Using Markov Chains to Assess Enrollment Policies

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Using Markov Chains to Assess Enrollment Policies

• Robert L. Armacost
• Higher Education Assessment and Planning
  Technologies
• Sandra Archer
• Interim Director, University Analysis and
  Planning Support
• Julia Pet-Armacost
• Assistant Vice President, Information, Analysis,
  and Assessment
• Jennifer Grey
• Research Associate
• University of Central Florida
• 2006 AIR Annual Forum
• May 18, 2006

Presentation available at http//uaps.ucf.edu
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Goals for Presentation

• General understanding of alternative enrollment
  modeling approaches
• Improved understanding of Markov chain concepts
• More knowledgeable about using Markov chains for
  enrollment planning
• More knowledgeable about potential use of Markov
  chains to examine enrollment policies

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The University of Central Florida
Stands for Opportunity

• Established in 1963 (first classes in 1968),
  Metropolitan Research University
• Grown from 1,948 to 45,000 students in 37 years
• 38,000 undergrads and 7,000 grads
• Ten colleges
• 12 regional campus sites
• 7th largest public university in U.S.
• 89 of lower division and 67 of upper division
  students are full-time

• Carnegie classification
• Undergraduate Professions plus arts sciences,
  high graduate coexistence
• Graduate Comprehensive doctoral (no medical)
  Medical school approved
• 92 Bachelors, 94 Masters, 3 Specialist, and 25
  PhD programs
• Largest undergraduate enrollment in state
• Approximately 1,200 full-time faculty 9,000
  total employees

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Why Do Enrollment Modeling?

• Predicting income from tuition
• Planning courses and curriculum
• Allocating resources to academic departments
• Long-term master planning
• Admissions policies
• How accurate do these predictions have to be?
• See Hopkins, David S. P. and Massy, William F.,
  Planning Models for Colleges and Universities,
  Stanford University Press, Stanford, CA, 1981 for
  additional information on enrollment planning

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Enrollment Flow Models

• Objective find simplest model that predicts
  future enrollment based on past enrollment levels
  and new students enrolling
• Methods
• Regression (REG)
• Grade progression ratio method (GPR)
• Cohort flow models (CF)
• Markov chain models (MC)
• Notation
• Nj(t) number of students in state j at time t
• fj(t) number of students enrolling in state j
  at time t
• j state indexstands for class level

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Regression Models

• Student inventory predicted returning students
  plus expected new students
• Prediction of returning students estimated by
  multivariate regression
• N(t) F Nj(t-1), fj(t-1), Nj(t-2), fj(t-2),
  f(t)

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Program Enrollment Projections
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Grade Progression Ratio

• Ratio of students in one class level at time t to
  students in next-lower class level at time t-1
• Assumes
• Students follow an orderly progression from one
  state to another
• All students in each state move on to next state
  in one time period or drop out of the system for
  good
• Very simple model
• Only good for year-to-year projections
• Data readily available
• Not usable in higher education

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Grade Progression Ratio
Year t-1
Year t

• Estimate the GPR from historical data
• aj-1,j(t) Nj(t)/ Nj-1(t-1)
• Apply GPR to current enrollment level to predict
  next time period enrollment

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Cohort Flow Models

• Adopt a longitudinal outlook
• Take account of students origins
• Consider students accumulated duration of stay
  at the university
• Students are grouped into cohorts at the time
  they enter the university

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Cohort Flow Models

• Based on cohort survivor fractions
• Enrollment in a given level is sum of products of
  survivor fraction and cohort size plus new
  students
• Estimate of returning students
• Cohorts typically defined for fall semester
• Extensive data analysis required to determine
  survivor fractions (retention)
• Combine with semester transition fractions to
  generate annual estimate

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Combined Cohort-Markov Model
Survivors
Transition
Transition
Transition
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Markov Chain

• Stochastic process
• Fluctuate in time because of random events
• System can be in various states
• Markov chain depicts movements between states
• Markov propertyeach outcome depends only on the
  one immediately preceding it
• Cross-sectional outlook
• Transition fraction
• pij fraction of students in class i in one
  period that can be found in class j in the
  subsequent time period

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State Transition
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Transition Matrix

• Unlike GPR, MC allows a repeat state in the
  following time period

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Markov Chain Flows
Year t-1
Year t
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State Transition Probabilities
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Grade Transition Fractions

• Transition matrix is super-diagonal
• favorable mathematical properties

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System Equations
New Freshmen
Freshmen continuing

• N1(t) p11N1(t-1)
  f1(t)
• N2(t) p12N1(t-1) p22N2(t-1)
  f2(t)
• N3(t) p23N2(t-1)
  p33N3(t-1) f3(t)
• N4(t)
  p34N4(t-1) p44N4(t-1) f4(t)
• Requires estimation of pij and fj(t)

Juniors continuing
Sophomores to Juniors
New Juniors
New students
Transition fraction
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Markov Chain Approach

• Requires more data to estimate transition
  fractions
• Data are less readily available
• Four state undergraduate model does not account
  for two-way flow of stopouts
• Add a vacation state
• Transition data extremely difficult to get
• Transition fractions do not depend on time in
  state

