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Predicting Grades

Yannick Meier, Jie Xu, Onur Atan, Mihaela van der Schaar

arXiv:1508.03865v2cs.LG

TL;DR

Large classes make individual monitoring difficult, and classroom predictions may have to rely on early assessment scores. The paper learns personalized, confidence-qualified prediction timing from past course years; on UCLA course data, it reports better accuracy and timeliness than benchmarks.

  • Problem

    The paper addresses how to identify students likely to do poorly early enough for remedial action when classroom instructors may have only course assessment scores.

    Source: §I · p. 1

  • Method

    The algorithm uses prior course-year assessment data to make individual grade predictions when a learned confidence threshold is met.

    Source: §I · pp. 1–2, E. Confidence-Learning Prediction… · p. 7

  • Results

    On UCLA digital signal processing course data, the algorithm outperformed benchmarks in accuracy and timeliness in regression and classification and was robust across instructors.

    Source: §V · p. 12

  • Takeaways & Limitations

    The authors conclude that early in-class exams can support timely identification of students likely to do poorly and early remedial action.

    Source: §V · p. 12

    The analyzed course data included no interventions, so the study does not evaluate their impact.

    Source: §I · p. 1

Abstract

from arXiv · show

To increase efficacy in traditional classroom courses as well as in Massive Open Online Courses (MOOCs), automated systems supporting the instructor are needed. One important problem is to automatically detect students that are going to do poorly in a course early enough to be able to take remedial actions. Existing grade prediction systems focus on maximizing the accuracy of the prediction while overseeing the importance of issuing timely and personalized predictions. This paper proposes an algorithm that predicts the final grade of each student in a class. It issues a prediction for each student individually, when the expected accuracy of the prediction is sufficient. The algorithm learns online what is the optimal prediction and time to issue a prediction based on past history of students' performance in a course. We derive a confidence estimate for the prediction accuracy and demonstrate the performance of our algorithm on a dataset obtained based on the performance of approximately 700 UCLA undergraduate students who have taken an introductory digital signal processing over the past 7 years. We demonstrate that for 85% of the students we can predict with 76% accuracy whether they are going do well or poorly in the class after the 4th course week. Using data obtained from a pilot course, our methodology suggests that it is effective to perform early in-class assessments such as quizzes, which result in timely performance prediction for each student, thereby enabling timely interventions by the instructor (at the student or class level) when necessary.

I. INTRODUCTION

Large classes make individual monitoring difficult, while early assessment signals vary in usefulness across students and years. The paper proposes personalized grade prediction from course assessment scores, with confidence-based timing and learning from past course data.

I. INTRODUCTION

  • Large classes can prevent instructors from tracking each student, while early identification may allow remedial or promotional support before the course ends.In classroom settings, the available data may be limited to scores from early assessments such as homework, quizzes, and midterms.Source: §I · p. 1
  • Prediction is challenging because early scores may be weakly related to final performance, assignments change across years, and students differ in how soon they become predictable.These differences motivate choosing prediction times for individual students rather than issuing predictions for the class at once.Source: §I · p. 1
  • The proposed algorithm predicts each student's grade at a personalized time, supports regression and classification, and accompanies predictions with confidence estimates.The paper also derives a bound on the probability that prediction error exceeds a desired value.Source: §I · p. 2

II. RELATED WORK

  • The paper uses only course assessment scores, addressing prior approaches that may depend on student information unavailable to instructors.Examples of unavailable data include prior GPA, performance in related courses, and self-efficacy.Source: §I · p. 2, §II · p. 2

I. INTRODUCTION

  • The paper evaluates its algorithm against established prediction methods on seven years of UCLA digital signal processing course data and uses a pilot course to examine course-design recommendations.The proposed course-design approach emphasizes early in-class assessments for timely prediction and intervention.Source: §I · p. 2

II. RELATED WORK

  • Unlike related course-prediction work, the algorithm uses past-year classes for training, chooses prediction timing per student using confidence, and supports regression as well as classification.The cited related approaches are described as relying on online-system logs, training data from the target course, and classification settings.Source: §II · p. 2

III. FORMALISM, ALGORITHM AND ANALYSIS

The section formalizes prediction of a student's final score or grade category before all assessments are complete. It defines a year-and-assessment timeline, assessment-score features, and assumptions for combining assessment results.

