3]

For each $\boldsymbol{\gamma}_{i}^{\ast}$ in a neighbourhood of
$\boldsymbol{\gamma}_{iX}$, there exists an integrable function
$\ddot{\boldsymbol{\psi}}(y,\boldsymbol{x})$ such that

$\displaystyle\left|\frac{\partial^{2}\psi_{ik}(y,\boldsymbol{x};\boldsymbol{\gamma}_{i})}{\partial\gamma_{ij}\partial\gamma_{il}}\right|\leq\ddot{\boldsymbol{\psi}}(y,\boldsymbol{x}),\
\forall k,j,l.$

  5. [R4]

The matrix

$\displaystyle\mathbb{\Gamma}_{i}(\boldsymbol{\gamma}_{i})=$ $\displaystyle\
\operatorname{\textsf{E}}\left(\frac{\partial\boldsymbol{\psi}_{i}(Y_{pi},\boldsymbol{X}_{p};\boldsymbol{\gamma}_{i})}{\partial\boldsymbol{\gamma}_{i}^{\top}}\right)$
$\displaystyle=$ $\displaystyle\
2\operatorname{\textsf{E}}\left(\frac{\partial\pi_{pi}}{\partial\boldsymbol{\gamma}_{i}}\left(\frac{\partial\pi_{pi}}{\partial\boldsymbol{\gamma}_{i}}\right)^{\top}-(Y_{pi}-\pi_{pi})\frac{\partial^{2}\pi_{pi}}{\partial\boldsymbol{\gamma}_{i}\partial\boldsymbol{\gamma}_{i}^{\top}}\right)$
$\displaystyle=$ $\displaystyle\
2\operatorname{\textsf{E}}\left(\frac{\partial\pi_{pi}}{\partial\boldsymbol{\gamma}_{i}}\left(\frac{\partial\pi_{pi}}{\partial\boldsymbol{\gamma}_{i}}\right)^{\top}\right)$

is finite and regular in a neighbourhood of $\boldsymbol{\gamma}_{iX}$.

  6. [R5]

The variance matrix

$\displaystyle\mathbb{\Sigma}_{i}(\boldsymbol{\gamma}_{i})=$ $\displaystyle\
\operatorname{\textsf{E}}\left(\boldsymbol{\psi}_{i}(Y_{pi},\boldsymbol{X}_{p};\boldsymbol{\gamma}_{i})\boldsymbol{\psi}_{i}^{\top}(Y_{pi},\boldsymbol{X}_{p};\boldsymbol{\gamma}_{i})\right)$
$\displaystyle=$ $\displaystyle\
4\operatorname{\textsf{E}}\left(\left(Y_{pi}-\pi_{pi}\right)^{2}\frac{\partial\pi_{pi}}{\partial\boldsymbol{\gamma}_{i}}\left(\frac{\partial\pi_{pi}}{\partial\boldsymbol{\gamma}_{i}}\right)^{\top}\right)$
$\displaystyle=$ $\displaystyle\
4\operatorname{\textsf{E}}\left(\pi_{pi}\left(1-\pi_{pi}\right)\frac{\partial\pi_{pi}}{\partial\boldsymbol{\gamma}_{i}}\left(\frac{\partial\pi_{pi}}{\partial\boldsymbol{\gamma}_{i}}\right)^{\top}\right)$

is finite for $\boldsymbol{\gamma}_{i}=\boldsymbol{\gamma}_{iX}$.

Specifically, Theorem 5.42 (van der Vaart, 1998, p. 68) implies that under the
conditions [R0]–[R5], the probability that the estimating equations,
$\frac{\partial\text{RSS}_{i}}{\partial\gamma_{ik}}=0$, have at least one root
tends to 1, as $n\rightarrow\infty$, and there exists a sequence
$\hat{\boldsymbol{\gamma}}_{i}$ (depending on $n$) such that
$\hat{\boldsymbol{\gamma}}_{i}\stackrel{{\scriptstyle[}}{{n}}\rightarrow\infty]{P}{\longrightarrow}\boldsymbol{\gamma}_{iX}$.
Moreover, the sequence $\hat{\boldsymbol{\gamma}}_{i}$ can be chosen to be a
local maximum for each $n$. Theorem 5.41 (van der Vaart, 1998, p. 68)
demonstrates that every consistent estimator $\hat{\boldsymbol{\gamma}}_{i}$
has asymptotically normal distribution, that is:

$\displaystyle\sqrt{n}\left(\hat{\boldsymbol{\gamma}}_{i}-\boldsymbol{\gamma}_{iX}\right)\stackrel{{\scriptstyle[}}{{n}}\rightarrow\infty]{\mathcal{D}}{\longrightarrow}\mathcal{N}\left(\boldsymbol{0},\mathbb{\Gamma}_{i}^{-1}(\boldsymbol{\gamma}_{iX})\mathbb{\Sigma}_{i}(\boldsymbol{\gamma}_{iX})\mathbb{\Gamma}_{i}^{-1}(\boldsymbol{\gamma}_{iX})\right),$

Clearly, the conditions [R0] and [R2] hold. To satisfy the condition [R1], one
must bound asymptote parameters $\boldsymbol{c}_{i}$ and $\boldsymbol{d}_{i}$
to open intervals. If the asymptote parameters are on the boundary of the
parameter space (e.g., $c_{i}=0$, $d_{i}=1$, and
$c_{i\text{DIF}}=d_{i\text{DIF}}=0$ in the group-specific 4PL model (3)), the
logistic regression model may be used instead. Additionally, the model (3)
with some of the parameters fixed (e.g., $d_{i}=1$ and $d_{i\text{DIF}}=0$),
may be considered analogously. Notably, the asymptotic properties derived here
will hold for these submodels. However, it is not possible to test whether the
full model or its submodel better fits the data using, for instance,
likelihood ratio test. Regarding the condition [R3], in this case, a
polynomial of $x$ of the fourth degree can be taken as an integrable
dominating function.

Furthermore, because $\boldsymbol{X}_{p}$ describes respondent characteristics
such as the standardised total score or gender, one can assume that their
range is bounded, so the partial derivatives
$\frac{\partial\pi_{pi}}{\partial\boldsymbol{\gamma}_{i}}$ are also bounded.
Thus, matrices $\mathbb{\Gamma}_{i}(\boldsymbol{\gamma}_{i})$ and
$\mathbb{\Sigma}_{i}(\boldsymbol{\gamma}_{i})$ are both finite, and the
condition [R5] holds. Finally, when the rows or columns of the matrix
$\mathbb{\Gamma}_{i}(\boldsymbol{\gamma}_{i})$ are linearly independent, the
matrix has a full rank and thus is regular, satisfying condition [R4]. For
instance, the singularity of the matrix may occur when $G_{p}=0,\forall p$ (or
$G_{p}=1,\forall p$), meaning that all respondents are from the reference (or
focal) group.

Therefore, all the assumptions [R1]–[R5] hold under the mild additional
conditions, so $\hat{\boldsymbol{\gamma}}_{i}$ has desired asymptotic
properties such as consistency and asymptotic normality of the item parameter
estimates.

##### Estimate of asymptotic variance.

The natural estimate of the asymptotic variance of
$\hat{\boldsymbol{\gamma}}_{i}$ is a sandwich estimator given by

$\displaystyle\frac{1}{n}\widehat{\mathbb{\Gamma}}_{in}^{-1}(\hat{\boldsymbol{\gamma}}_{i})\widehat{\mathbb{\Sigma}}_{in}(\hat{\boldsymbol{\gamma}}_{i})\widehat{\mathbb{\Gamma}}_{in}^{-1}(\hat{\boldsymbol{\gamma}}_{i}),$
(A1)

where

$\displaystyle\widehat{\mathbb{\Gamma}}_{in}(\hat{\boldsymbol{\gamma}}_{i})=$
$\displaystyle\
\frac{1}{n}\sum_{p=1}^{n}\left(\frac{\partial\boldsymbol{\psi}_{i}(Y_{pi},\boldsymbol{X}_{p};\hat{\boldsymbol{\gamma}}_{i})}{\partial\boldsymbol{\gamma}_{i}^{\top}}\right),$
$\displaystyle\widehat{\mathbb{\Sigma}}_{in}(\boldsymbol{\gamma}_{i})=$
$\displaystyle\
\frac{1}{n}\sum_{p=1}^{n}\left(\boldsymbol{\psi}_{i}(Y_{pi},\boldsymbol{X}_{p};\hat{\boldsymbol{\gamma}}_{i})\boldsymbol{\psi}_{i}^{\top}(Y_{pi},\boldsymbol{X}_{p};\hat{\boldsymbol{\gamma}}_{i})\right),$

with components of the matrix
$\widehat{\mathbb{\Gamma}}_{in}(\boldsymbol{\gamma}_{i})=\nabla^{2}\text{RSS}_{i}(\boldsymbol{\gamma}_{i})$
being

$\displaystyle\frac{\partial^{2}\text{RSS}_{i}(\boldsymbol{\gamma}_{i})}{\partial\gamma_{ik}\partial\gamma_{ij}}=$
$\displaystyle\
-2\sum_{p=1}^{n}\left\\{\left(Y_{pi}-\pi_{pi}\right)\frac{\partial^{2}\pi_{pi}}{\partial\gamma_{ik}\partial\gamma_{ij}}-\frac{\partial\pi_{pi}}{\partial\gamma_{ik}}\frac{\partial\pi_{pi}}{\partial\gamma_{ij}}\right\\},$
$\displaystyle\frac{\partial^{2}\text{RSS}_{i}(\boldsymbol{\gamma}_{i})}{\partial\gamma_{ik}^{2}}=$
$\displaystyle\
-2\sum_{p=1}^{n}\left\\{\left(Y_{pi}-\pi_{pi}\right)\frac{\partial^{2}\pi_{pi}}{\partial\gamma_{ik}^{2}}-\left(\frac{\partial\pi_{pi}}{\partial\gamma_{ik}}\right)^{2}\right\\}.$

###  A.2 Maximum likelihood

Asymptotic properties of the ML estimator can be shown under the set of the
following regularity conditions (van der Vaart, 1998, Theorems 5.41 and 5.42):

  * [R0∗]

The support set
$S=\left\\{y\in\mathbb{R}:f(y|\boldsymbol{x},\boldsymbol{\gamma}_{i})>0\right\\}$
does not depend on the parameter $\boldsymbol{\gamma}_{i}$.

