Boosted Decision Tree for Q-matrix Refinement Peng Xu Michel C. Desmarais PolytechniqueMontreal PolytechniqueMontreal [email protected] [email protected] ABSTRACT put of Q-matrix refinement algorithms leads us to pursue In recent years, substantial improvements were obtained in further along the line of using ensemble learning, or meta- the effectiveness of data driven algorithms to validate the learning techniques. In particular, a common approach is mapping of items to skills, or the Q-matrix. In the cur- to use boosting with a decision tree algorithm. This is the rentstudyweuseensemblealgorithmsontopofexistingQ- approachexploredinthecurrentstudy. matrixrefinementalgorithmstoimprovetheirperformance. We combine the boosting technique with a decision tree. 2. THREETECHNIQUESTOQ-MATRIX The results show that the improvements from both the de- VALIDATION cision tree and Adaboost combined are better than the de- Ourapproachreliesonmeta-learningalgorithmswhoseprin- cision tree alone and yield substantial gains over the best ciple in a general way is to combine the output of existing performanceofindividualQ-matrixrefinementalgorithm. algorithmstoimproveupontheindividualoraverageresults. Inourcase,theapproachcombinesadecisiontreetrainedon 1. INTRODUCTION theoutputofQ-matrixvalidationalgorithmswithboosting, A Q-matrix, as proposed by Tatsuoka (Tatsuoka, 1983), is a weighted sampling process in the training of the decision a term commonly used in the literature of psychometrics tree to improve its accuracy. In this section, we first de- andcognitivemodelingthatreferstoabinarymatrixwhich scribe the Q-matrix validation techniques before describing shows a correspondence between items and their latent at- thedecisiontreeandboostingalgorithms. tributes. Items can be questions or exercises proposed to students, and latent attributes are skills needed to succeed 2.1 minRSS these items. Usually, a Q-matrix is defined by a domain expert. However, this task is non trivial and there might The first Q-matrix refinement technique that serves as in- beerrors,whichinturnwillresultinerroneousdiagnosisof put to the decision tree is from Chiu and Douglas (2013). studentsknowledgestates(Rupp&Templin,2008;Madison Wenamethistechnique minRSS.Theunderlyingcognitive & Bradshaw, 2015). Therefore, better means to validate a modelbehindminRSSistheDINAmodel(seeDeLaTorre, Q-matrixisahighlydesirablegoal. 2009). Afairnumberofalgorithmshaveemergedinthelastdecade For a given Q-matrix, there is a unique and ideal response tovalidateanexpertgivenQ-matrixbasedonempiricaldata pattern for a given a student skills mastery profile. That (seeforeg.recentworkfromChen, Liu, Xu,&Ying, 2015; is, if there are no slip and guess factors, then the response delaTorre&Chiu,2015;Durand,Belacel,&Goutte,2015). pattern for every category of student profile is fixed. The Desmarais, Xu, and Beheshti (2015) showed that Q-matrix differencebetweentherealresponsepatternandtheidealre- refinement algorithms can be combined using an ensemble sponsepatternrepresentsameasureoffitfortheQ-matrix, learningtechnique. Theyusedadecisiontreeandtheresults typically the Hamming distance. Chiu and Douglas (2013) showasubstantialandsystematicperformancegainoverthe consideredamorerefinedmetric. Theideaisifanitemhas bestalgorithm,intherangeof50%errorreductionforreal a smaller variance (or entropy), then it should be given a data,eventhoughthebestalgorithmisnotalwaysthesame higher weight in measure of fit. A first step is to compute fordifferentQ-matrices. the ideal response matrix for all possible student profile, and then to find the corresponding student profiles matrix