An Analysis of Item Response Theory and Rasch Models Based on the Most Probable Distribution Method

Abstract

The most probable distribution method is applied to derive the logistic model as the distribution accounting for the maximum number of possible outcomes in a dichotomous test while introducing latent traits and item characteristics as constraints to the system. The item response theory logistic models, with a particular focus on the one-parameter logistic model, or Rasch model, and their properties and assumptions, are discussed for both infinite and finite populations.

Appendices

Appendix A. Derivation of Condition (24)

Inequalities (22) and (23) in the case of Equation (21) (and omitting the constant term μ for simplicity of notation) become

$$\begin{aligned} &\ln \biggl(\frac{g_{jk}!}{n_{jk}! (g_{jk} - n_{jk})!} \biggr)+\lambda h(\alpha_{j}, \delta_{k}, n_{jk})\\ &\quad \geq \ln \biggl(\frac{g_{jk}!}{(n_{jk}+1)! (g_{jk} - (n_{jk}+1))!} \biggr)+\lambda h(\alpha_{j}, \delta_{k}, n_{jk}+1), \\ &\ln \biggl(\frac{g_{jk}!}{n_{jk}! (g_{jk} - n_{jk})!} \biggr)+\lambda h(\alpha_{j}, \delta_{k}, n_{jk})\\ &\quad \geq \ln \biggl(\frac{g_{jk}!}{(n_{jk}-1)! (g_{jk} - (n_{jk}-1))!} \biggr)+\lambda h(\alpha_{j}, \delta_{k}, n_{jk}-1), \end{aligned}$$

so that expanding the logarithm and simplifying the common terms gives

$$\begin{aligned} &-\ln n_{jk}!-\ln(g_{jk} - n_{jk})! + \lambda h( \alpha_{j}, \delta _{k}, n_{jk})\\ &\quad \geq - \ln(n_{jk}+1)!-\ln(g_{jk} - n_{jk}-1)!+ \lambda h(\alpha_{j}, \delta_{k}, n_{jk}+1), \\ &-\ln n_{jk}!-\ln(g_{jk} - n_{jk})! + \lambda h( \alpha_{j}, \delta _{k}, n_{jk})\\ &\quad \geq - \ln(n_{jk}-1)!-\ln(g_{jk} - n_{jk}+1)!+ \lambda h(\alpha_{j}, \delta_{k}, n_{jk}-1). \end{aligned}$$

Since, now, by definition of factorial, ln(n+1)!=ln(n+1)+lnn!, the previous inequalities become

$$\begin{aligned} &-\ln(g_{jk} - n_{jk}) \geq -\ln(n_{jk}+1) + \lambda\bigl[h(\alpha _{j}, \delta_{k}, n_{jk}+1)-h(\alpha_{j}, \delta_{k}, n_{jk})\bigr], \\ &-\ln n_{jk} + \lambda\bigl[ h(\alpha_{j}, \delta_{k}, n_{jk}) - h(\alpha _{j}, \delta_{k}, n_{jk}-1)\bigr] \geq -\ln(g_{jk} - n_{jk}+1), \end{aligned}$$

which, by setting the finite and backward differences of h _jk as

$$\begin{aligned} \Delta h_{jk} = & h(\alpha_{j},\delta_{k},n_{jk}+1)-h( \alpha_j,\delta _{k},n_{jk}), \\ \nabla h_{jk} = & h(\alpha_{j},\delta_{k},n_{jk})-h( \alpha_j,\delta _{k},n_{jk}-1), \end{aligned}$$

can finally be rewritten as

$$\frac{g_{jk}-\exp(-\lambda\Delta h_{jk})}{1+\exp(-\lambda\Delta h_{jk})}\leq n_{jk} \leq \frac{1+g_{jk}}{1+\exp(-\lambda\nabla h_{jk})}, $$

which when divided by g _jk leads to condition (24).