  * [R1∗]

The true parameter $\boldsymbol{\gamma}_{iX}$ is an interior point of the
parameter space.

  * [R2∗]

The density
$f(y|\boldsymbol{x},\boldsymbol{\gamma}_{i})=y\log\left(\pi(\boldsymbol{x};\boldsymbol{\gamma}_{i})\right)+(1-y)\log\left(1-\pi(\boldsymbol{x};\boldsymbol{\gamma}_{i})\right)$
is twice continuously differentiable with respect to $\boldsymbol{\gamma}_{i}$
for each $(y,\boldsymbol{x})$.

  * [R3∗]

The Fisher information matrix $\mathbb{I}_{i}(\boldsymbol{\gamma}_{i})$ is
finite, regular, and positive definite in a neighbourhood of $\gamma_{iX}$.

  * [R4∗]

The order of differentiation and integration with respect to
$\boldsymbol{\gamma}_{i}$ can be interchanged for terms
$f(y|\boldsymbol{x},\boldsymbol{\gamma}_{i})$ and $\frac{\partial
f(y|\boldsymbol{x},\boldsymbol{\gamma}_{i})}{\partial\boldsymbol{\gamma}_{i}}$.

Overall, the conditions [R0∗] and [R2∗] hold. In this case, the regularity
condition for the ML estimator [R1∗] is the same as condition [R1] for the
NLS, so one must bound parameters of asymptotes to open intervals, as
discussed in Section A.1.

For the condition [R3∗], the Fisher information matrix takes the following
form:

$\displaystyle\mathbb{I}_{i}(\boldsymbol{\gamma}_{i})=\operatorname{\textsf{E}}\mathbb{I}_{i}(\boldsymbol{\gamma}_{i}|\boldsymbol{X}_{p})=\operatorname{\textsf{E}}\left(\frac{1}{\pi_{pi}\left(1-\pi_{pi}\right)}\frac{\partial\pi_{pi}}{\partial\gamma_{ik}}\frac{\partial\pi_{pi}}{\partial\gamma_{ij}}\right)_{k,j},$

which is a quadratic form and thus positive definite. Again,
$\boldsymbol{X}_{p}$ describes respondent characteristics, so one can assume
that their range is bounded, meaning partial derivatives
$\frac{\partial\pi_{pi}}{\partial\gamma_{ik}}$, making the Fisher information
matrix finite. Similarly, as in Section A.1, when the rows or columns of the
Fisher information matrix $\mathbb{I}_{i}(\boldsymbol{\gamma}_{i})$ are
linearly independent, the matrix has a full rank and thus is regular,
satisfying the condition [R3∗]. The singularity of the matrix could occur in
similar cases as for the matrix $\mathbb{\Gamma}_{i}(\boldsymbol{\gamma}_{i})$
described in Section A.1.

Finally, regarding the condition [R4∗], the order of differentiation and
integration can be interchanged by dominated convergence theorem, as far as
both $\frac{\partial
f(y|\boldsymbol{x},\boldsymbol{\gamma}_{i})}{\partial\boldsymbol{\gamma}_{i}}$
and
$\frac{\partial^{2}f(y|\boldsymbol{x},\boldsymbol{\gamma}_{i})}{\partial\boldsymbol{\gamma}_{i}\partial\boldsymbol{\gamma}_{i}^{\top}}$
are dominated by an integrable function. In this case a polynomial of $x$ of
the fourth degree can be taken as an integrable dominating function.

hogg2005introduction demonstrated that when the regularity conditions
[R0∗]–[R4∗] hold, there exists $n_{0}\in\mathbf{N}$ and a sequence
$\widehat{\boldsymbol{\gamma}}_{in}(n>n_{0})$ of solutions to the
corresponding likelihood equations such that

$\displaystyle\widehat{\boldsymbol{\gamma}}_{in}\stackrel{{\scriptstyle[}}{{n}}\rightarrow\infty]{P}{\longrightarrow}\boldsymbol{\gamma}_{iX},$

where $\boldsymbol{\gamma}_{iX}$ is a vector of true parameters. Because the
log-likelihood function is not strictly concave, the approach described does
not guarantee finding a unique solution of the corresponding likelihood
equations. Thus, there might be multiple solutions, each a local maximum.
However, there is one solution among them, which provides a consistent
sequence of estimators, whereas other solutions may not even be close to
$\gamma_{iX}$ and may not converge to it. Therefore, in practice, the key part
of estimating procedures is finding suitable starting values, preferably
easily calculated but consistent estimate of parameters. Furthermore, for this
consistent sequence of solutions it can be shown that

$\displaystyle\sqrt{n}\left(\widehat{\boldsymbol{\gamma}}_{in}-\boldsymbol{\gamma}_{iX}\right)\stackrel{{\scriptstyle[}}{{n}}\rightarrow\infty]{\mathcal{D}}{\longrightarrow}\mathcal{N}(\boldsymbol{0},\mathbb{I}^{-1}_{i}(\boldsymbol{\gamma}_{iX})).$

##### Estimate of asymptotic variance.

An estimate of the asymptotic variance of the item parameters
$\hat{\boldsymbol{\gamma}}_{i}$ is an inverse of the observed information
matrix, an inverse of the Hessian matrix as demonstrated here:

$\displaystyle\mathbb{I}_{in}^{-1}(\hat{\boldsymbol{\gamma}}_{i}|\boldsymbol{X},\boldsymbol{G})=\left(-\frac{1}{n}\frac{\partial^{2}l_{i}(\hat{\boldsymbol{\gamma}}_{i})}{\partial\boldsymbol{\gamma}_{i}\partial\boldsymbol{\gamma}_{i}^{\top}}\right)^{-1}.$
(A2)

##  Appendix B Tables

Table A1: Item parameter estimates with confidence intervals in parentheses by
four estimation methods using the simple 4PL model (1) with respect to sample
size

Parameter Method Sample size 500 1,000 2,500 5,000 $b_{0}=0$ NLS
$0.57(-1.50,2.65)$ $-0.01(-0.76,0.75)$ $0.00(-0.41,0.42)$ $0.00(-0.28,0.29)$
MLE $-0.00(-1.56,1.56)$ $-0.02(-0.72,0.67)$ $-0.00(-0.41,0.40)$
$0.00(-0.28,0.28)$ EM $-0.01(-1.56,1.55)$ $-0.02(-0.71,0.66)$
$-0.00(-0.41,0.40)$ $-0.00(-0.28,0.28)$ PLF $0.00(-1.07,1.07)$
$-0.02(-0.71,0.66)$ $-0.00(-0.41,0.40)$ $-0.00(-0.28,0.28)$ $b_{1}=1.5$ NLS
$4.32(-5.52,14.16)$ $1.81(-0.63,4.25)$ $1.58(0.85,2.30)$ $1.54(1.06,2.02)$ MLE
$2.29(-6.09,10.66)$ $1.77\hphantom{-}(0.27,3.27)$ $1.57(0.89,2.25)$
$1.53(1.07,1.99)$ EM $2.23(-6.12,10.58)$ $1.76\hphantom{-}(0.39,3.14)$
$1.58(0.90,2.26)$ $1.54(1.08,2.01)$ PLF $1.87(-1.68,\hphantom{0}5.42)$
$1.72\hphantom{-}(0.43,3.01)$ $1.58(0.89,2.27)$ $1.55(1.09,2.02)$ $c=0.25$ NLS
$0.25(0.00,0.55)$ $0.26(0.08,0.43)$ $0.25(0.14,0.36)$ $0.25(0.17,0.32)$ MLE
$0.25(0.00,0.52)$ $0.26(0.08,0.43)$ $0.25(0.14,0.36)$ $0.25(0.17,0.32)$ EM
$0.25(0.00,0.61)$ $0.26(0.08,0.43)$ $0.25(0.14,0.36)$ $0.25(0.18,0.32)$ PLF
$0.25(0.00,0.52)$ $0.26(0.08,0.43)$ $0.25(0.14,0.36)$ $0.25(0.18,0.32)$
$d=0.9$ NLS $0.89(0.62,1.00)$ $0.89(0.75,1.00)$ $0.90(0.81,0.99)$
$0.90(0.84,0.96)$ MLE $0.89(0.68,1.00)$ $0.90(0.74,1.00)$ $0.90(0.81,0.99)$
$0.90(0.84,0.96)$ EM $0.89(0.63,1.00)$ $0.89(0.75,1.00)$ $0.90(0.81,0.99)$
$0.90(0.84,0.96)$ PLF $0.90(0.67,1.00)$ $0.90(0.74,1.00)$ $0.90(0.81,0.99)$
$0.90(0.84,0.96)$
---

x
Note. NLS = nonlinear least squares, MLE = maximum likelihood estimation, EM =
expectation-maximisation algorithm, PLF = algorithm based on parametric link
function.

