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Green, Kathy e.; Kluever, Raymond C.
Component Identification and Item Difficulty of
Raven's Matrices Items.
Apr 91

17p.; Paper presented at the Annual Meeting of the
National council on Measurement in Education
(Chicago, IL, April 4-6, 1991).
Reports - Research/Technical (143) —
Speeches/Conference Papers (150)

MF01/PC01 Plus Postage.

Academically Gifted; "Children; Comparative Testing;
* Difficulty Level; Elementary Education; -Elementary
school students; item Response Theory; Mathematical
Models; Predictive Measurement; Preschool Education;
"Psychological Testing; Regression (Statistics); Test
Construction; "Test A ;ems

Rasch Model; *Ravens Coloured Progressive Matrices;
"Standard Progressive Matrices



ABSTRACT

Item components that might contribute to the
difficulty of items on the Raven colored Progressive Matrices (CPM)
and the Standard Progressive Matrices (SPM) were studied. Subjects
providing responses to CPM items were 269 children aged 2 years 9
months to 11 years 8 months, most of whom were referred for testing
as potentially gifted. A second sample containing 147 seventh-grade
students, drawn from J. K. Gallini's study n 1983. was used to
assess the utility of the equation developed using the first item
sample. CPM item characteristics were defined and rated. Rasch item
difficulties were used as the dependent variable with misfitting
items omitted. All 15 item characteristics were entered in a
regression equation using forced entry (multiple "R" of 0.90) and
stepwise entry (multiple "R" of 0.88). When the same predictors were
used with SPM items, the multiple "R" was 0.69. The poorest
prediction occurred for items containing characteristics (such as
line thickness) that were not captured by the coding system. The best
prediction occurred for items in which the orientation of the figure
or options was a crucial feature. Results are discussed with regard
to psychological processes and use of item characteristics to create
new test items. Two sample test items are included, and two tables
and an appendix present data on item difficulties. (SLD)



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COMPONENT IDENTIFICATION AND ITEM DIFFICULTY OF RAVEN'S MATRICES ITEMS



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Kathy E. Green and Raymond C. Kluever
University of Denver



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ABSTRACT

The purpose of this study was to identify and test item
characteristics that predict the difficulty of Raven's Colored and
Standard Progressive Matrices items. Colored Progressive Matrices item
characteristics were defined and rated; Rasch item difficulties were used
as the dependent variable with misfitting items omitted. The multiple R
was .90 (.88 using stepwise prediction). When the same predictors were
used with Standard Progressive Matrices items, the multiple R was .69.
Results are discussed with respect to psychological processes and to using
item characteristics to create new test items.



^ Paper presented at the Annual Meeting of the National Council on



Measurement in Education, Chicago, April 4-6, 1991.

Wc 2 BEST COPY AVAILABLE




INTRODUCTION



Figural reasoning items such as those developed by Raven (1965,
1985) have proven useful as nonverbal measures of fluid or analytic
intelligence (»g») . While matrices tests have been used for years
and while a number of researchers have examined the factor structure
of these tests, little attention has b.*n paid to empirical
examination of the item variations contributing to iter difficulty.
The purpose of this study was to identify item components that might
contribute to item difficulty and then to assess which components are
predictive of empirical item difficulty. The prediction equation
constructed was then used to predict difficulty of a second set of

matrices items.

The Raven Colored Progressive Matrices (CPM) and Standard
Progressive Matrices (SPM) have be ^ used as measures of a unitary
trait, although factor analyses suggest that 2-4 factors are
necessary to explain item intercorrelations (e.g., Carlson & Jensen,
1980; Schmidtke & Schaller, 1950). Analyses indicated that while a
mult if actor solution was needed, item responses yielded adequate fit
to a logistic model (the Rasch model: Green & Kluever, 1991). High
internal consistency coefficients have also been reported (e.g.,
Court & Raven, 1982; Raven, Court & Raven, 1986). If all test items
are reflective of a single latent trait, it should be possible to
identify ways in which items differ that make some items
systematically easier and others harder to answer. This process is
termed component identification. Identification of components would
prove useful in construction of additional test items and in
furtherance of our understanding of the construct being measured.



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Suggestions of bet?- method of analysis and of potential
components may be obtained from previous work with Knox Cube Block
Test items, paper-folding items, and progressive matrices items
(Carpenter, Just, & Shell, 1990; Gallini, 1983; Green & Smith, 1987;
Smith & Green, 1985; Ward & Fitzpatrick, 1973). Some of these item
types differ from matrices items but all are non-verbal. Analyses of
verbal analogies items have also been performed but these results
seem less relevant to the analysis of nonverbal items.