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Markov Model Considerations

• Number of states
• Estimation of transition fractions
• Data issues
• Potential use for policy evaluations
• Only generates headcount
• Useful for course planning
• Some use for allocating resources
• Limited use for tuition planning (SCH)

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Graduate Model Example

• Predict Masters enrollment at college level
• States
• Enrolled
• Enrolled after one semester stopout
• Stopout gt one semester
• Graduate

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Graduate Model Details
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Graduate Model States
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Graduate Model Data/Details

• Headcount data by college
• Queries created to pull individual enrollment
  by college by semester
• Enrolled
• Ea(yr)(S2(yr)?S3(yr)) (continued)
• Eb(yr)(S1(yr)?S3(yr)) (skipped a semester)
• Transitions
• Ta(yr)(S2(yr)?S3(yr))/Total S2(yr) (continued)
• Tb(yr)(S1(yr)?S3(yr))/Total S1(yr) (skipped a
  semester)
• Stopout/ins estimate by Graduate Studies
• New students estimate by Graduate Studies
• for model validation, using actuals

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Graduate Model Details

• Prediction formula Ta(yr-1)E a(yr) Tb(yr-1)E
  b(yr)NSO/IN
• Example
• Summer 1996
• (Sp95?Su95)/Sp95Sp96(Fa94? Su95)/Fa94Fa95NSu9
  6So/InSu96
• Fall 1996
• (Su95?Fa95)/Su95Su96(Sp95? Fa95)/Sp95Sp96NFa9
  6So/InFa96
• Spring 1997
• (Fa95?Sp96)/Fa95Fa96(Su95? Sp96)/Su95Su96NSp9
  7So/InSp97

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Example Transition Data
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Average Transition Data
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Graduate Model Results
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Graduate Model Results
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Graduate Model Summary

• Markov chain approach
• Data required are more specific
• Data are at the individual level
• Semester by semester enrollment
• Account for stopout semesters
• Model based on individual behavior
• More complex method
• Partially accounts for stopouts by using
  1-semester vacation state

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Enrollment Management Issues

• Can you find a recruitment policy that will
  maintain a constant size (or distribution of
  students) by level?
• If you have a desired distribution, can you get
  there from where you are?

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Barycentric Coordinates
B (0,1,0)
Möbius, 1827
P
A (1,0,0)
C (0,0,1)
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Policy to Maintain Constant Size

• N(t1) N(t) find f (new students)
• Equations to be satisfied
• Find fi such that
• N1 p11N1
  f1
• N2 p12N1 p22N2
  f2
• N3 p23N2 p33N3
  f3
• N4 p34N3
  p44N4 f4
• Maintainable region may be relatively small

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Maintainable Region Vertices
Solve N N P f N (I P) f N f (I
P)-1 Elements of N are given by rows of (I
P)-1 after rows are normalized
(I P)
(I P) -1
(I P) -1 rows normalized
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Maintainable Region
U (0,1,0)
(0,0.4,0.6)
Steady State
(0.286,0.286,0.428)
L (1,0,0)
G (0,0,1)
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Maintainable Region Implications
U (0,1,0)

• If you are at a given size and you are outside of
  the maintainable region, there is NO recruitment
  policy that will allow you to stay at that total
  enrollment and distribution
• You can change the maintainable region only by
  changing the transition fractions (e.g.,
  retention)

(0,0.4,0.6)
Steady State
(0.286,0.286,0.428)
L (1,0,0)
G (0,0,1)
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Other Constant Size Concerns

• Attainable region
• Set of distributions that can be attained in
  n-steps from any distribution using a set of
  recruitment policies
• Reachable region
• Set of distributions that can be reached in
  n-steps starting from any distribution in the
  maintainable region using a set of recruitment
  policies
• Considerations
• You may be someplace that you can never get to
  again
• You may want to go someplace that you will never
  be able to get to

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Example Attainable Region
U (0,1,0)
Steady State
L (1,0,0)
G (0,0,1)
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Summary

• Match model to decision/business question
• Need agreement among models at some level
• More involved models provide more detailed
  decision information
• Require more data
• More difficult to explain
• Markov chain models
• Use with cohort model to estimate enrollment
• Use aggregate model to evaluate management
  policies

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Session Assessment

• Please complete the assessment form for the
  session.
• THANK YOU FOR YOUR ATTENDANCE AND PARTICIPATION

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Questions
???

• Ms. Sandra Archer
• Interim Director, University Analysis and
  Planning Support
• University of Central Florida
• 12424 Research Parkway, Suite 215
• Orlando, FL 32826-3207
• 407-882-0287
• archer_at_mail.ucf.edu
• http//uaps.ucf.edu

• Dr. Julia Pet-Armacost
• Assistant Vice President, Information, Analysis,
  and Assessment
• University of Central Florida
• Millican Hall 377
• P. O. Box 163200
• Orlando, FL 32816-3200
• 407-882-0276
• jpetarma_at_mail.ucf.edu
• http//iaa.ucf.edu

• Contacts
• Dr. Robert L. Armacost
• Higher Education Assessment and Planning
  Technologies
• 602 Shorewood Drive, Suite 402
• Cape Canaveral, FL 32920
• 321-784-9921
• armacost_at_mail.ucf.edu