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DOCUMENT RESUME

ED 235 205

TM 830 618

AUTHOR

O'Brien; Francis J.; Jr;

TITLEA Derivation of the Sample .Multiple CorrelationFormula for Raw Scores; INSTITUTIONNational Opinion Research Center, New York, NY.

PUB DATE.

24 dun 83

NOTE64p.; For related document, see ED 223 429.

PUB TYPE

GuidesClassroom UseMaterials (For Learner)

(051)

EDRS PRICEDESCRIPTORS

IDENTIFIERSMF01/PC03 Plus_Postage.

correlation; Higher Education; InstructionalMaterials; *Mathematical Formulas; *Scores;*StatittiCS; *Supplementary Reading MaterialsLinear MOdelt; *Multiple Correlation Formula

ABSTRACT

This paper, a derivation of the multiple correlationformula for unstandardized (raw) scores, is the fourth in a series ofpublications. The purpose of these papers is to provide supplementaryreading for students of applied statistics; The intended audience isSocial science graduate and advanced undergraduate students familiarwith applied statistics; The minimum background for most of theexisting and forthcoming papers is knowledge of applied statisticsthrough rudimentary analysis of variance, and multiple correlation_and regression analysis; The unique feature of this set of papers_isdetailed proofs and derivations of important formulas and_ derivationswhich are not readily available in textbooksi journal articles; and.other similar sources. Each proof or derivation is presented in aclear, detailed and consistent fashion. When necessary, a review ofrelevant algebra is provided. Calculus is not -used or assumed. Thisseries seeks to address the needs of studentsto see a full,comprehensible Statement of a mathematical argument. (PN)

*Reroductions supplied by-EDRS are the best that can be made *from the original document.* A Derivation of the Sample Multiple Correlation Formula fz.)r Raw

Scores

Francis J. O'Brien, Jr., Ph.D.

Assistant Sampling Director

National Opinion Research

Center

1983

FRANCIS J. O'BRIEN, JR.

ALL RIGHTS RESERVEDNORC

Sampling Department

902 Broadway

New Ygrk,

NY 10010

June 24; 1983

"PERMISSION TO REPRODUCE THIS

MATERIAL HAS BEEN GRANTED BY

TO THE EDUCATIONAL RESOURCES

INFORMATION CENTER (ERIC).

U.SDEPABTJVIENT OF EDUCATION

NATIONAL INSTITUTE.OF EOUCATION.

EDUCATIONAL RESOURCES INFORMATION

CENTER (CRICI

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Table of Contents

Page

Introduction

1

Overview of Derivation

Bricf Review of RegreSion Analysis and Derivation for Two

Predictors5

Normal Equations

7

Multiple Correlation

12

Derivation

15

Derivation for Three Predictors

19

Derivation for p Predictors

28
Multiple Correlatibn for p Predictors and Derivation 30

Appendix A

:Normal Equations in Regression Analysis36

Introduction

36
Plan 39
Finding Normal Equations for the Two Predictor Model 40

Finding Normal Equations for p Predictors

44

Alternate Procedure

47

Example for Five Predictors

48

Appendix B: Errata for paper; ED 223 429

51

References

'" 52 list of Tables Fable Page

1; Descriptive Samplo Statistics

Normal Equations and Multiple Correlation Formula

:for Two Raw Score Predictors 16

3; Normal Equations and Multiple Correlation Formula

for Three Raw Score PredictOrS

4. Normal Equations and Multiple Correlation Formula

for p Raw Score Predictors 31
ii A Dorivation of the Sample Multiple Correlation Formula for Raw Scores

Francis J. O'Brien, Jr.; Ph.D.

Nat,ional Opinion

ResearchCenter, New York

Introduction

This paper is the fourth in a series, of publications.

The purpose

of these papers is to provide supplementary reading for students of applied statistics. (See O'Brien; 1982a; 1982b; 1982c).My intended audience is social science graduate and advanced undergraduate students familiar with applied statistica

The minimum background for most of

the existing and forthcoming papers is knowledge or applied statistics throul rudimentary analysis.of variance; and multiple correlation and regression analYSiS.

The unique feature of this set of

papers is detailed proofs and derivations of imnortant formulas and derivations Which are not readily available in textbooks, journal articles, and other similar sources. Each proof or derivation is presented in a clear; detailed and consistent fashion. When necessary, a review of.relevant algebra is provided.

