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MULTIPLE LINEAR REGRESSION VIEWPOINTS A publication of the Special Interest Group on Multiple Linear Regression of The American Educational Research Association. I VOLUME 4, NUMBER 1

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  • 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~

  • 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. TechnicalDocumenters 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 1a

    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:einitial 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 INDEPENDENTVARIABLESON 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

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    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

  • 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

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    +olxS+56

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    Model 3 Y a0

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    Model 4 Y1

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    Model 5 Y1

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    Model 6 Y1

    — a0

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    x7

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    Model 7 Y1

    = a0

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    56

    -fa7X7

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    +57

    Model 8 — aou+aqx4

    +a6

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    x7

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    +Eg

    Model 9 ‘fi. = aou+a4

    X4

    +a5

    X5

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    X7

    +a8

    X8

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    x9

    +Eg

    Model 10 Y1

    = aou+aixi+aSxS+o,x6

    +a6

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    +E10

    Model 11 Y1

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    Model 12 Y2

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    Model 13 Y2

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    Model 14 ‘~‘2 a0

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    Model 15 Y2 — aou+a2x2+El5

    Model 16 Y2

    = a0

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    Model 17 ‘~‘2 aou+aqX4

    +a5

    x5

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    Model 18 Y2

    = aoCC+a5

    x5

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    x7

    +a8

    x8

    +a9

    xg÷E18

    Model ~ = a0

    u+a4

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    Model 20 Y2

    au+a4

    x4

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    Model 21 ‘f2 — a0

    U+a4

    x4

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    x8

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    Model 22 ‘~‘2 a0

    u+a4

    x4

    +ax5

    +o6

    x6

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    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 626. 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 7715. 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 -5the 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 CC0 — C-I +15 tO CC 5+ cC

    x = 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 federalenforcement 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 tOblack 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 ~ ,~ tOO — 0 H 0 Di

    those 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 CC

    5 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 CCDC 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

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    a ‘,(vs-Co*111

    01 0.a—CC‘DC’SCm

    0C,

    CC

    DO0

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    CD

    Is0

    F’(CCCSCD

    05 “CID 0

    C’ —Cl

    (C

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    05 “C15 0

    C,Ci

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    ~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 americansand 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

  • 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 tosupport.

    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 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

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  • 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 NorthernColorado, 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