Theencouragingtheresultsobtainedbycombiningtheout- A given observed data. First, a squared sum of errors for eachitemk canbecomputedby N RSS = (r η )2 k ik− ik Xi=1 where r is the real response vector while η is the ideal re- sponse vector, and N is the number of respondents. Then, theworstfitteditem(highestRSS)ischosentoupdateits correspondentq-vector. Givenallpermutationsoftheskills foraq-vector,theq-vectorwiththelowestRSSischosento Proceedings of the 9th International Conference on Educational Data Mining 551 replace the original one. The Q-matrix is changed and the 3. DECISIONTREE wholeprocessrepeated,butthepreviouslychangedq-vector The three algorithms for Q-matrix refinement described in iseliminatedfromthenextiteration. Thewholeprocedure thelastsectionaretobecombinedtoyieldwithadecision terminates when the RSS for each item no longer changes. treetoobtainanimprovedrefinementrecommendation,and This method was shown by Wang and Douglas (2015) to furtherimprovedbyboosting. Wedescribethedecisiontree yieldgoodperformanceunderdifferentunderlyingconjunc- beforemovingontotheboostingmethod. tivemodels. Decision tree is a well-know technique in machine learning and it often serves as an ensemble learning algorithm to 2.2 maxDiff combine individual models into a more powerful model. It AkintominRSS,themaxDiffalgorithmreliesontheDINA usesasetoffeaturevariables(individualmodelpredictions) model. De La Torre (2008) proposed that a correctly spec- to predict a single target variable (output variable). There ified q-vector for item j should maximize the difference of are several decision tree algorithms, such as ID3 (Quinlan, probabilitiesofcorrectresponsebetweenexamineeswhohave 1986), C4.5 (Quinlan, 1993), CART (Breiman, Friedman, alltherequiredattributesandthosewhodonot. Anatural Stone, & Olshen, 1984). We used rpart function from the ideaistotestallq-vectorstofindthatmaximum,butthatis Rpackageofthesamename(Therneau,Atkinson,&Ripley, computationallyexpensive. DeLaTorre(2008)proposeda 2015). ItimplementstheCARTalgorithm. Thisalgorithm greedyalgorithmthataddsskillsintoaq-vectorsequentially. dividesthelearningprocessintotwophases. Thefirstphase Assumingδ representsthedifferencetomaximize,thefirst jl is for feature selection, or tree growing, during which the step is to calculate δ for all q-vectors which contains only jl feature variables are chosen sequentially according to Gini oneskillandtheonewithbiggestδ ischosen. Then,δ is jl jl impurity (Murphy, 2012). Then in the second phase, the calculatedforallq-vectorswhichcontainstwoskillsinclud- pruning phase, deep branches are split into wider ones to ing the previously chosen one. Again the q-vector with the avoidoverfitting. biggest δ is chosen. This whole process is repeated until jl no addition of skills increases δ . However, this algorithm jl Adecisiontreeisasupervisedlearningtechniqueandthere- requires knowing slip and guess parameters of the DINA fore requires training data. To obtain training data of suf- modelinadvance. Forrealdata,theyarecalculatedbyEM ficientsize,Desmaraisetal.(2015)usesyntheticdatafrom (ExpectationMaximization)algorithm(DeLaTorre,2009). Q-matrices generated by random permutations of the per- turbatedQ-matrix. Sincetheground-truthQ-matrixofsyn- 2.3 ALSC theticdataisknown,itbecomespossibletogeneratetrain- ALSC(ConjunctiveAlternatingLeastSquareFactorization) ing data containing the class label. The training set for isacommonmatrixFactorization(MF)algorithm. Desmarais decisiontreecantakethisform: andNaceur(2013)proposedtofactorizestudenttestresults Algorithmtargetprediction Otherfactors intoaQ-matrixandaskills-studentmatrixwithALSC. Target minRSS maxDiff ALSC ... 