Developing Scale Scores And Cut . - Center For Assessment
On Demand Assessments of Individual StandardsDeveloping Scale Scores and Cut Scores forOn-Demand Assessments of Individual StandardsWorking PaperApril, 2018Nathan Dadey1, Shuqin Tao2, and Leslie Keng112The National Center for theImprovement of Educational AssessmentCurriculumn AssociatesSuggested Citation:Dadey, N., Tao, S., & Keng, L. (2018, April). Developing Scale Scores and Cut Scores for OnDemand Assessments of Individual Standards. Paper to be presented at the annual meetingof the National Council of Measurement in Education: New York, NY.1
On Demand Assessments of Individual StandardsIntroductionNow, more than ever, assessments are being asked to fulfill an ever broadening range ofpurposes. Often, test users want an overall scale score, fine grained information on specific standards,as well as information on growth. Clearly, no one assessment can be expected to provide all suchinformation with the same level of precision, but a combination of assessments, carefully tailored,could. Using multiple assessments, however, poses a completely new challenge – integrating the resultsof multiple assessments into a coherent narrative about student learning. We believe there aremultitude of ways of doing so. The goal of this work is to examine a particular set of assessments to seewhether such a narrative can be told using one or more unidimensional reporting scales.Specifically, this work examines two types of interim assessments – a “general” assessment thatbroadly covers the fourth grade common core state standards (CCSS) in mathematics and a set of 31short “mini-assessments”, each of which covers a single 1 fourth grade math CCSS standard or substandard (e.g., 4.NF.B.4.C). These assessments differ not only in terms of content, but also in terms ofadministration and use. In one simple use case, students take the general assessment, receiveinstruction across multiple weeks, and then are assessed using one or more of the mini-assessments.However, this is just one of a variety of possible use cases, the uses of assessments, as well as the timingof their administration, vary across classroom, schools and districts. Given that these assessments arepart of an operational testing program that spans multiple states, variation in use and administration issubstantial. However, this is just the type of challenge systems of assessment are now facing – a widevariety of potential use cases paired with a diverse pattern of administrations, but still requiring soundmeasurement.The purpose of this paper is examine ways in which the results from the mini-assessments canbe modeled using psychometric methods – with an emphasis on the creation of one or moreunidimensional latent scales as well as associated cut points. Given the relatively novel nature of thiswork, we integrate considerations of various design decisions throughout the work. This work is guidedby two research questions:1. In what ways can the mini-assessments be scaled? Specifically, can and should the 31 miniassessments be:a. Placed onto a single unidimensional latent scale?b. Divided up and placed onto four unidimensional latent scales, corresponding to fourCCSS fourth grade mathematics domains?2. How can provisional cut scores be set on the mini-assessment total score scales?Addressing (1) entails investigating the dimensionality of the set of 31 mini-assessments. Typically,dimensionality is viewed as pertaining to a single assessment – here we extend the concept andmethods to the set of mini-assessments through a concurrent calibration approach. To address (2), we1Four of the mini-assessments break this rule, and cover a narrow range of standards instead of an individualstandard.2
On Demand Assessments of Individual Standardsdraw on information provided by the general assessments to create provisional cut scores. In doing so,we are attempting to increase the agreement between the mini-assessments and general assessment.We do note, however, that these provisional cut scores are meant to be revisited by content experts andadjusted as needed. Our ultimate aim in addressing these two research questions is to provide anexample that illustrates one way to tackle the thorny issues inherent in modeling results from this typeof distributed system of assessment.A System of Assessments: The Call and the State of the ArtThe concept of a carefully tailored combination of assessments is best reflected in the body ofliterature focused on the concept of a “system of assessments”. The idea for a system of assessmentscan be traced back at least to the seminal work of Pellegrino, Chudowsky and Glaser (2001), who outlinea plan for “coordinated systems of multiple assessments that work together, along with curriculum andinstruction, to promote learning” (p. 252, original emphasis). These systems of assessment are meant tooperate at multiple levels, “from the classroom through the school, district, and state” (p. 256). Workdetailing