Item Response Theory models (IRT, Lord & Novik, Reference Lord and Novik1968) and Rasch Models (RM, Rasch, Reference Rasch1960) are fundamental in psychology. Their derivations generally assume monotonicity, continuity and asymptotical behavior of the item response function, plus general criteria or additional requirements like local stochastic independence, sufficiency of statistics, conditional inference or specific objectivity. The need for a dense set of items to account for interval scales has also been debated (Fischer, Reference Fischer, Fischer and Molenaar1995a). For istance, the multidimensional Polytomous RM can be derived from sufficiency, while the Partial Credit model (PCM, Masters, Reference Masters1982) and the Rating Scale model (RSM, Andrich, Reference Andrich1978; Reference Andrich1982) from conditional inference (Fischer, Reference Fischer, Fischer and Molenaar1995b). Similarly, the conditional RM and the original RM can be derived requiring “measurement interchangeability” (Kelderman, Reference Kelderman, Fischer and Molenaar1995).
Derivations based on formal frameworks are relevant to gain insight on a model. The derivation based on conditional inference, for instance, describes both PCM and RSM as special cases of the Power Series Distribution. Similarly, the derivation based on items “interchangeability” (in respect to their relation to each other and with other variables) connects the RM to the Generalized Linear Models. Alternative derivations can also be given borrowing techniques from statistical physics: a derivation of the Polytomous RM based on the method of the steepest descent (Darwin & Fowler, Reference Darwin and Fowler1922a; Reference Darwin and Fowler1922b; Reference Darwin and Fowler1923) was given by Ebneth (Reference Ebneth1993) considering a testee answering a fictional series of tasks. The average numbers of equivalent task series, conditional to constraints, were used to obtain the probability. A derivation of the dichotomous RM, based on Boltzmann’s most probable distribution (MPD) method (see Huang, Reference Huang1987) was instead given by Noventa, Stefanutti, & Vidotto (Reference Noventa, Stefanutti and Vidotto2013) conceiving a test as a distribution of responses with several possible outcomes constrained by means of latent traits and item characteristics. The RM was derived as the distribution maximizing the number of possible outcomes.
Although these two approaches to RM are based on different techniques, they are related by the principle of Maximum Entropy (MaxEnt). Indeed, they both converge to the asymptotic probability distribution that corresponds to the condition of maximum ignorance about the system (Jaynes, Reference Jaynes1982).
In the present work, the derivation of the dichotomous RM given in Noventa et al. (Reference Noventa, Stefanutti and Vidotto2013) is extended to the Polytomous RM and its implications are discussed in light of the aforementioned literature, in particular, the Darwin-Fowler derivation (Ebneth, Reference Ebneth1993), the MaxEnt principle and the use of a statistical physics framework for understanding the rationale of IRT and RM.
A brief introduction to the MPD method and the MaxEnt principle
The “most probable distribution” method
The MPD is a combinatorial method suggested by Boltzmann in 1877 to derive the energy distribution of particles in a gas. Particles are allocated into energy levels
$\{ \varepsilon _k \} $
(like marbles in boxes) with occupation numbers
$\{ n_k \} $
under the constraints given by their total number N and energy E:
$$\sum\limits_k {n_k } = N\quad \quad \sum\limits_k {n_k \varepsilon _k } = E$$
Since particles can be distributed in several ways, there are different possible combinations (and sets of frequencies
$f_k = {{n_k } \over N}$
). Some of these combinations are also equivalent, since identical particles can be swapped. To understand which combination is more likely to be observed in Nature, multiplicity is defined as the number of ways in which a particular combination is realized:
$$W\left( {\left\{ {n_k } \right\}} \right) = {{N!} \over {\prod\nolimits_k {n_k !} }}$$