The number and complexity of components determine the difficulty
of an item. Potential components relevant to matrices items defined
by previous work include variables such as symmetry vs. assymmetry,
vertical or horizontal vs. diagonal axes, straight lines vs. curved
lines, size of cell attributes, number of dimensions of variation
(e.g., different line widths, different shapes), proportionate vs.
disproportionate change of size, shading, number of colors,
intersection vs. union of dimensions, rotation of elements, and
reflection of elements. Number, orientation, and figure type are the
problem descriptors used by Carpenter et al. (1990) in their analysis
of cognitive processes used by high and average performers on
matrices items. More difficult problems involve more figural
elements and/or more complex combination rules (Ward & Fitzpatrick,
1973) . If multiple rules are needed in problem solution, cognitive
management of rule construction and execution is taxed as well as the
mental processes used to construct the rules. Carpenter et al.
(1990) suggest that individual performance differences reflect
abilities to generate and maintain problem solving goals in working



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memory. Mulholland, Pellegrino, and Glaser (1980) found that errors
and response times increased when the number of operations needed to
solve geometric analogies increased. They also attributed this
performance decrement to an increased burden on working memory
created by the need to track more elements and transformations.

The skill with which individuals process information is dependent
on variables such as the kind(s) of cognitive processes involved, the
nature of the content, the complexity of processing required, and
one's previous experience with the task. Basic processing models
involving variations of the input-process-output systems are common
in the literature. One's experience with the content of the material
is reflected in some processing models. Other models are more
descriptive of the nature of the material and some models are more
concerned with the cognitive complexity required to solve a problem.

The Structure of Intellect (SOI) model (Guilford, 1959) lends
itself as a framework to systematic analysis of the content and
processes involved in solving matrices problems. Other relevant
models are simultaneous-sequential processing models and Horn and
Cattell's (1966) model of cognitive processing. Item difficulty and
item characteristics are related in this paper to these models as
indications of the cognitive processes involved, the nature of the
content, and its complexity.



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5



METHOD



Subjects providing responses to CPM items were 269 2-9 to 11-8
year-old children seen at a University Assessment Center for
individual testing through 1989. The majority of these children were
referred for testing as potentially gifted. Responses were fit to a
linear logistic model using BICAL (Wright, Mead, & Bell, 1980). Two
of the 36 CPM items misfit (both between and total fit >3.00) and so
were dropped from subsequent analyses. Logit item difficulty
obtained from BICAL was used as the dependent variable; the effective
n for this study was 34. A second sample was obtained from data
reported by Gallini (1983) who gave 30 of the 60 Standard Progressive
Matrices items to 147 seventh-grade students from an urban middle
school in the Southeast. Four of these 30 items were answered
correctly by all students so logit difficulties of one less than the
lowest logit value were assigned to these items. Three other items
misfit and were dropped from the analysis leaving an effective n of
27 for this sample. The second sample was used to assess the utility
of the equation developed using the first item sample.

Analyses were conducted using estimated Rasch item difficulties
regressed on component frequencies. Item calibration using tho Rasch
model provides a means to evaluate unidimensionality. Regression
analyses are of limited value unless the dependent variable assesses
a single trait (in this case, item difficulty). Modeling performance
on a set of items that are not well-defined can be pointless. Thus,
Rasch analysis with removal of misfitting items was used to refine



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the dependent measure, to provide reasonable assurance of
unidimensionality. Relationships among predictor variables was also
assessed since the presence of multicollinearity leads to varied
interpretation of results.

CPM items consist of an item "stem" which contains a figural
display with a missing piece, and 6 or 8 response options, one of
which completes the figural pattern. Stem characteristics assessed
were: vertical/horizontal orientation vs. other, symmetrical vs.
asymmetrical, progression vs. not, number of dimensions in the
pattern, straight lines vs. curved lines, number of lines or solids,
density of design, and color vs. black and white. Number of
dimensions and elements were coded a 0-3; all others were coded as
0-1. Response option characteristics assessed were: number of
distinct options (2-8), options contain progression, rotation,
reflection (0-1 for each), number of directions of options (e.g.,
horizontal, vertical, diagonal; 1-3), number of elements in the
design (1-3), and reversal between foreground and background (0-1).
Characteristics were rated independently by the two authors;
disagreements occurred for less clearly specified characteristics
such as "density of design." Disagreements were either resolved or
the category redefined. Figure 1 illustrates item component
categorization for two hypothetical items.



