Assessing Students using Multivariate Statistical Tools

1
Assessing Students using Multivariate Statistical
Tools

• A Statistics Senior Seminar Presented by
• Peter Butler

2
Overview

• Introduction
• Objectives and Questions
• Techniques Used, Application, and examples
• PCA
• Discriminent Analysis
• Logisitic Discrimination
• Tree Analysis
• Correspondence Analysis
• Results Discovered
• Conclusion

3
Introduction to Data and Variables to Consider

• Sexmale/female
• HSRHigh School Ranking
• ACT ACT Score
• English
• Math
• Reading
• Science
• Composition
• TypeType of Student
• NHSnew high school
• NASnew advanced standing
• IUTinter-university transfer
• NPSnew post secondary
• GPACumulative GPA
• TOTCRDTotal Number of Credits
• ProfessionalThose planning on going to a
  professional school
• DecidedStudents who are or are not decided upon
  their major

4
Two Important Questions

• Questions
• What are the characteristics of different types
  of students?
• How can we assess what we know about a student
  before they come to UMM, to tell how they will
  perform here?

5
Principal Components Analysis

• Basic Idea
• To describe the variation of a set of
  multivariate data in terms of a set of
  uncorrelated variables, each of which is a linear
  combination of the original variables.
• Purpose
• To see whether the first few components account
  for most of the variation in the original data.
• Summarize data with little loss of information.
• Reduce dimensionality, which can simplify future
  analysis.
• Algebraic Representation of Principal Component
  Analysis

6
PCA Relationships

• HSR ACTE ACTM
  ACTRD ACTSR ACTCOMP CUMGPA TOTCRD
• HSR 1.000
• ACTE 0.450 1.000
• ACTM 0.501 0.799 1.000
• ACTRD 0.399 0.859 0.749
  1.000
• ACTSR 0.434 0.832 0.838
  0.841 1.000
• ACTCOMP 0.481 0.901 0.871
  0.900 0.907 1.000
• CUMGPA 0.422 0.326 0.315
  0.297 0.269 0.327 1.00
• TOTCRD 0.287 0.184 0.187
  0.195 0.203 0.196 0.419 1.000
• Strong Relationships Between
• All ACT variables
• HSR and ACTM
• CUMGPA and HSR

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PCA Scree Plot

• Our scree plot shows
• Evidence that most
• Of our data from
• PCA can be explained
• By the first three
• Principal components.

8
Principal Component Analysis

• Component loadings
• 1 2 3
  4 5 6
  7 8
• HSR 0.605 -0.383 -0.579
  -0.380 -0.087 0.020 0.003
  0.005
• ACTE 0.920 0.165 0.048
  0.061 -0.141 -0.277 0.147
  0.045
• ACTM 0.900 0.130 -0.042
  -0.044 0.375 -0.090 -0.136
  0.052
• ACTRD 0.903 0.188 0.130
  0.067 -0.272 0.095 -0.198
  0.071
• ACTSR 0.919 0.194 0.098
  -0.029 0.091 0.241 0.196
  0.052
• ACTCOMP 0.960 0.174 0.045
  0.021 -0.009 0.012 -0.020
  -0.211
• CUMGPA 0.460 -0.683 -0.156
  0.544 0.028 0.027 0.009
  0.003
• TOTCRD 0.320 -0.742 0.515
  -0.284 0.007 -0.022 -0.004
  -0.004
• We can support evidence found in our scree plot
  with the data shown here. Notice how much higher
  the values are with the first few PCs, compared
  to the rest. This means that most of our data
  can be explained by these first few PCs.

9
Defining Classical Discrimination

• Definition
• Classical Discrimination is a process where we
  have groups that are known prior to investigation
  and the goal is to devise rules which can
  allocate previously unclassified objects or
  individuals into these groups in an optimal
  fashion.

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Fishers Linear Discriminant Function

• Only 60 years ago Fisher devised a solution to
  the discrimination problem for two groups with
  this linear function
• Where the ratio of the between-group variance of
  y to its within-group variance is maximized.

