MULTIPLE LINEAR REGRESSION VIEWPOINTSA publication of the Special Interest Groupon Multiple Linear Regressionof The American Educational Research Association.
I
VOLUME 4, NUMBER 1
Table of Contents
Acknowledgement I
Using Coefficients of Orthogonal Polynomials asPredictor Variables in Multiple RegressionW.K. Brookshire and J.T. Bolding 1
Testing an Hypothesis About a Single Polulation Meanwith Multiple Linear Regression — Keith A. McNeil . . . 7
Identification of Significant Predictors of Children’sAchievements and Attendance — Of elia Halasa 15
Application of Multiple Regression Analysis In Invest-igating the Relationship Between the Three Com-ponents of Attitude in Rosenberg and Hobland’sTheory for Predicating a Particular Behavior—Isadore Newman and Keith McNeil 23
Guidelines for Reporting Regression AnalysesJoe H. Ward, Jr 40
Reaction to Ward’s “Guidelines for Reporting RegressionAnalysis” and Some Alternatives — Keith McNeil 42
A Revised “Suggested Format for the Presentation ofMultiple Regression Analyses” — Isadore Newman 45
Business Meeting — Judy McNeil 48
Announcement of A Multiple Linear RegressionSymposium — Steve Spaner 51
1973 Membership List — Judy McNeil 52
T~T~9T.IflO~Si~ ‘STflO’[ •:1sSTflOTI ~S-T.InOssTw ~O A~TSJ~?A~U1L
J~S~>J ~ S~tpfl~ IBJOTA~T4O~
~ OA~~
~ ~/
54~~k o’~,Q~tvo~’~~(~y 73~4//~~f~943 colE
JQ 4-~‘S~Y3A/fY/’
ACKNOWLEDGKMENT
Chairman: Judy T. McNeil, Department of Guidance and Educa-tional Psychology, Southern Illinois University, Carbon-dale, Illinois 62901
Editor: Isadore Newman, Research & Design Consultant, TheUniversity of Akron, Akron, Ohio 44325
Secretary and Chairman-elect: James Bolding, Educational Founda-tions, University of Arkansas, Fayetteville, Arkansas72701
Cover by: David 0. Barr
Layout by: Edward Lasher
I wish to acknowledge
Assistant Dean caesar Carrino and
Dean Kenneth Sarker of the
University of Akron for their
support in putting out this
publication.
Isadore Newaan, EditorMultiple Linear Regressive Viewpoints
USING COEFFICIENTS OF ORThOGONAL POLYNOMIALSAS PREDICTORVARIABLES IN MULTIPLE LINEAR REGRESSION
William K. Brookshire and J. T. Bolding
Coefficients of OrthogonalPolynomials arepresentedby someauthors
(Snedecorand Cochran) as a meansof simplifying the computationrequired
in trend analysis. Linear regressionaddictswho are computeroriented can
still make good use of such coding in the analysisof complicateddesigns.
Considera two factor designwhere the factors are assusedto be
quantitative with levels selectedat equal intervals. Testing for main
effects and trend analysis canboth be simplified by the use of coefficients
of orthogonal polynomials as predictor vectors.
An exampleis presentedwhere factor A has two levels and factor B has
four equally spacedlevels. The data is taken from Kirk (1961) chapter 7,
Table #1
Data From Kirk Page 175
B1
B2
B3
B4
1~‘H iJAz ~
2 3 6 11
2. 3.
Since factor A only has two levels therewill only be a linear compo-
nent and the two levels of factor A will be coded -1 and ÷1. The assignment
X1
= +1 is given for scores in A1
. The assignmentX1
= -l is given for
scores in A2
.
The four levels of factor B will give rise to three components -
linear, quadratic, and cubic. The respective coefficients are foumd to be
as follows:
Linear Code Quadratic Code Cubic Code
Level 1 -3 +1 -1
Level 2 -1 -1 +3
Level 3 ÷1 —1 -3
Level 4 +3 +1 +1
Vector X2
is the linear component of factor B and is coded as follows:
-3 if the score is from B1
(column 1),
-l if the score is from B2
(column 2),
±1 if the score is from B3
(column 3), and
+3 if the score is from B4
(column 4).
Vectors X3
and X4
are similarly defined using the orthogonal ploynomial
coefficients for the quadratic and cubic components respectively.
There are three degrees of freedom associated with the interaction mean
square e. g., (2-l)(4-l). These three cognonents are defined as follows:
X5
A linear tines B linear,
A linear times B quadratic, and
= A linear times B cubic.
A condensedrepresentativeof the predictor vectors is given in Table 2.
The sum of squaresbetween rows, columns, or interaction can bepartitioned
into as many trend componentsas there are degreesof freedomfor the
respective variance estimate.
Table #2
CondensedRepresentationof Predictor Vectors
A1
B1
1 -3 1 -1 -3 1 -1
A1
B2 1
-1 -1 3 -1 -1 3
A1
B3
1 1 -1 -3 1 -1 -3
A1
B4
1 3 1 1 3 1 1
A2
B1
-1 -3 1 -1 3 -1 1
A2
B2 -1 -1 3 1 1 -3
A2
B3
-1 1 -1 -3 -1 1 3
A2
B4
-l L ~ 1 1 -3 -1 -1
Cell
Factor AReadership
xl
Terms for Factor B Interaction TermsLinear Quadratic Cubic = = X
7X2
X3
X4
X1
times X2
X1
times X3
X1
times
With the predictor vectors definedas above the test for main effects,
interaction, and trend analysisproceedsas outlined in Table 3.
Table #3- - (Continued)
Testing Cubic Trend Componentof BRestriction: A =0Model 7 Y=A
0U+i
1X~+A
2X
2+A
3X
3+A
5X
5÷A
6x6
+A7
X7
+E7 Restricted .9146 1/24 2.08 .1594 194
Testing Linear X Linear Trend ComponentRestriction: A =0Model 8 Y=A
0U+kX
1+A
2X
2÷A
3X
3+A
4X
4+A
6X
6+A
7X
7+E
8Restricted .8653 1/24 17.16 .0006 196
Testing Linear X Quadratic Trend ComponentRestriction: Ae=0Model 9 Y=A
0U+X.~X
1+A
2X
2+A
3X
3+A
4X
4÷A
5X
5+A
7X
7+E
9 Restricted .9082 1/24 4.05 .0527 196
Testing Linear X Cubic Trend ComponentRestriction: A =0Model 10 Y=A
0U+~
1X
1+A
2X
2+A
3X
3+A
4X
4+A
5x5
+A6
X6
+E10
Restricted .9086 1/24 3.92 .0563 196
Table #3
RegressionAnalysis of Main Effect and Trend
Model Model P.2
df F P Kirk’s Page
Full Model for All P TestModel 1 YA
0U+A
1X
3+A
2X
2+A
3X
3+A
4X
4+A
5X
5+A
6X
6+A
7X
7+E
1Full .9214
.0008 176
Testing Interaction EffectRestriction: A = A = A
70
Model 2 Y=~1J+~1
X1~
A2
X2
+A3
X3
+A4
X4
÷E2
Restricted .8392 3/24 8.38
Testing Colusm EffectRestriction: A
2=A
3A
4=0
Model 3 YA0
U+A1
X1
+A5
X5
+A6
X6
+A7
X7
÷E3
Restricted .0955 3/24 84.11 .0000 176
Testing Row EffectRestriction: A =0Model 4 Y=A
0U+A
2X
2+A
3X
3+A
4x4
+A5
X5
+A6
X6
+A7
X7
+E4
Restricted .9082 1/24 4.05 .0527 176
Testing Linear Trend Componentof BRestriction: A =0Model 5 Y=A
0U+~X
1+A
3X
3÷A
4X
4+A
5X
5+A
5X
6+A
7X
7+E
5Restricted .1363 1/24 239.87 .0000 193
Testing Quadratic Trend Component of BRestriction: A =0Model 6 YA
0U+~X
1÷A
2X
2+A
4X
4+A
5X
5+A
6X
6+A
7X
7+E
6Restricted .8875 1/24
:10.38 .0039 193
6.
7.
Draper and Smith (1966) discuss the use of orthogonal polynomials in
curve fitting. Mendeohall (1068) devotesmost of a chapter to the use of
orthogonal predictors including a section on orthogonal polynomials, and
their use in a “k-way classification” problem.
References
Draper, N. R. and Smith H. Applied Regression Analysis. New York: JohnWiley ~ Sons, Inc., 1966.
Kirk, Riger E. Experimental Desi~n: Procedures for the BehavioralSciences. Belnmnt, California: Brooks/Cole Publishing Company, 1969.
Mendenhall, William. Introduction to Linear Models and theAnalysis of Experiments. Belmont, California: Wodsworth PublishingCnepany, Inc., 1968.
Snedecor, George N. and William G. Cochran. Statistical Methods. Ames,Iowa: The Iowa State University Press, sixth ed., 1967.
Testing us :ynothes:s About a Single Population leanwith Ilultisle Linear Regression
Keith A. MclleilSouthern Illinois University at Carhondele
ABSTRACT
The recent emphasis on criterion referenced testing and on the explicit
stating of objectives Implies that lore researchers vi be testing hyDotheses
ahou+ a sinqie population fleas. The general iced regression procedure is one way
to test such an hypothesis. The appropriate regression models are eresented In
this paper.
The multiple linear regression procedure has been shown to be an eitrernely
flexible technique, encompassing both analysis of variance designs as well as
correlational designs (Bottenberg and Ifard, 1963; Kelly, Beggs, McNeil, Eicbel—
berger and Lyon, 1969; Will lams, 19701. Indeed, any hypothesis that requires
a least squares solution can be tested with the multiple linear regression approach,
with the exception of questions dealingwlth multiple dependent variables. Even
some of the non—parametric techniques have been accomplished with the general
linear model IIlcNei I and Morthland, 1971; Starr, 1971).
Of more importance though is the fact that multiple linear regression allows,
indeed, demands that the researcher state his research hypothesis. The flexibility
of the technique demands that the specific hyPothesis be stated by the user. The
specificity of the research hypothesis becomes quite clear when testing an
hypothesis about a single population mean. For example, the researcher may suspect
that the children in his school are, on the average, below the normal 10 mean.
Given that the “normal 10 mean” is 100, then the research hypothesis would be,
“The population of the school has a mean IC lower than the normal mean IC.”
6.7.
Draper and Smith (1966) discuss the use of orthogonalpolynomials in
curve fitting. Mendenhall (1968) devotesmost of a chapterto the use of
orthogonalpredictors including a section on orthogonal polynomials, and
their use in a k-way classification’ problem.
References
Draper, N. R. and Smith H. A lied Re ession Anal sis. New York: JohnWiley 6 Sons, Inc., 196
Kirk, Riger E. ~xpp~jmental Des4n: Procedures for the BehavioralSciences. Belmont, California: Brooks/ColePublishing Company, 1969.
Mendenhall, William. Introduction to Linear Models and the Cos~_god~pg~y~is of Experiments. Belmont, California: Wodswde~YJlPublishingCompany, Inc., 1968.
Snedecor,Coorge 9’. and William C. Cochran. Statistical Methods. Ames,Iowa: The Iowa State University Press, sixth ed., 1967.
Testing an lynothesis About a Single Population lean
with IIuitioie Linear Regression
Keith A. McNeii
Southern Ii inois University at Carhondeie
ABSTRACT
The recent emphasis on criterion referenced testing and on the explicit
stating of objectives implies that more researchers viii be testing hypotheses
shout sinqie population rear. The ‘-eneralized regression procedure is one way
to test such an hypothesis. The appropriate regression modeis are presented in
this paper.
The rnuitipie linear regression procedure has been shown to be am e~<treneiy
flexible technique, encorpassing both analysis of variance designs as well as
correlational designs (Bottenberg and Ward, i963; Kelly, Beggs, McNeil, Eicbei—
berger and Lyon, i969; Will iams, 1970). Indeed, any hypothesis that requires
a least squares solution cam be tested with the multiple linear regression approach,
with the exception of questions deaiingwith multiple dependent variables. Even
some of the non—parametric techniques have been accomplished with the general
linear model (McNeil and Morthiand, 1971; Starr, 1971).