A. Definitions and System Description

  • The task is to predict a student's final score or grade classification with a specified confidence before all performance assessments are complete.The course is assumed to recur over multiple years with only slight modifications.Source: §III · p. 3, A. Definitions and System Description · p. 3
  • The model indexes course years by y and assessment points by k, where k marks the point after an assessment has been graded.Each year has Y students and K assessments; the normalized score for student i on assessment k is denoted a_i,y,k.Source: A. Definitions and System Description · p. 3
  • At assessment k, a student's feature vector contains the normalized scores from assessments 1 through k, and the overall score is a weighted sum of assessments.The assessment weights are set by the instructor and sum to 1.Source: A. Definitions and System Description · p. 3
  • The setup assumes the number, sequence, and weights of assessments remain the same across years, an assumption the authors say can be relaxed for differing instructors and assessment schemes.The paper notes this extension in connection with its later analysis of courses taught by different instructors.Source: A. Definitions and System Description · p. 3
  • After assessment k, completed scores are known but the remaining score contribution must be estimated to obtain the overall-score prediction.The passage distinguishes the residual score from the predicted overall score.Source: A. Definitions and System Description · pp. 3–4
  • For each student, the formalism stores assessment-based feature vectors, residual scores at earlier assessment points, and the overall score.The notation also groups these data across students and completed course years.Source: A. Definitions and System Description · p. 4

B. Problem Formulation

The algorithm predicts each student’s overall score or class individually, choosing when to issue a prediction based on confidence and past students’ assessment results. Its confidence estimate supports an error-probability bound, while prediction accuracy improves with more neighbors and lower residual variance.

B. Problem Formulation

  • For each student, the algorithm chooses when to predict and estimates either an overall score or a binary classification using current assessment results and past students’ data.At each assessment, it may predict once confidence exceeds a user-defined threshold or wait for more results.Source: B. Problem Formulation · p. 4, C. Grade Prediction Algorithm,… · p. 4

C. Grade Prediction Algorithm, Regression Setting

  • Regression predicts only the unknown residual from future assessments, using nearby past students’ feature vectors and residual scores to estimate it.The known assessment scores are combined with the estimated residual to obtain the overall-score prediction.Source: C. Grade Prediction Algorithm,… · p. 4
  • The algorithm estimates neighborhood residual means and variances, then selects the candidate neighborhood with the highest confidence for the student.The confidence measure uses sample statistics; the authors connect its error-probability reasoning to Chebyshev’s inequality.Source: C. Grade Prediction Algorithm,… · p. 5
  • Prediction waits for the next assessment when the best neighborhood’s confidence is below the threshold; the algorithm adds student features and assessment results to its database.After the year ends, residual scores are calculated and added to the database for subsequent predictions.Source: C. Grade Prediction Algorithm,… · pp. 5–6
  • The theorem bounds the probability that a prediction’s absolute error exceeds ε; accuracy improves with more neighbors and lower residual variance.The bound is stated for normalized scores, and the authors expect residual variance to decrease as more assessment information becomes available.Source: C. Grade Prediction Algorithm,… · p. 6
  • The algorithm can retain only recent years of data, allowing faster adaptation to slowly changing students or course conditions.The passage presents this as an option for restricting the knowledge base.Source: C. Grade Prediction Algorithm,… · p. 6

D. Grade Prediction Algorithm, Classification Setting

In binary classification, the algorithm converts a predicted overall score into a class using a threshold between past grade groups. It modifies confidence for deciding when to predict and extends the approach to multiple classes.

D. Grade Prediction Algorithm, Classification Setting

  • Binary classification compares the predicted overall score with a threshold to classify students as doing well or poorly.For the B−/C+ example, the threshold is the midpoint of the average scores of past students in those grade groups.Source: D. Grade Prediction Algorithm,… · p. 6
  • The classification confidence reflects both neighborhood-score variance and the predicted score’s distance from the class threshold.Threshold distance affects the decision to issue a prediction, not the selection of the neighborhood.Source: D. Grade Prediction Algorithm,… · pp. 6–7
  • For multiple classes, the method uses ordered score thresholds to assign the predicted score to an interval and measures confidence by distance to the nearest threshold.The binary procedure also changes the returned output and uses modified confidence for the prediction decision.Source: D. Grade Prediction Algorithm,… · p. 7

E. Confidence-Learning Prediction Algorithm

The confidence-learning algorithm uses past years to choose a confidence threshold and prediction time for the next year. It targets a specified share of students predicted within a maximum average error.