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EXAMPLES OF ITEM CHARACTERISTIC CODING



Item 1



in in





Stem Variables:

Vertical/horizontal orientation: O-yes
Symmetrical : O-yes
Progression: 0-no progression

# dimensions in pattern: 1-pattern only
Straight lines vs. curved: O-straight

# elements: 1-lines

Color vs. black/white: 1-black/white

Option Variables:

# distinct options: 5

Options contain progression: 0-no

rotation: 1-yes
reflection: 1-yes (e.g., 1 and 5)

# option directions: 3 (vertical, horizontal, diagonal)
Reversal between foreground and background: 0-no



Item 2









Stem Variables:

Vertical/horizontal orientation: 1-diagonal
Symmetrical : 1-no

Progression: 1-yes (angle of line separation increases)

# dimensions in pattern: 2-size, orientation
Straight lines vs. curved: O-straight

# elements: 2 (lines, solids)
Color vs. black/white: l-black/white

Option Variables:

# distinct options: 6

Options contain progression: 1-yes (e.g., 2 and 6)

rotation: 1-yes (e.g., 4 and 5)
reflection: 1-yes. (e.g., 4 and 6)

# option directions: 2 (vertical, diagonal)
Reversal between foreground and background: 0-no



RESULTS



Table 1 presents the zero-order correlations of all
characteristics with item difficulty for CPM items. Item
characteristics were multicollinear. Table 2 presents significant
(p<.05) inter-characteristic correlations for CPM items. CPM items
were used as the basis for construction of a regression function.
All 15 item characteristics were entered in a regression equation
using two methods: forced entry and stepwise entry. With forced
entry, the multiple B was .90. With stepwise regression, number of
distinct options, reflection of one or more options, number of
dimensions in the stem, and number of directions of options
contributed significantly (p<.05) to prediction for a multiple g of
.88. The B 2 adjusted for number of cases was .74. Actual and
predicted item difficulties for CPM items are provided in Appendix A
as is the standardized regression function.

SPM item difficulty was predicted using the four characteristics
identified as significant predictors of CPM item difficulty. The
multiple B was .69. Actual and predicted item difficulties are
listed in Appendix A.

To assess the effect of nonlinearity, the squared and cubed
values of components were added to prediction equations. This
resulted in a small increase in prediction for CPM items (.88 to .91)
and in a similar increase in prediction for SPM items (.69 to .71).
When higher order terms were added, the variable representing number
of options dropped out of the function.



Table 1

Correlation of Item Characteristics with Ttem Difficulty.



Characteristic Cprrelatipn-CPM ££M_

Orientation 01

Symmetry 61

Progression 17

Number of Dimensions 66 61

Lines 22

Lines-Solids 68

Color 46

Progression "* 18

Rotation 28

Reflection 34 31

Number of Options 48 52

Density ~ 02

Number of Directions of Options 19 19

Reversal of Foreground and Background 02

Number of Elements 22



Table 2

Item Characteristic Intercorrelations-CPM

C6 C7 C8 CI 3 C14 ~ 16_ _~

C2: Symmetry 78 76 53

C4: Number of Dimensions 82 69

C6: Lines-Solids 61 38 41

C7: Color 45
C9: Rotation 34
C13: Density of Design -48

C8: Progression; C14: Number of Directions of Options; C16: Number of
Elements in Design

Note: Only item characteristics that were significantly correlated (p<.05)
are listed.



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-7-

The two items from each test for which prediction was the poorest
and best were reviewed. The poorest prediction occurred for items
containing characteristics (such as line thickness) that were not
captured by the coding system. The best prediction occurred for
items in which orientation of figure or of options was a crucial
feature.

DISCUSSION

Only tentative conclusions may be drawn from the results of this
study. Some variables that were found to be significant predictors
were correlated with other item characteristics. This collinearity
makes determination of which unique characteristics predict item
difficulty problematic.

Orientation of options (rotation and reflection) were both
significantly related to item difficulty with reflection somewhat
more highly correlated. Many progressive matrices items involve
orientation as well as design matching and number. Items tend to be
more difficult when several options are identical in shape/number but
are mirror images of each other. Rotation of an option away from a
vertical/horizontal orientation does not seem to pose as difficult a
problem as reversal in the same plane. For Raven's items, reflection
seems to require a finer discrimination than rotation. Rotation for
these items is likely to involve a disturbance to the orientation of
the figure which is relatively easy to identify as incorrect .



