Item Parameter Estimation IBock & Lieberman "1970#Bock & Aitkin "1981#1The plan:I. Using Ra# Bock$Lieberman ML, Normal Ogiveb# Bock$Lieberman ML, 2PLc# Bock$Aitkin %EM,& 2PLII. Using C++a# Bock$Aitkin %EM,& 3PL, Graded modelIII. Using Ra# Albert MCMC, Normal Ogive2Response PatternFrequencyResponse PatternFrequency11114201104111023010121101600111101161000380111101009110024001061010250001210017000020Stou!er & Toby "1951# Data, 4 Questions3m-3-3M-3M-3-2-2M-2M-2-1-1M-1M-100M0M011M1M122M2M233M3M3m0.20.2M0.2M0.20.40.60.6M0.6M0.60.81.01.0M1.0M1.0ThetaMThetaMThetaTMTMTm-3-3M-3M-3-2-2M-2M-2-1-1M-1M-100M0M011M1M122M2M233M3M3m0.00.0M0.0M0.00.50.5M0.5M0.51.01.0M1.0M1.0ThetaMThetaMThetaTMTMTm-3-3M-3M-3-2-2M-2M-2-1-1M-1M-100M0M011M1M122M2M233M3M3m0.00.0M0.0M0.00.50.5M0.5M0.51.01.0M1.0M1.0ThetaMThetaMThetaTMTMTm-3-3M-3M-3-2-2M-2M-2-1-1M-1M-100M0M011M1M122M2M233M3M3m0.00.0M0.0M0.00.50.5M0.5M0.51.01.0M1.0M1.0ThetaMThetaMThetaTMTMTTi(1i|θ) =1√2π!aiθ−ci−∞e−z22dzTi(1i|θ) = Φ[aiθ − ci]The normal ogive model for the positive "1# item response is or more compactly These are the four 'tted curves for the Stou!er$Toby data(the goal. 4m-3-3M-3M-3-2-2M-2M-2-1-1M-1M-100M0M011M1M122M2M233M3M3m0.00.0M0.0M0.00.10.1M0.1M0.10.20.2M0.2M0.20.30.3M0.3M0.30.40.4M0.4M0.4ThetaMThetaMThetaTMTMTm-3-3M-3M-3-2-2M-2M-2-1-1M-1M-100M0M011M1M122M2M233M3M3m0.00.0M0.0M0.00.50.5M0.5M0.51.01.0M1.0M1.0ThetaMThetaMThetaTMTMTm-3-3M-3M-3-2-2M-2M-2-1-1M-1M-100M0M011M1M122M2M233M3M3m0.00.0M0.0M0.00.10.1M0.1M0.10.20.2M0.2M0.20.30.3M0.3M0.30.40.4M0.4M0.4ThetaMThetaMThetaTMTMTPk=!∞−∞"nitems#i=1Ti(ki|θ)$φ(θ)dθFor any response pattern the probability is computed like this one "this is 1000; 38 respondents#: The population distribution is shown in blue, the trace lines in red, and the posterior in magenta. The area of the posterior is the probability of the response pattern. 5Pk=!∞−∞"nitems#i=1Ti(ki|θ)$φ(θ)dθL ∼!kPrkk! =!krklog(Pk)There are 16 response patterns k for four items; for each:The likelihood for the observed frequencies for all 16 patterns is:rkThe loglikelihood is:We seek the ML estimates of the parameters "a and c, buried in T, above#.6Estimation can be implemented in R with just that much information, with the integration approximated by rectangular quadrature "sums of heights at points# as it was to compute EAPs.7Bock & Lieberman "1970# did not have automagic minimizers with built$in numerical derivatives.So they needed the 'rst and second derivatives of Pk=!∞−∞"nitems#i=1Ti(ki|θ)$φ(θ)dθ! =!krklog(Pk)in which the item parameters are buried in speci'cally, inTi(1i|θ) = Φ[aiθ − ci]8After quite a bit of stu!, and ignoring side$trips, they arrived at ∂Pk∂ui= (−1)ki+1!∞−∞∂gi∂uiφ(gi)nitems$h=1,h#=iTh(kh|θ)φ(θ)dθin which u is either a or c, , andis either or $1, depending on whethera or c replaces u. gi= aiθ − ci∂gi∂uiθThat all goes into∂"∂ai= N!kpkPk∂Pk∂ai∂"∂ci= N!kpkPk∂Pk∂ciin which p is the observed proportion for k.and9Bock & Lieberman "1970# suggest approximating the matrix of second derivatives with ∂2"∂ui∂vi∼=−N!k1Pk∂Pk∂ui∂Pk∂viWe can make this work with Newton$Raphson directly, but not with R)s minimizer? 