1 EPSY 546: LECTURE 1 SUMMARY George Karabatsos. 2 REVIEW.

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EPSY 546: LECTURE 1SUMMARY

George Karabatsos

2

REVIEW

3

• Test (& types of tests)

REVIEW

4


• Item response scoring paradigms

REVIEW

5


• Item response scoring paradigms

• Data paradigm of test theory (typical)

REVIEW

6

DATA PARADIGM

X X X X

X X X X

X X X X

X X X X

T ota l

P erson

Score

X

X

X

X

X X X X T o ta l Item Score

p p p p

j J

j J

n n n j nJ

N N N j N J

n

N

j J

j J

11 1 2 1 1

2 1 2 2 2 2

1 2

1 2

1

2

1 2

1 2

:

,

,

Item d ifficu lty

P erso n s = 1 , . . . , ; Item s = 1 , . . . , ; C a teg o rie s 0 , . . . , n N j J k K pN K X

N Kj

j

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• Latent Trait Re (unidimensional)

REVIEW: Latent Trait

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• Latent Trait Re (unidimensional)

• Real Examples of Latent Traits

REVIEW: Latent Trait

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• Item Response Function (IRF)

REVIEW: IRF

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– Represents different theories

about latent traits.

REVIEW: IRF

11


– Dichotomous response:

Pj() = Pr[Xj = 1]

= Pr[Correct Response to item j | ]

REVIEW: IRF

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– Polychotomous response:

Pjk() = Pr[Xj > k | ]

= Pr[Exceed category k of item j | ]

REVIEW: IRF

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– Dichotomous or Polychotomous response:

Ej() = [Expected Rating for item j | ]

0 < Ej() < K

REVIEW: IRF

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IRF: Dichotomous items

.00

.50

1.00

-5 -4 -3 -2 -1 0 1 2 3 4 5

Theta

Pj(Theta)

15

IRF: Polychotomous items

.00

.50

1.00

-5 -4 -3 -2 -1 0 1 2 3 4 5

Theta

Pjk(theta)

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• The unweighted total score X+n

stochastically orders the latent trait

(Hyunh, 1994; Grayson, 1988)

REVIEW: SCALES

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• 4 Scales of Measurement

– Conjoint Measurement

REVIEW: SCALES

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• Conjoint Measurement

– Row Independence Axiom

REVIEW

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• Property: Ordinal Scaling and unidimensionality of (test score)

REVIEW

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INDEPENDENCE AXIOM (row)

ITEMSHard Easy

j = 1 2 3

3(i = 1)

P11 P12 P13

4(i = 2)

P21 P22 P23

TestScore

Group

5(i = 3)

P31 P32 P33

W1 Premise W1 Implication

21




• IRF: Non-decreasing over

REVIEW

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• IRF: Non-decreasing over • Models: MH, 2PL, 3PL, 4PL, True Score,

Factor Analysis

REVIEW

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2PL: P a bj j j

11

exp

.00

.50

1.00

-5 -4 -3 -2 -1 0 1 2 3 4 5

Theta

Pj(Theta)

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3PL: P c c a bj j j j j

1 11

exp

.00

.50

1.00

-5 -4 -3 -2 -1 0 1 2 3 4 5

Theta

Pj(Theta)

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4PL: P c d c a bj j j j j j

11

exp

.00

.50

1.00

-5 -4 -3 -2 -1 0 1 2 3 4 5

Theta

Pj(Theta)

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Monotone Homogeneity (MH)

.00

.50

1.00

-5 -4 -3 -2 -1 0 1 2 3 4 5

Theta

Pj(Theta)

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– Column Independence Axiom (adding)

REVIEW

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• Property: Ordinal Scaling and unidimensionality of both (test score) and item difficulty (item score)

REVIEW

29

ITEMSHard Easy

j = 1 2 3

3(i = 1)

P11 P12 P13

4(i = 2)

P21 P22 P23

TestScore

Group

5(i = 3)

P31 P32 P33

W1 Premise W1 Implication W2 Premise W2 Implication

INDEPENDENCE AXIOM (column)

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• IRF: Non-decreasing and non-intersecting over

REVIEW

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• IRF: Non-decreasing and non-intersecting over

• Models: DM, ISOP

REVIEW

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DM/ISOP (Scheiblechner 1995)

0.00

0.50

1.00

-5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8

Theta

Pr[

Co

rrect

Resp

on

se]

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– Thomsen Condition (adding)

REVIEW

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• Property: Interval Scaling and unidimensionality of both (test score) and item difficulty (item score)

REVIEW

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ITEMSHard Easy

j = 1 2 3

3(i = 1)

P11 P12 P13

4(i = 2)

P21 P22 P23

TestScore

Group

5(i = 3)

P31 P32 P33

Thomsen condition(e.g.,double cancellation)

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• IRF: Non-decreasing and parallel (non-intersecting) over

REVIEW

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• IRF: Non-decreasing and parallel (non-intersecting) over

• Models: Rasch Model, ADISOP

REVIEW

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RASCH-1PL: P j j

11

exp

.00

.50

1.00

-5 -4 -3 -2 -1 0 1 2 3 4 5

Theta

Pj(Theta)

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• 5 Challenges of Latent Trait Measurement

REVIEW

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• 5 Challenges of Latent Trait Measurement

• Test Theory attempts to address these challenges

REVIEW

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• Test Construction (10 Steps)

REVIEW

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• Test Construction (10 Steps)

• Basic Statistics of Test Theory

REVIEW

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• Total Test Score (X+) variance =

Sum[Item Variances]

+

Sum[Item Covariances]

REVIEW

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EPSY 546: LECTURE 2

TRUE SCORE TEST THEORY AND RELIABILITY

George Karabatsos

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TRUE SCORE MODEL• Theory: Test score is a random variable.