The logarithm of the multiplicity is known as Boltzmann’s Entropy. The gist of the MPD method is that the final probability distribution is associated to the combination that is realized in the maximum number of ways. To find such a probability the multiplicity (2), under the constraints (1), is maximed using the method of the Lagrangian multipliers:
$$\Lambda \left( {\left\{ {n_k } \right\}} \right) = \ln W\left( {\left\{ {n_k } \right\}} \right) + \lambda _1 \left( {N - \sum\limits_k {n_k } } \right) + \lambda _2 \left( {E - \sum\limits_k {n_k \varepsilon _k } } \right)$$
were
${\rm{\lambda }}_1 ,{\rm{\lambda }}_2 \in R$
. The difficulty in maximizing (3) rely on the integer nature of
$\left\{ {{\rm{n}}_{\rm{k}} } \right\}$
and in the factorials contained in the multiplicity. An approximate derivation, based on the fact that the limit
$N \to \infty $
is taken, requires Stirling’s approximation for factorials and a continuous approximation for
$\left\{ {{\rm{n}}_{\rm{k}} } \right\}$
. A better result, still in the continuity approximation, is obtained by replacing factorials with the Gamma function (Landsberg, Reference Landsberg1954). An elegant derivation was given by Clinton and Massa (Reference Clinton and Massa1972) avoiding both continuity and Stirling’s approximations. Since the present work relies on this derivation, it is useful to briefly sketch it for the present case: Clinton and Massa noticed that
${\rm{\Lambda }}(\{ {\rm{n}}_{\rm{k}} \} )$
is in a maximum if, moving a particle both in or out of an energy level, the following system of inequalities is satisfied:
$$\left\{ {\matrix{ {\Lambda \left( {n_1 , \ldots ,n_k , \ldots } \right) \ge \Lambda \left( {n_1 , \ldots ,n_k + 1, \ldots } \right)} \cr {\Lambda \left( {n_1 , \ldots ,n_k , \ldots } \right) \ge \Lambda \left( {n_1 , \ldots ,n_k - 1, \ldots } \right)} \cr } } \right.$$
Substitution of (3) in the system (4) leads to:
$$\left\{ {\matrix{ {\ln \left( {n_k + 1} \right) + \lambda _1 + \lambda _2 \varepsilon _k \ge 0} \cr {\ln \left( {n_k } \right) + \lambda _1 + \lambda _2 \varepsilon _k \le 0} \cr } } \right.$$
so that any occupation number (once divided by N) is bounded by:
$${{\exp \left( { - \lambda _1 } \right)} \over N}\exp \left( { - \lambda _2 \varepsilon _k } \right) - {1 \over N} \le {{n_k } \over N} \le {{\exp \left( { - \lambda _1 } \right)} \over N}\exp \left( { - \lambda _2 \varepsilon _k } \right)$$
Since these are actually the occupation frequencies, for the squeeze theorem the probability can be defined taking the limit
$N \to \infty $
where the terms 1/N vanishes:
$$p_k = \mathop {\lim }\limits_{N \to \infty } f_k = \mathop {{\rm{lim}}}\limits_{N \to \infty } {{\exp \left( { - \lambda _1 } \right)} \over N}\exp \left( { - \lambda _2 \varepsilon _k } \right)$$
The condition
$\sum\nolimits_k {{\rm{p}}_k } = 1$
is used to introduce a normalization term:
$${N \over {\exp \left( { - \lambda _1 } \right)}} = \sum\limits_k {\exp \left( { - {\rm{\lambda }}_2 \varepsilon _k } \right)} $$
and finally the probability is:
$$p_k = {{\exp \left( { - \lambda _2 \varepsilon _k } \right)} \over {\sum\nolimits_{\rm{k}} {\exp \left( { - \lambda _2 \varepsilon _k } \right)} }}$$
It is immediate the similarity of (8) with the Polytomous RM if the occupation numbers
$n_k $
are a measure of how many times a category response is chosen in N trials by a testee. Although with some differences in assumptions and methods, this line of reasoning is close to Ebneth’s (Reference Ebneth1993). His derivation, however, requires the concept of statistical ensemble, a collection of N copies of the system that, independently and randomly, assume all the possible states allowed to the system (Huang, Reference Huang1987). For instance, N copies of the entire gas. This concept is fundamental in statistical physics which is based on averages over ensembles, but it is not required in the MPD method (Landsberg, Reference Landsberg1954).
Boltzmann’s “molecular” perspective (the ensemble is the set of N particles) is followed in the present work, so that all the possible response patterns of the joint population of subjects and items are accounted (see section 3). Probability (8) is then derived as the most likely response pattern that can be observed in the population. The concept of MPD is well justified in a MaxEnt framework.
