ED 064 307 AUTHOR TITLE INSTITUTION SPONS AGENCY REPORT NO PUB DATE NOTE EDRS PRIC*4: DESCRIPTORS ABSTRACT DOCUMENT RESUME TM 001 426 Lord, Frederic M. Significance Test for a Partial Correlation Corrected for Attenuation. EducatioLal Testing Service, Princeton, N.J. Office of Naval Research, Washington, D.C. Personnel and Training Research Programs Office. RB-72-6 Jan 72 22p. MF-$0.65 HC-$3.29 *Correlation; Problem Solving; *Test Interpretation; *Test Results Correction for attenuation is important for partial correlations because not even the sign of the partial between true scores can be inferred safely from the partial between observed (fallible) scores. Methods for inferring the corrected partial are discussed. Unfortunately, the corrected partial will sometimes have an overwhelming sampling error. A significance test is developed that largely circumvents this problem in those casPs where it is enough to infer just the sign of the partial between true scores. (Author)
23
Embed
DOCUMENT RESUME ED 064 307 Lord, Frederic M. TITLE ... · DOCUMENT RESUME. TM 001 426. Lord, Frederic M. Significance Test for a Partial Correlation Corrected for Attenuation. EducatioLal
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
ED 064 307
AUTHORTITLE
INSTITUTIONSPONS AGENCY
REPORT NOPUB DATENOTE
EDRS PRIC*4:DESCRIPTORS
ABSTRACT
DOCUMENT RESUME
TM 001 426
Lord, Frederic M.Significance Test for a Partial Correlation Correctedfor Attenuation.EducatioLal Testing Service, Princeton, N.J.Office of Naval Research, Washington, D.C. Personneland Training Research Programs Office.RB-72-6Jan 7222p.
MF-$0.65 HC-$3.29*Correlation; Problem Solving; *Test Interpretation;*Test Results
Correction for attenuation is important for partialcorrelations because not even the sign of the partial between truescores can be inferred safely from the partial between observed(fallible) scores. Methods for inferring the corrected partial arediscussed. Unfortunately, the corrected partial will sometimes havean overwhelming sampling error. A significance test is developed thatlargely circumvents this problem in those casPs where it is enough toinfer just the sign of the partial between true scores. (Author)
CDI14N'i
C)facg
Cat
'CO
el)
C
C)
U.S. DEPARTMENT OF HEALTH,EDUCATION & WELFAREOFFICE OF EDUCATION
THIS DOCUMENT HAS BEEN REPRO-DUCED EXACTLY AS RECEIVED FROMTHE PERSON OR ORGANIZATION ORIG-INATING IT POINTS OF VIEW OR OPIN-IONS STATED DO NOT NECESSARILYREPRESENT OFFICIAL OFFICE OF EDU-CATION POSITION OR PO ICY
SIGNIFICANCE TEST FOR A PARTIAL CORRELATION
CORRECTED FOR ATTENUATION
Frederic M. Lord
RB-72-6
This research was sponsored in part by thePersonnel and Training Research ProgramsPsychological Sciences DivisionOffice of Naval Research, underContract No. N00014-69-C-0017
Contract Authority Identification NumberNR No.150-303Frederic M. Lord, Principal Investigator
Educational Testing ServicePrinceton, New Jersey
January 1972
Reproduction in whole or in part is permitted forany pirpose of the United States Government.
Approved for public release; distributionunlimited.
SIGNIFICANCE TEST FOR A PARTIAL CORREIATION
CORRECTED FOR ATTENUATION
Frederic M. Lord
RB-72-6
This research was sponsored in part by thePersonnel and Training Research ProgramsPsychological Sciences DivisionOffice of Naval Research, underContract No. N00014-69-C-0017
Reproduction in whole or in part is permitted forany purpose of the United States Government.
Approved for public release; distributionunlimited.
. _ .