12



-8-



Number of directions of options also was a significant predictor
for CPN items. This variable also assessed spatial orientation.
This variable did not significantly add to prediction for SPM items.

Number of distinct options was a significant predictor for CPM
items. All CPM items have six options and only two of the 30 items
have less than six unique options. These are the second and third
items on the test and are also among the easiest test items. SPM
items have six or eight options; again, only two items have options
that are not unique and these items are the easiest. This variable
served to identify extremely easy items.

Number of dimensions in the stem assessed the number of different
figural elements that needed to be considered in problem solution.
Possible elements were figural match, orientation, size, and number.
Items varying on more than one dimension were more difficult. This
variable was a measure of item complexity and was most predictive of
difficulty for both CPM and SPM items. This variable is similar to
number of transformations which previous researchers have also found
to be predictive of item difficulty.

While analysis of components does not describe the elementary
mental processes necessary to problem solution, we propose congruity
with certain processing models. Processing models may include
components such as the ability to attend, to encode information,
short- and long-term retention, the ability to retrieve infromation
from storage, certain cognitive processing skills, and an output
component. Task, content, and situational characteristics are

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important factors to consider in analyzing a specific processing
requirement .

Among models of cognitive processing, the Horn and Cattell (1966)
theory includes fluid and crystallized ability representing the
processing of new versus more familiar material. Success with
components of the CPM items probably places a high premium on fluid
abilities since solving matrices tasks is not common. At best, the
analysis leading to the correct choice for stimulus complex ".on may
parallel certain common everyday discrimination tasks but the CPM
content is certainly unique.

Analysis of this fluid ability for solving the CPM problems can
be in the context of a simultaneous and sequential processing scheme
proposed by several psychologists (Das, Kirby, & Jarman, 1975; Luria,
1973). Solution of the CPM problems seems to require good
simultaneous processing ability of a gestalt-like configuration with
differing components contributing to processing load as identified in
this study.

A model displaying greater detail of processing skills is
Guilford's Structure of Intellect (SOI) Model (1967). The Raven
designs probably represent the Figural Content as defined in the
model. Among the five Operations described in the model, Cognition
and Evaluation abilities appear to be most representative of the kind
of processing required to solve the CPM problems. The Products of
this model ranging from unitary components to complex abstracting
abilities requiring the reconfiguration of the stimulus material are
also evident in the Raven problem-solving requirements.

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Both the SOI model and the simultaneous processing-sequential
processing model have published tests based on those concepts. The
value of the component analysis in this study is the opportunity it
presents for construction of items that utilize component
characteristics in revisions of these tests to tap a defined range of
processing abilities. This has implications for test construction
and for test interpretation where manuals could provide guidelines
for this kind of interpretation.

There are several limitations of this study. First, all features
of items were not assessed nor were problem-solving process variables
assessed. Only fairly obvious observable features were included in
the analysis. Second, the regression analysis assumes that a linear
combination of variables predicts item difficulty. Even if the
appropriate components have been identified, they may relate to other
components and to item difficulty in a nonlinear fashion. Third,
individual children may use different strategies in solving matrices
items. The analyses performed implicitly assume that all subjects
use similar strategies and attend to similar aspects of problems.
This assumption may be overly simplistic. Finally, the size of the
samples used in calibrating items were smaller than those desirable
to establish highly reliable item statistics.



15



REFERENCES

Carpenter, p. A. , Just, M.A., & Shell, P. (1990). What one intelligence
test measures: A theoretical account of the processing in the Raven
Progressive Matrices Test. Psvcholooical Review. 22(3), 404-431.

carlsen, J.S., & Jensen, CM. (1980). The factorial structure of the
Raven Coloured Progressive Matrices Test: A reanalysis. Educational and
Psychological Mea surement . 4_0_, 1111-1116.

Das, J. P., Kirby, J.R. , & Jarman, R.F. (1975). Simultaneous and
successive synthesis: An alternative model for cognitive abilities.
Psychological Bulletin. &£, 87-103.

Gallini, J.K. (1983;. K Rasch analysis of Raven item data. Journal of
Experimental Education £2, 27-32.

Green, K.E., & Kluever, R.C. (1991). Structural properties of Raven's
Colored Progressive Matrices for a sample of gifted children. Perceptual
and Motor Skill s r 21, f-9-64.