Appendix B. Derivation of Equation (38) Applying Stirling Formula

The case of an infinite population can be described by the limits n _jk →∞, g _jk →∞ and g _jk −n _jk →∞. Under these limits, variations Δn _jk in the discrete variables n _jk are often considered to be negligible compared to the value of n _jk , namely Δn _jk ≪n _jk , so that it can be directly considered the limit Δn _jk →0.

In such a case the forward difference equation can be written as

$$\Delta \biggl[\sum_{jk} \ln \biggl( \frac{g_{jk}!}{n_{jk}! (g_{jk} - n_{jk})!} \biggr)+\lambda \biggl(\sum_{jk} h(\alpha_{j},\delta _{k},n_{jk})-\mu \biggr) \biggr]=0, $$

while the backward difference equation can be given by switching the operator Δ with ∇. Since, now, due to linearity, variation passes under the sign of summation, the previous equation becomes

$$\sum_{jk}\Delta \biggl[ \ln \biggl( \frac{g_{jk}!}{n_{jk}! (g_{jk} - n_{jk})!} \biggr)+\lambda \biggl(\sum_{jk} h(\alpha_{j},\delta _{k},n_{jk})-\mu \biggr) \biggr]=0, $$

and the summations can be dropped, leading to

$$\Delta \biggl[ \ln \biggl(\frac{g_{jk}!}{n_{jk}! (g_{jk} - n_{jk})!} \biggr)+\lambda \biggl( \sum_{jk} h(\alpha_{j},\delta _{k},n_{jk})-\mu \biggr) \biggr]=0, $$

which can be rewritten as

$$\Delta \biggl[ \ln g_{jk}!- \ln n_{jk}!- \ln(g_{jk} - n_{jk})!+\lambda \biggl(\sum _{jk} h(\alpha_{j},\delta_{k},n_{jk})- \mu \biggr) \biggr]=0. $$

The previous equation is usually approximated by means of the Stirling formula, lnN!≈NlnN−N, in order to get rid of the factorials. In passing, notice that the Stirling formula misses a term $\frac{1}{2}\ln 2\pi N$ with respect to the approximation (62). This is not a problem since this additional term vanishes in the limit for N→∞. Hence,

$$\begin{aligned} &\Delta\biggl[ g_{jk} \ln g_{jk}- g_{jk} - n_{jk} \ln n_{jk}+ n_{jk} - (g_{jk} - n_{jk}) \ln(g_{jk} - n_{jk})\\ &\quad {} + (g_{jk} - n_{jk}) + \lambda \biggl(\sum_{jk} h( \alpha_{j},\delta_{k},n_{jk})-\mu\biggr)\biggr]=0, \end{aligned}$$

which simplified gives

$$\Delta\biggl[ g_{jk} \ln g_{jk} - n_{jk} \ln n_{jk} - (g_{jk} - n_{jk}) \ln (g_{jk} - n_{jk}) + \lambda\biggl(\sum _{jk} h(\alpha_{j},\delta _{k},n_{jk})- \mu\biggr)\biggr]=0. $$

Since the forward difference implies n _jk →n _jk +Δn _jk (while the backward difference implies n _jk →n _jk −Δn _jk ), the equation becomes

$$\begin{aligned} &- (n_{jk}+\Delta n_{jk}) \ln(n_{jk}+ \Delta n_{jk}) + n_{jk} \ln {n_{jk}}\\ &\quad {} - (g_{jk} - n_{jk}-\Delta n_{jk}) \ln(g_{jk} - n_{jk}-\Delta n_{jk}) + \cdots\\ &\quad {}+ (g_{jk} - n_{jk}) \ln(g_{jk} - n_{jk})+ \lambda\bigl[h(\alpha _{j},\delta_{k},n_{jk}+ \Delta n_{jk})- h(\alpha_{j},\delta_{k},n_{jk}) \bigr]=0, \end{aligned}$$

which, dividing by Δn _jk , and taking the limit Δn _jk →0, gives

$$\ln \biggl(\frac{g_{jk} - n_{jk}}{n_{jk}} \biggr) + \lambda\partial ^{+}_{n_{jk}}h(\alpha_{j},\delta_{k},n_{jk}) = 0, $$