1 1 0 1 ... ALSCdecomposestheresultsmatrixR ofmitemsbyn m n studentsastheinnerproducttwosmalle×rmatrices: 0 0 1 0 ... ... ... ... ... ... R=Q S (1) ¬ ¬ Theotherfactorsconsideredtohelpthedecisiontreetoim- where Risthenegationoftheresultsmatrix(mitemsby provepredictionarethenumberofskillsperrow(SR),num- ¬ nstudents),QisthemitemsbykskillsQ-matrix,and Sis ber of skills per column (SC). Moreover, a feature named ¬ negationofthethemasterymatrixofkskillsbynstudents stickiness is introduced and makes a critical difference. It (normalizedforrowscolumnstosumto1). Bynegation,we measures the rigidity of cells under each validation meth- mean the 0-values are transformed to 1, and non-0-values ods. Stickiness represents the rate of a given algorithm’s to 0. Negation is necessary for a conjunctive Q-matrix. As false positives for a given cell of a Q-matrix. The rate is such, the model of equation (1) is analogous to the DINA measured by“perturbating”in turn each and every cell of modelwithoutaslipandguessparameter. theQ-matrix,andbycountingthenumberoftimesthecell is a false positive. The decision tree can use the stickiness The factorization consists of alternating between estimates factor as an indicator of the reliability of a given Q-matrix of S and Q until convergence. Starting with the initial ex- refinement algorithm suggested value for a cell. Obviously, pert defined Q-matrix, Q0, a least-squares estimate of S is ifacell’sstickinessvalueishigh,thereliabilityofthecorre- obtained: spondingalgorithm’srefinementwillbelower. ¬Sˆ0=(QT0 Q0)−1QT0 ¬R (2) Then,anewestimateoftheQ-matrix,Qˆ ,isagainobtained 1 bytheleast-squaresestimate: 4. BOOSTING Qˆ1=¬R¬SˆT0 (¬Sˆ0¬SˆT0)−1 (3) Twhitehcaunrortehnetrwmoerkta-elxetaernndinsgthteechidneiaquoefnuasminegdabdoeocsitsiinogn.tree And so on until convergence. Alternating between equa- tions (2) and (3) yields progressive refinements of the ma- Boosting(Schapire&Freund,2012)servesasameta-learning tricesQˆ andSˆ thatmorecloselyapproximateRinequa- technique for lifting a base learner. It operates on weights i i tion(1). ThefinalQˆ isroundedtoyieldabinarymatrix. of the loss function terms. For a training set of N samples i Proceedings of the 9th International Conference on Educational Data Mining 552 andagivenlossfunctionL,thegloballossis Table 1: Q-matrix for validation N Number of Loss= L(y,f(x)) Name Description i i Skills Items Cases Xi=1 QM1 3 11 536 Expertdrivenfrom Differentwaysofchoosinglossfunctionyielddifferentboost- (Henson,Templin,& ing algorithm. The most famous algorithm for boosting is Willse,2009) Adaboost(Freund&Schapire,1997),whichisespeciallyset QM2 3 11 536 Expertdrivenfrom forbinaryclassificationproblemandusesexponentialloss. (DeLaTorre,2008) QM3 5 11 536 Expertdrivenfrom Inour case, the baselearner isthe decisiontree. Adaboost (Robitzsch,Kiefer, trainsthedecisiontreeforseveraliterations,butwithadif- George,&Uenlue, ferent weighted training data for each iteration. That is, 2015) each time a decision tree is trained, the wrongly predicted QM4 3 11 536 Datadriven,SVD datarecordsinthecurrentiterationwillbeassignedhigher based weightsinthecomputationofthelossfunctionforthenext training iteration of the decision. The final output of Ad- aboost is a sgn function (sign function) of a weighted sum In order to use a standard performance measure, we define ofall“learners”trainedinthewholeprocedure(thedecision the following categories of correct and incorrect classifica- treewithdifferentweightsvectors). tionsasthenumberof: For a training set of N samples, the whole procedure for TruePositives(TP):perturbedcellcorrectlyrecov- • Adaboostisshownbelow(Murphy,2012): ered