The principle of Maximum Entropy
Information Entropy is a measure of uncertainty (Shannon, Reference Shannon1948). The higher the entropy of a system the more unpredictable is its state. The concept was inspired by Gibbs statistical Entropy:
$$S_I = - \sum\limits_k {p_k \log p_k } $$
and takes maximum value when the distribution is uniform. Connections between statistical entropy and information theory were higlighted by Jaynes (Reference Jaynes1957). In particular, he suggested the importance of a MaxEnt principle, briefly stated as “when we make inferences based on incomplete information we should drawn them from that probability distribution that has the maximum entropy permitted by the information we do have” (Jaynes, Reference Jaynes1982). Such a method is a generalization of the usual methods of statistical inference that allows the choice of different priors. For instance, the MaxEnt solution for a set of probabilities
$\sum\nolimits_k {p_k = 1} $
over some events
$x_k $
is the uniform distribution (Laplace’s principle of indifference). Adding some functions
$f_j \left( {x_k } \right)$
, whose expected values are constrained
$E\left[ {f_j (x_k )} \right] = a_j \in R$
, the result is the is the exponential family:
$$\max \left\{ {\left. {S_I } \right|f_j } \right\} \to p_k = {{\exp \left( { - \sum\nolimits_j {\lambda _j f_j \left( {x_k } \right)} } \right)} \over {\sum\nolimits_m {\exp \left( { - \sum\nolimits_j {\lambda _j f_j \left( {x_m } \right)} } \right)} }}$$
where
${\rm{\lambda }}_j \in R$
. On the one side, the MaxEnt distribution works as the inverse of the Darmois-Koopman-Pitman theorem and creates a model for which the data are sufficient statistics (Jaynes, Reference Jaynes1982). On the other side, it relates Gibbs and Boltzmann Entropies:
$$\mathop {\lim }\limits_{N \to \infty } {1 \over N}\log W \approx \mathop {\lim }\limits_{N \to \infty } \sum\limits_k {{{n_k } \over N}\log {{n_k } \over N}} = - \sum\limits_k {p_k \log p_k } = S_I $$
The probability distribution that maximizes entropy is numerically identical to the frequency that possesses the greatest multiplicity (Jaynes, Reference Jaynes1982). Boltzmann’s Entropy is the limit of Gibbs’ Entropy if probabilities are equal. Working on the concept of ensemble Gibbs formulation is however more general and allows to describe interacting particles (Jaynes, Reference Jaynes1965). Hence, the Boltzmann entropy seems to be a suitable description of RM since it requires independence between subjects and items.
Basic assumptions, definitions and notations
Measurement scale and equivalence classes
Given a set of subjects
${\rm{\nu }} \in \{ 1, \ldots ,s\} $
, a set of items
$i \in \{ 1, \ldots ,m\} $
, and a random variable, whose realizations are response categories
$X_{\nu i} = x_{\nu i} \in \{ 0, \ldots ,c\} $
, the response matrix of a test is
$\{ x_{\nu i} \} $
. In the dichotomous case,
$c = 1$
. A generalization of the response matrix to the population can be either a finite or an infinite matrix
$\{ x_{\nu i} \} $
with
${\rm{\nu }} \in S$
and
$i \in I$
, where S and I are the populations of all the subjects and items. The union
$P = S \cup I$
, endowed with a weak order relation
$ \mathbin{\lower.3ex\hbox{$\buildrel\prec\over {\smash{\scriptstyle\sim}\vphantom{_x}}$}} $
, allows comparisons between subjects and items. A common scale for latent traits and item parameters is a triple
$\left\langle {P,M,\phi } \right\rangle $
where
$\phi :\left\langle {P, \mathbin{\lower.3ex\hbox{$\buildrel\prec\over {\smash{\scriptstyle\sim}\vphantom{_x}}$}} } \right\rangle \to \left\langle {M, \le } \right\rangle $
, with
$M \subseteq R$
, is an homomorphism that preserves the weak order (see for instance Krantz, Luce, Suppes, & Tversky, Reference Krantz, Luce, Suppes and Tversky1971; Luce, Krantz, Suppes, & Tversky, Reference Luce, Krantz, Suppes and Tversky1990; Suppes & Zinnes, Reference Suppes, Zinnes, Luce, Bush and Galanter1963). Equivalence classes of all the subjects and of all the items possessing the same position on the measurement scale can be defined as:
$$S_\alpha = \left\{ {\nu \in S:\phi \left( {\rm{\nu }} \right) = \alpha \in A \subseteq M} \right\},\quad I_{\rm{\delta }} = \left\{ {i \in I:\phi \left( i \right) = {\rm{\delta }} \in D \subseteq M} \right\}$$
where
${\rm{\alpha }} \in A$
and
${\rm{\delta }} \in D$
are the values of the latent trait and of the item characteristic. Let j and k be the indexes spanning these sets, assuming they have at least countable cardinalities.
In order to build a Polytomous RM, thresholds are also needed. They are generally conceived as locations on the latent trait set that indicate a subject has exceeded a particular category response. Let
$r \in \left\{ {0, \ldots ,{\rm{c}}} \right\}$
be the index spanning the category responses, hence
${\rm{\tau }}_{kr} $
is the threshold value needed for scoring the category
$x_{jk} = r$
in an item of characteristic
${\rm{\delta }}_k $
. Thresholds appear then to be both levels of latent trait and item characteristic,
${\rm{\tau }}_{kr} \in {\rm{A}} \cap {\rm{D}}$
. In what follows, the situation is considered in which
${\rm{A}} = {\rm{D}}$
, so that there is always a match between levels of latent trait and of item characteristic.