where, by definition of a derivative,

$$\partial^{+}_{n_{jk}}h(\alpha_{j}, \delta_{k},n_{jk})=\lim_{\Delta n_{jk}\rightarrow0} \frac{h(\alpha_{j},\delta_{k},n_{jk}+\Delta n_{jk})- h(\alpha_{j},\delta_{k},n_{jk})}{\Delta n_{jk}}, $$

so that the forward finite variation Δ has been switched with the right derivative $\partial^{+}_{n_{jk}}$, thus giving

$$ n_{jk} = \frac{g_{jk}}{1 + \exp(-\lambda\partial^{+}_{n_{jk}}h(\alpha _{j},\delta_{k},n_{jk}))}. $$

(B.1)

A similar result can be obtained for the backward difference equation, so that the operator ∇ can be switched with the left derivative $\partial^{-}_{n_{jk}}$, giving

$$ n_{jk} = \frac{g_{jk}}{1 + \exp(-\lambda\partial^{-}_{n_{jk}}h(\alpha _{j},\delta_{k},n_{jk}))}. $$

(B.2)

Hence, both Equations (B.1) and (B.2) are satisfied when the function h(α _j ,δ _k ,n _jk ) is derivable so that both right and left derivatives exist and coincide, $\partial^{+}_{n_{jk}}h=\partial^{-}_{n_{jk}}h$ , thus giving Equation (38).

Appendix C. Other IRT Logistic Models

Other parameters in the logistic item response model can be introduced by modifying the constraint and the multiplicity in order to account for lucky guesses, careless errors, and single item discriminability.

3.1 C.1 Permutations

If in the (jk)th cluster there are l _jk lucky guesses and c _jk careless errors, it must be considered how n _jk −l _jk responses can distribute themselves inside g _jk −c _jk −l _jk cells in the matrix, so the total number of possible outcomes (15) becomes

$$W\bigl( \{n_{j k} \}\bigr)=\prod_{j k} \binom{g_{j k}-c_{jk}-l_{jk}}{ n_{j k}-l_{jk}} = \prod_{j k}\frac{(g_{j k}-c_{jk}-l_{jk})!}{(n_{j k}-l_{jk})! (g_{j k} - n_{j k}-c_{jk})!}. $$

3.2 C.2 Constraints

Items possessing different discriminating power can be accounted for by means of constraint (44),

$$\sum_{jk}\frac{n_{jk}}{\lambda}a_{k}( \alpha_j-\delta_k)= \mu, $$

where the coefficient a _k is the Birnbaum parameter (see Birnbaum in Lord & Novik 1968) and becomes just a dilation parameter for all the items when a _k =a for every item as in (44).

3.3 C.3 Derivation

For handiness of calculations, one can follow the same continuous derivation given in Appendix B (omitting the constant parameter μ for simplicity of calculation) in the case of n _jk →∞, g _jk →∞ and g _jk −n _jk →∞. Due to linearity, the finite variation Δ passes under the sign of summation, which gives

$$\sum_{jk} \Delta \biggl[\ln \biggl( \frac{(g_{j k}-c_{jk}-l_{jk})!}{(n_{j k}-l_{jk})! (g_{j k} - n_{j k}-c_{jk})!} \biggr)+\lambda \biggl(\frac {n_{jk}}{\lambda}a_{k}( \alpha_j-\delta_k) \biggr) \biggr]=0; $$

and the summation can be dropped, leading to

$$\Delta \biggl[\ln \biggl(\frac{(g_{j k}-c_{jk}-l_{jk})!}{(n_{j k}-l_{jk})! (g_{j k} - n_{j k}-c_{jk})!} \biggr)+ n_{jk}a_{k}(\alpha_j-\delta _k) \biggr]=0, $$

which can be rewritten as

$$\Delta\bigl[ \ln(g_{j k}-c_{jk}-l_{jk})!- \ln(n_{j k}-l_{jk})!- \ln {(g_{j k} - n_{j k}-c_{jk})!}+n_{jk}a_{k}( \alpha_j-\delta_k)\bigr]=0. $$