DOCUMENT CONTROL DATA- RSA D(Svcurity cMssilkation of title, body of abstract and indexing annotation must be entered when the overall report is classified)
I. ORIG1NA TING AC Ti VI TY (Corporate author)
Educational Testing ServicePrinceton, New Jersey 085)0
aa. REPORT SECURITY CLASSIFICATIONUnclassified
2h. GROUP
3. REPORT TITLE
SIGNIFICANCE TEST FOR A PARTIAL CORRELATION CORRECTED FOR ATTENUATION
4_ DESCRIPTIVE NOTES (Type of report and inclusive dates)
Technical Re ort5. Au THORISI (First name, middle initial, Mat name)
Frederic M. Lord
6. REPORT OATE
January 1972
71. TOTAL NO. OF PAGES
15
lb. NO. OF REFS
1761. CONTRACT OR GRANT NO.
NO001.449C0017b. PROJECT NO.
NR No. 150-303C.
(1.
9a, ORIGINATOR*SW:PORT NUAABER(S)
RB-72-6
9b. OTHER REPORT NO(S) (Any other numbers that may be as !goodthis report)
10. DISTRIBUTION STATEMENT
Approved for public release; distribution unlimited.
..
11. SUPPLEMENTARY NOTES 12. SPONSORING MILITARY ACTIVITY
Personnel and Training Research ProgramsOffice of Naval ResearchArlington, Virginia 22217
13. ABSTRACT
Correction for attenuation is important for partial correletions because noteven the sign of the partial between true scores can be inferred safely from the
partial between observed (fallible) scores. Methods for inferring the correctedpartial are discussed. Unfortunately, the corrected partial will sametimes havean overwhelming sampling error. A significance test is developed that largelycircumvents this problem in those cases where it is enough to infer just thesign of the partial between true scores.
DDFORM1 NOV 661473 (PAGE 1)
S/N 0101.807.6801 a Security Classification3ND PPSO
Security Classification1 KEV WORDS
Correlation
Mental test theory
Theory of errors
Errors of measurement
Testing hypotheses
DI) 11=1114173 (BACK)(PAGE 2)
411.- 4.
LINK A LINK S LINK C
ROLE WT LIU %VT ROLE WT
Security Classification
SIGNIFICANCE TEST FOR A PARTIAL CORRELATION
CORRECTED FOR ATTENUATION
Abstract
Correction for attenuation is important for partial correlations
because not even the sign of the partial between true scores can be
inferred safely from the partial between observed (fallible) scores.
Methods for inferring the corrected partial are discussed. Unfortunately,
the corrected partial will sometimes have an overwhelming sampling error.
A significance test is developed that largely circumvents this problem
in those cases where it is enough to infer just the sign of the partial
between true scores.
SIGNIFICANCE TEST FOR A PARTIAL CORRELATION
CORRECTED FOR ATTENUATION*
When each of two variables is contaminated by a third variable, it
often is important in studying the relationship between the first two to
"hold constant" or partial out the third. For example, scores on each of
two personality tests may be unavoidably contaminated by general intelli-
gence, by "attitude acquiescence," or by some other cognitive ability or
personality trait. If we want the correlation between the first two
personality traits: contaminating variables must be partialed out. For
numerous examples, see Stricker, Messick, and Jackson (1968).
The population partial p or sample partial r between xxy.z xy'z
and y with z "held constant" is by definition the correlation of the
residuals in the linear prediction of x from z with the residuals in
the linear prediction of y from z Since partial correlations are re-
garded with distrust by many behavioral scientists, it is worth pointing
out that if the re ression of x and on z is linear and if the con-
ditional distribution of the residuals is inde endent of z then the
partial pxy.z is numerically equal to the simple zero-order correlation
between x and y computed for all observations having an arbitrarily
chosen value z ; this zero-order correlation is the same no matter what
arbitrary value of z is chosen. Thus, in this case, use of the partial
correlation really eliminates the concomitants of variation in z If
the residuals are not distributed independently of z 0 the partial is a
11011,
*This research was aponsored in part by the Personnel and TrainingResearch Programs, Psychological Sciences Division, Office of NavalResearch, under Contract No. N00014-69-C-0017 Contract AuthorityIdentification NuMber NR No. 150-303, and Educational Testing Service.Reproduction in whole or in part is permitted for any purpose of theUnited States Government.