Of more importance though is the fact that multiple linear regression allows,
indeed, demands that the researcher state his research hypothesis. The flexibility
of the technique demands that the specific hynothesis be stated by the user. The
specificity of the research hypothesis becomes quite clear when testinq an
hypothesis about a single population mean. For example, the researcher may suspect
that the children in his school are, on the average, below the normal IC mean.
Given that the “normal 10 mean” is IOU, then the research hysothesis would be,
“The nopulation of the school has a mean IC lower than the normal mean IC.”
8. 9.
Stated symbolically, the research hypothesis would be:Jj ~ IOU where I
the population mean of the school, and IOU is the normal IC mean. The statistical
hypothesis used to test this hypothesis is “The population of the school has
a mean 1Q equal to that of the normal mean IC,” or symbolically: )~ = IOU.
Another example may be of some assistance. Consider a project uti I izinq
methods to reduce alienation. One of their objectives might be: After six
weeks of participation, the alienation mean score of the children in the project
will be less than five. Now if the project director is only Interested in how
the project works for the few children in the sroject, he simply needs to look
at the sample alienation mean to see if it is less than five. But a more
reasonable desire is to infer to the adequacy of the project, with the intent
of adopting it in other schools. With this desire, the project director wants
to Infer to a population of children. The research hypothesis in this case
would be: “After six weeks of Instruction, the alienation mean score in the
Dopulation will be less than five.” Symbol ical ly: Il < 5. The statistical
hypothesis is: “After six weeks of Instrsction, the alienation mean score in the
population will be five.” Symbolically, the statistical hynothesis isA1
= 5.
Traditional Solution
The traditional statistical solution to the kinds of hysotbeses being
discussed are sresemted as either a t test or a z test. Bloomers and Lindquist
119601 present a z test and their examsie is similar to the first example in
this paper. Since n z test is presented, the authors indicate that the test is
reserved for lame samples.
Glass and Stanley 119701 present the technique in terms of a t test; and
since the t test is sensitive to varyinm number of subjects, their formulation
provides the exact nrohabil ity values, whereas a a test will provide only a close
approximation. The data for the alienation research hypothesis discussed above
is presented in Table I and tested in Table 2. The resulting t and related F
values will be referred to later.
Regression Solution
The following regression solution also provides am exact probability value,
but since the formulation is applicable to all least sQuares procedures, it can
be argued that the regression formulation is preferred over the t test formulation.
To answer any research hypothesis on multiple linear regression, full and
restricted models must be constructed. The same F test formula is applicable to
all hypotheses, Providing that the unit vector is in both the full and restricted
models, If this is not the case, and the present solution Is not, then an
alternative formula for the F test must be used lBottenberq and Ward, 1963):
IESSr - ESS~I/In1
- m2
lFimi — m 1, IN - m I ______________________
~SS~l/lN - p1
1
where:
ESSr = error sum of squares in the restricted model
ESSf = error sum of squares in the full model
p1
= number of linearly independent vectors in the full model (number
of pieces of information in the full model I
m2
= number of linearly independent vectors in the restricted model(number of pieces of information In the restricted model)
The alienation example will now he formulated in regression models. The
research hypothesis: “After six weeks of instruction, the alienation mean score
in the population will be less than five” dictates a full model which must allow
the alienation mean to manifest itself:
10.
11.
a0
lJ + E1
where: Y1
= alienation scores;
U = ones for al I subjects; and
a0
= regression coefficient chosen so as to minimize the error sumof squares, or the sum of the squared elements In E
1, the error
vector
Readers fami liar with the regression techniaue will recognize this model as
“the unit vector model” yielding no differential predictability 1R2
= 01. The
one regression coefficient that nust be determined is s~, and this will be the
sample mean. The sum of the sqaured elements in E1
will he the EESf. The statis-
tical hypothesis implies the restriction that a0
= 5. Forcing this restriction
on the full model results in the following algebraic gyrations:
full model: Y1
= a0
U + E
restriction: a0
= 5
restricted model: V1
= 5U + E2
but since U = I for all subjects, SU is a constant, and subtracting
that constant from both sides yields the final form of the restricted
mode I
1Y1
— 5) = E2
The sum of the squared elements in E2
(or V — NJ) will be the ES$r. Note
that the full model utilizes one piece of information ithe unit vector), whereas
the restricted model utilizes no information, therefore, m1
= I and = 0. The
difference between m1
and m2
is one, being equal to the number of restrictions
made, and also being the degrees of freedom numerator for the F test. Table I
contains the intermediate values for the solution. The resultant F of 101.5 is
within rounding error of the t2
value of 102.4. The significance of the F must
he judged by referrim~ t’ tnhled values, ard simcv VbIs ~sv a Wirectional
hypothesis, one must use the 90th percentIle of F if his alpha was .05 and the
sample mean is in the hypothesized direction. If the alienation sample mean
was greater than 5, there would have been no need to go through the statistical
gyrations; it would have sufficed to report “not significant,” and then suggest
dropping the project. More thorough discussion of directional hypothesis testing,
within the context of multiple linear regression, can be found in IlcNei I and
Beggs 119711 and McNeil 119711.
It would appear that with the recent emphasis on criterion referenced testing
and on the explicit stating of objectives that more researchers will be turning
to the single population mean hypotheses preaemted In this paper, It Id hoped
that the regression formulation is utilized since it is general izable to other
least squares procedures. Researchers having access to computing facilities can
perform the required analysis quickly, as one computer run will provide all the
component values of the F test. The substitution of the numerical values into
the formula must be done by hand, but that isa small price to pay for the
utilization of the flexible multinle linear regression technique. Hypotheses
about a proportion could also be tested with the same full and restricted models.
The criterion vector in this case would be a dichotomous vector rather than a
continuous vector as in the alienative example.
Appendix A — Linear Setup to Achieve Intermediate Values
X12l = )Xll)—2.51”2.X131 = lxii l—5.Ol~2.
The 2.5 in the First data transformation statement reflects the observedsample mean, while the 5.0 in the second reflects the hypothesized sample mean.
13.12.
Table I
Numerical Solution for Regression Testing of anHypothesized Population MeanMean for Variable 2 will be ESS /N
Mean For Variable 3 wIll be ESS~/ii = a0
U + E1
E~ 1Y1
—5) 1Y1
—512
Calculation of F can be accomplished by using from this output: I I —1.5 2.25 —4 16
F1
141 = MEANVAR 3 — ~IEAF4VAR 2 I I —1.5 2.25 —4 16MEANVAR 2/IN—il
I I —1.5 2.25 —4 6
I I —1.5 2.25 —4 16
4 I 1.5 2.25 —I I
3 I .5 .25 —2 4
3 I .5 .25 —2 4
2 I —.5 .25 —3 9
—1.5 2.25 —4 16
4 I 1.5 2.25 —l I
2 I —.5 .25 —3 9
I =2.5 I 1.5 2.25 —4 16
3 I .5 .25 —2 4
2 I —.5 .25 —3 9
3 I .5 .25 —2 4
I I —1.5 2.25 —4 16
5 I 2.5 6.25 0 0
3 I .5 .25 —2 4
2 I —.5 .25 —3 9
3 I .5 .25 —2 4
3 I .5 .25 —2 4
3 I .5 .25 -2 4
d I 1.5 2.25 -I I
4 I 1.5 2.25 —I I
Lj
F1
23 = 101.5 ESSf = 34 ESSr = 184
14.15.
“’able 2
.umer’cul Spluton For Trad’~’ona “est’rain unotrosizea flopuI-,t’cn ‘ear
nor_~Ii From”’ ins arc mar e’ i77, “. 2’3IDENTIFICATION OF SIGNIFICANT PREDICTORSOF CHILDREN’S ACHIEVEMENT AND ATFENDANCE
1
tI,1 - I = -
where: “~
OFELIA HALASADivision of Research and Development
Cleveland Public SchoolsCleveland, Ohio
vor the al’enat’nn data:
= 1.21
5
= 24
— 2.5 —5.’) = —I,~.I2
- = 132.4 = -
Wloomers, ~‘. & Linacu’u, r~w• tIe’-em±ary ‘Tatistica “etho’Fs ‘n0
sycholoai’ andEducation. 3oston: ‘Iourhton ‘Ii’fI’n momnarv, 060.
Rottenbemn, P. N ‘nrd, J.I’. Analied ~ultipe I’near rerressir>r. Thchn’calocur”ertarv “’eport w, L— Th—’3—h, uS
7’l*b remsonrel ‘)esearch ,aloratory, ,ackisnd
Air morce ihase, Thxuv, I’l(I.
Glass, C... & Stanley, JO. Ststisticul “etbods ‘r Fducation and Csvcholocy.Englewood TIif+s, ‘,eu Jersey:
mrent’ce all, 3”).
nelly, r~,,•, deogs, 3.L .,“c’le’I, Y.v., richelperqe-, T, 4 [von, J.2
esearchec’cm in th= Ce~av’omeI bcisnces: ‘ t’r p Rermmss’o’
5rnroach. Tarloras e,
I inoL: Sout”emn III ‘p0k Lnivers’’rv Press, 191k.
‘Ac’,ei ,.A.,”To’-e ‘11k’ons “hou~ the sp5~
at’st’ca , o’t’rec’s ~stinn.”
Parer crusented yy CV, I 171.
Oc’Iei I, i.A., N leans, 7..,”’ “ec”ional Hvnotbeses vith the ‘‘u t’nle linearRepression Ansroacb.” °aner sresenTed at RICA, 97
“c’)ei ~, ‘k. N “or~ and,0
.,”br ‘‘a tip e :roar uppress’on Approac todqare
1”’o~e”s. ==nJK, ‘bee ‘!uJscm’m± IV”
Pram—, m, ,, ““re erarr’sgo F ‘ljk’ple ‘eoress’cr and Sr’Lstica nFerence: uIromisips Ipproach for vpposis Researchers.” Thp Amer’can Journal of
CInical lypnos’n, 13, 175—197, 97
‘liars, J.L,”t ‘epmnss’on ‘,ppoac to ExperimenTal Thn’””.” The n:rnul ~
Exper’mentn 0 Iucat’on, 39, 13—93, 197”.
The identification of variables other than the treatment process,
which is affecting the criterion measure variance has always been a problem.
Multiple regression techniques have been utilized to look at this problem
through an efficient linear equation by which scores may be combined to pre-
dict one’s level of performance on a criterion measure:
Y - b0
- b1
(X1
) • b2
(X2
) .... b~Xk
This is a fitted linear regression equation for a particular V response in
terms of the independent variables X1
X2
. . . .X~. It allows the investigator
to extract from several variables the main features of the relationships
hidden or implied.
To come up with reliable fitted values, it is necessary to include
as many “predictors” or independent variables. However, it is not only
realistically impossible in terms of cost and manpower, but the “overfitting”
of the regression equation may stabilize the residual mean square
Paper presented at the 1971 National Council on Evaluation andMeasurement Convention, February 5-7 at New York City.
Table 2
14.15.
l:unerical Solu-tion for Tradi-tional Testimo
of us ::vpotpesiZed Population ‘lean
Formula fron Glasn and Stamlan (1970, p. 273):IDENTIFICATION OF SICNIFICM4T PREDICTORSOF CHILDREN S ACHIEVEMENTAND ATFENDANCE
1
- =
Sx/~i—v,here: S~~
OFELIA HALASADivision of Research and Development
Cleveland Public Schools
Cleveland, Ohio
for the alienation data:
2.5—5.0 = —12.l2I .2l~
- = 102.4 = F~ ~, -
FF FR.LUCAS
Bloomers, P. 4 Lindpa)sS, 3.F. Elementary Statistical ‘lethods in Psycholoqa andEducation. Boston: llouyhton ‘lifflin Connanv, 960.
Rottenberm, R. S lard, 1.9. App) led multiple Hnear repression. Technical
Documenters Report PRL—TDR—A3—S, 857i7t3 “ersonnel Research Laboratory, LacklandAir Force Base, Texas, 963.
Glass, (0.V. & Stanley, J.C. Statistical Ilethods inFducation and Psycholocy.Enqiewood Cliffs, lieu Jersey: Prentice Ilal I, 1970.