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Classical Discrimination for Professional

• Group means
• no yes
• HSR 74.809 80.574
• ACTE 22.657 22.859
• ACTM 22.523 23.365
• ACTRD 24.289 24.201
• ACTSR 23.304 23.744
• ACTCOMP 23.437 23.701
• CUMGPA 2.933 2.842
• TOTCRD 113.148 120.385

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Classical Discrimination for Professional

• Canonical discriminant functions
• Constant 0.289
• HSR 0.027
• ACTE -0.000
• ACTM 0.108
• ACTRD -0.062
• ACTSR 0.019
• ACTCOMP -0.060
• CUMGPA -1.017
• TOTCRD 0.006

13
An Example of Classifying Students

• Canonical scores of group means
• no -0.075
• yes 0.305
• Now we calculate our
• y-value from Fishers
• equation.
• If ylt0.115 then Group 1
• If ygt0.115 then Group 2

14
Classifying a Student

• Evaluating Student 7
• -0.19lt0.115, so student 7 is classified in
• Group 1.

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Classical Discrimination for Decided

• Group means
• no yes
• HSR 67.940 78.591
• ACTE 20.743 23.343
• ACTM 20.777 23.322
• ACTRD 22.261 24.937
• ACTSR 21.408 24.047
• ACTCOMP 21.543 24.132
• CUMGPA 2.787 2.958
• TOTCRD 101.338 118.951

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Classical Discrimination for Decided

• Canonical discriminant functions
• Constant -3.860
• HSR 0.017
• ACTE 0.027
• ACTM -0.009
• ACTRD -0.029
• ACTSR 0.038
• ACTCOMP 0.072
• CUMGPA -0.059
• TOTCRD 0.004

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Classical Discrimination for type

• Group means
• IUT NAS NHS NPS
• CUMGPA 2.989 2.829 2.929 2.971
• TOTCRD 124.413 137.464 111.406 63.951
• HSR 54.356 63.039 78.887 75.933
• ACTE 18.169 19.573 23.383 23.457
• ACTM 18.068 19.303 23.459 22.314
• ACTRD 20.102 21.071 24.988 24.343
• ACTSR 19.000 20.277 24.099 23.267
• ACTCOMP 18.915 20.349 24.202 23.429

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Classical Discrimination for type

• Canonical discriminant functions
• 1 2 3 4
• Constant 2.649 0.688 -1.893 -1.274
• CUMGPA 0.090 0.944 0.813 0.219
• TOTCRD 0.010 -0.012 -0.000 0.000
• HSR -0.022 -0.007 -0.004
  -0.031
• ACTE -0.017 0.067 -0.211 -0.006
• ACTM -0.025 -0.028 0.198 -0.036
  
• ACTRD 0.010 0.018 0.094 0.111
• ACTSR -0.024 0.021 0.002
  0.063
• ACTCOMP -0.044 -0.141 -0.092
  -0.011

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Logistic Discrimination

• When we have non-normal cases or cases with
  binary variables we need a different approach for
  our analysis. This approach uses a logistic
  function to model the probability directly on an
  observation that is a member of each group.

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Logistic Discrimination

• With two groups the model is as follows
• The parameters for alpha in this model are
  estimated by maximum likelihood.

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Logistic Discrimination

• After estimation of the parameters, the
  allocation rule is to assign to Group 1 if,
• Assign to Group 2 if,

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Logistic Discrimination Decided

• Parameter Estimate S.E.
  t-ratio p-value
• 1 CONSTANT 0.543 0.154
  3.533 0.000
• 2 HSR -0.007
  0.001 -4.947 0.000
• 3 ACTE -0.012 0.012
  -0.972 0.331
• 4 ACTM 0.005 0.011
  0.456 0.649
• 5 ACTRD 0.013 0.011
  1.185 0.236
• 6 ACTSR -0.018 0.013
  -1.397 0.162
• 7 ACTCOMP -0.029 0.019
  -1.570 0.116
• 8 CUMGPA 0.012 0.051
  0.237 0.813
• 9 TOTCRD -0.002 0.001
  -3.710 0.000

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Interpreting Logistic Discrimination for Decided

• Constructing a logistic discriminant model for
  decided

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Applying Logistic Discrimination to a Student

• Student 11
• Since,
• We classify student 11 to Group 2.