-2-
sort of weighted average of the correlations between x and y for
fixed x averaged over the different fixed values of z .
An important prdblem arises because the present:e of errors of mea3ure-
ment in z can cause the partial correlation pxy.z to be negative when
otherwise it would be positive. If the errors of measurement are uncor-
related with each other and with all other variables, their effects can
be removed by correcting for attenuation each zero-order correlation
involving z This results, after a little algebra, in a formula given
and discusped by Stouffer (1936, eq. 9), Saunders (1951), Lord (1963,
Kelley, T. L. Crossroads in the mind of wan. Stanford, Calif.: Stanford
University Press, 1928.
Kelley, T. L. The reliability coefficient. Psychometrika, 1942, L 75-83.
Kelley, T. L. Fundamentals of statistics. Cambridge, Mass.: Harvard
University Press, 1947.
Kendall, M. G. & Stuart, A. The advanced theory of statistics. New York:
Hafner. Vol. I, 1958; Vol. II, 1961.
Lord, F. M. Elementary models for measuring change. In C. W. Harris (Ed.),
proirichane. Madison, Wis.: The University of
Wisconsin Press, 1963. Pp. 21-38.
Lord, F. M. & Novick, M. R. Statistical theories of mental test scores.
Reading, Mass.: Addison=Wesley, 1968.
Moran, P. A. P. On asymptotically optimal tests of composite hypotheses.
Biometrika, 1970) Ig.) 47-55.
Saunders, D. R. Partial reliability coefficients applied to the over-
achievement problem. Research Memorandum 51-16. Princeton) N. J.:
Educational Testing Service, 1951.
Spearman, C. The abilities of man. New York: Macmillan, 1927.
Stouffer, S. S. Reliability coefficients in a correlation matrix.
Psychometrika, June 1936, 11) 17 20.
Stricker, L. J., Messick, S., & Jackson, D. N. Desirability judgments
and self-reports as predictors of social behavior. Journal of
Experimental Research in Personality, 1968, 2.1 151-167.
2
DISTRIBUTION LIST
NAVY
4 Director, Personnel and TrainingResearch Programs
Office of Naval ResearchArlington, VA 22217
1 DirectorONR Branch Office495 Summer StreetBoston, MA 02210
1 DirectorONR Branch Office1030 East Green StreetPasadena, CA 91101
1 DirectorONR Branch Office536 South Clark StreetChicago, IL 60605
1 Office of Naval ResearchArea Office207 West 24th StreetNew York, NY 10011
1 CommanderOperational Test and Evaluation ForceU. S. Naval BaseNorfolk, VA 23511
6 DirectorNaval Research LaboratorYWashington, D. C. 20390ATTN: Library, Code 2029 (ONRL)
6 DirectorNaval Research LaboratoryWashington, D. C. 20390ATTN: Technical Information Division
12 Defense Documentation CenterCameron Station, Building 55010 Duke StreetAlexandria, VA 22314
1 Behavioral Sciences DepartmentNaval Medical Research InstituteNational Naval Medical CenterBethesda, MD 20014
1 ChiefBureau of Medicine and SurgeryCode 513Washington, D. C. 20390
1 ChiefBureau of Medicine and SurgeryResearch Division (Code 713)Department of the NayyWashington, D. C. 20390
1 Commanding OfficerNaval Medical NeuropsychiatricResearch Unit
San Diego, CA 92152
1 DirectorEducation and Training SciencesDepartment
Naval Medical Research InstituteNational Naval Medical CenterBuilding 142Bethesda, MD 20014