approaches to “balanced” (Gong, 2010), “comprehensive” (Perie, Marion, & Gong; 2009; Ryan,2010), and “next generation” (Darling-Hammond & Pecheone; 2010; Herman, 2010) assessment systemsfollowed. Also important is the work that examines a particular type of assessment system – the throughcourse assessment model (Bennett, Kane & Bridgeman, 2011; Ho, 2011; Kolen, 2011; Sabatini, 2011;Valencia, Pearson & Wixon, 2011; Way, McClarty, Murphy, Keng & Fuhrken, 2011; Wise, 2011; Zwick &Mislevy, 2011).The development and implementation of systems of assessment have been slow to start, buthas been gaining traction recently. The through course model appears to have had some success, withthe release of the Smarter Balanced interim comprehensive assessments and assessment blocks. Othersummative assessment vendors are beginning to respond as well, by providing assessments alongsidetheir summative offerings that are smaller in scope than the typical summative assessment and meantto be used within the academic year for purposes other than state accountability (e.g., interimassessments; cf., Perie, Marion, & Gong; 2009). These approaches do not fully meet the vision laid outby Pellegrino et al. (2001), but represent a significant improvement. Interestingly, whereas Pellegrino etal. (2001) mainly conceptualized the levels of the assessment in terms of educational units (e.g.,classroom, school, district or state) the work of Smarter Balanced and others focuses on providing shortassessments that can be used flexibly at multiple layers of the educational system based on userpreference. In this way, that work attempts to equip users with a set of assessments so that they canthen tailor the administration of a specific subset of assessments to match the theory of learningunderpinning instruction. Ideally, this tailoring should allow a set of general, curriculum neutralassessments to become curriculum relevant – tied to the scope and sequence of instruction. Whetherthis can be done successfully will likely require sustained effort from assessment users as well as acontinuous support from assessment developers.3
On Demand Assessments of Individual StandardsIntrinsically tied to the relevance educators and administrators draw meaning from suchassessments is how the results are reported. The Smarter Balanced interim comprehensiveassessments 2 are reported on the Smarter Balanced summative scale. This practice appears to beindicative of the current trend – to report the results of interim assessments on the scale of thesummative item bank, presumably by leveraging item parameters from the summative assessment(although Zwick & Mislevy’s 2011 approach is a variation on this, in that they suggest creating a scalethrough the application of the latent regression model used by the National Assessment of EducationalProgress).Our approach departs from this trend, instead of attempting to place the mini-summativeassessments on the scale of the general assessment, we investigate two ways in which scales can bedeveloped for just the set of mini-assessments. It is worth noting, however, that even though currentpractice is to report interim results on the scale of the summative item bank, the results of each interimassessment are generally provided in isolation. That is, the information produced from theseassessments is often left in separate silos – never integrated into a holistic picture of what studentsknow and can do and made easily available for practical use. In our work, we touch on this issue,partially, by drawing on the results of the general assessment to set preliminary cut scores on the miniassessments.MethodsMeasuresMini-assessments. The mini-assessments are short assessments meant to provide school anddistrict educators and administrators with information about student mastery on individual contentstandards, or a grouping of similar standards, throughout the academic year. For example, districtadministrators often assign sets of mini-assessments to provide aggregate information at specific pointsduring the academic year, which they then use to drill down in order to find specific areas of weakness(e.g., specific standards, grades, teachers or schools) to which they can provide targeted support. Thus,the mini-assessments are intended to be aggregated and used at the school and district levels. Like thegeneral assessments, the mini-assessments are computer administered, but unlike the generalassessment, the mini-assessments are not adaptive and are not currently scaled using an item responsetheory (IRT) model. Both the assessments and mini-assessments provide instant score reporting,although the mini-assessments are not currently scaled. The mini-assessments and the generalassessment do not have items in common. There are also no common items among the miniassessments. Key design features of the mini-assessments include: Configurable. Administrators can choose to group multiple mini-assessments into an“assignment”, which are then administered by educators within an administrator-definedwindow. Assignments often function as end-of-unit quizzes.2The interim assessment blocks are not, and instead primarily reported in terms of three categories - abovestandard, near standard, and below standard.4