True Negatives (TN): non perturbed cell left un- • Initializeω =1/N changed i fori=1toM do FalsePositives (FP):nonperturbedcellincorrectly • Fittheclassifierφ (x)tothetrainingsetusingweights recovered m w FalseNegatives(FN):perturbedcellleftunchanged • N Computeerr = i=1ωiI(y˜i6=φm(xi)) We give equal weight to perturbed and unperturbed cells m P N ωi and use the F1-score, or F-score for short. The F-score is Computeα =log[(1 iP=er1r )/err ] definedas m − m m setω ω exp[α I(y˜ =φ (x)] precision recall i← i m i6 m i F =2 · endfor · precision+recall M return f(x)=sgn( αmφm(x)) In which precision is calculated by the model accuracy on m=1 non-perturbatedcellwhilerecalliscalculatedbythemodel P accuracyonperturbatedcell. In which M is the number of iterations (10 in our experi- ment), ω is the weight for i-th data, I() is the indicator i · 6. DATASET function,y˜ 1, 1 istheclasslabeloftrainingdata,and φ (x)isthie∈d{ecis−ion}treemodelinourcase. Forthesakeofcomparison,weusethesamedatasetsasthe m ones used in Desmarais et al. (2015). Table 1 provides the basicinformationandsourceofeachdataset. Boosting has had stunning empirical success (Caruana & Niculescu-Mizil,2006). Moredetailedexplanationandanal- 7. RESULT ysis of boosting can be found in Bu¨hlmann and Hothorn TheresultsofapplyingAdaboostoverthedecisiontree(DT) (2007) and Hastie, Tibshirani, and Friedman (2009). The arereportedintable2forsyntheticdataandTable3forreal Adaboostalgorithmwasimplementedinthisexperimentto data. Theindividualresultsofeachalgorithmarereported improvetheresultsobtainedbyDesmaraisetal.(2015). The (minRSS,maxDiff,andALSC),alongwiththedecisiontree resultsarereportedinsection7. (DT) and the boosted decision tree (BDT). Different im- provementoverbaselinesarereportedas: 5. METHODOLOGYANDPERFORMANCE CRITERION DT %Gain: the Decision Tree (DT) improvement • overthebestofthethreeindividualalgorithm(minRSS, To estimate the ability of an algorithm to validate a Q- maxDiff,ALSC) matrix,weperturbatea“correct”Q-matrixandverifyifthe algorithmisabletorecoverthiscorrectmatrixbyidentify- BDT %Gain: Boosted Decision Tree improvement ingthecellsthatwereperturbatedwhileavoidingtoclassify • over the DT performance, which corresponds to the unperturbatedcellsasperturbated. Inthisexperiment,only gainwegetfromboosting. oneperturbationisintroduced. Forsyntheticdata,the“cor- rect”matrixisknownandistheoneusedinthegeneration of the data. For real data, we assume the expert’s is the Let us focus on the F-Score which is the most informative correctone,albeititmaycontainerrors. sinceitcombinesresultsoftheperturbedandnonperturbed Proceedings of the 9th International Conference on Educational Data Mining 553 Table 2: Results for synthetic data Table 3: Results for real data Individual Ensemble Individual Ensemble S ff ) ) S ff ) ) minRS maxDi ALSC DT %Gain BDT%Gain minRS maxDi ALSC DT %Gain BDT%Gain QM ( ( QM ( ( Accuracy of perturbated cells Accuracy of perturbated cells 1 0.809 0.465 0.825 0.946 (69.4%) 0.951 (9.2%) 1 0.485 0.167 0.515 0.758 (50.0%) 0.758 (0.0%) 2 0.069 0.259 0.359 0.828 (73.2%) 0.903 (43.5%) 2 0.345 0.093 0.564 0.618 (12.5%) 0.764 (38.1%) 3 0.961 0.488 0.953 1.000(99.7%) 1.000 (0.0%) 3 0.212 0.091 0.364 0.818 (71.4%) 0.818 (0.0%) 4 0.903 0.489 0.853 0.956 (54.3%) 0.971 (33.9%) 4 0.394 0.111 0.576 0.576 (0.0%) 0.818 (57.1%) X 0.685 0.425 0.747 0.933 (74.2%) 0.956 (21.7%) X 0.359 0.115 0.505 0.692 (33.5%) 0.789 (23.8%) Accuracy of non perturbated cells Accuracy of non perturbated cells 1 0.970 0.558 0.387 0.990(65.1%) 0.990 (0.0%) 1 0.435 0.670 0.418 0.606 ( 19.4%) 0.606 (0.0%) − 2 0.987 0.529 0.431 