Probability
An important debate in IRT concerns the source of randomness. In the stochastic subject view, probability explains variations due to the person or to the test situation, in the random sampling view, probability is the proportion of subjects with the same latent trait giving positive answers (Moleenar, Reference Molenaar, Fischer and Molenaar1995). Another important debate concerns whether latent traits and item characteristics are on an ordinal level or on a metric continuum (Michell, Reference Michell1990). In the former case, different subjects and items might be associated to the same non-metric scale value, in the latter, the probability of subjects and items to possess the same values would be zero. Interestingly, the MPD method accommodates all the previous perspectives. Equivalence classes (12) are defined independently on whether they describe an ordinal ranking or a coarse-grained description of a continuum (i.e, classes due to limited precision in measurement). Indeed, the MPD method moves from occupation numbers, so it is not important whether they result from resampling a subject or from different testees. In the most general perspective the response might be rewritten as
$x_{jkr}^{\nu it} $
where the supra-indexes refer to subjects, items, and repetitions, while sub-indexes refer to latent traits, item characteristics, and category responses. Since debating about the existence of subjects with the same latent trait and item with the same characteristic is pointless in the MPD framework,
$\nu ,i,t$
will be dropped.
Let
$X_{jk} $
be the response variable corresponding to a subject with latent trait
${\rm{\alpha }}_j $
, answering to an item with characteristic
${\rm{\delta }}_k $
, and let
$x_{jk} = r \in \left\{ {0, \ldots ,c} \right\}$
be its realization. The matrix
$\{ x_{jk} \} $
can be ordered by increasing levels of latent trait and item characteristic and then partitioned into different blocks characterized by a couple (
${\rm{\alpha }}_j ,{\rm{\delta }}_k $
). The number of responses within each block is given by a set of numbers
$\{ n_{jkr} \} $
. If
$N_{jk} = \sum\nolimits_{r = 0}^c {n_{jkr} } $
is the total number of cells in the specific jk-th block, then the ratio
$n_{jkr} /N_{jk} $
gives the proportion of responses for the r-th category (the number of subjects giving a response r to a certain class of items or the proportion of responses of a single subject to a single item depending on the view). The probability of drawing from the population a testee with latent trait
$\alpha _j $
, answering r (whose threshold is
$\tau _{kr} $
) to an item with a characteristic value of
$\delta _k $
, is then:
$$P\left( {\left. {X_{jk} = r} \right|\alpha _j ,\delta _k ,\tau _{kr} } \right) := \mathop {\lim }\limits_{N_{jk} \to \infty } {{n_{jkr} } \over {N_{jk} }}$$
where the limit accounts for an infinite population. The law of total probability becomes:
$$\sum\limits_{r = 0}^c {P\left( {\left. {X_{jk} = r} \right|\alpha _j ,\delta _k ,\tau _{kr} } \right)} = 1$$
The most probable distribution for a Polytomous test
Permutations
Multiplicity can be derived consideing that the total number of ways in which
$n_{jkr} $
responses can fill the
$N_{jk} $
cells of the block is given by the binomial coefficient:
$$W_{jkr} = \left( {\matrix{ {N_{jk} } \cr {n_{jkr} } \cr } } \right) = {{N_{jk} !} \over {n_{jkr} !\left( {N_{jk} - n_{jkr} } \right)!}}$$
Once the response category
$r = 0$
has been filled, there is room left for
$N_{jk} - n_{jk0} $
responses in the other categories, and so on. Hence:
$$W_{jk} = \left( {\matrix{ {N_{jk} } \cr {n_{jk0} } \cr } } \right) \times \left( {\matrix{ {N_{jk} - n_{jk0} } \cr {n_{jk1} } \cr } } \right) \times \ldots \times \left( {\matrix{ {N_{jk} - \sum\nolimits_{r = 0}^{c - 2} {n_{jkr} } } \cr {n_{jkc - 1} } \cr } } \right) \times 1$$
that, after some algebra, yields:
$$W_{jk} = {{N_{jk} !} \over {\prod\nolimits_{r = 0}^c {n_{jkr} !} }}$$
so that all the possible ways in which all the blocks can be filled is given by:
$$W\left( {\left\{ {n_{jkr} } \right\}} \right) = \prod\limits_{jk} {W_{jk} } = \prod\limits_{jkr} {{{N_{jk} !} \over {n_{jkr} !}}} $$
that is exactly multiplicity (2) but generalized to the joint population of subjects and items.