Table A2: Item parameter estimates with confidence intervals in parentheses by
four estimation methods using the group-specific 4PL model (3) with respect to
sample size

Parameter Method Sample size 500 1,000 2,500 5,000 $b_{0}=0$ NLS
$0.17(-5.37,5.72)$ $0.12(-1.62,1.86)$ $0.01(-0.62,0.63)$ $-0.01(-0.42,0.40)$
MLE $0.10(-9.01,9.21)$ $0.07(-6.59,6.73)$ $-0.00(-0.60,0.60)$
$-0.01(-0.42,0.39)$ EM $0.06(-2.15,2.26)$ $0.09(-1.69,1.86)$
$-0.00(-0.61,0.60)$ $-0.02(-0.43,0.40)$ PLF $-0.04(-2.81,2.73)$
$-0.01(-1.17,1.15)$ $-0.00(-0.63,0.63)$ $-0.02(-0.43,0.40)$ $b_{1}=1.5$ NLS
$2.97(-13.55,19.49)$ $2.73\hphantom{0}(-2.53,\hphantom{0}7.99)$
$1.62(0.18,3.06)$ $1.57(0.85,2.29)$ MLE $2.71(-13.13,18.55)$
$2.61(-30.03,35.24)$ $1.60(0.57,2.64)$ $1.56(0.88,2.24)$ EM
$2.49\hphantom{0}(-2.75,\hphantom{0}7.72)$
$2.66\hphantom{0}(-4.62,\hphantom{0}9.93)$ $1.61(0.51,2.72)$ $1.56(0.81,2.32)$
PLF $2.00\hphantom{0}(-2.18,\hphantom{0}6.18)$
$1.85\hphantom{0}(-0.83,\hphantom{0}4.53)$ $1.60(0.38,2.81)$ $1.57(0.77,2.38)$
$b_{2}=-1$ NLS $-1.35\hphantom{0}(-7.49,\hphantom{0}4.79)$ $-1.19(-3.11,0.73)$
$-1.02(-1.77,-0.27)$ $-1.00(-1.50,-0.50)$ MLE $-1.27(-27.09,24.55)$
$-1.15(-7.85,5.55)$ $-1.02(-1.73,-0.31)$ $-1.00(-1.49,-0.52)$ EM
$-1.14\hphantom{0}(-3.94,\hphantom{0}1.67)$ $-1.16(-3.07,0.76)$
$-1.02(-1.74,-0.30)$ $-1.00(-1.49,-0.51)$ PLF
$-1.01\hphantom{0}(-4.09,\hphantom{0}2.06)$ $-1.04(-2.40,0.32)$
$-1.01(-1.76,-0.27)$ $-0.99(-1.49,-0.50)$ $b_{3}=0.5$ NLS
$-0.21\hphantom{0}(-18.50,\hphantom{0}18.07)$
$-0.46\hphantom{0}(-6.01,\hphantom{0}5.09)$ $0.50(-1.09,2.09)$
$0.51(-0.34,1.36)$ MLE $-0.01(-107.94,107.93)$ $-0.37(-33.05,32.31)$
$0.50(-0.72,1.71)$ $0.51(-0.29,1.32)$ EM
$-0.07\hphantom{00}(-6.10,\hphantom{00}5.96)$
$-0.43\hphantom{0}(-7.84,\hphantom{0}6.98)$ $0.49(-0.83,1.81)$
$0.51(-0.40,1.42)$ PLF $0.36\hphantom{00}(-4.52,\hphantom{00}5.24)$
$0.34\hphantom{0}(-2.70,\hphantom{0}3.38)$ $0.50(-0.97,1.96)$
$0.49(-0.46,1.45)$ $c=0.25$ NLS $0.23(0.00,0.69)$ $0.24(0.00,0.52)$
$0.24(0.07,0.41)$ $0.25(0.14,0.36)$ MLE $0.23(0.00,0.76)$ $0.25(0.00,0.56)$
$0.24(0.06,0.42)$ $0.25(0.14,0.36)$ EM $0.23(0.00,0.68)$ $0.25(0.00,0.78)$
$0.24(0.08,0.41)$ $0.25(0.14,0.36)$ PLF $0.23(0.00,0.71)$ $0.24(0.00,0.57)$
$0.24(0.07,0.41)$ $0.25(0.13,0.37)$ $c_{\text{DIF}}=-0.15$ NLS
$-0.13(-0.60,0.35)$ $-0.14(-0.43,0.15)$ $-0.14(-0.32,0.05)$
$-0.14(-0.26,-0.03)$ MLE $-0.13(-0.67,0.42)$ $-0.14(-0.46,0.18)$
$-0.14(-0.32,0.05)$ $-0.14(-0.26,-0.03)$ EM $-0.14(-0.70,0.42)$
$-0.14(-0.68,0.39)$ $-0.14(-0.31,0.03)$ $-0.15(-0.26,-0.03)$ PLF
$-0.13(-0.63,0.37)$ $-0.14(-0.48,0.20)$ $-0.14(-0.32,0.04)$
$-0.15(-0.27,-0.02)$ $d=0.9$ NLS $0.90(0.51,1.00)$ $0.89(0.68,1.00)$
$0.90(0.77,1.00)$ $0.90(0.81,1.00)$ MLE $0.91(0.52,1.00)$ $0.90(0.65,1.00)$
$0.91(0.77,1.00)$ $0.90(0.81,1.00)$ EM $0.91(0.56,1.00)$ $0.89(0.56,1.00)$
$0.90(0.77,1.00)$ $0.90(0.80,1.00)$ PLF $0.91(0.53,1.00)$ $0.90(0.54,1.00)$
$0.91(0.74,1.00)$ $0.90(0.78,1.00)$ $d_{\text{DIF}}=0.1$ NLS
$0.06(-0.37,0.49)$ $0.09(-0.16,0.33)$ $0.08(-0.08,0.24)$ $0.08(-0.02,0.19)$
MLE $0.06(-0.43,0.55)$ $0.09(-0.22,0.40)$ $0.08(-0.09,0.25)$
$0.09(-0.03,0.20)$ EM $0.06(-0.42,0.55)$ $0.09(-0.30,0.47)$ $0.08(-0.10,0.26)$
$0.09(-0.05,0.22)$ PLF $0.06(-0.43,0.55)$ $0.09(-0.35,0.52)$
$0.08(-0.13,0.30)$ $0.09(-0.06,0.24)$
---

x
Note. NLS = nonlinear least squares, MLE = maximum likelihood estimation, EM =
expectation-maximisation algorithm, PLF = algorithm based on parametric link
function.