E. Confidence-Learning Prediction Algorithm

  • The objective balances the share of students predicted by time k against their average absolute prediction error, with only two of these quantities independently specified.The passage notes that predicting everyone with zero error requires waiting until the course ends.Source: E. Confidence-Learning Prediction… · p. 7
  • The algorithm learns the minimum prediction time and confidence threshold needed to predict a target share of students within a maximum average error.The target share and maximum error are user-specified, while the prediction time and confidence threshold are dependent quantities.Source: E. Confidence-Learning Prediction… · p. 7
  • For each year, it evaluates available past data across confidence thresholds, selects the threshold meeting the objective, and applies it to the following year.For the first year, it uses a user-defined starting threshold because no training data are yet available.Source: E. Confidence-Learning Prediction… · p. 7

IV. EXPERIMENTS

The experiments section introduces the dataset and evaluation of the algorithm in regression and binary classification settings; results for classifications with more than two categories are omitted because of space limitations.

IV. EXPERIMENTS

  • The experiments evaluate regression and binary classification performance against other prediction methods, but omit results for classifications with more than two categories.The section also introduces the data and explains application of the algorithm to the dataset.Source: §IV · p. 7

A. Data Analysis

The section describes seven years of UCLA digital signal processing course data and examines how performance assessments relate to overall and final-exam scores. Correlations vary across years, while the midterm is strongly associated with both outcomes.

A. Data Analysis

  • Approximately 700 students’ scores and final letter grades from seven years of the course form the dataset; annual enrollment ranged from 30 to 156.The course ran for 10 weeks, with seven homework assignments, a midterm, a project, and a final exam.Source: A. Data Analysis · p. 8
  • The final exam had the strongest average correlation with overall score, followed by the midterm; correlations varied greatly across years.The project’s average correlation with overall score was no higher than homework’s, despite accounting for 15% of the score.Source: A. Data Analysis · p. 8
  • The midterm had the highest correlation with final-exam score among assessments other than the overall score.The authors suggest that both the midterm and final are in-class exams, unlike the take-home assessments.Source: A. Data Analysis · p. 8

B. Our Algorithm

The algorithm uses normalized assessment histories and weighted distances to identify similar students, adapts neighborhood sizes, and preprocesses past-year data when course structure changes.

B. Our Algorithm

  • Assessment scores are centered on their year’s assessment mean and scaled by the standard deviation of that year’s overall scores, making scores comparable across years.The authors note that assessment maxima and difficulty can differ by year; this normalization expresses scores relative to same-year classmates.Source: B. Our Algorithm · p. 8
  • A weighted distance over students’ assessment-score feature vectors gives more influence to assessments with greater course weight.The vectors contain scores for assessments taken up to the current point, in the order they occurred.Source: B. Our Algorithm · p. 8
  • Neighborhood radii are selected by neighbor count: the smallest includes three students, and each subsequent neighborhood adds at least one neighbor.This avoids neighborhoods with only two students, because sample variance becomes more accurate with more samples.Source: B. Our Algorithm · p. 8
  • Past-year assessments are combined, duplicated, reordered, or matched by assessment type and topic so their sequence aligns with the current course.Combined assessments use past assessment weights; duplicated assessments give that topic more weight in selecting similar students. Predictions use current-year weights and may draw on several preprocessed past years.Source: B. Our Algorithm · pp. 8–9

C. Benchmarks

The section compares the proposed approach with several prediction methods and illustrates cases where it correctly classifies students that logistic regression misclassifies. The examples involve changing performance patterns across assessments.