Constraints
The first constraint that must be taken into account is given by the total number of cells in each block, that must sum up to the total number N of cells in the response matrix. It follows that:
$$N = \sum\limits_{jk} {N_{jk} } = \sum\limits_{jkr} {n_{jkr} } $$
The second constraint depends on the fact that the number of possible outcomes defined by the multiplicity (17) depends on
${\rm{\alpha }},{\rm{\delta }},$
and
${\rm{\tau }}$
. The number of responses
$n_{jkr} $
is conditional to latent traits, item characteristics and thresholds. The constraint can be modeled as an implicit function of
${\rm{\alpha }}_j ,{\rm{\delta }}_k ,{\rm{\tau }}_{kr} $
and
$n_{jkr} .$
Namely,
$H\left( {\left\{ {n_{jkr} } \right\},\left\{ {{\rm{\alpha }}_j } \right\},\left\{ {{\rm{\delta }}_k } \right\},\left\{ {{\rm{\tau }}_{kr} } \right\}} \right) = {\rm{\mu }}$
, with
${\rm{\mu }} \in R$
. However, to avoid any interaction between different response categories in different blocks, additive independence is assumed:
$$H\left( {\left\{ {n_{jkr} } \right\},\left\{ {\alpha _j } \right\},\left\{ {\delta _k } \right\},\left\{ {\tau _{kr} } \right\}} \right) = \sum\limits_{jkr} {h\left( {n_{jkr} ,\alpha _j ,\delta _k ,\left\{ {\tau _{kt} } \right\}_{t \le r} } \right)} = \mu $$
Notice that, in each block, constraints likely depend on all the thresholds
$\left\{ {{\rm{\tau }}_{kt} } \right\}_{t \le r} $
that precede the one associated to the r-th category. Any
$h_{jkr} := h\left( {n_{jkr} ,{\rm{\alpha }}_j ,{\rm{\delta }}_k ,\left\{ {{\rm{\tau }}_{kt} } \right\}_{t \le r} } \right)$
is a generic constraint for the r-th category in the jk-th cluster, and is assumed to be a monotonic function of its arguments.
Derivation of the most probable distribution
The MPD can be derived by maximizing the Boltzmann Entropy given by (17) under the effect of constraints (18) and (19). As in section (2) this extremality problem under external constraints is reduced with Lagrangian multipliers method to the unconstrained maximization of the function:
$$\Lambda \left( {\left\{ {n_{jkr} } \right\}} \right) = \ln W\left( {\left\{ {n_{jkr} } \right\}} \right) + \lambda _1 \left( {\sum\limits_{jkr} {n_{jkr} - N} } \right) + \lambda _2 \left( {\sum\limits_{jkr} {h\left( {n_{jkr} ,\alpha _j ,\delta _k ,\left\{ {\tau _{kt} } \right\}_{t \le r} } \right) - \mu } } \right)$$
where
${\rm{\lambda }}_1 ,{\rm{\lambda }}_2 \in R$
and the sets of
$\left\{ {{\rm{\alpha }}_j } \right\},\left\{ {{\rm{\delta }}_k } \right\},\left\{ {{\rm{\tau }}_{kr} } \right\}$
enter in the equation as parameters. As noticed, the variables
$n_{jkr} $
are positive integers so the previous equation is not differentiable. Since the derivation follows exactly the step given in section (2), apart from some minor changes, full proof is given in Appendix A. There is however a point worth to be mentioned: not all the shape of the constraint
$h_{jkr} $
do define a probability. A unique solution can be achieved when the constraints are linear functions of
$n_{jkr} $
, that is
$h_{jkr} = n_{jkr} f_{jkr} $
where
$f_{jkr} : = f({\rm{\alpha }}_j ,{\rm{\delta }}_k ,\left\{ {{\rm{\tau }}_{kt} } \right\}_{t \le r} )$
is a generic functions of latent traits, item characteristics and thresholds (addition of constants or functions unrelated to
$n_{jkr} $
does not affect the multiplicity, see Appendix A). Interestingly, linear constraints are related to averages of latent traits, thresholds and item characteristics over all blocks and category responses. They are indeed the expected values of the sufficient statistics for the exponential family as in equation (10).