1 Technical Reference Library :
Naval Medical Researe: InstituteNational Naval Medical CenterBethesda, MD 20014
1 Chief of Naval TrainingNaval Air StationPensacola, FL 32508
ATTN: Capt. Allen E. McMichael
1 Mr. S. FriedmanSpecial Assistant for Research & Studies
OASN (MA)The Pentagon, Room 4E794Washington, D. C. 20350
1 ChiefNaval Air Technical TrainingNaval Air StationMemphis, TN 38115
1 Chief of Naval Operations (0p-98)Department of the NavyWashington, D. C. 20350
ATTN: Dr. J. J. Collins
3 Technical DirectorPersonnel Research DivisionBureau of Naval PersonnelWashington, D. C. 20370
3 Technical Library (Pers-11B)Bureau of Naval PersonnelDepartment of the NavyWashington, D. C. 20360
3 Technical DirectorNaval Personnel Research and
Development LaboratoryWashington Navy Yard, Building 200Washington, D. C. 20390
3 Commanding OfficerNaval Personnel and Training ResearchLaboratory
San Diego, CA 92152
1 ChairmanBehavioral Science DepartmentNaval Command and Management DivisionU. S. Naval AcademdLuce HallAnnapolis, MD 21402
1 SuperintendentNaval POstgraduate SchoolMonterey, CA 93940ATTN: Library (Code 2124)
1 Commanding OfficerService School CommandU. S. Naval Training CenterSan Diego, CA 92133
1 Research Director, Code 06
Research and Evaluation DepartmentU. S. Naval Examining Center
Building 2711 - Green Bay Area
Great Lakes, IL 60088
ATTN: C. S. Winiewicz
1 CommanderSubmarine Development Group TwoFleet Post OfficeNew York, NY 09501
1 Mr. George N. GraineNaval Ship Systems Command (SHIP 0311)
Department of the NavyWashington, D. C. 20360
I Head, Personnel Measurement StaffCapital Area Personnel Service OfficeBallston Tower #2 Room 1204801 N. Randolph StreetArlington, VA 22203
1 Dr. A. L. SlafkoskyScientific Advisor (Code AX)Commandant of the Marine CorpsWashington, D. C. 20380
I Dr. James J. Regan, Code 55Naval Training Device CenterOrlando, FL 32813
ARMY
1 Behavioral Sciences DivisionOffice of Chief of Research andDevelopment
Department of the ArmyWashington, D. C. 20310
1 U. S. Army Behavior and SystemsResearch Laboratory
Commonwealth Building, Roam 2391320 Wilson BoulevardArlington, VA 22209
1 Director of ResearchUS Army Armor Human Research UnitATTN: LibraryBldg 2422 Morande StreetFort Knox, KY 40121
1 Commanding OfficerATTN: LTC CosgroveUSA CDC PASAFt. Benjamin Harrison, IN 46249
1 DirectorBehavioral Sciences LaboratoryU. S. Army Research Institute of.Environmental Medicine
Natick, MA 01760
1 Division ofWalter Reed
ResearchWalter ReedWashington,
NeuropsychiatryArmy Institute of
Army Medical CenterD. C. 20012
21
-2-
1 Dr. George S. Harker, DirectorExperimental Psychology DivisionU. S. Army Medical ResearchLaboratory
Fort Knox, KY 40121
AIR FORCE
1 AFHRL (TR/Dr. G. A. Eckstrand)Wright-Patterson Air Force BaseDayton, Ohio 45433
I AMU, (TRT/Dr. Ross L. Morgan)Wright-Patterson Air Force Base
Dayton, Ohio 45433
1 AFSOR (NL)1400 Wilson BoulevardArlington, VA 22209
1 Lt, Col. Robert R. Gerry, USAFChief, Instructional Technology ProgramsResources & Technology Division(DPTBD DCS/P)The Pentagon (Room 4c244)Washington, D. C. 20330