On Demand Assessments of Individual Standards Administered as Needed. Administrators choose when, and how often, mini-assessments aregiven. There is a recommended a calendar with a suggested administration schedule thatmatches the scope and pacing of their provided curriculum, but users may deviate from thisschedule.Short. The mini-assessments each contain 6 to 10 items, all of which are machine scorable. Eachmini-assessment is made up of selected response and a variety of polytomous item types (e.g.,ordered-list, cloze drop down).Multiform. Within each subject and grade-level, there are approximately 30 standards-basedassessments, grouped into two forms (A and B), resulting in about 60 assessment forms pergrade. There are no common items across mini- forms for the same standard(s), nor are therecommon items across any two mini-assessments.Open. Students, educators and administrators can see the items making up each form, as wellas student responses to each item.It is worth noting that each of the mini-assessments was developed in isolation from the others. That is,the items were written specifically for each mini-assessment and the classical test statistics used forquality control were computed using only the data from the items within the given mini-assessment.Therefore, the quality control process was not designed to ensure undimensionality at the domain oroverall mathematics levels, which can be a byproduct of processes designed to increase reliability.General assessment. The general assessment is a computer adaptive assessment meant tobroadly assess fourth grade mathematics. Results from the general assessment are reported on avertical scale that spans kindergarten to twelfth grade. The current instantiation of the vertical scale wascreated in 2015 using a concurrent calibration of approximately 9.5 million assessment administrationsfrom operational data from a prior version of the assessment. The Rasch model was used with maximumlikelihood estimation to produce parameter estimates. For fourth grade mathematics, the maximumnumber of possible items a student could be administered is 66 and the stopping rule is based onsatisfying item minimums and maximums for each of four CCSS mathematics domains. Like the miniassessments, the general assessment includes selected response items and a number of polytomousitems, including short constructed response, text highlight, drag-and-drop, multiple-correct response,coordinate grid, and number line items. The recommended administration pattern of the generalassessment is three times a year, once in fall, then again in winter and finally in spring. However, theassessment can be given as frequently as educators and administrators choose.In addition to the vertical scale score, student reports on the general assessment also havenumber of other scores, including four CCSS domain subscores and a set of “indicator classifications”.The four domains upon which subscores are reported are: Operations and Algebraic Thinking, Numberand Operations (which includes both Numbers and Operations in Base 10 and Numbers and Operations Fractions), Geometry, and Measurement and Data. Each subscore is created by using the itemparameter estimates from the overall vertical scale, but just for the items aligned to the given domain. Astudent’s domain subscore on Operations and Algebraic Thinking, for example, is based just on the5
On Demand Assessments of Individual Standardsitems that he or she took that are aligned to standards within the Operations and Algebraic Thinkingdomain.The domain subscores also serve as the basis for set of reported scores – the indicatorclassifications. These dichotomous indicator classifications are meant to signal whether a student needsadditional instruction on a given standard, sub-standard or grouping of standards or sub-standards.There are approximately 30 indicators for fourth grade mathematics and generally align to the samestandards assessed by the mini-assessments. As explained in detail in Appendix A, the indicatorclassifications are defined for each student by, essentially, comparing his or her relevant domain scoreto the difficulty the items aligned to that indicator’s standard(s). Since each student can receive adifferent set of items, indicators are only reported when a student receives six or more items within anindicator. Students identified as needing additional instruction are provided with content basedrecommendations for improving, which were generated by content experts through examination ofeach indicator’s items. An example of the reporting of indicator classifications is provided below inFigure 1, in which each row corresponds to an indicator classification.Figure 1. Example Student Report of Indicator Classifications.DataDuring the 2016-2017 academic year (August 2016 to July 2017) 101,966 students had a score on one ormore of the mini-assessment forms. There are 31 mini-assessments in two forms (A and B), for a total of62 assessment forms. The number of administrations per mini-assessment form ranges fromapproximately 3,000 to 47,000 with a mean of approximately 12,000 and a median of approximately8,000. There is no mini-assessment form that all students take. The Form A mini-assessments of earlierstandards (within a suggested instructional sequence) are taken by larger numbers of students than theother mini-assessments. The number of mini-assessment administrations, including re-tests, taken per6
