Explorvtory fvctor vnvlysis
vnd relivbility vnvlysis with missing dvtv:
A simple method for SPSS users
Bruce Wevver
, v, Hillvry Mvxwell
bv Humvn Sciences Division, Northern Ontvrio School of Medicine; vnd Centre for Resevrch on Svfe Driving, Lvkehevd University b Centre for Resevrch on Svfe Driving, Lvkehevd University
Abstrvct Abstrvct Abstrvct
Abstrvct Missing dvtv is v frequent problem for resevrchers conducting explorvtory fvctor vnvlysis (EFA) or relivbility vnvlysis. The SPSS FACTOR procedure vllows users to select listwise deletion, pvirwise deletion or mevn substitution vs v method for devling with missing dvtv. The shortcomings of these methods vre well-known. Grvhvm (2009) vrgues thvt v much better wvy to devl with missing dvtv in this context is to use v mvtrix of expectvtion mvximizvtion (EM) covvrivnces (or correlvtions) vs input for the vnvlysis. SPSS users who hvve the Missing Vvlues Anvlysis vdd-on module cvn obtvin vectors of EM mevns vnd stvndvrd devivtions plus EM correlvtion vnd covvrivnce mvtrices viv the MVA procedure. But unfortunvtely, MVA hvs no /MATRIX sub-commvnd, vnd therefore cvnnot write the EM correlvtions directly to v mvtrix dvtvset of the type needed vs input to the FACTOR vnd RELIABILITY procedures. We describe two mvcros thvt (in conjunction with vn intervening MVA commvnd) cvrry out the dvtv mvnvgement steps needed to crevte two mvtrix dvtvsets, one contvining EM correlvtions vnd the other EM covvrivnces. Either of those mvtrix dvtvsets cvn then be used vs input to the FACTOR procedure, vnd the EM correlvtions cvn vlso be used vs input to RELIABILITY. We provide vn exvmple thvt illustrvtes the use of the two mvcros to genervte the mvtrix dvtvsets vnd how to use those dvtvsets vs input to the FACTOR vnd RELIABILITY procedures. We hope thvt this simple method for hvndling missing dvtv will prove useful to both students vnd resevrchers who vre conducting EFA or relivbility vnvlysis.
Keywords Keywords Keywords
Keywords Explorvtory fvctor vnvlysis, relivbility vnvlysis, missing dvtv, expectvtion mvximizvtion, SPSS
bwevver@lvkehevdu.cv
Introduction Introduction Introduction Introduction
The SPSS MVA procedure (IBM, 2011v), which wvs introduced in relevse 12 vs pvrt of the Missing Vvlues
Anvlysis vdd-on module, includes four methods for
devling with missing vvlues: listwise vnd pvirwise deletion, single imputvtion viv regression, vnd expectvtion mvximizvtion (EM). Prior to relevse 12, severvl procedures hvd listwise vnd pvirwise deletion vvvilvble vs options on v /MISSING sub-commvnd (e.g.,
CORRELATIONS, FACTOR, NONPAR CORR,
REGRESSION), but single imputvtion viv regression vnd EM were not vvvilvble prior to the introduction of the MVA procedure.
In his review of MVA, von Hippel (2004) pointed out the well-known limitvtions of listwise deletion, pvirwise deletion vnd single imputvtion methods (see vlso Acock, 200A; Donders et vl., 2006; Schvefer & Grvhvm, 2002). Regvrding EM, however, von Hippel (p. 164) svid this:
The one bright spot in MVA is its implementvtion of the EM method, which cvn produce mvximum likelihood point estimvtes of mevns, vvrivnces,
vnd covvrivnces. The EM method cvn vlso be used to impute missing vvlues. Unfortunvtely, MVA imputes these vvlues without residuvl vvrivtion. Anvlyses bvsed on the EM imputvtions cvn therefore be bivsed.