1 Headquarters, U. S. Air ForceChief, Personnel Research and AnalysisDivision (Amen)
Washington, D. C. 20330
1 Personnel Research Division AFHRL)Lackland Air Force BaseSan Antonio, TX 78236
1 Headquarters, Electronics Systems DivisionATTN: Dr. Sylvia Mayer/MCDSL. G, Hanscom FieldBedford, MA 01730
1 Director of Manpower ResearchOASD (M&RA) (M&RU)Room 3D960The PentagonWashington, D. C.
OTHER GOVERNMENT
1 Mk. Joseph J. Cowan, ChiefPsychological Research Branch (B-1)U. S. Coast Guard Headquarters400 Seventh Street, S. W.Washington, D. C. 20591
1 Dr. Alvin E. Goins, ChiefPersonllity and Cognition Research SectionBehavioral Sciences Research BranchNational Institute of Mental Health5454 Wisconsin Avenue, Room 10A01Bethesda, MD
1 Dr. Andrew R. MolnarComputer Innovation in Education SectionOffice of Computing ActivitiesNational Science FoundationWashington, D. C. 20573
MISCELLANEOUS
1 Dr. Richard C. AtkinsonDepartment of PsychologyStanford UniversityStanford, CA 94305
1 Dr. Bernard M. BassUniversity of RochesterMhnagement Research CenterRochester, NY 14627
1 Dr. Lee R. BeachDepartment of PsychologyUniversity of WashingtonSeattle, WA 98105
1 Dr. Mats BjorkmanUniversity of UmeaDepartment of PsychologyUmea 6, SWEDEN
1 Dr. Kenneth E. ClarkUniversity of RochesterCollege of Arts & SciencesRiver Campus StationRochester, NY 14627
1 Lawrence B. Johnson .
Lawrence Johnson & Aemaciates, Inc.2001 "S" St. N. W.Washington, D. C. 20037
1 Dr. E. J. McCormickDepartment of PsychologyPurdue UniversityLafayette, IN 47907
1 Dr. Robert GlaserLearning Research and Development
CenterUniversity of PittsburghPittsburgh, PA 15213
1 Dr. Albert S. Glickmap:.American Institutes for Research8555 Sixteenth StreetSilver Spring, MD 20910
1 Dr. Bert GreenDepartment of PsychologyJohns Hopkins UniversityBaltimore, MD 21218
1 Dr. Duncan N. HansenCenter for Computer Assisted InstructionFlorida State UniversityTallahassee, FL 32306
1 Dr. Richard S. HatchDecision Systems Associates, Inc.11428 Rockville PikeRockville, MD 20852
. 22
-3-
1 Dr. M. D. HavronHuman Sciences Research, Inc.Westgate Industrial Park7710 Springhouse RoadMcLean, VA 22101
1 Human Resources Research OrganizationLibrary300 North Washington StreetAlexandria, VA 22314
1 Human Resources Research OrganizationDivision #3Post Office Box 5787Presidio of Monterey, CA 93940
1 Human Resources Research OrganizationDivision #4, InfantryPost Office Box 2086Fort Henning, GA 31905
1 Human Resources Research OrganizationDivision #5, Air DefensePost Office Box 6021Fort Bliss, TX 77916
1 Human Resources *Research OrganizationDivision #6, Aviation (Library)Post Office Box 428Fort Rucker, ALA 36360
1 Dr. Robert R.' MhokieHuman Factors Researa, Inc.Santa Barbara Research Park6780 Cortona DriveGole:Aa CA 93017
1 Mr. Lasi Petrullo2431 North Edgewood StreetArlington, VA 22207
1 Psychological AbstractsAmerican Psychological Association1200 Seventeenth Street, N. W.Washington, D. C. 20036
1 Dr. Diane M. Ramsey-KleeR-K Research & System Design3947 Ridgemont DriveMalibu, CA 90265
1 Dr. Joseph W. RigneyBehavioral Technology LaboratoriesUniversity of Southern CaliforniaUniversity ParkLos Angeles, CA 90007
1 Dr. George E. RowlandRowland and Company, Inc.Post Office Box 61Haddonfield, NJ 08033
1 Dr. Robert J. SeidelHuman Resources Research Organization300 N. Washington StreetAlexandria, VA 22314
1 Dr. Arthur I SiegelApplied Psychological ServicesScience Center404 East Lancaster AvenueWayne, PA 19087