On Demand Assessments of Individual Standardsstudent range from 1 to 80, with a median of 6 and a mean of 7.6. There were 667 unique assignments(i.e., combinations of mini-assessments) administered 461,299 times. The number of mini-assessmentswithin each assignment ranged from 1 to 9, with a median of 3. The number of times an assignment wasadministered ranged from 1 to 15,886 with a median of 93 and a mean of 692.We are able to match general assessment data to 91,440 of the students taking miniassessments. Our matching process uses the results from the general assessment that was administeredclosest to each mini-assessment in question. For example, if a student took the general assessmenttwice – once in the fall and again in the spring – we would use the fall administration for a miniassessment taken in the fall. However, if the student took another mini-assessment in the spring, wewould use the spring administration of the general for the analysis of that mini-assessment. Note,however, that we implemented this matching approach using the actual date of the administration,instead of the administration window.Analytic ApproachResearch question 1. Research question one asks whether the set of fourth grade mathematicsmini-assessments can be placed onto a single unidimensional latent scale, or can be divided into CCSSdomains and then scaled to produce four unidimensional latent scales 3. We also entertained the idea ofcreating separate scales for each mini-assessment, but maintaining so many scales across multiple yearsseemed unfeasible. In addition, a finding that the mini-assessments could be scaled unidimensionally atthe overall or domain level would render the need to investigate the individual assessment level moot.The key criterion we use to examine whether or not any particular scaling is defensible is that ofunidimensionality, as examined through a principle components analysis (PCA) of the standardized itemresiduals produced from a concurrent calibration of the mini-assessments. To conduct this concurrentcalibration, we create a single person by item response matrix across all 62 mini-assessment forms andthen apply the Rasch model (Rasch, 1960). This process produces a single scale spanning the 62assessments, and we repeat this process within each domain to produce the domain-level scales (scalingeach domain separately, ignoring the items from other assessments that are not within the domain). Weconduct these calibrations in WINSTEPS (Linacre, 2016.) We then examine the standardized itemresiduals produced using these scales to determine whether the unidimensional scaling has adequatelycaptured variation in student responses – that is, there are no remaining secondary dimensions presentwithin the residuals. We also examine person and item fit using unweighted and weighted meansquared fit statistics.To create the matrices for concurrent calibration, we pool across testing occasions. In instanceswere a student took a mini-assessment more than once, we use the item responses from the finaladminstration. In creating this pooled item response matrix, we are eschewing traditional approaches toscale creation – we have neither common items nor common persons. In terms of the latter, althoughwe have the same students, these students are generally not taking the mini-assessments at the same3Another alternative we have not explored is to apply a multidimensional IRT model to the data.7
On Demand Assessments of Individual Standardspoints in time - posing questions around the applicability of a common person design. Our approach isto ignore differences in administration and scale the item responses across the 62 mini-assessmentforms. This dataset represents a best-case scenario for detecting dimensionality - if we do not detectdimensionality in data where time is clearly a factor, we suspect we would not detectmultidimensionality elsewhere (e.g., under an ideal common person design).Research question 2. Research question two asks about the ways in which cut scores can be seton the mini-assessments. Specifically, the goal is to develop two cut scores for each mini-assessmentform that classify students into three categories – Beginning, Progressing and Proficient. A prioriqualitative descriptions