Table A3: Item parameters estimates with standard errors in parentheses by four estimation methods using the simple 4PL model (1) for the Anxiety dataset Item | Method | $b_{0}$ | $b_{1}$ | $c$ | $d$
---|---|---|---|---|---
R1 | NLS | $-0.64(0.20)$ | $2.59(0.31)$ | $0.00(0.01)$ | $0.94(0.03)$
| MLE | $-0.76(0.23)$ | $3.01(0.40)$ | $0.00(0.03)$ | $0.94(0.04)$
| EM | $-0.76(0.17)$ | $3.01(0.36)$ | $0.00(0.01)$ | $0.94(0.04)$
| PLF | $-0.89(0.21)$ | $2.69(0.38)$ | $0.00(0.03)$ | $0.98(0.03)$
R2 | NLS | $-0.76(0.21)$ | $2.39(0.33)$ | $0.00(0.01)$ | $0.91(0.04)$
| MLE | $-0.86(0.24)$ | $2.89(0.45)$ | $0.00(0.03)$ | $0.90(0.05)$
| EM | $-0.86(0.20)$ | $2.89(0.42)$ | $0.00(\text{N\kern-1.00006pt/\kern-1.00006ptA})$ | $0.90(0.05)$
| PLF | $-1.01(0.23)$ | $2.57(0.42)$ | $0.00(0.03)$ | $0.96(0.05)$
R3 | NLS | $-1.43(0.22)$ | $2.41(0.27)$ | $0.00(0.01)$ | $1.00(0.02)$
| MLE | $-1.55(0.24)$ | $2.69(0.28)$ | $0.00(0.02)$ | $1.00(0.02)$
| EM | $-1.55(0.24)$ | $2.69(0.28)$ | $0.00(0.02)$ | $1.00(0.02)$
| PLF | $-1.55(0.24)$ | $2.69(0.28)$ | $0.00(0.02)$ | $1.00(0.02)$
R4 | NLS | $1.50(0.32)$ | $4.13(0.55)$ | $0.00(0.02)$ | $0.95(0.02)$
| MLE | $1.17(0.47)$ | $3.87(0.82)$ | $0.00(0.05)$ | $0.98(0.04)$
| EM | $0.88(0.20)$ | $3.42(0.31)$ | $0.00(0.05)$ | $1.00(0.01)$
| PLF | $0.88(0.20)$ | $3.42(0.31)$ | $0.00(0.05)$ | $1.00(0.01)$
R5 | NLS | $-1.55(0.26)$ | $2.47(0.31)$ | $0.00(0.02)$ | $0.99(0.03)$
| MLE | $-1.63(0.27)$ | $2.87(0.33)$ | $0.00(0.03)$ | $0.97(0.02)$
| EM | $-1.63(0.15)$ | $2.87(0.25)$ | $0.00(\text{N\kern-1.00006pt/\kern-1.00006ptA})$ | $0.97(0.02)$
| PLF | $-1.63(0.27)$ | $2.87(0.33)$ | $0.00(0.03)$ | $0.97(0.03)$
R6 | NLS | $-0.79(0.23)$ | $2.23(0.25)$ | $0.00(0.03)$ | $1.00(0.02)$
| MLE | $-0.82(0.20)$ | $2.42(0.28)$ | $0.00(0.03)$ | $1.00(0.01)$
| EM | $-0.82(0.20)$ | $2.42(0.28)$ | $0.00(0.03)$ | $1.00(0.01)$
| PLF | $-0.82(0.20)$ | $2.42(0.28)$ | $0.00(0.03)$ | $1.00(0.01)$
R7 | NLS | $2.82(0.49)$ | $5.01(0.71)$ | $0.00(0.03)$ | $0.95(0.02)$
| MLE | $2.40(0.52)$ | $4.60(0.84)$ | $0.00(0.07)$ | $0.97(0.02)$
| EM | $1.75(0.24)$ | $3.70(0.36)$ | $0.00(0.07)$ | $1.00(0.01)$
| PLF | $1.75(0.24)$ | $3.70(0.36)$ | $0.00(0.07)$ | $1.00(0.01)$
R8 | NLS | $-0.42(0.23)$ | $2.00(0.42)$ | $0.00(0.04)$ | $0.88(0.05)$
| MLE | $-0.45(0.25)$ | $2.05(0.45)$ | $0.00(0.06)$ | $0.88(0.05)$
| EM | $-0.45(0.18)$ | $2.05(0.25)$ | $0.00(\text{N\kern-1.00006pt/\kern-1.00006ptA})$ | $0.88(0.05)$
| PLF | $-0.46(0.25)$ | $2.04(0.44)$ | $0.00(0.06)$ | $0.88(0.05)$
R9 | NLS | $0.34(0.29)$ | $2.26(0.34)$ | $0.00(0.03)$ | $0.85(0.04)$
| MLE | $0.41(0.33)$ | $2.52(0.45)$ | $0.00(0.07)$ | $0.84(0.04)$
| EM | $0.41(0.28)$ | $2.52(0.39)$ | $0.00(\text{N\kern-1.00006pt/\kern-1.00006ptA})$ | $0.84(0.04)$
| PLF | $0.19(0.30)$ | $2.25(0.41)$ | $0.00(0.07)$ | $0.89(0.04)$
R10 | NLS | $-1.39(0.19)$ | $2.65(0.31)$ | $0.00(0.01)$ | $0.98(0.03)$
| MLE | $-1.60(0.22)$ | $3.06(0.34)$ | $0.00(0.02)$ | $0.98(0.03)$
| EM | $-1.60(0.15)$ | $3.06(0.29)$ | $0.00(\text{N\kern-1.00006pt/\kern-1.00006ptA})$ | $0.98(0.03)$
| PLF | $-1.60(0.22)$ | $3.07(0.34)$ | $0.00(0.02)$ | $0.98(0.03)$
R11 | NLS | $0.31(0.24)$ | $2.41(0.40)$ | $0.00(0.04)$ | $0.89(0.04)$
| MLE | $0.27(0.26)$ | $2.47(0.48)$ | $0.00(0.07)$ | $0.90(0.04)$
| EM | $0.27(0.24)$ | $2.47(0.35)$ | $0.00(\text{N\kern-1.00006pt/\kern-1.00006ptA})$ | $0.90(0.04)$
| PLF | $0.11(0.25)$ | $2.27(0.45)$ | $0.00(0.07)$ | $0.93(0.04)$
R12 | NLS | $1.96(0.35)$ | $4.04(0.49)$ | $0.00(0.03)$ | $0.93(0.02)$
| MLE | $2.00(0.39)$ | $4.26(0.64)$ | $0.00(0.07)$ | $0.93(0.02)$
| EM | $0.99(0.28)$ | $2.80(0.56)$ | $0.00(0.08)$ | $1.00(0.00)$
| PLF | $0.99(0.28)$ | $2.80(0.56)$ | $0.00(0.08)$ | $1.00(0.00)$
R13 | NLS | $-0.31(0.21)$ | $2.07(0.27)$ | $0.00(0.03)$ | $0.94(0.04)$
| MLE | $-0.51(0.25)$ | $2.03(0.38)$ | $0.00(0.05)$ | $0.99(0.06)$
| EM | $-0.51(0.25)$ | $2.03(0.38)$ | $0.00(0.05)$ | $0.99(0.06)$
| PLF | $-0.50(0.25)$ | $2.04(0.38)$ | $0.00(0.05)$ | $0.98(0.06)$
R14 | NLS | $0.59(0.22)$ | $2.84(0.35)$ | $0.00(0.03)$ | $0.96(0.02)$
| MLE | $0.34(0.19)$ | $2.62(0.27)$ | $0.00(0.06)$ | $1.00(0.01)$
| EM | $0.34(0.19)$ | $2.62(0.27)$ | $0.00(0.06)$ | $1.00(0.01)$
| PLF | $0.34(0.19)$ | $2.62(0.27)$ | $0.00(0.06)$ | $1.00(0.01)$
R15 | NLS | $-0.77(0.19)$ | $2.20(0.30)$ | $0.00(0.02)$ | $0.96(0.04)$
| MLE | $-0.98(0.20)$ | $2.24(0.24)$ | $0.00(0.03)$ | $1.00(0.02)$
| EM | $-0.98(0.20)$ | $2.24(0.24)$ | $0.00(0.03)$ | $1.00(0.02)$
| PLF | $-0.98(0.20)$ | $2.24(0.24)$ | $0.00(0.03)$ | $1.00(0.02)$
R16 | NLS | $2.44(0.32)$ | $4.70(0.52)$ | $0.00(0.04)$ | $0.98(0.01)$
| MLE | $2.49(0.30)$ | $4.90(0.61)$ | $0.00(0.07)$ | $0.98(0.01)$
| EM | $2.03(0.18)$ | $4.23(0.10)$ | $0.00(0.06)$ | $1.00(\text{N\kern-1.00006pt/\kern-1.00006ptA})$
| PLF | $2.03(0.18)$ | $4.23(0.10)$ | $0.00(0.06)$ | $1.00(\text{N\kern-1.00006pt/\kern-1.00006ptA})$
R17 | NLS | $-2.49(0.35)$ | $2.12(0.31)$ | $0.00(0.01)$ | $0.97(0.06)$
| MLE | $-2.75(0.37)$ | $2.54(0.33)$ | $0.00(0.02)$ | $0.93(0.06)$