10a1c1a2c2a3c3a4c4a14041c133328483a210$3$22526c2$4$4$10$10329797a3$11$2$232$1$11818c3$4$4$11$11$1$1$17$17108787a42122126632661515c4$2$2$9$922$17$1722$18$19$21$226262Comparing the information matrices, R)s numerical hessian "red# vs. Bock & Lieberman)s SSCP approximation "blue#:11The logistic model "%2PL&# can be substituted for the normal ogive with minimal changes.In the R, with the change in functions "normal to logistic# we also change to slope$threshold parameterization.12m-3-3M-3M-3-2-2M-2M-2-1-1M-1M-100M0M011M1M122M2M233M3M3m0.00.0M0.0M0.00.50.5M0.5M0.51.01.0M1.0M1.0ThetaMThetaMThetaTMTMTm-3-3M-3M-3-2-2M-2M-2-1-1M-1M-100M0M011M1M122M2M233M3M3m0.00.0M0.0M0.00.50.5M0.5M0.51.01.0M1.0M1.0ThetaMThetaMThetaTMTMTm-3-3M-3M-3-2-2M-2M-2-1-1M-1M-100M0M011M1M122M2M233M3M3m0.00.0M0.0M0.00.50.5M0.5M0.51.01.0M1.0M1.0ThetaMThetaMThetaTMTMTm-3-3M-3M-3-2-2M-2M-2-1-1M-1M-100M0M011M1M122M2M233M3M3m0.00.0M0.0M0.00.50.5M0.5M0.51.01.0M1.0M1.0ThetaMThetaMThetaTMTMTNormal ogive "magenta, solid curves# vs.Logistic"cyan, dashed curves#for the Stou!er$Toby data13Direct maximization of the likelihood does not scale to "many# tens of items and some multiple of "many# tens of item parameters.Bock & Aitkin "1981# proposed a solution to this problem.14m-3-3M-3M-3-2-2M-2M-2-1-1M-1M-100M0M011M1M122M2M233M3M3m0.00.0M0.0M0.00.51.01.0M1.0M1.0ThetaMThetaMThetaTMTMTBock & Aitkin "1981# in pictures(repeat EM$EM$EM… until convergence:E$step "for each item# creates pseudo$data table:M$step "for each item# 'ts the trace line"s#:Quadrature points:$3.0$2.5…0.00.5…3.0r10.070.43…36.1432.93…0.53r00.371.32…7.453.67…0.00115Pk=Q!q=1"nitems#i=1Ti(ki|θq)$φ(θq)ri1q=!krkki"#nitemsi=1Ti(ki|θq)$φ(θq)Pkri0q=!krk(1 − ki)"#nitemsi=1Ti(ki|θq)$φ(θq)PkThe quadrature$based computation of the probability of each response pattern is:The expected frequency correct/positive/1 at each quadrature point is:and the expected frequency for the r0 row is:16m-3-3M-3M-3-2-2M-2M-2-1-1M-1M-100M0M011M1M122M2M233M3M3m0.00.0M0.0M0.00.51.01.0M1.0M1.0ThetaMThetaMThetaTMTMTOnce the pseudo$data table is created, likethe M$step "for each item# 'ts the trace line"s# using the pseudo$data table as real data, and ML. Such a problem is formally identical to probit or logit analysis for the %Constant Method& saltiness data.Quadrature points:$3.0$2.5…0.00.5…3.0r10.070.43…36.1432.93…0.53r00.371.32…7.453.67…0.001Standard errors are not %free,& however.17a1b1a2b2a3b3a4b4a13326b1$30$183622a210$10817b240$50$1$16346a310$101000610b360$7000$130007659a420$1010$2010$3047b420$7020$16010$2101289070Comparing the information matrices for the 2PL, R)s numerical hessian for Bock$Lieberman)s direct ML "red# vs. assembly of M$steps, rate$adjusted