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Logistic Discrimination Professional

• Parameter Estimate S.E.
  t-ratio p-value
• 1 CONSTANT 1.474 0.188
  7.847 0.000
• 2 HSR -0.013 0.002
  -6.166 0.000
• 3 ACTE -0.003 0.014
  -0.185 0.853
• 4 ACTM -0.043 0.013
  -3.235 0.001
• 5 ACTRD 0.022 0.013
  1.691 0.091
• 6 ACTSR -0.010 0.015
  -0.707 0.480
• 7 ACTCOMP 0.032 0.028
  1.134 0.257
• 8 CUMGPA 0.406 0.058
  6.938 0.000
• 9 TOTCRD -0.002 0.001
  -3.646 0.000

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Logistic DiscriminationType

• Choice Group IUT
• Parameter Estimate S.E.
  t-ratio p-value
• 1 CONSTANT 1.916 1.619
  1.183 0.237
• 2 HSR 0.050 0.018
  2.732 0.006
• 3 ACTE -0.086 0.229
  -0.375 0.708
• 4 ACTM 0.243 0.190
  1.280 0.200
• 5 ACTRD 0.068 0.201
  0.340 0.734
• 6 ACTSR -0.009 0.228
  -0.040 0.968
• 7 ACTCOMP -0.121 0.238
  -0.510 0.610
• 8 CUMGPA 0.614 0.621
  0.989 0.323
• 9 TOTCRD -0.016 0.009
  -1.709 0.087

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Logistic Discrimination Type

• Further analysis showed that NAS and NPS had no
  significant variables.
• NHS
• HSR, p-value0.000
• ACTM, p-value0.047
• TOTCRD, p-value0.000

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Tree Analysis

• Tree modeling
• An exploratory technique for uncovering structure
  in data.
• Uses a series of classification rules that are
  derived from the data by a procedure known as
  recursive partitioning and the result is a
  classification tree.

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Tree Analysis Decided
30
Interpreting the Decided Tree

• -First we select a group, say the first red box
  group on the left.
• If a students ACTCOMP is less than 8, then (s)he
  is classified as an undecided student.
• -If a students ACTCOMP is greater than 8 and
  TOTCRD is greater than 63.5, then (s)he is is
  classified as a decided student.

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Tree Analysis Professional
32
Tree Analysis Type
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Interpreting the Type Tree

• Again, we select a group first (say the first red
  group on the left). A student is classified as a
  NAS here if their ACTCOMP is less than 8 and
  TOTCRD is greater than 31.5.

34
Correspondence Analysis

• Correspondence Analysis is a technique for
  displaying the associations among a set of
  categorical variables in a type of scatter plot
  or map.
• Objectives
• To reveal characteristics of the data.
• To generate hypotheses from the data to then
  test.

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Correspondence Analysis (cont)

• Correspondence Analysis can be viewed as a method
  for decomposing the chi-squared statistic for a
  contingency table into components corresponding
  to different dimensions.

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Correspondence Analysis

• From our correspondence plot we can see GPA,
  Total Credits, HSR, and ACTM are the most
  significant variables.
• Most of the ACT variables are correlated, but not
  as significant.

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Basic Results and Conclusion

• HSR plays the largest role in determining how
  well a student will perform at UMM.
• If a student is decided upon their major they are
  more likely to perform better at UMM, however we
  didnt show enough evidence to make any
  conclusions about professional.
• We could predict what numerical variables had a
  significant effect on type, but more analysis was
  required to tell if type had a significant effect
  on how well a student will do here.

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References

• References
• (1) Everitt and Dunn
• 2001 Applied Multivariate Data Analysis. Arnold
  MPG Books
• (2) D. Wright
• 2002 Exploring Multivariate Statistics
• University Press
• (3) M. Hveisen
• 2004 Discrimination Models. McGraw Hill
• (4) All graphics and some analysis were done by
  SYSTAT 11

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THANK YOU!

• Engin Sungur
• and
• Jon E. Anderson