| EM | $-2.75(0.22)$ | $2.54(0.28)$ | $0.00(\text{N\kern-1.00006pt/\kern-1.00006ptA})$ | $0.93(0.06)$
| PLF | $-2.75(0.35)$ | $2.54(0.32)$ | $0.00(0.02)$ | $0.93(0.06)$
R18 | NLS | $1.49(0.44)$ | $3.14(0.43)$ | $0.00(0.04)$ | $0.89(0.03)$
| MLE | $1.51(0.39)$ | $3.37(0.50)$ | $0.00(0.08)$ | $0.90(0.03)$
| EM | $1.51(0.35)$ | $3.37(0.47)$ | $0.00(\text{N\kern-1.00006pt/\kern-1.00006ptA})$ | $0.90(0.03)$
| PLF | $0.90(0.28)$ | $2.59(0.42)$ | $0.00(0.08)$ | $0.97(0.02)$
R19 | NLS | $-1.38(0.19)$ | $2.42(0.26)$ | $0.00(0.01)$ | $1.00(0.03)$
| MLE | $-1.55(0.21)$ | $2.80(0.30)$ | $0.00(0.02)$ | $0.99(0.02)$
| EM | $-1.55(0.21)$ | $2.80(0.30)$ | $0.00(0.02)$ | $0.99(0.02)$
| PLF | $-1.55(0.21)$ | $2.80(0.30)$ | $0.00(0.02)$ | $0.99(0.02)$
R20 | NLS | $-0.80(0.18)$ | $2.72(0.36)$ | $0.00(0.02)$ | $0.98(0.02)$
| MLE | $-0.93(0.19)$ | $2.83(0.36)$ | $0.00(0.03)$ | $0.99(0.02)$
| EM | $-0.93(0.19)$ | $2.83(0.36)$ | $0.00(0.03)$ | $0.99(0.02)$
| PLF | $-0.93(0.19)$ | $2.84(0.36)$ | $0.00(0.03)$ | $0.99(0.03)$
R21 | NLS | $-0.14(0.32)$ | $1.72(0.37)$ | $0.00(0.04)$ | $0.78(0.07)$
| MLE | $-0.06(0.36)$ | $1.95(0.46)$ | $0.00(0.07)$ | $0.76(0.07)$
| EM | $-0.06(0.36)$ | $1.95(0.47)$ | $0.00(0.07)$ | $0.76(0.07)$
| PLF | $-0.12(0.35)$ | $1.89(0.45)$ | $0.00(0.07)$ | $0.77(0.07)$
R22 | NLS | $1.22(0.24)$ | $3.98(0.41)$ | $0.00(0.02)$ | $0.97(0.01)$
| MLE | $1.16(0.28)$ | $4.12(0.59)$ | $0.00(0.05)$ | $0.98(0.02)$
| EM | $0.95(0.20)$ | $3.77(0.36)$ | $0.00(0.05)$ | $1.00(0.01)$
| PLF | $0.95(0.20)$ | $3.77(0.36)$ | $0.00(0.05)$ | $1.00(0.01)$
R23 | NLS | $1.31(0.33)$ | $3.20(0.43)$ | $0.00(0.03)$ | $0.93(0.02)$
| MLE | $1.06(0.31)$ | $3.01(0.50)$ | $0.00(0.07)$ | $0.96(0.03)$
| EM | $0.70(0.21)$ | $2.49(0.28)$ | $0.00(0.07)$ | $1.00(0.01)$
| PLF | $0.70(0.21)$ | $2.49(0.28)$ | $0.00(0.07)$ | $1.00(0.01)$
R24 | NLS | $1.50(0.30)$ | $4.19(0.65)$ | $0.00(0.03)$ | $0.93(0.02)$
| MLE | $1.52(0.33)$ | $4.38(0.69)$ | $0.00(0.05)$ | $0.93(0.02)$
| EM | $1.52(0.33)$ | $4.38(0.53)$ | $0.00(0.00)$ | $0.93(0.02)$
| PLF | $0.93(0.25)$ | $3.48(0.51)$ | $0.00(0.05)$ | $0.98(0.01)$
R25 | NLS | $3.57(0.70)$ | $4.63(0.67)$ | $0.00(0.07)$ | $0.92(0.02)$
| MLE | $3.30(0.60)$ | $4.40(0.78)$ | $0.00(0.14)$ | $0.93(0.02)$
| EM | $3.30(0.59)$ | $4.41(0.72)$ | $0.00(\text{N\kern-1.00006pt/\kern-1.00006ptA})$ | $0.93(0.02)$
| PLF | $2.59(0.47)$ | $3.59(0.64)$ | $0.00(0.14)$ | $0.97(0.01)$
R26 | NLS | $2.74(0.56)$ | $4.97(0.72)$ | $0.00(0.02)$ | $0.94(0.02)$
| MLE | $2.55(0.45)$ | $4.96(0.68)$ | $0.00(0.06)$ | $0.95(0.02)$
| EM | $1.50(\text{N\kern-1.00006pt/\kern-1.00006ptA})$ | $3.45(\text{N\kern-1.00006pt/\kern-1.00006ptA})$ | $0.00(0.06)$ | $1.00(\text{N\kern-1.00006pt/\kern-1.00006ptA})$
| PLF | $1.50(\text{N\kern-1.00006pt/\kern-1.00006ptA})$ | $3.45(\text{N\kern-1.00006pt/\kern-1.00006ptA})$ | $0.00(0.06)$ | $1.00(\text{N\kern-1.00006pt/\kern-1.00006ptA})$
R27 | NLS | $1.35(0.29)$ | $4.05(0.57)$ | $0.00(0.03)$ | $0.96(0.02)$
| MLE | $0.93(0.19)$ | $3.64(0.33)$ | $0.00(0.05)$ | $1.00(0.01)$
| EM | $0.93(0.19)$ | $3.64(0.33)$ | $0.00(0.05)$ | $1.00(0.01)$
| PLF | $0.93(0.19)$ | $3.64(0.33)$ | $0.00(0.05)$ | $1.00(0.01)$
R28 | NLS | $2.78(0.42)$ | $5.21(0.79)$ | $0.00(0.04)$ | $0.97(0.01)$
| MLE | $2.60(0.38)$ | $5.06(0.71)$ | $0.00(0.06)$ | $0.98(0.01)$
| EM | $2.60(0.37)$ | $5.06(0.55)$ | $0.00(0.00)$ | $0.98(0.01)$
| PLF | $2.46(0.33)$ | $4.88(0.64)$ | $0.00(0.06)$ | $0.98(0.01)$
R29 | NLS | $-0.07(0.22)$ | $3.57(0.59)$ | $0.00(0.01)$ | $0.91(0.03)$
| MLE | $-0.13(0.27)$ | $4.08(0.64)$ | $0.00(0.03)$ | $0.91(0.04)$
| EM | $-0.57(0.19)$ | $2.98(0.28)$ | $0.00(0.04)$ | $1.00(0.01)$
| PLF | $-0.57(0.19)$ | $2.98(0.28)$ | $0.00(0.04)$ | $1.00(0.01)$
Table A4: Item parameters estimates with standard errors in parentheses by four estimation methods using the group-specific 4PL model (3) for the Anxiety dataset Item | Method | $b_{0}$ | $b_{1}$ | $b_{2}$ | $b_{3}$ | $c$ | $c_{\text{DIF}}$ | $d$ | $d_{\text{DIF}}$
---|---|---|---|---|---|---|---|---|---
R1 | NLS | $-0.68(0.32)$ | $2.69(0.44)$ | $0.08(0.40)$ | $-0.21(0.61)$ | $0.00(0.02)$ | $0.00(0.03)$ | $0.93(0.05)$ | $0.01(0.07)$
| MLE | $-0.73(0.36)$ | $3.39(0.61)$ | $-0.09(0.48)$ | $-0.83(0.83)$ | $0.00(0.04)$ | $0.00(0.07)$ | $0.92(0.06)$ | $0.07(0.09)$
| EM | $-0.73(0.22)$ | $3.39(0.49)$ | $-0.14(0.26)$ | $-0.95(0.53)$ | $0.00(\text{N\kern-1.00006pt/\kern-1.00006ptA})$ | $-0.00(\text{N\kern-1.00006pt/\kern-1.00006ptA})$ | $0.92(0.02)$ | $0.08(0.02)$
| PLF | $-0.73(0.31)$ | $3.39(0.51)$ | $-0.14(0.40)$ | $-0.95(0.56)$ | $0.00(0.04)$ | $0.00(0.07)$ | $0.91(\text{N\kern-1.00006pt/\kern-1.00006ptA})$ | $0.09(\text{N\kern-1.00006pt/\kern-1.00006ptA})$
R2 | NLS | $-0.80(0.36)$ | $2.47(0.64)$ | $0.07(0.44)$ | $-0.18(0.73)$ | $0.00(0.02)$ | $0.00(0.03)$ | $0.87(0.08)$ | $0.08(0.09)$
| MLE | $-0.70(0.48)$ | $3.52(0.96)$ | $-0.29(0.54)$ | $-1.17(1.00)$ | $0.00(0.04)$ | $0.00(0.06)$ | $0.81(0.08)$ | $0.19(0.09)$