of these categories are: Beginning. The student is not progressing well in the standard and would most likely benefitfrom review of concepts and skills that are prerequisite to understanding the conceptsembodied in the standard.Progressing. The student is progressing towards mastery of the concepts embodied in thestandard, but would most likely benefit from more practice on these concepts.Proficient. The student has mastered the concepts embodied in the standard – therefore he orshe needs little or no additional instruction on the concepts within that standard.With a single test or limited number of assessments, cut scores are generally set using judgmentalprocedures involving panels of experts (cf., Cizek & Bunch, 2007). However, with 62 mini-assessmentforms, such a standard setting process is daunting. Moreover, the classifications from the miniassessments are meant to signal whether students need additional instruction on a given standard, asare the indicator classifications from the general assessment. Thus, there is the potential fordisagreement from the two different types of assessment – for example a mini-assessment couldindicate that a student is does not need instruction on the standard(s), but a later administration of thegeneral assessment could indicate that the student does need instruction.For these reasons we use a method for setting cut scores that relies on the indicatorclassifications. Specifically, we use quantile regression to predict performance on the relevant generalassessment indicator using the total scores from a mini-assessment, controlling for the difference inadministration (in days). We then evaluate the resulting regression function to select a cut point foreach assessment that is meant to differentiate between the Progressing and Proficient levels. Doing soentails making a number of decisions, decisions for which there is little empirical guidance. Below we listthe steps we used to create the preliminary cut scores, as well as detailing the underlying motivation forthe steps and decision points:1. Create the probability of mastering the corresponding indicator.For each student, create the probability of mastering the corresponding indicator based on hisor her domain score from the closest administration of the general assessment. This probabilityis computed as,exp(𝜃𝜃𝑗𝑗 𝑏𝑏𝑖𝑖 )𝑃𝑃 𝑋𝑋𝑖𝑖𝑖𝑖 1 (1)1 exp(𝜃𝜃𝑗𝑗 𝑏𝑏𝑖𝑖 )8
On Demand Assessments of Individual Standardswhere 𝑃𝑃 𝑋𝑋𝑖𝑖𝑖𝑖 1 is the probability of student j mastering indicator i, 𝜃𝜃𝑗𝑗 is student j’s thetaestimate for domain i from the closest general assessment administration and 𝑏𝑏𝑖𝑖 is, essentially,an aggregate item difficulty that determines whether a student is classified as mastering or notmastering the given indicator. This difficulty value is derived through a multistep process, asdescribed in Appendix A, and represents the domain theta value associated with obtaining 67%of the possible raw points on the items aligned to the indicator on the general assessment –adjusting 𝜃𝜃𝑗𝑗 for the aggregate difficulty of the items within the indicator.This application of (1) treats each indicator as an item within the familiar Rasch function. Thisapplication also departs slightly from the way in which the indicator classifications are reportedon the general assessment – that is, as dichotomous statements about whether students havemastered a particular standard and thus do, or do not, need instruction. Instead, we use aprobability of mastery – to avoid losing information on student performance on the indicator.An alternative would be to use the indicator classifications directly and therefore predict theclassifications via logistic regression.2. Conduct quantile regression.Perform a quantile regression in which the probabilities of mastery produced in step 1 arepredicted by the mini-assessment raw scores, controlling for the difference in administrationsbetween the mini-assessments and the general assessments (in days) 4. The quantile regressionsare implemented using the quantreg R package (Koenker, 2017) in R version 3.4.0 and estimatedfor the 10th, 20th, 90th quantiles.3. Evaluate the quantile regression.Select a cut point by evaluating the quantile regression functions, with an emphasis ondetermining what mini-assessment raw score corresponds to 𝑃𝑃 𝑋𝑋𝑖𝑖𝑖𝑖 1 0.67, which is thevalue that is used to define the indicator classifications on the general assessment, and𝑃𝑃 𝑋𝑋𝑖𝑖𝑖𝑖 1 0.50, which mirrors the approach in Rasch modeling to report item difficulties interms of a response probability of 0.50. In addition, we also focus on the 50th quantile, whichprovides one indication of how the “typical” student performs. However, we do not treat thesevalues as set in stone, nor do we a priori define the specific quantile to be evaluated.These resulting cuts are meant to be provisional and subject to content expert review.4There is a linear negative association between 𝑃𝑃 𝑋𝑋𝑖𝑖𝑖𝑖 1 and the difference between administrations betweenthe assessment administrations (ranging from -0.29 to -0.04 across mini-assessment forms, with a mean of -0.18).The difference variable is defined as Date of Mini-Assessment minus the Date of the General Assessment, so thelater the general assessment is administered after the mini-assessment, the higher 𝑃𝑃 𝑋𝑋𝑖𝑖𝑖𝑖 1 is (and vice versa).This association is not present between the difference variable and the mini-assessment scores. Likely, this patterncan be explained by the fact that scores on the general assessment domain subscales generally increase across theyear, whereas the scores on the mini-assessment generally do not (barring the very beginning and end of the year).9
