In SICSresearch/IRTpp: Estimating IRT Parameters using the IRT Methodology

Statistic of Maria Orlando and David Thissen

1. The statistic that calculates orlando_itemf( ) is:

$S-X_j^2=\sum\limits_{k=1}^{p-1} N_k\dfrac{(O_{jk}-E_{jk})^2}{E_{jk}(1-E_{jk})}\sim {\chi^2}_{[(p-1)- (number\ \ of \ \ parameters \ \ considered)]}$

Where:

The index $k$ concerns to the classic score $k=1,2,...,p-1$; $p$ is the number of items
The index $j$ concerns to the ítem $j$ ($j$ is fixed but arbitrary)
For the $3PL$ model, the number of parameters considered is 3, for the $2PL$ model, the number of parameters considered is 2, and for the $1PL$ model, the number of parameters considered is 1

\ 1.1 Elements of statistic

1.1.1 expected frequencies \

$E_{jk}=\dfrac{\int T_jS_{k-1}^{*j}\Phi(\theta) \ \ \partial\theta}{\int S_{k}\Phi(\theta) \ \ \partial\theta}$

Where:

\ ( i ) The integral is approximated by methods of quadrature about equally spaced intervals, in the range of $\theta$ ($- 6\leq\theta\leq6$)

( ii ) $S_{k-1}^{* j}$ is the accumulated likelihood score $k$ - 1 without considering the item $j$.

( iii ) $S_k$ is the accumulated likelihood score $k$, considering all the items.

( iv ) $T_j$ is the probability of responding correctly to the item $j$

1.1.1 observes frequencies \

$O_{jk}=\dfrac{N_{jk}}{N_k}$

Where:

\ ( i ) $N_k$ is the total of individuals with score $k$.

( ii ) $N_{jk}$ is the total of individuals with score $k$. that answer correctly to the item $j$.

\ 2. The null hypothesis that is judged is:

$H_0$: The model adjusts to data

\ 3. Interpretation practice

$H_0$ is rejected if the p-value that returns orlando_itemf( ) is less than $\alpha$, where $\alpha$ is the level of significance (generally, $\alpha=5\%$)

Statistic of Fritz Drasgow, Michael V. Levine and Esther A. Williams, asessing the goodness of fit of individuals or of ítems

1. The statistic that that calculates z3_itemf( ) or z3_personf is:

$Z_3=\dfrac{l_0-E_3(\hat{\theta}_d)}{\sigma_3(\hat{\theta}_d)} \sim N(0,1)$

1.1 Elements of statistic

1.1.1 Indice ($l_0$) \

$l_0=\sum\limits_{j=1}^{n}u_jlog[P_j(\hat{\theta}_d)]+(1-u_j)log[1-P_j(\hat{\theta}_d)]$

Where:

( i ) $n$ is the number total of the items.

( ii ) $\hat{\theta}_d$ is the estimation maximum likelihood of laten trait, associated with the pattern of response $d$.

( iii ) $P_j(.)$ is the likelihood of correctly answering at item j, dado un trazo y los parámetros del ítem $j$.given the parameters of the items and a laten traits

( iv ) $l_0$ is the maximum of the logarithm of the likelihood of a reponse pattern $d$.

1.1.2 expected value conditional ($E_3$)

$E_3(.)$ is the expected value conditional of the random variable:

$X_3(t)=\sum\limits_{j=1}^{n}u_jlog[P_j(t)]+(1-u_j)log[1-P_j(t)]$ therefore.

$E_3(t)=E(X_3(t)/\theta=t)=\sum\limits_{j=1}^{n}u_jlog[P_j(t)]+(1-u_j)log[1-P_j(t)]$

1.1.3 standard deviation conditional ($\sigma_3$)

Similarly you have that:

$\sigma_3^2(t)=VAR(X_3(t)/\theta=t)=\sum\limits_{j=1}^{n}P_j(t)(1-P_j(t))\left[\dfrac{P_j(t)}{1-P_j(t)}\right]^2$

\ 2. The null hypothesis that is judged is:

$H_0$: The model adjust to data

\ 3. Interpretation practice

$H_0$ is rejected if the observed value of the statistic that returns z3_itemf or z3_personf is less than -1.7, or greater than 1.7 (generally), the statistic is N(0,1) under $H_0$, then then the user can choose a different level of significance.