| EM | $-0.70(0.27)$ | $3.52(0.65)$ | $-0.29(0.31)$ | $-1.17(0.68)$ | $0.00(\text{N\kern-1.00006pt/\kern-1.00006ptA})$ | $-0.00(\text{N\kern-1.00006pt/\kern-1.00006ptA})$ | $0.81(0.03)$ | $0.19(0.02)$
| PLF | $-0.95(0.34)$ | $3.01(0.54)$ | $-0.04(0.42)$ | $-0.66(0.59)$ | $0.00(0.04)$ | $0.00(0.06)$ | $0.88(0.03)$ | $0.12(0.02)$
R3 | NLS | $-1.26(0.30)$ | $2.30(0.39)$ | $-0.32(0.45)$ | $0.33(0.57)$ | $0.00(0.02)$ | $0.00(0.03)$ | $0.97(0.05)$ | $0.03(0.05)$
| MLE | $-1.42(0.34)$ | $2.64(0.52)$ | $-0.23(0.51)$ | $0.26(0.70)$ | $0.00(0.04)$ | $0.00(0.05)$ | $0.97(0.07)$ | $0.03(0.07)$
| EM | $-1.42(0.33)$ | $2.64(0.39)$ | $-0.23(0.50)$ | $0.26(0.60)$ | $0.00(0.04)$ | $-0.00(0.05)$ | $0.97(0.02)$ | $0.03(0.01)$
| PLF | $-1.42(0.33)$ | $2.65(0.39)$ | $-0.23(0.50)$ | $0.25(0.60)$ | $0.00(0.04)$ | $0.00(0.05)$ | $0.97(0.02)$ | $0.03(0.01)$
R4 | NLS | $1.32(0.55)$ | $3.58(0.76)$ | $0.40(0.68)$ | $1.29(1.15)$ | $0.00(0.03)$ | $0.00(0.05)$ | $0.94(0.04)$ | $0.02(0.04)$
| MLE | $0.77(0.29)$ | $3.08(0.40)$ | $1.02(0.61)$ | $2.12(1.30)$ | $0.00(0.09)$ | $0.00(0.11)$ | $1.00(0.01)$ | $-0.05(0.03)$
| EM | $0.77(0.29)$ | $3.08(0.39)$ | $0.21(0.39)$ | $0.70(0.61)$ | $0.00(0.09)$ | $-0.00(0.11)$ | $1.00(0.01)$ | $0.00(0.01)$
| PLF | $0.77(0.28)$ | $3.08(0.39)$ | $0.21(0.37)$ | $0.70(0.58)$ | $0.00(0.09)$ | $0.00(0.11)$ | $1.00(0.01)$ | $-0.00(\text{N\kern-1.00006pt/\kern-1.00006ptA})$
R5 | NLS | $-1.54(0.39)$ | $2.29(0.37)$ | $-0.04(0.54)$ | $0.39(0.63)$ | $0.00(0.02)$ | $0.00(0.03)$ | $1.00(0.03)$ | $-0.02(0.05)$
| MLE | $-1.67(0.37)$ | $2.61(0.39)$ | $0.01(0.56)$ | $0.47(0.64)$ | $0.00(0.04)$ | $0.00(0.06)$ | $1.00(0.03)$ | $-0.05(0.04)$
| EM | $-1.67(0.21)$ | $2.61(0.30)$ | $0.01(0.30)$ | $0.47(0.48)$ | $0.00(0.00)$ | $-0.00(0.01)$ | $1.00(0.03)$ | $-0.05(0.04)$
| PLF | $-1.67(0.40)$ | $2.61(0.41)$ | $0.01(0.61)$ | $0.47(0.67)$ | $0.00(0.04)$ | $0.00(0.06)$ | $1.00(0.03)$ | $-0.05(0.04)$
R6 | NLS | $-0.46(0.32)$ | $1.98(0.25)$ | $-0.76(0.47)$ | $0.87(0.57)$ | $0.00(0.04)$ | $0.00(0.05)$ | $1.00(0.03)$ | $-0.01(0.04)$
| MLE | $-0.46(0.30)$ | $2.26(0.35)$ | $-0.79(0.44)$ | $0.91(0.67)$ | $0.00(0.06)$ | $0.00(0.07)$ | $1.00(0.03)$ | $-0.03(0.04)$
| EM | $-0.46(0.30)$ | $2.26(0.36)$ | $-0.79(0.44)$ | $0.91(0.67)$ | $0.00(0.06)$ | $-0.00(0.07)$ | $1.00(0.03)$ | $-0.03(0.04)$
| PLF | $-0.46(0.30)$ | $2.26(0.36)$ | $-0.79(0.44)$ | $0.91(0.67)$ | $0.00(0.06)$ | $0.00(0.07)$ | $1.00(0.03)$ | $-0.03(0.04)$
R7 | NLS | $3.81(1.02)$ | $6.40(1.57)$ | $-1.66(1.12)$ | $-2.30(1.70)$ | $0.00(0.04)$ | $0.00(0.07)$ | $0.92(0.03)$ | $0.06(0.03)$
| MLE | $3.99(0.97)$ | $6.79(1.55)$ | $-2.22(1.02)$ | $-3.07(1.63)$ | $0.00(0.08)$ | $0.00(0.13)$ | $0.92(0.03)$ | $0.08(0.03)$
| EM | $1.72(0.36)$ | $3.68(0.52)$ | $0.05(0.46)$ | $0.04(0.71)$ | $0.00(0.09)$ | $-0.00(0.14)$ | $1.00(0.01)$ | $0.00(0.01)$
| PLF | $1.72(0.35)$ | $3.68(0.49)$ | $0.05(0.44)$ | $0.04(0.67)$ | $0.00(0.09)$ | $0.00(0.14)$ | $1.00(0.01)$ | $0.00(\text{N\kern-1.00006pt/\kern-1.00006ptA})$
R8 | NLS | $-0.37(0.33)$ | $2.09(0.80)$ | $-0.08(0.46)$ | $-0.15(0.94)$ | $0.00(0.07)$ | $0.00(0.10)$ | $0.86(0.09)$ | $0.03(0.11)$
| MLE | $-0.40(0.38)$ | $2.12(0.70)$ | $-0.09(0.51)$ | $-0.13(0.91)$ | $0.00(0.08)$ | $0.00(0.11)$ | $0.86(0.08)$ | $0.03(0.11)$
| EM | $-0.40(0.29)$ | $2.11(0.39)$ | $-0.09(0.37)$ | $-0.13(0.52)$ | $0.00(0.00)$ | $-0.00(\text{N\kern-1.00006pt/\kern-1.00006ptA})$ | $0.86(0.07)$ | $0.03(0.10)$
| PLF | $-0.44(0.38)$ | $2.07(0.67)$ | $-0.05(0.51)$ | $-0.09(0.89)$ | $0.00(0.08)$ | $0.00(0.11)$ | $0.88(0.08)$ | $0.02(0.11)$
R9 | NLS | $0.45(0.43)$ | $2.23(0.38)$ | $-0.19(0.57)$ | $0.17(0.73)$ | $0.00(0.05)$ | $0.00(0.07)$ | $0.89(0.06)$ | $-0.07(0.08)$
| MLE | $0.50(0.46)$ | $2.45(0.60)$ | $-0.16(0.65)$ | $0.29(0.93)$ | $0.00(0.10)$ | $0.00(0.13)$ | $0.89(0.06)$ | $-0.08(0.09)$
| EM | $0.50(0.39)$ | $2.45(0.52)$ | $-0.16(0.56)$ | $0.28(0.79)$ | $0.00(0.01)$ | $-0.00(0.01)$ | $0.89(0.06)$ | $-0.08(0.09)$
| PLF | $0.31(0.42)$ | $2.23(0.54)$ | $-0.19(0.59)$ | $0.22(0.83)$ | $0.00(0.11)$ | $0.00(0.13)$ | $0.92(0.06)$ | $-0.07(0.08)$
R10 | NLS | $-0.80(0.33)$ | $3.17(1.57)$ | $-1.02(0.45)$ | $-0.21(1.65)$ | $0.00(0.03)$ | $0.00(0.03)$ | $0.89(0.10)$ | $0.11(0.10)$
| MLE | $-0.98(0.39)$ | $3.62(1.15)$ | $-1.06(0.55)$ | $-0.37(1.24)$ | $0.00(0.03)$ | $0.00(0.05)$ | $0.89(0.09)$ | $0.11(0.09)$
| EM | $-0.98(0.24)$ | $3.63(0.60)$ | $-1.06(0.33)$ | $-0.37(0.68)$ | $0.00(0.01)$ | $-0.00(\text{N\kern-1.00006pt/\kern-1.00006ptA})$ | $0.89(0.02)$ | $0.11(0.01)$
| PLF | $-1.22(0.26)$ | $3.01(0.42)$ | $-0.81(0.45)$ | $0.24(0.57)$ | $0.00(0.03)$ | $0.00(0.04)$ | $0.97(0.01)$ | $0.03(0.01)$
R11 | NLS | $0.42(0.38)$ | $2.50(0.78)$ | $-0.18(0.49)$ | $-0.17(0.89)$ | $0.00(0.07)$ | $0.00(0.09)$ | $0.85(0.06)$ | $0.07(0.07)$
| MLE | $0.49(0.45)$ | $2.72(0.89)$ | $-0.30(0.56)$ | $-0.31(1.03)$ | $0.00(0.10)$ | $0.00(0.13)$ | $0.84(0.07)$ | $0.09(0.08)$
| EM | $0.49(0.44)$ | $2.72(0.63)$ | $-0.30(0.50)$ | $-0.30(0.73)$ | $0.00(0.00)$ | $-0.00(0.00)$ | $0.84(0.06)$ | $0.09(0.08)$