Kelly, F.J., Bemps, DL., ‘IcUell, K.A., Eichelher~er, T., S Lyon, 1. ResearchDenies in the Behavioral Sciences: ‘lultiple Repression Anproach. Carbondale,iiTinoi~T Southern Illinois University Press, 1959.
McNeil, K.A., “Some Notions About the Use ef Statistical hypothesis testinc.”Paper presentea at 5()PA 1971.
McNeil, 4.5., 5 Bepps, 0.L .,“Directional Hypotheses with the ‘lultiple linearRepression Approach.” Paper presented at ALPS, 1971.
lIcliell, 4.A.S ‘lorthiand, F.,”The ‘(ultiple Linoar Repression Approacs to CA)Square Problems.” Unpublished ‘danuscript. 1971.
Starr, F.l:., “The Remarriaqe of ‘lultiple Repression and Statistical Inference: APromising Approach for Hypnosia Researchers.” The American Journal of
Clinical hypnosis, IS, 175—197, 1971.
Dill laos, J.l).,”A Repression Approach to Experimental flesien.” Th~Journal ofExperimental E:lucation, 39, 83—90, 197(1.
The identification of variables other than the treatment process,
which is affecting the criterion measure variance has always been a problem.
Multiple regression techniques have been utilized to look at this problem
through an efficient linear equation by which scores may be combined to pre-
dict one’s level of performance on a criterion measure:
Y b0
- b1
(X1
) * b2
(X2
) .... bk Xk
This is a fitted linear regression equation for a particular Y response in
terms of the independent variables X1
X2
.. . .X~. It allows the investigator
to extract from several variables the main features of the relationships
hidden or implied.
To come up with reliable fitted values, it is necessary to include
as many “predictors” or independent variables. However, it is not only
realistically impossible in terms of cost and manpower, but the “overfitting”
of the regression equation may stabilize the residual mean square
1 Paper presented at the 1971 National Council on Evaluation and
Measurement Convention, February 5-7 at New York City.
S = 1.21x
17.
16.
The stepwiue regnessuenanalysis (draper a::d 5:-i ph, 1960) appears
to represent a compromisebetween too many and too few variables, and
allows for the selection of the best regrensiem emoutiop. It jnvols’es
reexamination at entry stage of the regression of the variables incorporated
into the model in previnus stages, A variable wRich nay have been the best
single variable to enter at an early stage may be- superflous becauseof the
relationships between it and the variables now in regression. Thus, partial
F criterion for each variable in the regression at any stage is evaluated and
compared with a preselected percentage point of the appropriate F distribution,
This provides a judgment on the contribution made by each variable as though
it had been the most recent variable entered regardless of its point of entry
into the model, Asy s’ariable which has a non-significant contribution is
removed from the model. This process is continued until no more variables
will be admitted or rejected,
- The procedure oay be briefly sunanarized as follows:
- 1, The procedure starts with a pimple correlation matrixand enters into a regression the variable most highlycorrelated with the criterion, and finds the first
order linear regression equation:
A
Y = f(X1
)
2. Partial correlations of the other variables not inregression with the criterion are then calculated.Mathen:atically, the partial correlations representcorrelations between the residuals from the firstorder linear regression and the residual from anotherregression not yet performed:
f~(X~)
The Xj with the highest partial correlation with Y(criterion) is now selected, e.g. Xl, and a secondregression equation is performed.
3. Given the regression equatioT: of:
A
Y f(X1
, X2
)
the procedure then exsmii:es the contribution X~would have made if X
2had been entered first and
entered second, If the partial F value exceededthe established level of significance, it is retained.This procedure is continued until contribution ofother variables to the criterion variance becomesnon_significant.
Results obtained from regression ~naly’si5 take the form of correla-
tion coefficients and regression coefficients along with standard errors of
the regression coefficients. The regression coefficient gives the estimated
effects of the independent variable which is significantly related to the
criterion. A standard error estimated for mach significant coefficient gives
some indication of the confidence that can be placed in this coefficient.
The multiple correlation coefficient (6) indicates how well the data fit the
model. A square of this correlation (R2
) indicates the per cent of variation
of the dependent variable or criterion that could be attributed to the indepen-
dent variable or variables.
The stepwise regression technique was utilized recently in the
evaluation of a federally_funded project at first and second grades to answer
the following question:
1. Are there factors othar than treatment effects whichare influencing children’s level of achievement andattendance?
18.
19.
Ten regression analyse-; we;:: rey ;-:ilh ~ foll::ipp dcpe;:dent
(criteria) and independent variailes:
oat Variab 1�a
At First Grade - COOP Primary (129) Post Scores
ListeuingWord Apaly’sis
MathReading
At t on dan ce
At Second Grade - COOPPrimary (231) Post Scores
ListeningWord Analysis
MathReading
Attendance
Independent Variables
Number of Children in the FamilyOrdinal RankMobility RateDuration of Project ParticipationPre-Test Score
Attendance
1. Pre—test score shooed significant effects on achieve-ment at firsL ard second eades. Tie higher tl:e
initial score, the higher was the level of achieve-ment at the er:d of the year, At first grnde, criterionvariable which nay he artmib:;tcd te this variableranged from 5% to 29%. At second grade, predictablevariance ranged from 7% to 29%.
2. Attendance of first grade children was a function of theOrdinal Rank, Mobility Rate, and Duration of ProjectParticipation. The older, the lass mobile the child, andthe longer the duration of Project participation, thel:igher was his school attendance. Approximately 163 of’variance of attendance nay be attributed to the combinedeffects of these three variables.
3. Attondance of second grade children was a function ofNumber of Children and duration of Project Participation.
The more children in the family, and the longer theDuration of Project Participation, the higher was t]:eattendance, Predictable variance of attendance was 21%
Identification of variables with significant influences on the
criterion measures has the following advantages:
1. Statements on treatment effects can be made with ahigher level of confidence as they are less subjectto contamination problems.
2, Variables from a larger initial set can be reducedto a smaller but more meaningful set which hasimplications in terms of economy, tine, manpower,and expenditures,
3. Future data gathering procedures can result inhigher predictive accuracy with subsequent samplingunits.
~(p~ications
Findings
Most of the regression coefficients which give the estimated effects
of the different predictors failed a statistical test of significance. Of
the six predictors, the pre-test score evidenced consistent significant
contributions to the criterion variance. The per cent of predictable variance,
however, indicates that a significant proportion of the variance remains
unaccounted for (Tables 1 and 2):
TABLE 1
STEPOOWN REGRESSIONANALYSIS OF SIX INDEPENDENTVARIABLES
ON FIVE DEPENDENTVARIABLES
Dependent Variables R2
R
Regression Coefficients
Children RankMobility
RateDuration ofParticipatio:
Ftc-Test
Score(October)
Attendance(1969-1970)
COOPPrimary Test (l2B)
0.26 0.51
0,17 0.41
0.29 0.54
0.05 0.23
0.16 0.40
I
-
-
-
-
-
-
-
-
-
-
-
-
-
~2.2l”
-
~
-
0,70~
0.49*
0.54=
0,64*
0.2l~
-
0.04**
-
-
Listening
Word Analysis
Mathematics
Reading
Attendance
* p<.OOl
p<.Ol
- mi.05
a.; .~.o a Sc,.~00
a 0*,
O ‘0 acc ‘2 ~ra ‘, Cv -1ca so
co o~z-‘--‘Ca .v~ 0
o cc5- 2’~ ~1* P-o c., *,Z Ca
0-’ oa- .me cc
~0 ~J,t O,4~-e ‘0 ‘0‘0 51. 9,0
51 oa-a‘0 ~‘ti C”-’~ -2°O ‘00o ‘Co 0-C-_ mc
U) ‘0 >‘.‘0 ‘0.0
~ ~ ‘~-~z a ma- C
~a- a- a< a-..a- 0 ‘-‘‘0
~O --4’a- 0 mm COO
-e a’s, to~, ~- 2~’2
~‘ ~ U)
a-a .~
= ‘-‘0 ,e0*0
-a- -em me..0,0 410 -.-,4J04 ‘-‘0 Ct~
op — a ‘0p.. p a,
to -~4 ‘U>a—’ .0* 00~ Ice ,a.-,
Ia C L~
0a ~, ,
o ‘0 ‘0~
a ‘a ,*C a-
5,a--4a--ca-a.I,.:5144Cz‘US)a-a-44.44Cm:5<
a-04
5,10 E’..44~ 0144< CW. opa-
‘-Icop 0.5-10
zo.0—z5)05,‘Ua-La
5.1444.-a,
0 0
Ca-‘s—~0’Ca-a ‘0Pa-p—a:-.-’
41‘000a aa-5—00
Q4.4a 00.
Os O5 a-
‘aa-tpnçasd~spasi seas pOgpnes eq ac umsopa apoe-sDesapj
‘lopasopaq uaTna-g~,asd a go uo-gcaipa.nd ape ug seuauodmoa asape go dipsuoge
_agaaape dpnes oe soLo uoTesS-geseAupsipe go asodundap3 ‘eoepsodsrp
asuodsa~ pus aspeaaggo ‘asgetudos :seeadea aaepe esoag Do go sespsuoa
apnegeeo US DopD isoaqe a paeuasaed (0961) pua-psomj pus 8eaquaeo~
‘peon soo apneteeo go oogegupgap
gauegsuanrgpun us cope ass ‘(zcGI) .nauperp so pans seapeo pus ‘sapasaa
°eape go ms-teteTua cocoa auo ‘eogasapaq agqoasaasqo eappaid usa
5a,nleuue5eeanbapnegeeo cope aeolesuolaepoe se-noggasnoia,snu uaaq easop
a-nape ‘aaneo.aaeg-[apne-teeaape pasapasac(9E61) ~-n-na-raa-i aau-rg
S’IVNOI.LVS ~IMYNOIILD000XJ.NI
‘iogASpaq aeasapeup—uou pus aeompeuç
peoq uog agoes aaspeappaed Doom ape sos gopeua.zaggppgO.Pogasapaq ape ‘pasn sa.aneoasiaaepe ape go ceo ‘uogu,ap
—eq aeompeup—uouso; aaspeappasdeou enq eopsopaqaeomcgeu-ggo asteappeud ATeuoapgruWps eq oe punog sos euauoomoa
aegepudea apj, ‘ssogasopaq aeOi,Teup.uou uope lapeas acorn-zeusso; as-gea-speed eq oe dga
9gg aeon,ala oogeaoeaeu-t eoaup
1ape
pus apneeeo go seuauodsnoa cue-na-gp ape cope puno; see;
ci ‘sasod aasseagpasd ToleuacaTp .apape pus aprle-geeo geseuauodtnoa ape uaaseaq dppsuopeegas ape aee8pesaeu-g oe peon
a-paLo sanb~upaae spswgauo seeaeg pam uopssa.nbae aidreinpi
XOV8ISCV
,çepseanarun ~souptgi u.aapenog ‘1-tagaw peTeRpus
ue~s~go ieslaasufl ‘uoetsa1
ga.aepos1
C‘a
‘0 0’4’ ,*C CC.0 0-C
UU
~‘E‘a 0’50 05.0
C0 5.5 .,a
-0,5a a-a-ss o0 51a-
.vC‘a , , , I
a-
C
a‘UCO
, , IaCON
CO0
a- N a- ~ NS UI 04
dood
a-~
o
‘j4
p ~N 0 (N C
d ~ ~04
c
.nsgasOpag~‘°T’°~Te°d 0 8ueapaa~ an; dieapj spUoIsoH pus 8saquaso~ a.2a-(5
‘215a.~oa‘U040.aC
10
~UU
~a4.-p., 2a a Cs a- aB 55 — -e‘u C 15 41a -.., o a so5. o < C C
0 a -ea- P ‘U .0 ‘0O a a p aO -‘0 0 ‘0 aLI .44 Ia 51 a-
Up apnep~~co SeuauodeioD aasp~ ape uaeseag dipsueieoiaog aip
Wugeo9-tesaasul up sstgouy uogssa-n1a~ agdpelnw go uoreoai~ddy
a,Uz‘0
‘UCSIpp<
000
~Jv A0.0,0,
a a art CD CC 0 C-’
CC Cl Cl (0CD 0. a 0. 00CD — CD COCC. CD 5 ~ CDCC 05 CC CD•CD 00 CC CDCl — — OSCD Cl
o a 0 o 0• . . • .N) 0 N) C Ni— Di CO Di Di
aN)
FFFFa Di Cii Di aa -j a a -j
a
—
— I I J IC
(CCD
0.ClCD
051
CO
I I I
aCCCD‘C.
a0aa
CCC-’
ODCD-
.~
C-fl
~CD
———..——~—-—---.-----—-.———-—--
C.