Statistic of Steven P. Reise, to evaluate the fit of items

1. The statistic that that calculates x2_itemf( ) is:

$X^2=\sum\limits_{k=1}^{G} N_k\dfrac{(O_{ik}-E_{ik})^2}{E_{ik}(1-E_{ik})}\sim\chi^2_{[G-(# \ \ param. \ \ considerados)]}$

1.1 Elements of statistic

1.1.1 Numer of groups(G) \

The latent trait range is partitioned in $G$ groups, for example using percentiles.

1.1.2 expected frequencies($E_{ik}$)

Where:

( i ) $E_{ik}$ It refers to the expected frequency of individuals in the Group $k$, that you respond correctly to the item $i$.

( ii ) $E_{ik}$ It is a measure representative of the group, for example, the median or average.

1.1.3 observed frecuencies($O_{ik}$)

$O_{jk}=\dfrac{N_{jk}}{N_k}$

Where:

( i ) $N_k$ It is the total number of individuals belonging to the Group $k$.

( ii ) $N_{jk}$ It is the total number of individuals in the Group $k$, that you respond correctly to the item $j$.

\ 2. The null hypothesis that is judged is:

$H_0:$ The item modeled adjust to data

\ 3. Interpretation practice

$H_0$ is rejected if the p-value simulated that returns x2_itemf is less than $\alpha$, where $\alpha$ is the level of significance (generally, $\alpha=5\%$)

Envelopes: David Thissen; Howard Wainer, to evaluate the fit of items

$\zeta_l=(a_l,b_l,c_l) \sim N(\hat{\zeta_l},I^{-1}(\hat{\zeta}_l))$ when $N\rightarrow\infty$, $N$ the number of individuals who respond to the test; $I$ is the information matrix, and $l$ the item $l$

Algorithm :

Generate a sequence over the domain of the latent trait $\theta$.
plot the item characteristic curve $l$.
Generate a sample of size $k$ of the distribution of $\zeta_l$: $(\zeta_{l1},...,\zeta_{lk})\sim N(\hat{\zeta_l},I^{-1}(\hat{\zeta_l}))$.
For every $\zeta_{li}, i = 1, 2,..., k$ is calculated the probability that an individual with latent trait equal to a fixed point in the sequence generated in step 1, correctly answer the item $l$.
After obtaining the $k$ odds, setting a level of error, say 5\ %. Are graphed percentiles 25 and 75, this procedure is performed for each and every one of the points of the sequence generated at the beginning of the algorithm

\ 2. The null hypothesis that is judged is:

\ $H_0$: the item modeled adjust to data

\ 3. Interpretation practice

\ If bands do not contain completely the characteristic curve the item does not fit

AIC, Akaike Information Criterion, to compare models

$AIC=-2L+2h$

1.1 Elements of statistic

\subsubsection{Log-likelihood(L)}

1.1.1 Marginal Log-likelihood (L) \

$L=p(u_{j.} /\zeta,\eta)=\int_\mathbb{R}log[p(u_{j.} /\theta,\zeta)]g(\theta/\eta)\partial \theta$ \ \ Que corresponde a la marginal de $(u_{j.},\theta)$ \

Where:

$u_{j.}$ It refers to the pattern of the individual's response $j$.
$p(u_{j.} /\theta,\zeta)=\prod\limits_{l=1}^{I}p(u_{jl} /\theta,\zeta)$, with $l=1,2,...,I$ the ítem $l$.
$g(\theta/\eta)\sim N(0,1)$ It is a priori for the distribution $\theta$.

1.1.2 Number of estimated parameters (h) \

$h$, It is the total number of estimated parameters.

3. Interpretation practice

By its construction, the AIC is a measurement of mismatch between the model and the data,so lower values expected.

BIC, Bayesian Information Criterion, to compare models

$BIC=-2L+h*logN$

1.1 Elements of statistic

\subsubsection{Log-likelihood(L)}

1.1.1 Marginal Log-likelihood (L) \

$L=p(u_{j.} /\zeta,\eta)=\int_\mathbb{R}log[p(u_{j.} /\theta,\zeta)]g(\theta/\eta)\partial \theta$ \ \ Corresponding to the marginal of $(u_{j.},\theta)$ \

Where:

$u_{j.}$ It refers to the pattern of the individual's response $j$
$p(u_{j.} /\theta,\zeta)=\prod\limits_{l=1}^{I}p(u_{jl} /\theta,\zeta)$, with $l=1,2,...,I$ the ítem $l$.
$g(\theta/\eta)\sim N(0,1)$ It is a priori for the distribution $\theta$.

1.1.2Number of estimated parameters (h) \