X+n Observed Test Score of person n,

Tn True Test Score (unknown)

en Random Error (unknown)

X T en n n

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TRUE SCORE MODEL

• The Observed person test score X+n is a random

variable (according to some distribution) with mean Tn = E(X+n) and variance 2(X+n) = 2(en).

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TRUE SCORE MODEL

• The Observed person test score X+n is a random

variable (according to some distribution) with mean Tn = E(X+n) and variance 2(X+n) = 2(en).

• Random Error en = X+n – Tn

is distributed with

mean E(en) = E(X+n–Tn) = 0,

and variance 2(en) = 2(X+n) .

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TRUE SCORE MODEL• True Score:

Tn true score of person n

E (Xn) expected score of person n

s Possible score s {0,1,…,s,…,S}

pns Pr[Person n has test score s]

T E X spn n nss

S

1

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TRUE SCORE MODEL• 3 Assumptions:

1) Over the population of examinees, error has a mean of 0. E[e] = 0

2) Over the population of examinees, true scores and error scores have 0 correlation.

[T, e] = 0

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TRUE SCORE MODEL• 3 Assumptions:

3) For a set of persons, the correlations of the error scores between two testings is zero.

[e1, e2] = 0

– “Two testings”: when a set of persons take two separate tests, or complete two testing occasions with the same form.

– The two sets of person scores are assumed to be randomly chosen from two independent distributions of possible observed scores.

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TRUE SCORE ESTIMATION

T XnT

Xn X X

2

2

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T XnT

Xn X X

2

2

T Xn

n

N X

N

1

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T XnT

Xn X X

2

2

e X Tn n n

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0 12

2

T

X

is test reliability.

The proportion of variance of observed scores that is explained by the variance of the true scores.

T XnT

Xn X X

2

2

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TEST RELIABILITY

is the error of measurement.

T

X

X T

X

2

2

2 2

21

X T2 2

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TEST RELIABILITY

is the standard error of

measurement.

(random error)

X T2 2

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TEST RELIABILITY

is the standard error of

measurement.

(random error)

Estimated ((1–)*100)% confidence interval around the test score:

X T2 2

X Zn X T /

22 2

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TEST RELIABILITY

• It is desirable for a test to be Reliable.

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TEST RELIABILITY

• Reliability – the degree to which the respondents’ test scores are consistent over repeated administrations of the same test.

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TEST RELIABILITY


• Indicates the precision of a set of test scores in the sample.

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TEST RELIABILITY


• Indicates the precision of a set of test scores in the sample.

• Random and systematic error can affect the reliability of a test.

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TEST RELIABILITY


• Test developers have a responsibility to demonstrate the reliability of scores obtained from their tests.

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ESTIMATING RELIABILITY

C ronbach sJ

J

jj

J

X

'

=

1

1

2

1

2

j2

X

2

Estimated item variance

Estimated total test score variance

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ESTIMATING RELIABILITY

C ronbach sJ

J

iji j

J J

X

'

/

=

1

21 2

2

ij2

X

2

Estimated covariance between

items i and j

Estimated total test score variance

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OTHER FORMS OF RELIABILITY

• Test-Retest Reliability:

The correlation between persons’ test scores over two administrations of the same test.

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OTHER FORMS OF RELIABILITY

• Split-Half Reliability (using Spearman-Brown correction for test length):

AB Correlation between scores of Test A and Test B

2

2

1

A B

A B

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TEST VALIDITY

• VALIDITY: A test is valid if it measures what it claims to measure.

• Types: Face, Content, Concurrent, Predictive, Construct.

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• Face validity: When the test items appear to measure what the test claims to measure.

• Content Validity: When the content of the test items, according to experts, adequately represent the latent trait that the test intends to measure.

TEST VALIDITY

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• Concurrent validity: When the test, measuring a particular latent trait, correlates highly with another test that measures the same trait.

• Predictive validity: When the scores of the test predict some meaningful criterion.

TEST VALIDITY

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• Construct validity: A test has construct validity when the results of using the test fit hypotheses concerning the nature of the latent trait. The higher the fit, the higher the construct validity.

TEST VALIDITY

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RELIABILITY & VALIDITY

• Up to a point, reliability and validity increase together, but then any further increase in reliability (over ~.96) decreases validity.

• For e.g., when there is perfect reliability (perfect correlations between items), the test items are essentially paraphrases of each other.

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RELIABILITY & VALIDITY “If the reliability of the items were increased to

unity, all correlations between items would also become unity, and a person passing one item would pass all items and and another failing one item would fail all the other items. Thus all the possible scores would be a perfect score of one or zero…Is the dichotomy of scores the best that would be expected for items with equal difficulty?”

(Tucker, 1946, on the attenuation paradox)

(see also Loevinger, 1954)

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1 EPSY 546: LECTURE 1 SUMMARY George Karabatsos. 2 REVIEW.

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