-——---~(C0CDCOoCC,
C-’
~50501CCl005
NJ I I I I*C
ICC’-“00CD
C-’CDCD
0~00IOC
C-C
O ‘1o C 0 C (CC/SO• • . 00(0 Ca /0 Di Oil 00o C-Il 10 10 CCOCD* * * * 05(05 * Cl Ct
a-Caa
aaaa
Ca0
liz
00~a-sa aa —
all-C a
a
050
a aaa
aaaz-CCa.aa.
aa
‘o a aV t~r~000
22.CCCaa
‘1
SCCC-C‘~0
CD
a
0
aCDC-CDCD
CCCClCC
CD
a.05CtIn
23.
Application of Multiple Regression Analysis in Investigating
the Relationship Between the Three Components of Attitude in
Rosenberg and Hovlands Theory for Predicting a Particular Behavior
Isadore Newman, University of Akronand
Keith McNeil, Southern Illinois University
ABSTRACT
Multiple regression and factor analysis techniques wereused to investigate the relationship between the componentsof attitude and their differential predictive power. Itwas found that the different components of attitude andthe linear interaction are more likely to be predictive forintimate rather than non—intimate behaviors. The cognitivecomponent was found to be significantly predictive of
intimate behavior but not predictive for non—intimate be-havior. Out of the three measures used, the behavioraldifferential was the most predictive scale for bothintimate and non—intimate behavior.
INTRODUCTION AND RATIONALE
Since La Pierre (1934) reviewed the attitude literature, there
have been numerous efforts to demonstrate that attitude questionnaires
can predict observable behavior. One major criti’cism of the La Pierre
review, and others such as Ruthner (1952), was that an unidimensional
definition of attitude was used.
Rosenberg and Hovland (1960) presented a theory that an attitude
consists of at least three aspectsC cognitive, affective and response
disposition. The purpose of this investigation was to study the rela-
tionship of these components in the prediction of a particular behavior.
The attitude chosen to be studied was racial prejudice.—a.5-.rC1000a sCmCO.0CCCOODi (C05
25.
24.
The theoretical model that is the basis of the three component
theory of attitude infers that an individual interprets and gives
meaning to a stimulus in reference to three aspects (factors);
cognitive, affective an/C response disposition. These three dimensions
are likely to interact with each other and take on differential weights
in producing an individual’s response. These weights should be thought
of as being determined by the particular stimulus and the context in
which the stimulus is presented. This model represents the position
that a stimulus only acquires meaning through the individual’s inter-
pretation and that these three components may have different relation-
ships for different stimuli.
METHOD
~ The Ss were 308 students from Southern Illinois University.
Since 10% of the posulatiom of students at Southern Illinois University
is black, the sample was chosen so that it would contain approximately
the same racial proportions.
~ An attitude questionnaire was designed to
measure the three components as defined. The definitions used in con-
structing the scales were:
£2dOALy!.V2nsonen~ consists of such things as thinking,
perceiving, remembering and the beliefs that a person
holds towards an object; ±ncluding stereotypes.
deals with the likes and dislikes
a person has towards an object. Included would be
his evaluation of an object and his emotional feelings
towards that object.
response disposition: consists of all behavioral dis-
positions associated with the attitude. This component
is usually operationally defined in terms of a social
distance scale or a behavioral differential scale.
Three scales were constructed to measure these components. The
Subjective Perception Rating Scale (SPRS) was used to measure the
cognitive dimension. Measurement was then based upon the subjective
rating of items by Ss in the following examples:
The percent of white Americans who are exploiting blacks
iso 0%, 5¾, 10¾, 20¾, 25¾, 30¾, 35%, 407,, 457 100%.
The percent of blacks who are in favor of intermarriage
between whites and blacks is: 0%, 5%, 10%, 15%. .100%.
The above scales were constructed to measure the social perception of
the Is responding to it. The affective component was operationally
defined by seven semantic differential (SD) scales, employing bipolar
adjectives which loaded high on the evaluative factor of the SD.
Osgood, Suci and Tannenbaum (1957) presented evidence that the evaluative
component of the SD is a measure0
f IlAttitude.CI Williams and Robinson
(1967) presented evidence that the evaluative factor of the SD was
capable of assessing racial Ilattitudesll in children. The evaluative
factor of the SD is very similar to what has been defined as the
affective component of an attitude.
The response disposition of an attitude was operationally defined
by the use of four behavioral differential (SD) scales (Triandis, 1964).
Ostrom (1969) suggested that such a scale may be the most sensitive in
measuring- the response disposition component of an attitude (for a
more detailed description of the scales and the rationale for their
selection, see Newman, 1971).
26.27.
The SPRS, SD and BD scales comprised the attitude questionnaire
used in this study. The scales were thgn factor analyzed to make
sure they were tapping separate components. The results of the
factor analysis confirmed the belief that the three scales were
measuring separate dimensions (see Tables 1 and 2).
The instrument used in the study consisted of two parts. Part I
contained eight semantic differential scales, four behavioral differ-
ential scales, and a subjective perception rating scale. Part II,
which was administered exactly one week later, consisted of three
separate conditions. In Condition I, one third of the Ss were randomly
chosen to receive an article entitled, “Militants Aren’t the Brave
Blacks,” and were told that the author of the article was a prominent
white statesman. After reading the article, the Ss were asked to rate
the author on his fairness, whether or not they would elect him to
political office, if they would want him as a roommate, etc. Another
one third of the is were randomly chosen for Condition II. This
condition was exactly the sane as Condition I, except that the author
was proported to be a prominent black statesman. The final one third
of the Ss were given Condition lit, which differed only in that the
Ss were given no information concerning the author’s race (see Newman,
1971, for a more detailed description of the scales used).
The ratings of the author of the article were factor analyzed
using a principle component solution with l’s in the diagonal, a
varimax rotation, which had an arbitrary cut off point of an eigen-
value ~ 1 (Nummally, 1967). Pactor scores were computed for each S.
RESULTS
The ratings of the author of the article were factor anlayzed
and resulted in a two factor solution. Pactor I, Political—Evaluation
accounted for 23% of the trace and Factor II, Intimate—Social
Response Tendency (‘f2
) accounted for 21% of the trace (see Table 3).
Twenty—two regression equations were calculated, eleven for each
of these criterion, and are presented in Table 4. It was found that
Model 1, using all available information —— knowledge of author’s
race, the Ss’ factor scores on the semantic differential concepts,
on the behavioral differential concepts and on the subjective per-
ception rating scale, and the linear interaction between all of these
variables —— was found to be significant at p-. .00004, accounting for
11¾ of the criterion variance of ‘fl However, the same variables,
when used to predict the second criterion, ‘f2 (Model 12), was found
to be significant at p~.OOOOl, accounting for 25% of the variance
(see Tables 5 and 6).
It was found that knowledge of race did not account for a
significant amount of variance in predicting Criterion 1, but was
significant in predicting Criterion 2. The interaction between the
components of attitudes was found to be nonsignificant in predicting
Criterion 1, p-..6
34
, while the interaction in predicting Criterion 2
just missed being significant, p=.OS6. It was also found that the
SPRS accounted for a significant amount of variance in prediction
‘f2 (p<.0Ol) above the other variable of Model 17, but found to be
nonsignificant in predicting Y. The behavioral differential scale
28.
29.
was found to be tie single best independent predictor for Crcierion 1
and Criterion 2. The results of these and other questions are
presented in Tables 5 and 6.
SUSDIARY AND DISCUSSION
The surpose of this study was to investigate the predictive
power of an attitude questionnaire which was constructed on the basis
of Rosenberg and Hovland’s (1960) three component theory of attitude,
and to investigate some of the relationships between these components.
The behavior predicted was an independent rating of an unnamed author
whose one—page article was road by all Ss. The ratings of the author
were factor analyzed, producing a two factor solution. The factor
solution was used to obtain two factor scores, ‘fi and
Eleven regression models were calculated to predict each
criterion. Model 1 was capable of accounting for ill of the variance,
which was significant. The component that accounted for the most
independent amount of variance was response disposition (see Table 4).
Model 12 accounted for 25.1% of the variance, which was signif-
icant (pi.00001) in predicting Y2
- It was also found that in both
cases response disposition (behavioral differential) was better able
to predict the two criteria than the other two components.
In predicting 12. knowledge of the author’s race was found to
account for a significant amount of the variance, however this
information was not found to be significant in predicting Y1
. The
cognitive component accounted for 5.9% of the variance in predicting
“2~ but was nonsignificant in predicting Y1
.
Factor II (Intimate—Social Response Tendency), which was
criterion Y2
, was more predictable and consistent with the Sa’
responses to the rating of the author than was Factor I
Political Evaluation). A possible explanation of this outcome is
that there are different prejudices on some continuum of intimacy.
It is likely that the less intimate prejudices are more susceptible
to social pressure than the more intimate types of prejudice.
In general, it was found that the components of attitude
differentially predicted behavior that may be classified as evaluative
behavior and intimate behavior. It was also clearly demonstrated
that multiple regression analysis has the desired flexibility to.
determine complex functional relationships.
This study confined itself to additive and multiplicative
linear relationships. Another area of investigation is the nonlinear
relationships between components of attitude and their predictive
ability. For example, in addition to investigating the linear
component of affect, one may be interested in looking at affect2
or
affect3
(the authors of this paper are now in the process of analyzing
such data).
One major limitation of this study was that another questionnaire
was used as the criterion behavior, rather than observation of actual
behavior, and any inferences made from this study must keep this in
mind.
30.
31.
22 Regressions Models Used In This Study
Model 1 Y1
= aou+alxl-l-a2
x2
±a3
x3
+a/Ix4
+olxS+56
x6
+a7
x7
÷a8
x8
÷a9
xg÷E -
Model 2 ‘fi aou+alxl+a2
x2
+a3
x3
+a4
x4
+a5
x5
+a6
x6
+s7
x7
+a8
x8
+52
Model 3 Y a0
u+a1
x1
+E3
Model 4 Y1
-a0
u+a2
x2
+74
Model 5 Y1
a0
u+a3
x3
+E5
Model 6 Y1
— a0
u+a4
x4
+a5
x5
+a6
x,+a7
x7
+a8
x8
+ a9
xg+E6
Model 7 Y1
= a0
u+a5
x5
+a6
56
-fa7X7
+ a8
x5
+a9
x9
+57
Model 8 — aou+aqx4
+a6
xg+a7
x7
+a8
x8
+agx9
+Eg
Model 9 ‘fi. = aou+a4
X4
+a5
X5
+a7
X7
+a8
X8
+a9
x9
+Eg
Model 10 Y1
= aou+aixi+aSxS+o,x6
+a6
x5
+a9
x9
+E10
Model 11 Y1
o0
u+a1
x1
+ +a7
x7
+agxg+E11
Model 12 Y2
— aou+alxl+a2
x2
+a3
x3
+a4
x4
+aSxS+a6
x6
+a7
x7
+aSXS+a9
x9
÷E
Model 13 Y2
00
0+alxl+a2x2+a3
x3
+a4
x4
+aSxS+a6
x6
+s7
x7
÷aBxS+Fl3
Model 14 ‘~‘2 a0
u+a1
x1
+E14
Model 15 Y2 — aou+a2x2+El5
Model 16 Y2
= a0
u+a3
x3
+E16
Model 17 ‘~‘2 aou+aqX4
+a5
x5
+a6
x5
+a7
x7
+a8
x8
+a9
x9
+E17
Model 18 Y2
= aoCC+a5
x5
+agx6
+a7
x7
+a8
x8
+a9
xg÷E18
Model ~ = a0
u+a4
x4
+a6
x6
+a7
x7
+a8
x8
+a9
x+E
Model 20 Y2
au+a4
x4
+a5
x5
+07
x7
+a8
x8
+ax+E
Model 21 ‘f2 — a0
U+a4
x4
+a5
x5
+a6
x6
+a8
x8
+a9
x9
+E2
Model 22 ‘~‘2 a0
u+a4
x4
+ax5
+o6
x6
+a7
x7
±a9
x+E
Model 99 The nodel that accounts for zero variance
TABLE (‘3
Varimax Factor Solution of the II ond SD ScalesRotluc the AC:thor of the Article
TABLE #4
IPolitical—Evaluation
IIIctimate—lociul
ResponseTendoocv
1. Invite this person to my hone 662. Defend his rights if they were jeopardized 483. Admire the ideas of this person 73
4. Exclude from ~ny ncighbcrhc-od 665. Take person into hone if a riot victim 62
6. Participate in a discussion with 577. Want as a member of my church 578. Elect this person to o political office 709. Accept as a close kin by marriage 69
10. Want my child to go to school with ‘ 6911. Be alone with this person 6512. Want as a roommate 42 5213. Fair — Unfair 8414. Worthless — Valuable 77
15. Good — Bad 8216. Far — Near17. Boring — Interesting 5218. Unfamiliar — Familiar
19. Believable — Unbelievable 5120. Important — Unimportant 6321. Superficial — Profound 52
NOTE: Only factor loading of an .40 and above have been reportedand decimal points have been omitted. Factor I, which
accounted for 23% of the trace was used to obtain thecriterion factor scores (Y
1). Factor III which accounted
for 21% of the trace was used to obtain the criterion factor
scores (Y2
) -
(cont.)