Since in the population limit the Stirling approximation can be applied to the logarithm of a factorial, the difference formula becomes

$$\begin{aligned} &\Delta\bigl[ (g_{j k}-c_{jk}-l_{jk}) \ln(g_{j k}-c_{jk}-l_{jk})- (g_{j k}-c_{jk}-l_{jk}) - (n_{j k}-l_{jk}) \ln(n_{j k}-l_{jk})\\ &\quad {}+ (n_{j k}-l_{jk}) + \cdots- (g_{jk} - n_{jk}-c_{jk}) \ln(g_{jk} - n_{jk}-c_{jk})\\ &{}\quad + (g_{jk} - n_{jk}-c_{jk}) + n_{jk}a_{k}( \alpha_j-\delta_k)\bigr]=0, \end{aligned}$$

which simplified gives

$$\begin{aligned} &\Delta\bigl[ (g_{j k}-c_{jk}-l_{jk}) \ln(g_{j k}-c_{jk}-l_{jk}) - (n_{j k}-l_{jk}) \ln(n_{j k}-l_{jk})+ \cdots\\ &\quad {}+ (n_{j k}-l_{jk}) - (g_{jk} - n_{jk}-c_{jk}) \ln(g_{jk} - n_{jk}-c_{jk}) + n_{jk}a_{k}(\alpha_j-\delta_k)\bigr]=0. \end{aligned}$$

Consider, now, n _jk as a continuous variable, as has already been done in Appendix B, so that the finite variation Δ can be switched into the derivative $\partial_{n_{jk}}$ thus giving

$$- \ln(n_{jk}-l_{jk}) - \frac{n_{jk}-l_{jk}}{n_{jk}-l_{jk}} + \ln(g_{jk} - n_{jk} - c_{jk}) + \frac{g_{jk} - n_{jk}- c_{jk}}{g_{jk} - n_{jk}- c_{jk}} + a_{k}(\alpha_j - \delta_k) =0, $$

which, simplifying, is

$$\ln \biggl(\frac{g_{jk} - n_{jk} - c_{jk}}{n_{jk}-l_{jk}} \biggr) + a_{k}( \alpha_j - \delta_k) = 0, $$

so that

$$\frac{g_{jk} - n_{jk} - c_{jk}}{n_{jk}-l_{jk}} = \exp\bigl(a_{k}(\delta_k - \alpha_j)\bigr), $$

which can be rewritten as

$$g_{jk} - c_{jk} + l_{jk}\exp \bigl(a_{k}(\delta_k - \alpha_j)\bigr) =n_{jk}\bigl(1+ \exp\bigl(a_{k}(\delta_k - \alpha_j)\bigr)\bigr), $$

or equivalently as

$$g_{jk} - (l_{jk}+ c_{jk}) + l_{jk}\bigl(1+\exp\bigl(a_{k}(\delta_k - \alpha _j)\bigr)\bigr) = n_{jk}\bigl(1+ \exp \bigl(a_{k}(\delta_k - \alpha_j)\bigr) \bigr), $$

which gives

$$n_{jk} = l_{jk}+\frac{g_{jk}-(c_{jk}+l_{jk})}{1 + \exp(a_{k}(\delta_k - \alpha_j))}. $$

Since the proportion c _jk /g _jk gives the probability of a careless error $P^{C}_{jk}$, and the proportion l _jk /g _jk gives the probability of a lucky guess $P^{L}_{jk}$, dividing the previous equation by g _jk returns the most probable distribution for the number of cells filled in a cluster,

$$P\bigl(X^{\nu i}_{jk}=1;\alpha_j, \delta_k\bigr)=P^{L}_{jk}+\frac{1 -(P^{L}_{jk}+P^{C}_{jk})}{1+ \exp({\lambda a_{k}(\delta_k - \alpha_j)})}, $$

which is exactly model (47), if one defines $b_{k}=P^{L}_{jk}$ and $c_{k}=1-P^{C}_{jk}$. Notice that the choice of omitting the index j is just a simplification in which lucky guesses and careless errors are considered to depend only on the item difficulty.