The probability resulting from maximizing (20) is then:
$$P\left( {\left. {X_{jk} = r} \right|\alpha _j ,\delta _k ,\tau _{kr} } \right) = {{\exp \left( {\lambda _2 f\left( {\alpha _j ,\delta _k ,\left\{ {\tau _{kt} } \right\}_{t \le r} } \right){\rm{}}} \right)} \over {\sum\nolimits_{r = 0}^c {\exp \left( {\lambda _2 f\left( {\alpha _j ,\delta _k ,\left\{ {\tau _{kt} } \right\}_{t \le r} } \right){\rm{}}} \right)} }}$$
since
${\rm{\lambda }}_1 $
becomes a normalization term
${\rm{\lambda }}_2 $
and is a scale factor. As it can also be easily noticed, PCM and RSM can be obtained from equation (21) setting the appropriate constraint (19) to:
$$H_{RSM} \left( {\left\{ {n_{jkr} } \right\},\left\{ {\alpha _j } \right\},\left\{ {\delta _k } \right\},\left\{ {\tau _{kr} } \right\}} \right) = {1 \over {\lambda _2 }}\sum\limits_{jkr} {n_{jkr} } \left( {\sum\limits_{t = 0}^r {\left( {\alpha _j - \left( {\delta _k - \tau _{kt} } \right)} \right)} } \right)$$
$$H_{PCM} \left( {\left\{ {n_{jkr} } \right\},\left\{ {\alpha _j } \right\},\left\{ {\delta _k } \right\},\left\{ {\tau _{kr} } \right\}} \right) = {1 \over {\lambda _2 }}\sum\limits_{jkr} {n_{jkr} } \left( {\sum\limits_{t = 0}^r {\left. {\left( {\alpha _j - \tau _{kt} } \right)} \right)} } \right)$$
Similar constraints can be given for any form of the Polytmous RM (Rasch, Reference Rasch1961).
Discussion
In the present work Boltzmann’s MPD method was used to obtain the Polytomous RM. This methodology was chosen for sake of simplicity: first, although methods like the steepest descent are preferable (Jaynes, Reference Jaynes1982; Schrödinger, Reference Schrödinger1946), they lack the intuitivity of the MPD method. Darwin-Fowler’s method, indeed, requires the concept of ensemble and knowledge of complex analysis. The MPD method instead needs few assumptions, is based on simple algebra and appears to be more suited for exploratory and introductory purpouses (Landsberg, Reference Landsberg1954). Second, “the Boltzmann MPD method and the Darwin-Fowler method lead to the same result in the limit
$N \to \infty $
” (Jaynes, Reference Jaynes1982; Schrödinger, Reference Schrödinger1946). They are equivalent in light of the MaxEnt principle, since the maximization of the total number of outcomes can be replaced by the maximization of the number of ways in which equally probable outcomes, conditional to the constraints, can be realized. Third, but not less important, it is argued that Boltzmann Entropy is a sufficient framework to describe the RM since it requires the independence of its basic elements, that is, subjects and items. The concept of ensemble, meaning a set of mental copies of the system assuming all its possible microstates, is however essential to extend the model to account for interactions. The concept however needs to be defined in the framework of psychology.
A possible solution was given by Ebneth (Reference Ebneth1993), in the stochastic subject view, considering a testee undergoing a series of fictional tasks as the ensemble. In the present work a different perspective is higlighted, that defines the concept of ensemble in relation to the idea of a test as a collection of responses of different subjects to several items. Following the ideas of section (3.2), a more general approach enclosing both stochastic subject view and random sampling view might be given considering an ensemble as the set of all the response matrices in the joint population of subjects and items.
From the perspective of statistical physics, different ensembles are possible: the microcanonical is the one in which states are equally likely since they possess the same energy; the canonical is the one in which some states are more likely than others since energy can vary; a grand-canonical is the one in which the total number of basic elements may also vary. Most importantly, the descriptions given by the different ensembles are equivalent in an infinite population, in which fluctuations are negligible, so that most of the microstates have the same average values of energy and of basic elements.
The equivalent of probability (21) can be derived as the probability associated to a canonical (C) or a grand-canonical (GC) statistical ensemble (Huang, Reference Huang1987):
$$P_C \left( {H_i } \right) = {{\exp \left( { - \beta H_i } \right)} \over {\sum\nolimits_i {\exp \left( { - \beta H_i } \right)} }}\quad ,\quad P_{GC} \left( {H_i } \right) = {{\exp \left( { - \beta \left( {\sum\nolimits_j {\mu _j N_{ij} - H_i } } \right)} \right)} \over {\sum\nolimits_i {\exp \left( { - \beta \left( {\sum\nolimits_j {\mu _j N_{ij} - H_i } } \right)} \right)} }}$$
In the case of a gas of particles, β is the Boltzmann factor associated to the temperature, Hi
is the Hamiltonian (a function describing the energy) of the i-th microstate, Nij
is the number of particles of the j-th species in the i-th microstates and
${\rm{\mu }}_{ij} $
is the chemical potential describing the energy related to exchanges of particles. In the case of a test a parallel might be to associate to β a discrimination factor, Hi
to a function (the constraints) describing latent traits, item characteristics and thresholds, Nij
might be the number of subjects in the j-th category and
${\rm{\mu }}_{ij} $
a related function describing changes in the number of subjects in a category (or in a formal similarity with the multidimensional polytomous RM they might be associated to person’s value and weight parameter in the j-th latent trait).