32.33.
TABLE 84 (coOt.)
;,~era: y1
Tnelutriu~iplefac”recoce_~fl,eaul}Iii I a mw a .0 HR -5
the article. — — I — —
— u, Ii Ii II C-I C-) Ci II 0 C/Cl a a II +~
= The 2nd principle factor scores of the author of ‘~ ~‘ °n 0 0 Ci (4 0 00 0 Cithe article. ~ ~. ~‘ ~ ~ ~‘ ~8 ~‘ ~ a
o 0 5.1 0 0 Di 0 0 0 — 0 I-’ 0 0 + ~-‘ 00 11 0 CC 5 (4 Di 110 Ii OaiC 0. 0/C
x1
liftheacthcrofthearticlewasidcntifiedas “~ ~ ~ ~
black, 0 otherwise. m ~“ ~ : ~ °
te~u s of toe ride5-
srenti as ~o 0Chita, I ooer,’ise. “ ~
* — Si cC ± II ‘0 L.I CC
0 — C-I +15 tO CC 5+ cCx = 1 if no information was given about the race of the ~-
author, 0 otherwise. :~ a- r, C-i,+1’- I: +0 C-I CC CC
x4
= factor scoreu over the S.D. concept of federal
enforcement of open housing, NAACP, bussing, white -~ 0’ aand black civil rights activists, of the Ss who took +
Part II, reading the article and rating the ast:or.
x = factor scores on the S.D. scale over the concools ofblack Presidents and the Slack Panthers, for those Ss ~. ~
who took Part II. -----. —-———- __________ _____________ a
x = factor scores of those is on the SPRS who took Part II6 of the quesliennaire. ~c ~ ~, ~
1.1 Cl ‘I CC 0. o‘S IC Cl (I (C I-’
x7
~actor scores on the ED scale over toe concepts of 00 C’ C, 00 e tO
black and white persons who favor civil rights, for ~. ~. ~. 4.those Ss who took Part II.
x = factor scores on the ED scales over the concepts of . . . . . . . . . . 08 black and white persons who oppose civil rights, for ~ g ~ ,~ tO
O — 0 H 0 Dithose who took Part II. Is
x = (x4
*x5
*x6
*x7
*x8
) interaction between the components
of attitude. 80 Di 0 0 0 Cl
u Unit vector. 0 0
‘5
K1
through E22
— Error terms for Model 1 through Model 22, 0
respectively. 05
a1
— Partial regression weight.
0 0 a0- 0
0”
TABLE #6
Models, F—Ratings and R2
For Predicting The Ratings Of The Author (Y2
)
Models Models R2
df F P
Model 12 Y2
a0
u±5ixl±alx23
ml~4
x4
’~S5S+0656 Full .251
+a7
x7
+a8
x8
±a9
x9
8~E12
Restriction: a1
”a2
=a3
=aq=a5
a6
a7
’ 8/300 102.9 .00001
Model 90 ‘fl” a0
u±E0
Restricted .000
F del 12 Y~= a0
u +a1
x1
± agxg+R12
Full .251
Rcetnictilln: a~”O(Interaction) 1/300 3.66 .056
1’ dcl 3 Y9
“ 50
u+u1
x1
+a5
a2
±08
x3
+E13
Restricted .242
P:ldel 14 P2
a0
u+a1
x1
±E12
FCC11 .021
Resrrfttinn: s—O (Black) 1/307 6.5 .011
Model 99 Y.~ a0
u±E0
Restricted .000
.__
NIdel 15 Y~ - 50
11+a2
x2
±E15
Foil .025
Ilmettoicrion. 512_O (White) 1/307 7.88 .005
Mode; 99 V., — a~u+E0
Restricted .000
Md,:cl 16 Yn - a0
u±a3
x3
±E16
Full - .003
/S31~Q ~ ~ 1/307 .1 ~
(cont - )
TABLE 8 5 (coot.)
Model 6 ‘f1 a0
u±a4
xq+s5
x5
+a5
x6
+a7
x7
-C-a8
x8
+agx9
+E6
Restrictier’: 54=0 (affect1
)
Model 7 Y1
= a0
u+a5
x5
±a6
x6
+a7
x7
+a8
x8
±a9
x9
±E7
Full
Restricted
.099
.089
1/303 3.119 .078
Model 6 Y1
au+a4
x4
+ a9
x9
+E6
‘
Restrictient a5
—0 (affect2
)
Model B Y1
au+a4
x4
+a6
x6
+a7
x7
+a8
x8
±a9
x9
+R8
Full
Restricted
.099
.098
1/303 .243
•
.622
Model 6 Y~ a0
u +54
x4
+ agxg±E6
-
Restriction: a6
=0 (cognitive)
Model 9 Y~- a0
u+a4
x4
±a5
x5
+a7
x7
+a8
x8
+a9
x9
+Eg
Full
Restricted
.099
.098
1/303 .045 .831
Model 6 ‘f1 a0
s~a4
X4
+ a9
xg+Eg
Restriction: a_=0 (response disposition)
Model 10 ‘fi= a0
u+a1
s:4
+a5
x5
+a6
x6
+a8
x8
+agxg+R10
Full
Restricted
.099
.022
1/303 25.713 .001
Model 6 Y1
= a0
0+a4
x4
±9
x9
+E6
Full .099
Restriction: a8
0 (response disposition2
) 1/303 .614 .433
Model 11 Y1
” a0
u+a4
x4
+a5
x5
+a6
x6
+a7
x7
+a9
x9
+E1
, Restricted .097
NOTE: The probability values (P) that are reported are for a two tail test of significance
(see Table #4 for description of variables).
36.37.
*401~toC..I’,
0. 1/.CD ID
.5 tO .5NJ 15 I/i
CC C-’ CCo l~C 0
~. -~.o 0 CC5 -C--
CC -- CCa- ~-+ +
CC CC -5cC-a -
IC II -ci, 0 -+ -CI .. CC
Os a so1-4 II CS+05 +CC0 SC
1) Di
+CC C,
DC S
+ 5COO
Ni ‘5
N a0 00. CC.0 5
-4 a -C(.1 is Ni
‘SCC I-. CC CCo Il 10 0
C 0 ± CC0 1.C-C-
_+ Di +CCCC (I
>0L~O 1-CC— CCa a Os1115 Cl+A °
00 CC5 CC
+ ±CC C, CC
‘0 5 5550 >1
ICC 0. 55±5 +“Cs
-C
±CCa-
14
+
CC
±SI
N 040o 0
0 Di
1-C 11 -.01.1 15 I-i
CC 0. CCo (1 0
;~_ ~, ~-CC 0 CC
0— 10 -N-- 1<
+ ±0 CC -
CC C -S~
CC
110(4Di o 0
4- Is +0 0 Cl
+ -4CC S
04
+:5
0
N00.
.SR
C-CCC Ci
o ‘S
a- SC-I
±CC_a
CC II
CC CC
± ~‘1CC 11
±.—
CC
DC
±C.,
a a0. 0.
.550 ‘.0
Ni 15 I-i
C, C-
CC — 5o (1 e
i— ¾-CC 0 CC
Iii 0 0-:4 .-
L~ .1-+ ±CC CC CC
CC II CCa 0 5.’+ ±
CC CCCC CC CC+
CC 15 DI
CCC’:-;
+ ‘-.4-a CC
DC IC‘0~ ~C5 CC
0 Cl
±
a0*4C.,
0*1*0*
~01-1*1C/ia0.0-“CC0* CD~
0
CC0* —
0C- Is‘DCC
C-C 05
•15
0.
CC-C-’, C-I
ID
a ‘,(vs-Co*111
01 0.a—CC‘DC’SCm
0C,
CC
DO0
CC
CD
Is0
F’(CCCSCD
05 “CID 0
C’ —Cl
(C
S0,
05 “C15 0
C,Ci
(C
~0.
05 “C1> 0
C, —IC
Cl
is0.
tO “C15 0
C, —C,
(I
50.
05 ‘C5 CDC, H‘I
‘1
50.
Ni Li0 Ni
55 0 Di Li NJ 0 5.) 00 Ni Li Ni Os Ni I--’ Ni
0 0 Li0Li
0Li
10 0 Di 55 C-il
REFERENCES
Goldschmid, ML. (Ed.)1970 Black americans and white racism, theory and research.
New York: Holt, Rinehart, and Winston.
Green, B.1965 Attitude measurement. In G. Lindzey (ed.) Handbook of
social psychology. Los Angeles: Addison—Wesley.
Grigg, A.1959 A validity study of the SD technique. Journal of
Clinical Psychology, 15, 179—181.
Isaac, Paul D.1970 Linear regression, structural, relations, and measurement
error. Psychological Bulletj~74
, 3, 213—218.
Jackson, D.N. and Messick, S. (Eds.)
1967 Problems in human assessment. New York: McGraw—Hill.
Kelly, P.2., Beggs, D.L. , McNeil, E.A., Eichelberger, T. , and Lyon, 3.1969 Research design in the behavioral sciences: Multiple
regression approach. Carbondale, Illinois: SouthernIllinois University Press.
Kelly, GA.1958 Man’s construction of his alternatives. In Gardner
Linzey (ed.). Assessment of human motivation. New York:
Grover Press.
Krech, D. , Crutchfield, R.S., and Ballachey, EL.1962 Individuals in society. New York: McGraw—Hill, Ch. 5,6,
and 7.
Kuthner, B.1952 Verbal attitudes and overt behavior involving racial
prejudice. Journal of Abnormal and Social PsyChQ~gy, 41,649—652-
La Pierre, R.T.
1943 Attitudes vs. action, Social Forces. 13, 203—247.
Likert, R.1932 A technique for the measurement of attitudes.
of~y~lo, 140.
Lion, L.S.1965 Verbal attitudes and overt behavior: A study of racial
discrimination. Social Forces, 1965, 43, 334—364.
00 0
a
Mann, J.H.1959 The relationship between cognitive, affective, and
behavioral aspects of racial prejudice. Journal of
Social Psychology, 49, 223—228.
McGuire, W,J.1969 The nature of attitudes and attitude change. In
G, Lindzey and E. Aronson (Edsj. Handbook of Social2gygjyj
5°gy~ Los Angeles: Addison—Wesley.
Newman, I.1971 A multivariate approach to the construciton of an
attitude battery. Unpublished doctoral dissertation,Southern Illinois University,
Osgood, C.E., Suci, G.J., and Tannenbaum. P.H,1967 The measurement of meaning. Chicago: University of
Illinois Press.
