Solved – Similarities and differences between IRT model and Logistic regression model

Question

Despite the basic similarities like both of these model the probability of success rather than modelling the response variable directly; I believe that there are more reliable answers which point out the differences and similarities between these models.

One difference being, in logistic one can use different type and different number of independent variables; whereas in IRT model we just have only one independent variable that is ability.

One more similarity : To estimate the parameters in logistic we use Maximum likelihood approach. In IRT also we use marginal maximum likelihood as one of the parameter estimating approach.

So can anyone please state out the statistical/ mathematical differences in these two models?

Answer 1

Have a look at Section 1.6 ("The linear regression perspective") in De Boeck and Wilson (2008) Explanatory Item Response Models (http://www.springer.com/de/book/9780387402758) and Formann, A. K. (2007), (Almost) Equivalence between conditional and mixture maximum likelihood estimates for some models of the Rasch type, In M. von Davier & C. H. Carstensen (Eds.), Multivariate and mixture distribution Rasch models (pp. 177-189), New York: Springer.

In short: IRT models are generalized nonlinear mixed effects models:

• the score $Y_{pi}\in\left\{ 0,1\right\} $ of a student $p$ to an item $i$ is the dependent variable,
• given a randomly sampled student's trait, e.g. $\theta_{p}\sim N\left(\mu,\sigma^{2}\right)$, the responses are assumend to be independent Bernoulli distributed,
• given $\theta_{p}$, the predictor $\eta_{pi}=\textrm{logit}\left(P\left(Y_{pi}=1\right)\right)$ is a linear combination of item characteristics $$\eta_{pi}=\sum_{k=0}^{K}b_{k}X_{ik}+\theta_{p}+\varepsilon_{pi},$$
• let $X_{ik}=-1,$ if $i=k$, and $X_{ik}=0$, otherwise - thus obtain the Rasch model $$P\left(Y_{pi}=1\mid\theta_{p}\right)=\frac{\exp\left(\theta_{p}-b_{i}\right)}{1+\exp\left(\theta_{p}-b_{i}\right)};$$

Note that IRT models are extended towards different aspects:

• With respect to discriminatory power (2PL) and guessing ratio (3PL) of an item $$ P\left(Y_{pi}=1\mid\theta_{p}\right)= c_i+(1-c_i)\frac{\exp\left(a_{i}\left(\theta_{p}-b_{i}\right)\right)}{1+\exp\left(a_{i}\left(\theta_{p}-b_{i}\right)\right)} $$
• With respect to polytomous scores $$ P\left(Y_{pi}=k\mid\theta_{p}\right)=\frac{\exp\left(a_{ik}\theta_{p}-b_{ik}\right)}{\sum_{k=0}^{K}\exp\left(a_{ik}\theta_{p}-b_{ik}\right)} $$
• With respect to known student characteristics constituting the population (e.g., sex, migration status) $$ \theta_{p}\sim N\left(\mathbf{Z}\boldsymbol{\beta},\sigma^{2}\right), $$
• With respect to construct dimensionality $$ P\left(Y_{pi}=1\mid\theta_{p}\right)=\frac{\exp(\sum_{d}a_{id}\theta_{pd}-b_{i})}{1+\exp(\sum_{d}a_{id}\theta_{pd}-b_{i})},\quad\theta_{p}\sim N^{d}\left(\boldsymbol{\mu},\Sigma\right) $$
• With respect to discrete skill classes (continuous distributions can be easily approximated by discrete ones) $$ P\left(Y_{pi}=1\mid\theta_{p(l)}\right)=\frac{\exp(\theta_{p(l)}-b_{i(l)})}{1+\exp(\theta_{p(l)}-b_{i(l)})},\quad\theta_{p(l)}\in\left\{ \theta_{p(1)},\dots,\theta_{p(L)}\right\} $$

(taken from the useR!2015 slides for the R package TAM)