Green, K.E., & Smith, R.M. (1987). A comparison of two methods of
decomposing item difficulties. Journal of_ Educational Statistics . 12,
369-381.

Guilford, J. P. (1959). The three faces of intellect. T he American
Psychologist , 2A, 469-479.

Horn, J.L., & Cattell, R. B. (1966). Refinement and test of the theory of
fluid and crystallized intelligence. Journal of Educational Psychology .
57. 253-370.

Luria, A.R. (1973). The worki ng brain: An introduction to
neurops ychology . London: Penguin Books.

Mulholland, T.M., Pellegrino, J.W., & Glaser, R. (1980). Components of
geometric analogy solution. Cognitive Psychology . XZ, 252-284.

Raven, j.c. (1965). Guide to using the Coloured Progressive Matrices
Sets A. Ab. B. Dumfries, Scotland: Grieve.

Raven, J.C. (1985). Standard Progressive Matrices Sets A.B.C.D. and E.
London: H.K. Lewis & Co.

Raven, J.C, Court, J.H., & Raven, J. (1977). Coloured Progressive
Matrices. London: H.K. Lewis.

Schmidtke, A., & Schaller, S. (1980). Comparative study of factor
structure of Raven's Coloured Progressive Matrices. P erceptual and Motor
Skills . 5J,, 1244-1246.

Smith, R.M., & Green, K.E. (1985). Components of difficulty in
paper- folding tests. Paper presented at the annual meeting of the
American Educational Research Association, Chicago.



ward, J., & Fitzpatrick, T.F. (1973). Characteristics of matrices
items. Perceptual and Motor Skills. 3j£, 987-993.



Wright, B.D., Mead, R.J., & Bell, S.R. (1980). BICAL: Ca librating items
with the Rasch model (Research Memorandum 23C) . Chicago: University of
Chicago, Statistical Laboratory, Department of Education.



APPENDIX A

Actual and Predicted Item Difficulties for CPM and SPM Items



Difficulty Difficulty



CPM Item


Lpqit


predicted


SPM Item




Predicted


RA1


-2 . 83


-1 . 87


RA1


— 5 • lo


— 2 tCl


RA2


-4 . 66


A C £

—4 • 56


RA2


— o . lo


— D • i.o


RAJ


—3 . 97


—4 • 10


KA4


— 3 • UU


mm "> Q £


RA4


"3 .40


—2 • / 1


KAD


"•DO




RAo


—2 • 59


—2 .71


KAo


-L • X3


— mm .30


KAo






KA /


— D ■ J.O


Z . J O


KA /


• Z 3


— J. . UJ


KAo


— D . ID


mm OA


KAo




• 4 3


KDl


mm *> A

— 46 .43


• O O


KA9






ADZ




Z iDl


DM ft


• /o




Kd3


— • /4


J. . U4


KA11


O CI


A 1

.43


RB6


J.. /X






-5 • 4S J






^ • Z L


mm 1 QQ

z too


KAol




— X .03




* Oo


Z • OO


DAD*)


"J. .3D


— J. • U3


KC3


mm

** • 3*c


— O QQ

~ Z • oo


RAB3


-1 » 32


.43


RC4


— . 20


-2 .88


RAB4


.21


.15


RC5


1.56


1.42


RAB5


.21


.99


RC6


1.37


.22


RAB6


1.04


-1.87


RD1


-3.44


-1.69


RAB7


.54


.99


RD2


-2.21


-1.69


RAB8


1.88


2.11


RD3


-.59


-1.69


RAB9


1.74


.15


RD5


.94


-2.88


RAB11


1.16


.15


RD6


2.90


-2.88


RBI


-4.66


-2.71


RE1


.03


1.42


RB2


-.81


-1.87


RE2


.56


.22


RB3


-.30


-.69


RE3


1.97


1.67


RB4


-.19


.15


RE4


2.02


2.13


RB5


1.02


.43


RES


4.05


1.88


RB6


.99


1.61








RB7


1.23


1.61








RB8


2.73


3.91








RB9


2.37


2.73








RB10


1.83


1.89








RB11


2.80


2.73








RB12


4.26


2.73









Zy ■ . 17 (foptions) + .43 (reflection) + .75(#dimensions) +
. 29 (# option directions) + e

Zy ■ . 31 (#options) + .18 (reflection) + . 39 (#dimensions) +
.07(# option directions) + e

1 /




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