C. Benchmarks

  • Benchmarks include recent-assessment and weighted-score predictions, 7-neighbor k-NN, OLS regression, logistic regression, and SVMs.Logistic regression replaces linear regression in classification settings.Source: C. Benchmarks · p. 9
  • Three students classified incorrectly by logistic regression were classified accurately by the proposed algorithm after the midterm.The case-study table reports their assessment scores through the midterm and their true classifications.Source: C. Benchmarks · p. 10, Table II · p. 10
  • The examples contrast students whose initially good homework performance deteriorated by the midterm with predictions based on their average early scores.The authors say the algorithm is likely to have learned such patterns from past data, while logistic regression predicts from the average performance of the first four assessments.Source: C. Benchmarks · p. 10

D. Results

The algorithm trades prediction timeliness against accuracy, learns from prior course years, and remains effective when trained on data from a different instructor.

1) Performance Comparison with Benchmarks in Regression Setting:

  • After homework 4 on average, the algorithm's error E is up to 65% lower than linear regression.Benchmark predictions used fixed assessment times, while the algorithm’s curve averages students’ individually selected prediction times.Source: Figure 4 · p. 10, 1) Performance Comparison with… · p. 10

2) Learning across Years and Instructors in Regression Setting:

  • Across prediction times, training on all five prior years gives the highest performance for predicting year 7, despite performance varying by training year.The comparison uses the algorithm to predict year 7 from different combinations of years 1–5.Source: 2) Learning across Years and… · pp. 10–11, Figure 5 · p. 11
  • Training on a different instructor's years 1–5 remains effective for predicting instructor 1's year 7 class, although same-instructor training is slightly more accurate.The authors describe this as robustness to instructor differences.Source: Figure 6 · p. 11, 2) Learning across Years and… · p. 11

3) Performance Comparison with Course Containing Early Quizzes in Regression Setting:

Early quizzes can support timely grade prediction, and the algorithm improves on logistic regression for binary classification from homework 4 onward. Its results also support early in-class evaluations and remedial action.

3) Performance Comparison with Course Containing Early Quizzes in Regression Setting:

  • Across two consecutive years, using prior-year data to predict overall scores, the EE103 quiz-course algorithm predicted the same percentage of students by a given week with up to 22% smaller cumulative average prediction error.In EE103, the increase in the share of students predicted was larger in weeks with quizzes than in weeks without them.Source: 3) Performance Comparison with Course… · p. 11

4) Performance Comparison with Benchmarks in Classification Setting:

  • Through homework 4, the algorithm and logistic regression have similar accuracy; from homework 4 onward, the algorithm performs significantly better, especially in recall.Even with full information, neither method reaches 100% accuracy; the paper attributes this to changes in the mapping from overall scores to letter grades.Source: Figure 8 · p. 12, 4) Performance Comparison with… · p. 12

5) Decision Time and Accuracy in Classification Setting:

V. CONCLUSION

  • In-class exams predict overall performance better than homework, and early in-class evaluations support early identification of students likely to do poorly and remedial action.The conclusion also reports that predictions outperform benchmarks in accuracy and timeliness in regression and classification, and remain robust across instructors.Source: §V · p. 12
  • Future work includes applying the algorithm in practice and investigating how much timely interventions based on its predictions improve student performance.The paper also proposes repeated predictions to monitor predicted-grade trends after an intervention.Source: §V · p. 12

APPENDIX

The appendix proves a prediction-error bound by separating neighborhood selection, estimation of the neighborhood mean, and variation in an individual student's residual score. It bounds these sources using concentration inequalities and a lemma on selecting the wrong neighborhood.

APPENDIX

  • Fact 1 names the Chernoff-Hoeffding bound for independent random variables bounded in [0,1] with expected value μ, and defines their sample mean as (X1 + ... + Xn)/n.The passage says a proof of Fact 1 can be found in Hoeffding’s paper.Source: APPENDIX · p. 13
  • The probability of selecting the wrong neighborhood is bounded using the empirical Bernstein bound for sample standard deviation error.The lemma defines the selected neighborhood, the optimal neighborhood, and the quantities used in its probability bound.Source: APPENDIX · p. 13
  • The prediction error has three sources: selecting the wrong neighborhood, estimating its residual mean inaccurately, and individual residual variation around that mean.The proof derives a bound for each source.Source: APPENDIX · p. 13
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