Probabilities (24) are however conceptually different from (21). Boltzmann’s MPD method is based on different assumptions, and probability (21) describes the proportion of subjects in a response category (or the propensity distribution) for a given item difficulty. Probabilities (24) are instead descriptions of all the subjects in all the category responses of all the items. Hence, they do not provide distinct distributions for distinct blocks of the responses matrix, but the probability of an entire response matrix. The concept of ensemble allows then to introduce interactions whereas the Boltzmann approach requires independency: only if local stochastic independence holds the probabilites (24), by introducing an additive structure of the Hamiltonian like in constraint (19), can be decomposed into a product of probabilities (21). Similarly, the difference between a dichotomous and a polytomous RM is the absence of local independence between the category responses. Such a perspective generalizes RM to those situation in which local independence does not hold and interactions are required.
It is also important to notice that probability distributions (24) describe equilibrium solutions that satisfy the MaxEnt principle and account for the maximum uncertainty. As in the MPD method, they decribe a system in the microstate having the highest probability, that is a system whose entropy is in a maximum, or in other words, for which the knowledge of the observer is minimum. Exponential families are indeed MaxEnt solution in presence of linear constraints on the expected value of their sufficient statistics. For instance, probability (21) is the one that maximizes multiplicity (17) under linear constraints on expected values of latent traits, item characteristics and thresholds (see section 4.3). Interestingly, multiplicity (17) describes a series of independent categorical distributions (over r categories) each related to a block
$j,k$
, and within each block there are Njk
trials. The multinomial distribution (with fixed number of trials and unknown probabilities) in itself is an exponential family whose inverse parameter mapping is the generalization of the logistic function, called softmax function, that corresponds to equation (21). These concepts can also be traced back in other derivations: the fundamental one is based on sufficiency (Fischer, Reference Fischer, Fischer and Molenaar1995b). The derivation based on conditional inference (Fischer, Reference Fischer, Fischer and Molenaar1995b) connects the RM to the Power Series Distribution, that is an exponential family that contains the binomial, the geometric, and the Poisson distributions (Patil, Reference Patil and Patil1965). Even the derivation based on “measurement interchangeability” (Kelderman, Reference Kelderman, Fischer and Molenaar1995) might have a parallel on the commutative nature of constraint (19) that allows the possibility of switching items.
Such a line of reasoning suggests that MaxEnt principle provides a rationale behind IRT models and RM in the framework of psychology (notice also that RM, rather than pairwise comparisons or psychometric logistic curve depends on which population is considered). RM would be the most suitable description of a test under the constraints previously described. Other IRT models related to exponential families would be descriptions of systems with different combinatorial natures, that is, situations in which a different state of information is available. Notice however that MaxEnt works with noiseless data, the complete case needs a full Bayesian approach (Jaynes, Reference Jaynes1982).
Some final remarks. First, this result does not hold for a finite population, in which there is not a unique definition of probabilty (21). In such a case, fluctuations are not negligible and distributions describing different states should be considered. Second, such an approach to IRT and RM appears to separate the rationale behind the models from the problem of their measurement type. The derivation was indeed given independently on the nature of the measurement scale, given by the triple
$\left\langle {P,M,\phi } \right\rangle $
, and whose type depends on the admissible transformation of the homomorphism ɸ that maps subjects and items into latent traits and item characteristics as in the definition of equivalence classes (12). The MaxEnt principle indeed appears to justify the rationale behind the models, but does not grant a type of measurement that should be ascertained considering the permissible transformations allowed to the homorphism ɸ, by a specific constraint. This approach was applied in Noventa et al. (Reference Noventa, Stefanutti and Vidotto2013) to derive the measurement scale for the dichotmous RM under the requirement of a constraint satisfying specific objectivity. As a result, the metric degraded from an interval scale to an ordinal one depending on the cardinality of the equivalence classes (12). Finite equivalence classes granted indeed only an ordinal type, unless the constraint satisfied the axioms of conjoint measurement. Interestingly, this finding parallells the results of non-parametric item response models (see for instance, Karabatsos, Reference Karabatsos2001) in which cancellation axioms, up to the last empirically testable finite order, are required to obtain ordered-metric scales for respondents and items.The higher the cardinality of the sets of items, subjects, and category responses, the more the scales approximate an intervale scale.
In conclusion, the method of the MPD is a pratical and combinatorial way to derive the RM moving from assumptions of independence. The concept of ensemble and a statistical physics approach are however needed to generalize the system out of independence. Most of all, the MaxEnt principle suggests that RM and IRT models, might possess a deeper rationale in entropy. They would describe the distribution of responses that is more likely to find during an experiment.