| PLF | $0.06(1.80)$ | $2.15(3.79)$ | $0.14(1.83)$ | $0.27(3.83)$ | $0.00(0.21)$ | $0.00(0.23)$ | $0.92(0.44)$ | $0.01(0.45)$
R12 | NLS | $1.87(0.45)$ | $3.82(0.52)$ | $0.17(0.72)$ | $0.53(1.08)$ | $0.00(0.04)$ | $0.00(0.07)$ | $0.97(0.02)$ | $-0.06(0.04)$
| MLE | $1.88(0.42)$ | $4.03(0.65)$ | $0.22(0.75)$ | $0.55(1.37)$ | $0.00(0.11)$ | $0.00(0.14)$ | $0.98(0.02)$ | $-0.07(0.04)$
| EM | $1.48(0.40)$ | $3.45(0.69)$ | $-0.77(0.43)$ | $-0.96(0.63)$ | $0.00(0.11)$ | $-0.00(0.14)$ | $1.00(0.00)$ | $0.00(0.01)$
| PLF | $1.48(0.38)$ | $3.46(0.64)$ | $-0.65(0.51)$ | $-0.74(0.85)$ | $0.00(0.11)$ | $0.00(0.14)$ | $1.00(0.00)$ | $-0.01(0.02)$
R13 | NLS | $-0.21(0.31)$ | $2.18(0.37)$ | $-0.18(0.42)$ | $-0.17(0.54)$ | $0.00(0.04)$ | $0.00(0.05)$ | $0.96(0.04)$ | $-0.03(0.07)$
| MLE | $-0.39(0.30)$ | $2.16(0.35)$ | $-0.22(0.47)$ | $-0.19(0.65)$ | $0.00(0.07)$ | $0.00(0.10)$ | $1.00(0.03)$ | $-0.02(0.11)$
| EM | $-0.39(0.30)$ | $2.16(0.34)$ | $-0.27(0.42)$ | $-0.28(0.44)$ | $0.00(0.07)$ | $-0.00(0.10)$ | $1.00(0.03)$ | $0.00(0.04)$
| PLF | $-0.39(0.30)$ | $2.16(0.35)$ | $-0.24(0.47)$ | $-0.23(0.65)$ | $0.00(0.07)$ | $0.00(0.10)$ | $1.00(0.03)$ | $-0.01(0.11)$
R14 | NLS | $0.45(0.31)$ | $2.66(0.48)$ | $0.26(0.44)$ | $0.32(0.69)$ | $0.00(0.05)$ | $0.00(0.07)$ | $0.96(0.03)$ | $-0.01(0.04)$
| MLE | $0.26(0.29)$ | $2.54(0.40)$ | $0.14(0.39)$ | $0.15(0.54)$ | $0.00(0.09)$ | $0.00(0.12)$ | $1.00(0.02)$ | $0.00(0.02)$
| EM | $0.26(0.29)$ | $2.54(0.39)$ | $0.14(0.39)$ | $0.15(0.53)$ | $0.00(0.09)$ | $-0.00(0.12)$ | $1.00(0.01)$ | $0.00(0.02)$
| PLF | $0.26(0.29)$ | $2.54(0.39)$ | $0.14(0.39)$ | $0.15(0.53)$ | $0.00(0.09)$ | $0.00(0.12)$ | $1.00(0.01)$ | $0.00(0.02)$
R15 | NLS | $-0.92(0.32)$ | $1.98(0.33)$ | $0.24(0.40)$ | $0.40(0.59)$ | $0.00(0.03)$ | $0.00(0.03)$ | $0.97(0.06)$ | $-0.01(0.08)$
| MLE | $-1.08(0.31)$ | $2.09(0.34)$ | $0.18(0.40)$ | $0.28(0.47)$ | $0.00(0.04)$ | $0.00(0.06)$ | $1.00(0.04)$ | $0.00(0.05)$
| EM | $-1.08(0.31)$ | $2.09(0.32)$ | $0.18(0.40)$ | $0.28(0.45)$ | $0.00(0.04)$ | $-0.00(0.06)$ | $1.00(0.03)$ | $0.00(0.04)$
| PLF | $-1.08(0.31)$ | $2.09(0.32)$ | $0.18(0.40)$ | $0.28(0.45)$ | $0.00(0.04)$ | $0.00(0.06)$ | $1.00(0.03)$ | $0.00(0.04)$
R16 | NLS | $2.47(0.55)$ | $4.90(0.89)$ | $-0.03(0.68)$ | $-0.38(1.09)$ | $0.00(0.04)$ | $0.00(0.08)$ | $0.97(0.02)$ | $0.01(0.02)$
| MLE | $2.37(0.41)$ | $4.92(0.81)$ | $0.24(0.60)$ | $-0.03(1.20)$ | $0.00(0.09)$ | $0.00(0.14)$ | $0.99(0.01)$ | $-0.00(0.02)$
| EM | $1.99(0.31)$ | $4.34(0.57)$ | $0.09(0.48)$ | $-0.24(0.89)$ | $0.00(0.09)$ | $-0.00(0.14)$ | $1.00(\text{N\kern-1.00006pt/\kern-1.00006ptA})$ | $0.00(0.00)$
| PLF | $1.99(0.31)$ | $4.34(0.57)$ | $0.09(0.44)$ | $-0.24(0.78)$ | $0.00(0.09)$ | $0.00(0.14)$ | $1.00(\text{N\kern-1.00006pt/\kern-1.00006ptA})$ | $-0.00(\text{N\kern-1.00006pt/\kern-1.00006ptA})$
R17 | NLS | $-2.67(0.51)$ | $2.43(0.48)$ | $0.31(0.70)$ | $-0.53(0.63)$ | $0.00(0.01)$ | $0.00(0.02)$ | $0.93(0.09)$ | $0.07(0.11)$
| MLE | $-3.17(0.45)$ | $3.14(0.59)$ | $0.64(0.65)$ | $-1.05(0.69)$ | $0.00(0.01)$ | $0.00(0.04)$ | $0.89(0.09)$ | $0.11(0.11)$
| EM | $-3.17(0.41)$ | $3.14(0.58)$ | $0.64(0.49)$ | $-1.04(0.68)$ | $0.00(\text{N\kern-1.00006pt/\kern-1.00006ptA})$ | $-0.00(0.00)$ | $0.89(0.09)$ | $0.11(0.14)$
| PLF | $-3.17(0.50)$ | $3.15(0.61)$ | $0.64(0.68)$ | $-1.05(0.70)$ | $0.00(0.01)$ | $0.00(0.04)$ | $0.89(0.08)$ | $0.11(0.11)$
R18 | NLS | $1.64(1.24)$ | $3.17(1.20)$ | $-0.18(1.32)$ | $0.14(1.31)$ | $0.00(0.06)$ | $0.00(0.08)$ | $0.87(0.08)$ | $0.04(0.09)$
| MLE | $1.55(0.88)$ | $3.28(0.87)$ | $-0.02(0.98)$ | $0.27(1.16)$ | $0.00(0.13)$ | $0.00(0.17)$ | $0.88(0.06)$ | $0.02(0.07)$
| EM | $1.57(0.70)$ | $3.31(0.88)$ | $-0.05(0.82)$ | $0.23(1.07)$ | $0.00(0.00)$ | $-0.00(0.00)$ | $0.88(0.06)$ | $0.02(0.07)$
| PLF | $1.07(0.54)$ | $2.71(0.60)$ | $0.21(0.66)$ | $0.53(0.91)$ | $0.00(0.13)$ | $0.00(0.17)$ | $0.94(0.04)$ | $-0.01(0.05)$
R19 | NLS | $-1.12(0.29)$ | $2.43(0.44)$ | $-0.48(0.41)$ | $0.17(0.61)$ | $0.00(0.01)$ | $0.00(0.02)$ | $0.98(0.05)$ | $0.02(0.06)$
| MLE | $-1.39(0.29)$ | $2.77(0.36)$ | $-0.33(0.42)$ | $0.14(0.59)$ | $0.00(0.03)$ | $0.00(0.04)$ | $1.00(0.02)$ | $-0.02(0.03)$
| EM | $-1.39(0.29)$ | $2.77(0.36)$ | $-0.33(0.43)$ | $0.14(0.59)$ | $0.00(0.03)$ | $-0.00(0.04)$ | $1.00(0.02)$ | $-0.02(0.04)$
| PLF | $-1.39(0.29)$ | $2.77(0.36)$ | $-0.33(0.43)$ | $0.14(0.59)$ | $0.00(0.03)$ | $0.00(0.04)$ | $1.00(0.02)$ | $-0.02(0.04)$
R20 | NLS | $-0.94(0.27)$ | $2.58(0.50)$ | $0.22(0.37)$ | $0.16(0.69)$ | $0.00(0.02)$ | $0.00(0.03)$ | $0.98(0.04)$ | $0.00(0.05)$
| MLE | $-1.09(0.28)$ | $2.76(0.41)$ | $0.29(0.39)$ | $0.11(0.65)$ | $0.00(0.03)$ | $0.00(0.05)$ | $1.00(0.02)$ | $-0.01(0.04)$
| EM | $-1.09(0.28)$ | $2.76(0.40)$ | $0.26(0.39)$ | $0.02(0.57)$ | $0.00(0.03)$ | $-0.00(0.05)$ | $1.00(0.02)$ | $0.00(0.02)$
| PLF | $-1.09(0.28)$ | $2.76(0.35)$ | $0.26(0.39)$ | $0.02(0.36)$ | $0.00(0.03)$ | $0.00(0.05)$ | $1.00(\text{N\kern-1.00006pt/\kern-1.00006ptA})$ | $-0.00(\text{N\kern-1.00006pt/\kern-1.00006ptA})$