Rosenberg, M.J. and Hovland, C.I.l96D Cognitive, affective and behavioral components of
attitudes. In CI. Rowland and M.J. Rosenberg (Eds.)Attitude organization and change, New Haven, YaleUniversity Press.
Rummel, R,J.1970 Applied factor analysis. Evanston: Northwestern
University Press.
Scott, W.A,1969 Attitude neasurement. In G. Lindzey and E. Aronson
(Edo,) Handbook of social psychology. Los Angeles:
Addison—Wesley.
38.
Triamdis, HG.1961 A note on Rokeach’s theory of prejudice. Journal of
Abnormal and Social Psychology, 62, 184—186,
Triandis, HG, and Davis, E.1965 Race and belief as determinants of behavioral intentions.
Journal of Personality and Social Psychology, 2, 715—725.
Veldnan, D,J.1967 FORTRANprogramming for the behavioral sciences, New
York: Holt, Rinehart, and Winston.
Williams, JR. and Roberson, JR.1967 A method for assessing racial attitudes in preschool
children. Educational and Psychological Measurement, 27,671—689.
Smith, CR,, Williams, 1. and Willis, R.H.1970 Race, sex and belief as determinants of friendship and
acceptance. In M,L, Goldschmid (Ed.), Black americans
and white racism~ theory and research. New York: Holt,Rinehart, and Winston, 327—328.
Suci, G.J.1952 A multidimensional analysis of social attitudes with
special reference to ethnocentrism. Unpublished doctoraldissertation. University of Illinois.
Triandis, H.C.1964 Exploratory factor analysis of the behavioral component
of social attitude. Journal of Abnormal and Social
~9yg)Thlggy, 68, 420—430.
39.
40.
r-~--DvNar~.,Jr.
Air Force kasanResourcesLaboratoryBrooks Air ForceBase, Texas
(BitE input from Earl Jenningsand Bob Bottenherg)
The follosing items sight be considered for inclusion in a report of
regressionanalyses:
Title: Name of Analysis
1. General Coniaents
this section includes general information about the data and theanalyses. (e.g. description of sample, population, number of observa-tions) This can be whatever seems appropriate to the writer.
2. Regression Analysis Discussion
This section can include (1) natural language statements of thehypotheses, (2) identification of the assumed model, (3) hypothesesin terms of assunedmodel, (4) identification of the restricted model,and (5) results of the test. The numberingwithin this section (2.1,2.2, . . .) should correspond to the model comparison in 4.2 below.
3. Vector Definitions
Vector Number Definit ion
p
4. Analyses
4.1 5~del Specification and nuasaryof Results
D~del Criterion PredictorNumber Vector(Y) Vectors SSE R
2B Ply P145 SEST
40A
4.2 Model Comparisons
Comparison
Nusber Assumed Restricted R2
a ~ NIVA NIVR DFJ DF2 F P
5. RegressionComputer Output
Contains detailed computeroutput of Models and F-Tests if appropriate
for reporting.
Notation:
SSL = Sum of Squaresof Error Vector
B2
Squaredmultiple correlation coefficient
B Multiple Correlation Coefficient
NOV Number of linearly independentvectors in the predictor vectors(See Ward and Jennings - Introduction to Linear Models, Co 5, p 77)
ISIS Error Mean Square SSE~noectors-
SEaT Standarderror of estimate=
R2
a Squaredmultiple correlation for assumed (or full) model
R2
r Squaredmultiple correlation for restricted model
NIVA Number of linearly independentvectors in the assumedmodelpredictor vectors
NIVR = Msiiber of linearly independentvectors in the restricted modelpredictor vectors
DF1 = NOVA - NIVR
DF2 Dimension of Vectors (i.e. number of observations) - NOVA
F = F - statistic
P = Probability
41.
.xampic oi ~uidclines fur Reporting RegressionAnalyses
Analysis of Problems From Chapters4 and 6 of Ward and Jennings
1. General Consents
The data are artificial, representing (N = 20) observations oftyping - performance on studentswho are describedas freshman,sophomore,junior, or senior. Seep 58-59 of Ward and Jennings,Introduction to Linear Models.
2. RegressionAnalysis Discussion
2.1 (1) Is it appropriateto say that the levels of typing
performancefor freshman,sophomores,juniors, and seniors areequal?
(2) The assumedmodel is
a1
X(2) + a2
x(3)+ 53 x(4) * a4
x(5) + E(1
)
(3) Thehypothesis is
Ii (fr) E (soph) Ii (jr) E (sr)or
a2
— a3
= 54 ac
(4) The restricted model is
= a~U *
(5) The result of the test (see Section 4.9) indicates thatthere is a statistically significant difference (p <.0006)betweenthesefour groups.
2.2 (1) Is the amount of changein typing performancefor eachyear changein grade level constantfor all grade levels?
(2) The assumedmodel is:
x(1) a1
X(2) + 52 x(3) + 53 x(4) * a4
~(S) * ~(l)
(3) The hypothesisis:
a~- a1
=a3
- a2
a4
-a3
w1or
defining al = w0
* ~1 then the hypothesis is
*
= + l0w~w
0* 11w
134 w
0+ 12w
1
41 .A
(4) The restricted model is:
x11) w0
u * w1
x~6~* ~(3)
(5) The result of the test (seeSection 6.7) indicates that thehypothesis is reasonable.
3. Vector Definitions
tector Number ___________
6
Definitions
Typing performancein words/mm.1 if student is freshman1 if student is sophomore1 if student is junior1 if student is seniorgradeof st~1ent (9, 10, 11, 12)
4.
8.1 Model Specification and Sussnaryof Results
Model Criterion Predictor
Number Vector(Y) Vectors SSE R2
1 1 2 5 1996.8 .6554
2 1 U 5795.2 0
3 1 U,6 2000.6 .6548
4.2 Model Comparisons
Comparison Assumed Restricted 2 2Model Model Model Rp ~ NOVA Nll1( DF1 DF2 F P
1 1 2 .6554 0 4 1 3 16 10.1 .0006
2 1 3 .6554 .6548 4 2 2 16 .015 .9847
S. RegressionComputer Output
Results of detailed computer outputs (see p. 263 of Ward and Jennings).
Reactions to Wand’s “Guioelines for Reporting RegressionAnalyses,” and Some Alternatives
Keith ScNeil
Wand’s proposedguidelines needdiscussion by DIG members in a number of
places
(1) There is not enough emphasis upon the statement of the question tie
researcher wants to establish, and the statistical hypothesis employed to test
that suestion.
(2) there is extraneousregression information, which is rot desired by
most researchers.
(3) Do allowance is madefor alpha, and the decision regarding hypotheses
is non given enough play=-the guidelines makeregression important for its own
sake (rightfully so for SIG members, but not for common researchers) rather than as
a too: for answering the res0
archer’s5
uestion.
(N) The encouragementof a ‘natural languagestatement,” one that the
researcher must state in his own language is welcomed, but tie statement is
nothing more than a “nall oypothesis ,“ which is usually not what the researcher
is wanting to establish, the following guidelines I propose include both a
research and a statistical hynothesis. (Those concerned about directional
hypothesis testing realize that the sanestatistical (null) hypothesis serves
both the directional and non-directional research hypotneses.)
(5) Under Model Speoificanions, n.e criterion vector is referred to as
when in fact in is an “X”. DDE, R, MDV, EMS, SEST are all, with the possible
exception of SEST, not usually of interest to researchers.
(6) Coder Model Comparisons,DIVA and NIVR are excess information.
R NW~1S
.8096 4 124.80
0 1 305.01
.8092 2 111.15
501ST
11.17
17.5
10. 54
4~A 42.
(4) The restricted model is:
x(1
) u x~6~
* ~(3)
(5) The result of the test (see Section 5,7) indicates that the
hypothesis is reasonable.3. Vector Definitions
Vector Number ____________
3
6
Definitions
Typing performance in words/mm.1 if student is freshman1 if student is sophomoreI if student is junior1 if student is seniorgrade of student (9, 10, 11, 12)
4.
4,1 Model Specification and Suiaeary of Results
Model Criterion Predictor
Number Vector(Y) Vectors SSE R2
R
1 1 2,..., S 1996.8 .6554 .8096
2 1 U 5795.2 0 0
3 1 U,6 2000.6 .6548 .8092
4.2 ~oarjsons
Comparison Assumed Restricted 2 2Model Model Model ~_ ~,&_ Nfl/A
1 1 2 .6554 0 4
2 1 3 .6554 .6548 4
5. Regression Computer Output
Results of detailed computer outputs (see p. 263 of Ward and Jennings).
Reactions to Ward~s “Guidelines for Reporting Regression
Analyses,” and lime Alternatives
beith McNeil
Ward’s proposedguidelines needdiscussion by SIG members in a number of
(1) There is not enougnemphasis upon tie slitement of the question the
researonerwants to establisn, and the statiStioal tsyrothesis eftployed to test
that question.
(2) jhsre is extraneousregression information, whicO is not desired cv
most researchers.
(3) No allowance is madefor alpha, and the decision regarding hypotheses
is not given enoughplay~-the guidelines makeregression important for its own
sake (rightfully so for SIG members, but not for coimson researchers) rather than as
tool for answwring the researcher’s question.
(L,) The encouragementof a natural language statement,” one tnat the
researchermust state in his own language is aelcomed,but the statement is
nothing more than a “null hypothesIs,’ which is usually not what the researcher
is wanting to establish, the following guidelines I propose include both a
research and a statistical hypothesis. (Those concernedabout directional
hypothesis testing realize that the sane statistical (null) hypothesis serves
both the directional and non-directional research hypotheses.)
(5) Under Model Specificacions, toe criterion vector is referred to as
when in fact it is an “X”. SE, R, NIV, EMS, SEST are all, with the possible
exception of SEST, not usually of interest to researchers.
(5) Under Model Comparisons, NIVA and NIVR are excess information.
NW R4S
4 124.80
1 305.01
2 111.15
SF.ST
11.17
17.5
10.54
NIVR DF1 DF2 F P
1 3 16 10.1 .0006
2 2 16 .015 .9847
43.
Suggested Guidelines for Reporting Regression Analyses
Statement of the research hypothesis — that whici the researcher .s hoping to
support.
Statement of the statistical ~Ipq~esis.
Statement of - the risk (probanility) the researcher is willing to make in
rejecting a true statistical hypothesis.Formulation of the f..ll model — all variables must be InpIie~ unasbiguously by
the research hypothesis.
Statement of the restrictions implied by the statistical hypothesis.
Formulation of the restricted model - reflecting the statistical hjpothesis.
Definition of the vectors.
Reporting of the probability (p) of calculateo F occurring by chance alone andcomparisonof that p with the preset alpha level, in order for the researcherto makea decision:
I. If p~ alpha, then reject statistical hypothesis and accept researchhypothesis.
2. If p > alpha, fail to reject statistical hypothesis and fail to acceptresearch hypothesis.
An xample Following the Above Guidelines
DIrectional Research Hypothesis: For some poPula’-Ion, Method A Is better ncr.Method B on the criterion Y
1.
StatistIcal Hypothesis: For some population, Method A and Method B are equallyeffective on the criterion Y
1.
lull Model: Y1
c a5
U t a1
G1
± + —-
Iesrlctions: ~l °
Restricted Model: “ a0
U m
where: = uritenion
U = Ifor all subjects;
01 1 if subject In Method A, zero otherwise;
02 = 1 if subject in Method B, zero otherwise; and
a0
, a1
, and a2
are least squares weighting coefficients calculated
so as to minimize the sum of the squared values in the error vectors,
and E2
.
F= 222 p<.000l
Decision: Since the weight a1
‘> a2
as hynothesized and p L. alpha, reject thestatistical hypothesis and hold as tenable the research hypothesis.
45.
A Revised Suggested Format for the Presentation
of Multtole Regression Analysis
Isadore Newman
University of Akron
In an earlier issue I suggested a format for presenting
the results of multiple regression analysis. Since then,
a committee, chaired by Joe Ward, was apnointed by the Puitiple
Reeression Soecial luterest Grout. At the last meeting in
New Orleans, Ward discussed his suggested guide lines, Selth
YcNeil has also made suggestions for the oresentation of results
of multiple regression analysis.