Calculus is not used or assumed.

As a former instructor of applied statistics on the graduate level, I know that many students are very capable of understanding the proofs and deriliations preSented in these paperS

My experience has been

that many students desire to see a full, comprehensible statement of a mathematical argument.

This series seeks to address such needs.

The present paper is a companion work to an earlier paper (O'Brien,

1982c).

Each is a derivation of the multiple correlation formula for the linear model. The first paper formulated a detailed derivation of the

1multiple correlation formula for standard (z) scores.The present paper

is a derivation of the multiple correlation formula for unstandardized (raw) scores. Readers should find each paper interesting and informative. 1- TYpographical errors appeared in this paperFor the readers convenience, corrections are summarized in Appendix B of the present paper. The author would be grateful if other errors in that paper or the present paper were communicated to him. The two paperS taken together are meant to be preparatory reading for a related paper. 1

Overview of Derivation

In this paper we will present a derivation of the linear multiple correlation fortuld for raw scores.

The basic objective is to derive

this formula for one raw score criterion (dependent variable) and lany finite number of raw score predictors (independent variables); Let us first state the formula we will derive and introduce the notation used The linear Multiple correlation between one criterion and p predictors can be expressed as: RY.

1,x ,...,x,,...,x

P=hitSy S1 +g+v 2

br-,S S + ...+ b rS SJ YJyP YP y p TJriting the right hand side in summation notation: RY.X x 2,".,.XXE'13.r:S S1J YJ Y -where: Y

Y.A ix

x-x=multiple correlation of rawscores, the observed raw score criterion to be predicted; xl,x2,...,xj,...,xp raw score predictors of the criterion, 1

Forthcoming with the expected title:"A Derivation of the UnbiasedSample Standard Errorof Estimate: the General Case."It will appear in ERIC.

3. b'ill ...,b ,... b_--slope coefficients oregression weight-Si 1 2IP r ,rr=prOdUct moment cri:erionifiredie.:or correlationS,

1vi,..., vp

S

I;S-,0....S

j...,SP,--standard deviations of the predictors`, S the standard deviation of the criterion.y ThiS IS the formula that is derived in this paper:

We will

first present a deriVation for the simplest multivariate case: one criterion and two predictors.A derivation is then presented for three predictors The latter derivation is a useful exercise because it allows a review of the logic and procedures used in the derivation.

In addition, it

will motivate the use of summation when the algebra becomes complex. The deriVation is then presented for the general case of 0 (finite) predictors. An integral part of this paper is' Appendix'A.In that appendix, a method is presented for finding the "normal equations" in regression analysis for raw score linear models. Prior to starting the derivation for two predictors, let us outline the plan which will be followed in the derivations.

The stepswe

willuseare: 1.

State the regression model

2. derive the normal. equations (see Appendix A) T. define the multiple correlation 4. apply rules of covariance and variance algebra to simplify the definitional form of the multiple correlation formula 5. substitute the normal equations into the multiple -correlation formula 6. simplify. We will refine these steps to suit a particular application: 4.

Brief Overview of Regression Analysis and

Derivation for Two Prediccor5

In this section we will review the baSic concepts, logic and our notation for regression analysis.

Introductory applied statistics

textbooks can he consulted for more detailed information on regression analysis theory. See, for example, Lindeman,et_al., 1982.The intention in this section is to review the rationale of regression analysis; The primary use of statistical regression analysis is controlled prediction and explanation of. quantatativedata.

The basic principle

that-lay behind regression analysis involves aelecting a general mathematical functicn that beSt matches the underlying form of variables over which one desires to exercise p ictability.Assume one i5 attempting to predict one raw Score criterion by use OT two raw scorepredictorS.

Assume further that the relationship between each

predictor and the

Criterion is linear in form

The mathematical function most

often selected to obtain the best linear "fit" for these conditions is provided by the following equation: Y where:

Y+ 6-X. +

x2 the predicted (not actual or observed) criterion, a, ),b2=constants_to be selected by the "least.squares";procedure; a = the Slope intercept, and b- and b. = slope coefficient ter-MS, 1quotesdbs_dbs35.pdfusesText_40

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