R21 | NLS | $0.03(0.45)$ | $1.84(0.54)$ | $-0.31(0.61)$ | $-0.10(0.79)$ | $0.00(0.06)$ | $0.00(0.07)$ | $0.84(0.09)$ | $-0.11(0.14)$
| MLE | $0.03(0.48)$ | $1.95(0.67)$ | $-0.19(0.70)$ | $0.13(0.94)$ | $0.00(0.11)$ | $0.00(0.13)$ | $0.84(0.10)$ | $-0.14(0.13)$
| EM | $0.03(0.47)$ | $1.95(0.64)$ | $-0.19(0.61)$ | $0.13(0.77)$ | $0.00(0.09)$ | $-0.00(0.05)$ | $0.84(0.10)$ | $-0.14(0.13)$
| PLF | $0.02(0.48)$ | $1.95(0.67)$ | $-0.26(0.69)$ | $0.05(0.93)$ | $0.00(0.11)$ | $0.00(0.13)$ | $0.84(0.10)$ | $-0.12(0.13)$
R22 | NLS | $1.26(0.39)$ | $4.28(0.78)$ | $-0.05(0.51)$ | $-0.57(0.90)$ | $0.00(0.03)$ | $0.00(0.05)$ | $0.96(0.02)$ | $0.01(0.03)$
| MLE | $0.94(0.31)$ | $4.04(0.57)$ | $0.30(0.48)$ | $-0.06(0.98)$ | $0.00(0.07)$ | $0.00(0.11)$ | $1.00(0.01)$ | $-0.02(0.03)$
| EM | $0.93(0.28)$ | $4.02(0.52)$ | $0.06(0.39)$ | $-0.47(0.70)$ | $0.00(0.07)$ | $-0.00(0.10)$ | $1.00(0.01)$ | $0.00(0.01)$
| PLF | $0.93(0.28)$ | $4.02(0.51)$ | $0.06(0.37)$ | $-0.47(0.66)$ | $0.00(0.07)$ | $0.00(0.10)$ | $1.00(0.01)$ | $-0.00(\text{N\kern-1.00006pt/\kern-1.00006ptA})$
R23 | NLS | $1.33(0.45)$ | $3.41(0.57)$ | $-0.06(0.66)$ | $-0.47(0.84)$ | $0.00(0.04)$ | $0.00(0.07)$ | $0.93(0.03)$ | $0.01(0.05)$
| MLE | $1.34(0.43)$ | $3.63(0.71)$ | $-0.61(0.52)$ | $-1.27(0.78)$ | $0.00(0.09)$ | $0.00(0.15)$ | $0.94(0.04)$ | $0.06(0.04)$
| EM | $0.67(0.31)$ | $2.64(0.48)$ | $0.06(0.43)$ | $-0.28(0.57)$ | $0.00(0.09)$ | $-0.00(0.15)$ | $1.00(0.01)$ | $0.00(0.02)$
| PLF | $0.67(0.28)$ | $2.64(0.36)$ | $0.06(0.35)$ | $-0.28(0.33)$ | $0.00(0.09)$ | $0.00(0.15)$ | $1.00(0.00)$ | $0.00(\text{N\kern-1.00006pt/\kern-1.00006ptA})$
R24 | NLS | $1.12(0.30)$ | $3.83(0.63)$ | $0.86(0.74)$ | $0.96(1.83)$ | $0.00(0.05)$ | $0.00(0.09)$ | $0.98(0.02)$ | $-0.09(0.04)$
| MLE | $1.21(0.34)$ | $4.11(0.82)$ | $0.87(0.73)$ | $0.94(1.57)$ | $0.00(0.08)$ | $0.00(0.11)$ | $0.98(0.02)$ | $-0.09(0.04)$
| EM | $0.94(0.26)$ | $3.67(0.50)$ | $-0.43(0.36)$ | $-1.12(0.57)$ | $0.00(0.07)$ | $-0.00(0.11)$ | $1.00(\text{N\kern-1.00006pt/\kern-1.00006ptA})$ | $0.00(\text{N\kern-1.00006pt/\kern-1.00006ptA})$
| PLF | $0.94(0.29)$ | $3.67(0.62)$ | $-0.30(0.43)$ | $-0.86(0.82)$ | $0.00(0.07)$ | $0.00(0.11)$ | $1.00(0.01)$ | $-0.01(0.02)$
R25 | NLS | $4.11(0.97)$ | $5.23(1.09)$ | $-0.96(1.28)$ | $-1.06(1.27)$ | $0.00(0.10)$ | $0.00(0.15)$ | $0.91(0.03)$ | $0.03(0.04)$
| MLE | $3.71(1.59)$ | $4.83(2.01)$ | $-0.45(1.72)$ | $-0.42(2.22)$ | $0.00(0.20)$ | $0.00(0.29)$ | $0.92(0.04)$ | $0.02(0.05)$
| EM | $3.71(1.58)$ | $4.83(1.86)$ | $-0.45(1.71)$ | $-0.42(2.03)$ | $0.00(0.00)$ | $-0.00(0.00)$ | $0.92(0.04)$ | $0.02(0.05)$
| PLF | $2.43(0.68)$ | $3.35(0.84)$ | $0.28(0.93)$ | $0.44(1.26)$ | $0.00(0.19)$ | $0.00(0.29)$ | $0.97(0.02)$ | $-0.00(0.03)$
R26 | NLS | $3.88(1.63)$ | $6.56(2.46)$ | $-1.77(1.71)$ | $-2.41(2.54)$ | $0.00(0.05)$ | $0.00(0.07)$ | $0.93(0.03)$ | $0.01(0.04)$
| MLE | $2.87(0.81)$ | $5.42(1.18)$ | $-0.74(0.96)$ | $-1.03(1.44)$ | $0.00(0.09)$ | $0.00(0.13)$ | $0.97(0.03)$ | $-0.02(0.04)$
| EM | $2.15(0.39)$ | $4.43(0.60)$ | $-1.00(0.49)$ | $-1.49(0.74)$ | $0.00(0.09)$ | $-0.00(0.14)$ | $1.00(0.01)$ | $0.00(0.01)$
| PLF | $2.15(0.48)$ | $4.43(0.74)$ | $-1.00(1.07)$ | $-1.49(1.83)$ | $0.00(0.09)$ | $0.00(0.14)$ | $1.00(0.01)$ | $-0.00(0.04)$
R27 | NLS | $1.45(0.39)$ | $4.27(0.81)$ | $-0.23(0.58)$ | $-0.48(1.10)$ | $0.00(0.04)$ | $0.00(0.05)$ | $0.96(0.02)$ | $-0.00(0.03)$
| MLE | $1.08(0.28)$ | $3.91(0.53)$ | $-0.26(0.38)$ | $-0.46(0.68)$ | $0.00(0.07)$ | $0.00(0.10)$ | $1.00(0.01)$ | $0.00(0.01)$
| EM | $1.08(0.27)$ | $3.91(0.52)$ | $-0.26(0.37)$ | $-0.47(0.67)$ | $0.00(0.07)$ | $-0.00(0.10)$ | $1.00(0.01)$ | $0.00(0.01)$
| PLF | $1.08(0.27)$ | $3.91(0.52)$ | $-0.26(0.37)$ | $-0.47(0.67)$ | $0.00(0.07)$ | $0.00(0.10)$ | $1.00(0.01)$ | $0.00(0.01)$
R28 | NLS | $2.57(0.65)$ | $5.02(1.15)$ | $0.39(0.85)$ | $0.33(1.58)$ | $0.00(0.05)$ | $0.00(0.09)$ | $0.96(0.02)$ | $0.02(0.03)$
| MLE | $2.12(0.44)$ | $4.54(0.79)$ | $0.92(0.71)$ | $1.00(1.39)$ | $0.00(0.08)$ | $0.00(0.13)$ | $0.98(0.02)$ | $-0.01(0.02)$
| EM | $1.80(0.38)$ | $4.07(0.74)$ | $1.24(0.67)$ | $1.47(1.37)$ | $0.00(0.09)$ | $-0.00(0.14)$ | $1.00(0.00)$ | $-0.03(0.02)$
| PLF | $1.80(0.38)$ | $4.07(0.74)$ | $1.24(0.68)$ | $1.47(1.37)$ | $0.00(0.09)$ | $0.00(0.14)$ | $1.00(0.00)$ | $-0.03(0.02)$
R29 | NLS | $0.22(0.30)$ | $4.08(0.65)$ | $-0.64(0.41)$ | $-1.10(0.87)$ | $0.00(0.02)$ | $0.00(0.03)$ | $0.92(0.04)$ | $0.02(0.05)$
| MLE | $0.39(0.34)$ | $4.90(0.89)$ | $-1.07(0.46)$ | $-1.69(1.05)$ | $0.00(0.05)$ | $0.00(0.07)$ | $0.91(0.03)$ | $0.06(0.06)$
| EM | $-0.32(0.28)$ | $3.20(0.50)$ | $-0.46(0.38)$ | $-0.25(0.60)$ | $0.00(0.05)$ | $-0.00(0.08)$ | $1.00(0.01)$ | $0.00(0.02)$
| PLF | $-0.32(0.25)$ | $3.20(0.31)$ | $-0.46(0.33)$ | $-0.25(0.11)$ | $0.00(0.05)$ | $0.00(0.08)$ | $1.00(\text{N\kern-1.00006pt/\kern-1.00006ptA})$ | $-0.00(\text{N\kern-1.00006pt/\kern-1.00006ptA})$