EL ACCESO A LOS SERVICIOS DESDE LA PERSPECTIVA DE LAS PERSONAS JÓVENES

An easy descriptive approach to check the local independence is to check the point-biserial correlation be- tween responses and response times, with or without conditioning on the the estimated ˆθ and ˆτ ’s. If condi- tional correlation decreases significantly to near 0, one can conclude that the local independence assumption

might be satisfied. To calculate the conditional correlation, for each item, we first grouped examinees into K relatively homogeneous groups via k-means clustering, with the number of clusters, K, determined such that the number of examinees within each group was at least 5. We then calculated the point-biserial correlation within each group then obtained the mean for each item. By doing so, the weighted mean squared corre- lation, averaged over all items weighted by the sample size for each item, decreased from 0.13 to 0.07. In other words, the conditional correlation between item responses and RTs were nearly ±0.27. The correlation was still not 0 because either the ˆθ and ˆτ estimation might contain measurement error; or some items were answered by as few as 6 test takers. Furthermore, within each “homogeneous” cluster, the θ and τ value could still vary substantially.

A more rigorous way to check the conditional independence assumption is via hypothesis testing. The conditional independence in Equation (2.7) can equivalently be expressed as

f (tij|yij, τi) = f (tij|τi), yij= 0, 1 (4.2)

for all i and j. According to van der Linden and Glas (2010), this assumption is preferred over the alternative, f (yij|tij, θi) = f (yij|θi) for all i and j. Two reasons were given. First, we only need to check whether the

two conditional distributions of Tij given Yij = 0 or 1 are equal rather than checking the equality of an

entire family of distributions of Yijgiven the continuous measure of Tij= tij. Second, the estimation of τiis

typically more accurate than the estimation of θi (see Tables 2.1 and 3.2) because the continuous response

time data is expected to contain more information than the binary response accuracy data. If this assumption is violated, the response time model could be modified as

hj(tij) = h0j(t) exp(βjτi+ λjuij), (4.3)

thus the assumption check reduces to check the significance of λj for item j. The null hypothesis would be

H0: λj= 0,

whereas the alternative hypothesis is H1 : λj 6= 0. Similar to van der Linden and Glas (2010), we assumed

the item parameters, including βj and h0j, were pre-calibrated. Thus, for a given item, the parameters that

need to be estimated are τ = (τ1, ..., τN) and λj. The likelihood can be rewritten as

l(τ , λj) = log ½_YN i=1 · f (tij|τi, βj, λj) J Y l=1;l6=j f (tij|τi, βj) ¸¾ (4.4)

where

f (tij|τi, βj, λj) = Hj(tij) expβjτi+λjuijexp− exp

βj τi+λj uij_H j(tij)_,

and

f (til|τi, βl) = Hj(tij) expβjτiexp− exp

βj τi_H j(tij)_.

Typically there are three ways of conducting this hypothesis test:

1. Likelihood ratio test: One has to fit both the null model and the alternative model in (4.3), and compare the differences between the two likelihoods.

2. Wald test statistic: One can construct the test statistics as χ2 ₌ (ˆλj)2

var(ˆλ), and this involves estimating

the unknown parameter λj. It becomes slightly complicated when the latent covariates τ need to be

estimated as well.

3. Lagrange Multiplier (LM) test: This test is relatively easy to compute because only the null model needs to be estimated. Specifically, assume the null model has parameters η1 and the alternative

model has additional parameters η2. Assume the hypothesis is H0: η2= 0 against H1: η26= 0. The

LM test is defined as

LM (η) = h(η)0H(η, η)−1h(η) |η1= ˆη,η2=0, (4.5)

where h(η) = ∂ ln L(η;x)_∂η , and H(η, η) denotes an observed information matrix with elements h(ηp, ηq) =

− ∂2

∂ηp∂ηq ln L(η; x). The maximum likelihood estimate of η1 is ˆη1 , and x represents the data. One

apparent advantage of LM test is the unknown parameter η₂ does not need to be estimated, and thus the LM statistic is generally straightforward to calculate. The LM statistic follows a chisquare distribution with degree of freedom equal to the number of components in η2.

As suggested in van der Linden and Glas (2010), we will adopt the LM statistic to check the local indepen- dence assumption. For item j, the LM statistic is constructed as

LM (λj) =

h(λj)2

h(λj, λj) − H(τ , λj)0H(τ , τ )−1H(τ , λj)

|τ =ˆτ ,λj=0, (4.6)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 50 100 150 200 250 Significance Probability of LM(λ_j) Frequency

LM test of conditional independence

Figure 4.14: Lagrange Multiplier probability for conditional independence

item. By plugging in the likelihood function in Equation (4.4) into (4.6), we have

−Hii(τ , τ ) = −β2 jHj(tij) exp (βjτi+ λjuij) − J X l=1;l6=j β2 jHj(tij) exp (βjτi) −h(λj) = nj X i=1 uij− nj X i=1 Hj(tij) exp(βjτi+ λjuij)uij −h(λj, λj) = − nj X i=1 u2 ijHj(tij) exp(βjτi+ λjuij) −Hi(τ , λj) = −Hj(tij) exp(βjτi+ λjuij)βjuij

In the calculation, replace τi with its corresponding MLE that were obtained by maximizing the likelihood

function in (4.4). For the 620 items in the item bank, the LM(λj)’s are presented in Figure (4.14). Only

43 items have probabilities significant at 5% level. This again supported our conclusion that this local independence between θ and τ assumption was satisfied.