On Demand Assessments of Individual StandardsResultsResearch Question 1.The results of the principle components analyses of the residuals from the Rasch model aresummarized in Table 1 below and graphically in Appendix C. At most, the largest principle componentaccounted for about 2.0% of the unexplained variance – indicating that there are no sizable factorsunaccounted for by the model. In addition, for the domain scaling, the percentages of items displayingmisfit (values less than 0.75 and greater than 1.33) based on the unweighted mean squared fit statistics(i.e., infit) ranged from 1% to 6% across the domains. Similarly, the percent of items displaying misfitbased on the unweighted mean squared fit statistics ranged from 11% to 22%.Table 1. Eigenvalues and Corresponding Percentages of Variance Accounted for From the PrincipleComponents Analyses of Rasch Residuals (Domain Scaling).ComponentDomain1234Operations & Algebraic Thinking1.35 (1.1%)1.31 (1.0%)1.24 (1.0%)1.21 (0.9%)5--Numbers & Operations - Base Ten1.51 (1.2%)1.46 (1.2%)1.34 (1.1%)1.30 (1.0%)1.27 (1.0%)Numbers & Operations - Fractions1.71 (0.9%)1.54 (0.8%)1.48 (0.8%)1.44 (0.8%)1.37 (0.7%)Measurement & Data1.59 (1.1%)1.53 (1.1%)1.47 (1.0%)1.43 (1.0%)1.36 (0.9%)Geometry1.42 (1.8%)1.40 (1.8%)1.32 (1.6%)1.24 (1.6%)1.17 (1.5%)Table 2. Summary of Mean Squared Fit Statistics (Domain 020.223%8%72Numbers & Operations Base 2Numbers & Operations 03%7%108Measurement & 6DomainOperations & an% of Items 0.750%% of Items 1.331%#Items72Research Question 2.For most of the mini-assessments and quantile functions, the possible cut points were quitehigh. For example, Figure 2 provides multiple plots for the first mini-assessment, 1A, which is aligned to4.NBT.A.1. The first plot is across all of the data available and shows that only the higher quantiles (60,70, 80 and 90) intersect the line that corresponds to a Probability of Indicator Mastery of 67%. Thisresult is partially an interaction between the changing student mastery probabilities across the year and10
On Demand Assessments of Individual Standardsthe patterns of administration of the mini-assessment 5. That is, student mastery probabilities aregenerally lower at the beginning of the year than later on, and when the matching mini-assessments aregenerally administered towards the earlier parts of the year, the resulting regression relationships willshow the total score that corresponds to a given Probability of Indicator Mastery is higher than if themini-assessments are generally administered evenly across the year or towards the end of the year. Thesecond plot in Figure 2 attempts to illustrate this point by using only data from the second half of theyear. Whereas the 50th quantile regression line did not reach a Probability of Indicator Mastery of 67%when computed using all the data, it did when based on data from the latter half of the year.Given these trends, the question of how to set the cut score becomes a question of what datashould be used. One option is to simply use the data as is, producing the patterns mentioned above. Asecond option is to produce a different pattern of administrations through weighting, resampling orrestricting the data window. For example, one might re-sample so that each of the recommended majoradministration windows (beginning of fall, middle of year and end of spring) are equally represented.Such an approach may be preferable, as the administration patterns of the mini-assessments do varyquite a bit from assessment to assessment, and thus doing so could insure some uniformity across thecuts. However, this approach is also not without issue – some assessments of standards that come laterin the recommended instructional sequence (e.g., mini-assessments 32A and 32B) have almost noadministrations within the first window.We provide evaluations of the 50th quantile at 𝑃𝑃 𝑋𝑋𝑖𝑖𝑖𝑖 1 0.50 and 0.67, as well as 0.25, inAppendix D for both the overall sample as well as sample that attempts to balance t
The Smarter Balanced interim comprehensive assessments. 2: . practice is to report interim results on the scale of the summative item bank, the results of each interim assessment are generally provided in isolation. That is, the information produced from these . The interim assessment bl
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