I have since revised my origional format and I am now
Presenting it, All of these suggestions should be considered,
I believe it is imoortant to have a standard format which
will reduce some ambiguity regarding the symbols used and
the Interpretation of multiole regression tables. This,
believe, will enhance our ability to promote further use of
multiple regression through better communicating the results
in the most concise and easily interoretable form.
Si 5o oa. a.a ae~
C~0 9
C-C 5<
cD~ 9
5 50 0
Di ci+ +5 5
SM 9C1’< 5<
+
+~
Si~<
+.3
5<SM
+
Si
x iiio 0 w0. 0- -oa a o— D~ Ct
0~— 4D~ a
aCC C-C
9’ 0. 5
.3 5 ‘-C0 0 --
C C+ + 5a~e5 5 owo
C-_C’ CDC5< 5< Ceo
— SM C~C-5
D+ + C)D~C-a a oa.a
C’) -~ C C5< 5< 595
9 -~ Ca—+ + -CamC-C 5 aSa
~o SM maP-C 5< 3m~—
C--.) SM Ca+ + 0S C
3
-5- -0 mmm5< 5< awe-,
-5~ sO CD a+ + seeS ~C Ct~5
SM -~ OnP-< 0 CC C-C
SM 5< PC-C-Ct— 000
+ 0as
Si + sa
CC-i -CCC-,
5C-C90CC-9a5
S
Cs
‘-<-CCOO.‘——ao -~~t
_a ~a00
as Cmm—o am
C--C- 005<O Ca a -~C-,’-:00-C’SeaC’ S
55ssHC-o C-~ acas‘-<5a a.
5—9 aCD ama
Ciamoa.
-DC Si CDo 0 ‘Ca. a.CD 0 0— C--’ C-,
9C-3M C5<5<
3sa~ aU 5 Ct
0CD C
0 + 9a0~-39 5 C<C-<C-<Q+ —. 5005 ~ eea~o
9’ — ‘-<aP-C + wasa
a’ S 99Cm+ ~C-C Sa C’5 5< 0 55
~‘) Ct5C-’5< + 0’-650
5- 5 0.3 0+ -9 CO-s5 5< .DC-C-C-
‘a -~.3 Ca-C--C5< + 5<-O
SM .3 -‘OCC-,+ 5- mm-Pm-
P-c ps—aSi 5-
+ 95 ‘-<moC
SM aSCt’5< C-ts.-.~,
CM a55<~-+ SC--. ISi ~t- O.m
a C CDS a a p.a w -~o ~,m S0 0aa500• OCt
0<00Si
aDi
Si
I-C
C-C
0SiCCC
Si0a0
CCCaCCC
-359Ct0
a0-U
C-C
a.-C
C)‘1
0C
0aa90
-15‘U
a9C--’aS
Caaa.
46.
PC
0
S0a-a
C-a
a00
-t--C
47.
TABLE II
THE COPPLETE REGRESSION NODEL S
WHICH REFLECTS THE ENPIRICALLY TESTED FUNCTIONAL REi~TIONSHIPS
= a~U+a1
X1
+a_X2
= a3
X + ... a10
X1~
+ E
where:
= the criterion, posttest score in reading comprehension
a0
, a1
, S3~ ,,, a10
= partial regression weights;
U = the unit vector (a “1’~ for each samole);
X1
= 1 if S was in the 1-ulti-Nedia Reading Program,
zero otherwise;
X = 1 if S was in the traditional basal text reading
2 program, zero otherwise;
= 1 if S were male, zero otherwise;
Xa = 1 if S were female, zero otherwIse;
= pretest raw score in reading comprehensIon measuredby The Ohio Survey Test;
X — 1 if S were male and in the Fulti-Pedia Readina7 Program, zero otherwise;
= 1 if 5 were female and in the Fulti-Nedia ReadingProgram, zero otherwise;
X_ = 1 if S were male and in the traditional basal textreading program, zero otherwise;
X10
— 1 if S were female and in the traditional basaltext reading prograir, zero otherwise;
S = Error vector, difference between predicted score andactual score
Si C’)a C5 C-’a C-0
0C-C-aa.
Si Sia CS —
C-,Cj
0CC-aa-
Si Sia CS C--.C, —Cj
C)Cta0
Si000<
(IC
CDC—C
a-0<~
SM SM0 -.
~‘
5-
SM SM0 0
C’-.C
SM
SM SM0 0
—.C
SM
0C-
0SM
0SM
I’-~Sia
0
SMSO
C0’
—0 0
9) 005 9
~—-~
:~CSM
-
~
Newman, Isadore. Nulticle ~ Regression Viewpo_in~Vol. 2, No. I~5, Parch 1q72; Special Interest Grouppublication of AERA.
48.
BUSINESS MEETING NOTES
The annual business meeting of the AERA Special Interest Group
on Multiple Linear Regression was held on February 28, 1973 duringthe 1973 AERA Annual Meeting in New Orleans. 1972—73 ChairmanBill Connett presided.
Old business:
A. Joe Ward, chairman of the committee to develop guidelinesfor reporting regression analyses, reported on a suggestedformat and invited comments on it from the Viewpointsreaders.
B. Dues were collected.
N~wbusiness:
A. The meeting was turned over to 1973—74 chairman, Judy McNeil.
B. Election was held for the Office of Secretary, Chairman—elect.James Bolding of the University of Arkansas was elected.
C. The membership expressedappreciation for the years of servicegiven to the SIG by John Williams serving as editor and expresseda desire to find soother individual and institution to take overthe burden. Isadore Newmanof the University of Akron accepted
the position.
D. The membership approved a proposal to combine the responsibilitiesof Chairman and program chairman beginning with this year.
Interaction Hours
A social interaction party was held for the SIG on the evening
of February 28 in New Orleans.
Fpr
Membership:
49.
50.
Dues ($1.00) for membership in the AERAMultiple LinearRegression Special Group were due as of the New Orleans Annualmeeting (1973—1974). If you did not pay your $1 at New Orleanssend it to the new Secretary: James Bolding, EducationalFoundations, University of Arkansas, Fayetteville, Arkansas, 72701
Since the paper presentedby Keith and Judy McNeil at the AERA:SIG
session was some 30 pages long, it wifl not be reprinted in Viewpoints.
Anyone desiring a copy should write to Keith McNeil, Department of Guidance
and Educational Psycholo~, Southern Illinois University, Carbondale,
Illinois 62901,
I51.
Steve Spaner is proud to announce that his MLR symposium was acceptdd byDiv. 5 (Measurenent and Evaluation) of the APA for presentation Thursday,
August 30, 1973 from 10—12 AM at the 1973 APA Convention in Montreal,Canada. The following is the list of participants and their presentations
(abstracts are available from Steve):
The application of multiple linear regression (MLR) to research evaluation
Steven B. Spaner, University of Missouri—St. Louis, St. Louis, Mo.
Participants:
Joseph Liftik, Services for Traffic Safety, Boston, Mass. Theapplication of MLR in alcoholism diagnosis.
Jack Byrne, Westinghouse Research Laboratories, Pittsburgh, Pa.
An evaluation of first grade reading: a multiple linearregression analysis.
Judy T. McNeil and Keith A. McNeil, Southern Ill inois University,Carbondale, Ill. A regression analysis of the functionalrelationship between mother—infant physical contact and
infant development.
Isadore Newman and Gerald J. Blumenfeld, The University of Akron,Akron, Ohio. The use of multiple regression in evaluatingalternative methods of scoring multiple choice tests.
Thomas E. Jordan and Steven 0. Spaner, University of Missouri —
St. Louis, St. Louis, Mo. An AID—4 analysis of antecedentsto internal locus of control at age 5.
Samuel R. Houston and William E. Connett, University of Northern
Colorado, Greeley, Col. The use of judgment analysis incapturing student pol icies of rated teacher effectiveness.
Discussants:
Francis J. Kelly, Southern Illinois University, Carbondale, Ill.
William AdrianSpecial Assistant to the Chancellor
University of DenverDenver, Colorado 80210
PD Joel AgerPsychologyWayne State UniversityDetroit, Michigan 48202
Earl A. Alluisi, DirectorPerformanceResearchLaboratoryUniversity of LouisvilleLouisville, Kentucky 40208
PD Richard ArakakiHawaii State Dept. of EducationHonolulu, Hawaii
PD Arnold G. Ashburn1804 Sabine CourtCollege Station, Texas 77840
Sylvia Auton8 Department of Educational Mess.
and StatisticsUniversity of MarylandCollege Park, Maryland
PD Richard L. SaleInstitute for Social ResearchFlorida State UniversityTallahassee, Florida 32306
Paul Barbuto, Jr.Box 37Teachers CollegeColumbia UniversityNew York, New York 10027
PD Richard BeesonResearchMethodologySt. Louis University221 N. GrandSt. Louis, Missouri 63103
Donald L. BeggsDepartnent of Guidance & Ed. Psych.Southern Illinois UniversityCarbondale, Illinois 62901
PD Katherine Bmnis
S.W. Cooperative Ed. LaboratoryAlbuquerque, New Mexico
PD William BeusseAFHRLWright Patterson AFBMaryland
E.H. BlekkingBox 12524University StationGainesville, Florida 32601
James BoldingEducational FoundationsCollege of EducationUniversity of ArkansasFayetteville, Arkansas 72701
PD Marvin BossFaculty of EducationUniversity of OttawaOttawa, Ontario
Robert Bottenberg4014 FaworidgeSan Antonio, Texas 78229
* Michael A. BrebngerDepartment of Ed. Psych.University of CalgaryCalgary 44, AlbertaCanada
PD William K. BrookahireNorth Texas State UniversityP.O. Box 13841
Denton, texas 76203
K. Brown
Apt. 159 Villa de Palmer5811 Atlantic Blvd.
Jacksonville, Florida 32202
* Robert L. Brownlee
CTB/McGraw—HillDel Monte Research ParkMonterey, California 93940
p~ Gave BuckholtCEMFEL10646 St. Charles Rock Rd.st Ann, Mo. 63074
PD Mel BuckleyColumbia Public SchoolsP. 0, Box 550
Columbia, Mississippi 39429
Leigh Burstein744 Coleman Avenue, 4EMenlo Park, CA 94025
Dale CarlsonOffice of Program EvaluationDepartment of Education721 Capitol MallSacramento, CA 95814
Leonard S. CahenEducational Testing ServiceDivision of Psychological Studies
Princeton, NT 08540
Donald J. Cegala241 Dalton CourtTallahassee, Florida 32304
Gary J. ColesAssociate Research Scientist
P. 0. Box 113Palo Alto, CA 94302
Arnold J. ColtvetIowa Central Community College330 Avenue MFort Dodge, Iowa 50501
William E. ConnettDepartment of Research and
Statistical MethodologyUniversity of Northern Colorado
Greeley, Colorado 80631
PD Frank CompestineAIMS CollegeGreeley, Colorado 80631
* Robert J. Coldiron
Educational Research AssociateDepartment of Public Instruction
Harrisburg, PA 17126
PD John Convey
Florida State UniversityTallahassee, Florida
PD Louise CormanResearch Institute for Educational Problems
12 Maple Avenue
Cambridge, Mass. 02139
PD Hal CorsonResearch Associate
Miami Dade Junior College11011 S. W. 104 St.Miami, Florida 33156
* Paul I. Costa
Harvard UniversityDepartment of Social RelationsWilliam James HallCambridge, Mass. 02138
PD Laura R. CraneChicago Board of EducationDepartment Govt. Funded, Fm. 1130228 N. LaSalleChicago, Illinois
PD Carl CrosswhiteUniversity of No. Colorado
Greeley, Colorado 80631
PD William Diffley8 Boxwood LaneNorwich, Coon. 06360
* William DonaldsonDelaward Rd. Piney RidgePine Grove Mills, PA 16868
* Vern Dravland
Coordinator of Educational ResearchUniversity of LethbridgeLethbridge, AlbertaCanada,
* Charles G. Eberly
Office of Evaluative Services
239 S. Kedzie HallMichigan State UniversityEast Lansing, Michigan 48823
PD Tony EichelbergerLearning Research and Development Center
160 N. CraigUniversity of PittsburghPittsburgh, PA 15243
Patricia ElmoreCounseling and Testing CenterSouthern Illinois University
Carbondale, Illinois 62901
NEMBERSHIPOP SPECIAL INTEREST GROUPMULTIPLE LINEAR REGRESSION
52. I 53.
54. 55.