The present work has been supported by a grant from the Center for Assessment, University of Verona, Director Prof. Giuseppe Favretto. We would also like to thank the anonymous reviewers of the journal for their keen insight into the manuscript and their unvaluable suggestions.
Appendix A- Derivation of equation (21)
Following the derivation of Clinton and Massa (Reference Clinton and Massa1972), equation (20) yield two inequalities:
$$\Lambda \left( {n_{1r} , \ldots ,n_{jkr} ,\lambda _1 ,\lambda _2 } \right) \ge \Lambda \left( {n_{1r} , \ldots ,n_{jkr} + 1,\lambda _1 ,\lambda _2 } \right)$$
$$\Lambda \left( {n_{1r} , \ldots ,n_{jkr} ,\lambda _1 ,\lambda _2 } \right) \ge \Lambda \left( {n_{1r} , \ldots ,n_{jkr} - 1,\lambda _1 ,\lambda _2 } \right)$$
In particular, dropping the sums and the indices jk for simplicity of notation, and dropping also the costant terms
${\rm{\mu }},N$
since thay cancel out, they become:
$$\ln \left( {{{N!} \over {n_r !}}} \right) + {\rm{\lambda }}_1 n_r + {\rm{\lambda }}_2 h(n_r , \ldots ) \ge \ln \left( {{{N!} \over {(n_r + 1)!}}} \right) + {\rm{\lambda }}_1 (n_r + 1) + {\rm{\lambda }}_2 h(n_r + 1, \ldots )$$
$$\ln \left( {{{N!} \over {n_r !}}} \right) + {\rm{\lambda }}_1 n_r + {\rm{\lambda }}_2 h(n_r , \ldots ) \ge \ln \left( {{{N!} \over {(n_r - 1)!}}} \right) + {\rm{\lambda }}_1 (n_r - 1) + {\rm{\lambda }}_2 h(n_r - 1, \ldots $$
so that, expanding the logarithm and simplifying the common terms become:
$$\ln (n_r + 1) \ge {\rm{\lambda }}_1 + {\rm{\lambda }}_2 [h\left( {n_r + 1, \ldots } \right) - h\left( {n_r , \ldots } \right)]$$
$$ - \ln n_r \ge - {\rm{\lambda }}_1 - {\rm{\lambda }}_2 [h\left( {n_r , \ldots } \right) - h\left( {n_r - 1, \ldots } \right)]$$
Once defined the forward and backward finite difference equations:
$\Delta h_r = h\left( {n_r + 1, \ldots } \right) - h\left( {n_r , \ldots } \right)$
$$\nabla h_r = h\left( {n_r , \ldots } \right) - h\left( {n_r - 1, \ldots } \right)$$
the previous inequalities (divided by N) yield upper and lower bounds in the proportion of responses given by the subjects to the r-th category into the jk-th block:
$${{\exp \left( {{\rm{\lambda }}_1 + {\rm{\lambda }}_2 {\rm{\Delta }}h_r } \right)} \over N} - {1 \over N} \le {{n_r } \over N} \le {{\exp \left( {{\rm{\lambda }}_1 + {\rm{\lambda }}_2 \nabla h_r } \right)} \over N}$$
In an infinite population, namely in the limit
$N \to \infty $
, becomes:
$${{\exp \left( {{\rm{\lambda }}_1 + {\rm{\lambda }}_2 {\rm{\Delta }}h_r } \right)} \over N} \le {{n_r } \over N} \le {{\exp \left( {{\rm{\lambda }}_1 + {\rm{\lambda }}_2 \nabla h_r } \right)} \over N}$$
so that frequencies are bounded by exponentials with different arguments, in contrast to equation (6). A unique definition of probability is given only if the condition
${\rm{\Delta }}h_r = \nabla h_r $
holds. This is a functional equation that can be solved as a second-order linear homogeneous recurrence relation with constant coefficients (Aczel, Reference Aczel1966). Solutions have the shape
$h_r = n_r f_r + g_r $
with fr
and gr
generic functions of latent traits, thresholds and item characteristic values. However, the term gr
cancels out and can be set equal to zero. Squeeze theorem and definition of probability (15) then yield:
$$P\left( {\left. {X_{jk} = r} \right|\alpha _j ,\delta _k ,\tau _{kr} } \right) = \mathop {\lim }\limits_{N \to \infty } {{\exp \left( {\lambda _1 + \lambda _2 f_r } \right)} \over N}$$
only if the limit converges to a finite value. This can be obtained by normalizing through the law of total probability (14), indeed substitution of (A5) in (14) yields:
$${{\rm{N}} \over {\exp \left( {\lambda _1 } \right)}} = \sum\limits_r {\exp \left( {\lambda _2 f_r } \right)} $$
So that once inserted (A6) in equation (A5) it yields as result probability (21).