0.989 (20.5%) 0.996 (59.1%) 2 0.875 0.929 0.110 0.956 (37.9%) 0.966 (21.4%) 3 0.950 0.258 0.736 0.994 (88.9%) 1.000(100.0%) 3 0.661 0.830 0.219 0.785 ( 26.2%) 0.752 ( 15.1%) − − 4 0.966 0.559 0.391 0.997 (92.2%) 0.998 (19.2%) 4 0.520 0.889 0.148 0.546 ( 308.7%) 0.658 (24.7%) − X 0.968 0.476 0.486 0.993 (65.3%) 0.996 (49.4%) X 0.623 0.829 0.224 0.723 ( 79.1%) 0.746 (8.0%) − F-score F-score 1 0.882 0.507 0.527 0.968 (72.4%) 0.970 (7.4%) 1 0.459 0.267 0.461 0.673 (39.4%) 0.673 (0.0%) 2 0.128 0.348 0.392 0.902 (83.8%) 0.947 (46.1%) 2 0.495 0.168 0.184 0.751 (50.6%) 0.853 (40.9%) 3 0.955 0.337 0.831 0.997 (93.5%) 1.000(100.0%) 3 0.321 0.164 0.273 0.801 (70.7%) 0.784 ( 8.7%) − 4 0.934 0.522 0.536 0.976 (64.0%) 0.984 (33.6%) 4 0.448 0.198 0.235 0.560 (20.3%) 0.730 (38.5%) X 0.725 0.429 0.571 0.961 (78.4%) 0.975 (46.4%) X 0.431 0.199 0.288 0.696 (45.25%) 0.760 (17.8%) cellsoftheQ-matrix. Forsyntheticdata,theerrorreduction However,wefindstrongdifferencesbetweentheQ-matrices. of boosting over the gain from the decision tree is substan- Foreg.,QM2benefitsofimprovementscloseto50%(QM2), tially improved for all Q-matrices. The range of improve- while QM1 has a null improvement for real data and only ment is from 7% to 100%. For real data, two of the four 7.4% for synthetic data. In that respect, the boosting does Q-matrices show substantial improvements of around 40%, notprovideagainthatisassystematicastheoneobtained whereas the other two show no improvements, even a de- fromtheDTwhichispositiveforallmatrices. crease of 8.7% for Q-matrix 3 which is characterized by a single skill per item. However, let us recall that we assume Animportantadvantageofthemeta-learningapproachout- the expert Q-matrices are correct, which may be over opti- lined here is that it can apply to any combination of algo- mistic. Violationof this assumption could negatively affect rithmstovalidateQ-matrices. Futureworkcouldlookinto someoftheQ-matricesscoresforrealdata. combiningmorethanthethreealgorithmsofthisstudy,and addnewalgorithmsthatpotentiallyoutperformthem. And Note that QM3 has an inconsistent 100% gain from boost- if the current results generalize, we would expect to make ingwithsyntheticdatacomparedtoasmalllossisobtained supplementarygainsoveranyofthem. with real data. The value of 100% should be taken cau- tiously because the F-score difference is measured close to Moreover, the Q-matrices used in this research are quite the boundary of 1 and therefore the result of only a few small in size. The performance of boosted decision tree on casesinoursample. Nevertheless,thefactthataveryhigh largerQ-matrixandlargerdatasetwouldalsobeofinterest. F-scoreisobtainedforsyntheticdatacomparedtorealdata doesraiseattentionandmightberelatedtothefactthatit However,besidestheQ-matrix-basedalgorithmsmentioned istheonlysingleskillperitemmatrix. above, there are other frameworks for knowledge tracing ordomainmodeling, especiallywhendealingwithdynamic data. Forexample,thereareLearningFactorAnalysis(Cen, 8. DISCUSSION Koedinger,&Junker,2006),WeightedCRP(Lindsey,Kha- This study shows that the gain obtained from combining jah,&Mozer,2014),HMM-basedBayesianKnowledgeTrac- theoutputofmultipleQ-matrixrefinementalgorithmswith ing (Corbett & Anderson, 1994; Lindsey et al., 2014) and adecisiontreecanbefurtherimprovedwithboosting. The other HMM-based models (Gonza´lez-Brenes, 2015). Com- results for synthetic data show an F-score error reduction parisonwiththeseframeworksarealsolefttofuturework. fromboostingovertheDTscoreofcloseto50%onaverage for all four Q-matrices, and a 18% reduction for real data. 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