PD Claire B. ErshartHofstra UniversityHempstead, N.Y.
PD Avigdore FarineFaculty of EducationUniversity of B~ntreal90 Vincent D’IndyMontreal 193, Camada
PD Garrett F. Foster405 EducatiomFlorida State UniversityTallahassee, Florida 32306
Paul Games+20 Social SciencesUniversity Park
Peonsylvania 16802
Beatrice GreenEast StreetStockbridge, Massachusetts
5Robert G, Gardner1305 6th StreetGreeley, Coloradu
PD Vincent GreaveyBoston CollegeBoston, Mass.
Hugh Greenup5406 Rhea AvenueTarzana, California 91356
Willa GuptaData Analyst, UCLAEducational Preschool Language1868 Greenfield AvenueLos Angeles, Calif. 90025
Ofelia HalasaDiv. of Research and Dew.Cleveland Public SchoolsCleveland, Ohio
Robert E. HaleResearch Methodology211 N. GrandSt. Louis University
St. Louis, Mo. 63103
Irma Half terVice—Pres, Analytic Studies
DePaul University25 East Jackson Blvd.
Chicago, Illinois 60604
Ronald S. Halinski
Department of EducationIllinois State UniversityNormal, Ill. 61761
Marvin H. NalldorsonSchool of BusinesaUniversity of No. Colorado
Greeley, Colorado 80631
PD Joe B. HansenEducation Service CenterRegion 13Austin, Texas
* Joseph Harrison
Math Coordinator, Exp. im Higher Ed.13200 St. Louis Avenue
East St. Louis, Illinois 62201
* Beatrice Harris
Yeshiva University55 Fifth Avenue
New York, New York 10003
Jack R. HaynesDept. of PsychologyBox 13587, N.T. Station
Denton, Texas 76203
* James D. Hennes
Program Evaluation CenterUniversity of Missouri Medical Center201 Lewis HallColumbia, Missouri 65201
Letjtia Neil204 Briarton LaneCrystal City, Missouri 63109
Beatrice HeimerlDept. of Research and Statistical
MethodologyUniversity of No. Colorado
Greeley, Colorado 80631
PD Dennis HemAugustana CollegeSouix Falls, South Dakota 57102
PD John HemneterIndiana UniversityBloomington, Indiana
Veron L. Hendrix
221 Horton HallUniversity of MinnesotaMinneapolis, Minnesota 55455
Thomas L. HickDirector, Child Study CenterCampus SchoolState University CollegeNew Paltz, New York 12561
Vynce A. Hines1220 S.W. Nimth RoadGainesville, Florida 32601
James H. HoggeSchool of EducationBox 512George Peabody CollegeNashville, Tennessee 37200
Samuel K. Houston
Dept. of Research & StatisticalMethodology
University of No. ColoradoGreeley, Colorado 80631
Carl J. Huberty325 AderholdUniversity of Georgia
Athens, Georgia 30601
Jack I.Hoffamn4845 Jerome AvenueSkokie, Illinois 60076
Brad Huitema
Psychology Dept.Western Michigan UniversityKalamazoo, Michigan 49001
Janet Carol Hyde1048 M. Graduate HouseWest Lafayette, Indiana 47906
Earl JenningsSchool of EducationSutton Hall 6University of TexasAustin, Texas 78712
Paul JonesResearch & Development DivisionAmerican College Testing ProgramP.O. Box 168
Iowa City, Iowa 52240
Thomas F. JordanBehavioral Studies & Research
University of Missouri ‘~St. LouisSt. Louis, Missouri 63121
PD Daniel C. KauDept. of Educational Psych
University of Illinois
Urbana, Illinois 61801
Francis J. KellyDept. of Guidance & Ed. Psych.
Southern Illinois University
Carbondale, Illinois 62901
~‘ F.J. King403 Education BuildingInstitute of Human LearningFlorida State UniversityTallahassee, Florida 32306
Alan C. Klaas219 Carbondale Mobile HomesCarbondale, Ill. 62901
J.A. Klock2226 Mercer Circle So.Jacksonville, Fla. 32317
Janus B. KoplyayPersonnelResearchDivisionAirforce Human Resource LaboratoryLackland AFB, Texas 78236
PD Richard L. Kohr500 Winand DriveHarrisburg, Pa. 17109
PD Harold V. KnightDirector, Education ResearchBox 98, Southern StationUniversity of Southern Mississippi
Hattiesburg, Misa. 39401
* Patricia R. Knox
6350 N. Lake DriveMilwaukee, Wisconsin 53217
Conrad C. Krauft2371 Ora DriveFayetteville, Arkansas 72701
PD Reynold J. KruegerUniversity of IllinoisUrbana, Illinois 61801
Albert K. KurtzApartment 418Park Knowles ApartmentsWinter Park, Florida 32789
W.L. Landrum3 Chisolm St.
Charleston, South Carolina 29401
PD Rex L. Leonard
Box 5221 So. StationHsttiesburg, Miss. 39401
James 0. LindenDepartment of PsychologyPurdue UniversityLafayette, Indiana .79 7
PD Jam Lokan
Research CenterOttawa Board of Education
330 Gilnour St.Ottawa 4, Ontario Canada
Frank Many2220 Piedmont AvenueBerkeley, California 94702
R.A. MartinCollege of EducationUniversity of Toronto371 Bloor St. N.
Toronto, Canada
PD Gerald R. Martin1 TIES Project
1925 West County Rd. B—Z
St. Paul, Minn. 55113
V. Rutledge McClaran1722 W. OakDenton, Texas 76201
PD John McClureWest. Va. Department of Education
Charleston, West Virginia
John N. McConnell
1415 S. Wolf Road, a 205Wheeling, Illinois 60090
William J, McCormick
Bureau of Evaluation & ResearchState Education Building721 Capitol MallSacramento, Calif. 95614
Garnet L. McDiarmidOntario Institute fur Studies i.jEd.
Toronto, OntarioCanada
Norman S. McEachron
Room Mo—201Stanford Research Institute333 Ravensvood Avenue
Menlo Park, Calif. 94025
56.
PD Lawrence McNallyBoard of Cooperative Educational ServicesDivision of Research & Development125 Jericho TurnpikeJericho, New York 11753
PD Keith McNeilDept. of Guidance & Ed. PsychSouthern Illinois UniversityCarbondale, Illinois 62901
PD Judy McNeilDept. of Guidance & Ed. Psych.Southern Illinois UniversityCarbondale, Illinois 62901
W.V. Meredith1231 WE. 14th CourtDeerfield Beach, Florida 33441
Paul F. Merrill1A Tully BuildingFlorida State iniversityTallahassee, Florida 32306
PD Tom MuseAbt Assoc.55 W’neeler St.Cambridge, Mass. 02l3B
PD Isadore NewmanDept. of Educational FoundationsUniversity of AkronAkron, Ohio 44325
Ronald L. NuttallAssociate Professor
Institute of Human SciencesBoston College
Chestnut Hill, Mass. 02167
George H. Olson2325 West Pensacola
Apt. 103
Tallahassee, Florida 32304
PD Raleigh PegramDallas Independent Schools3700 Rose AvenueDallas, Texas
Dan N. Perkuchin
Department of SociologyBowling Green State UniversityBowling Green, Ohio 43403
Vincent J. Piraino11791 Birchwood Lane
Franklin, Wisconsin 53132
PD John T. PohlmanTesting CenterSo. Illinois UniversityCarbondale, Illinois 62901
Marjorie Powell20 Dartmouth PlaceBoston, Mass. 02116
Thomas W. PyleDept. of Psychology
Eastern Washington State CollegeCheney, Washington 99004
PD Fred PyshDept. of Ed. PsychologyUniversity of CalgaryCalgary, Alberta
Canada
Nambury S. Raju301 Plainfield RoadL+ Grange, Illinois 60525
PD Phillip RamseyDept. of PsychologyHofstra UniversityHampstead, Nen York 11550
5Nichclas F. RayderDirector of Evaluation
Far West Ed. Lab1 Garden Circle, Hotel ClaremontBerkeley, California 94705
Cheryl L. ReedRt. 9, Box 72N. Lafayette, Indiana
James A. ReynoldsRitenour Colsolidated School District2420 Woodson Road
Overland, Missouri 63114
Carolyn F. RitterComputer and Data Processing Center
Carter HallUniversity of Northern Colorado
Greeley, Colorado 80631
Emmett A. RitterEducational AdministrationMcKee 419
University of Northern ColoradoGreeley, Colorado 80631
Bruce C. Rogers
Measurement and StatisticsCollege of EducationUniversity of MarylandCollege Park, Mo. 20742
Bob Rosemier
Educational Administration & ServicesNorthern Illinois University
DeKalb, Illinois
Nolan F. Russell
4507 Berkeley StreetHarrisburg, Pennsylvania 17109
PD Gerald Schluck
4037 North MonroeTallahassee, Florida 32301
PD Terry SchurrBall State UniversityMuncie, Indiana
* Donald F. Senter
Research DirectorEducational Developmental Laboratories, Inc.
Huntington, New York 11743
* 0. Suthern Sins, Jr.Dean, Student AffairsUniversity of GeorgiaAthens, Georgia 30601
Ping Eec Sin
23 East 17th St.Apt. 1C
Brooklyn, New York 11226
PD Donald N. Smith3009 Amherst RoadMuncie, Indiana 47304
PD Steve SoanerBehavioral Studies & Research
University of Missouri — St. LouisSt. Louis, Missouri 63121
* John C. Soderstrum
P.O. Box 13677University StationGainesville, Florida 32601
Pay H. StarrDept. of PsychologySouthern Illinois UniversityEdwardsville, Illinois 62025
57.
58.
Alan D. StewartAssociate in Education Research
State Education Dept.Albany, New York 12224
* Gary C. Stock
Candler HallUniversity of GeorgiaAthens, Georgia 30601
* Eric Strohmeyer
Reid HallMontana State UniversityBozeman, Montana 59715
David E. SuddickUniversity of GeorgiaTesting & Evaluation CenterAthens, Georgia 30601
* Steve Teglovic, Jr.
School of BusinessUniversity of No. ColoradoGreeley, Colorado 80631
* Jerome mayer
Director of Testing & ResearchUnion CollegeLincoln, Nebraska 68506
Donald L. Thomas76 Loretta AvenueApartment 2Fairborn, Ohio 45324
PD Bonnie Traf tonDept. of Guidance & Educ. Psych.Southern Illinois UniversityCarbondale, Illinois 62901
PD Margaret TrikalskyUniversity of No. ColoradoGreeley, Colorado 80631
Norman Uhl407 Landerwood LaneChapel Hill, North Carolina 27514
Michael R. Vitale4551 Likini St.Honolulu, Hawaii
Timothy 0. DevaneyDenton State SchoolBox 368Denton, Texas 76202
PD Karen VroeghInstitute for Juvenile Research
1140 5. Pauline St.Chicago, Illinois 60612
PD Joe H. Ward, Jr.Southwest Educational Laboratory
167 East Arrowhead DriveSan Antonio, Texas 78228
* William B. Ware
College of EducationUniversity of FloridaGainesville, Florida 32601
PD G. Leighton Wasem512 Oak StreetChatham, Illinois 62629
* Billy—Belle Weber
605 Washington PlaceEast St. Louis, Illinois 62205
* Bill Webster
Research & Evaluation
Dallas Independent School District3700 Rosa AvenueDallas, Texas 75204
Donald WellsPsychology Dept.University of Tennessee at MartinMartin, Tennessee 38237
George D. White1588 Mokulua DriveKailua, Hawaii 96734
John 0. WilliamsBureau of Educational ResearchUniversity of North DakotaGrand Forks, North Dakota 58201
PD Makonnen YimerUniversity of IllinoisUrbana, Illinois 61801
Virginia ZachertRock House — Jacks Creek
Route 1, Box 28Good Hope, Georgia 30641
* means: has not paid dues since 1971
PD means: paid up member