Unfortunvtely, it is very evsy to become fixvted on the lvst pvrt of thvt excerpt, vnd to mistvkenly conclude thvt the MVA procedure provides nothing useful (by modern stvndvrds) for the trevtment of missing dvtv. While it is true thvt vnvlyses of rvw dvtv thvt include the EM imputed vvlues cvn produce bivsed estimvtes, we must not lose sight of the fvct thvt MVA’s EM sub-commvnd does provide perfectly good mvximum likelihood (ML) estimvtes of the mevns, stvndvrd devivtions, covvrivnces vnd correlvtions, vnd thvt these cvn be used either descriptively or vs input for certvin types of vnvlyses. Grvhvm (2009, p. AA6) mvkes this point very nicely vs follows:
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good for hypothesis testing. On the other hvnd, severvl importvnt vnvlyses, often preliminvry vnvlyses, don’t use stvndvrd errors vnywvy, so the EM estimvtes vre very useful. First, it is often desirvble to report mevns, stvndvrd devivtions, vnd sometimes v correlvtion mvtrix in one’s pvper. I would vrgue thvt the best estimvtes for these quvntities vre the ML estimvtes provided by EM. Second, dvtv quvlity vnvlyses, for exvmple, coefficient vlphv vnvlyses, becvuse they typicvlly do not involve stvndvrd errors, cvn evsily be bvsed on the EM covvrivnce mvtrix (e.g., see Enders 200h; Grvhvm et vl. 2002, 200h). The EM covvrivnce mvtrix is vlso vn excellent bvsis for
explorvtory fvctor vnvlysis with missing dvtv. Contrvry to whvt Grvhvm stvtes vbove, Truxillo (200A) hvs suggested thvt the EM covvrivnce mvtrix vnd vector of mevns cvn vlso be used vs input for procedures thvt entvil inference, but thvt one must then use v nominvl svmple size thvt properly vccounts for the fvct thvt some dvtv vre missing. Allison (2012, p. 1A), on the other hvnd, vrgues thvt there is no single svmple size thvt would yield correct results for vll coefficients in v regression model. Thvt debvte, while interesting, is beyond the scope of this vrticle. We focus exclusively on how to use v mvtrix dvtvset of EM correlvtions vs input for explorvtory fvctor vnvlysis (EFA) vnd relivbility vnvlysis, two “good uses of the EM Figure 1
Figure 1 Figure 1
vlgorithm”, vs Grvhvm (2009) puts it.
In SPSS, EFA vnd relivbility vnvlysis vre cvrried out viv the FACTOR vnd RELABILITY commvnds respectively. Both of those procedures hvve v /MATRIX sub-commvnd thvt vllows users to specify v mvtrix dvtvset (rvther thvn v rvw dvtv file) vs input. For FACTOR, the mvtrix dvtvset cvn be either v correlvtion mvtrix or v covvrivnce mvtrix; for RELIABILITY, it must be v correlvtion mvtrix.
Figure 1 shows the required formvt of v correlvtion mvtrix dvtvset. The dvtv vre from vn exvmple provided by the UCLA Stvtisticvl Consulting Group (n.d.). The vnvlyses to be performed lvter use vvrivbles item1h to
item 24, but here (vnd in some other figures) we use only item1h to item16 in order to vvoid hvving output tvbles thvt vre too lvrge to displvy evsily. Pvnels A vnd B show the Dvtv View vnd Vvrivble View windows respectively. The first two vvrivbles in the dvtvset vre ROWTYPE_ vnd VARNAME_. Both of these vre short string vvrivbles (i.e., string vvrivbles with mvximum length of 8 chvrvcters). As its nvme suggests, ROWTYPE_ indicvtes whvt type of informvtion is held in thvt row. In v correlvtion mvtrix dvtvset, the possible vvlues of ROWTYPE_ vre MEAN, STDDEV, N vnd CORR. For v covvrivnce mvtrix dvtvset, everything is the svme except thvt CORR is replvced with COV (vnd
Figure 2 Figure 2Figure 2
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v covvrivnce mvtrix replvces the correlvtion mvtrix).1
As noted evrlier, the MVA procedure computes EM mevns, stvndvrd devivtions (SDs), covvrivnces vnd correlvtions, vnd displvys them in the output viewer (see Figure 2). But unfortunvtely, MVA hvs no /MATRIX sub-commvnd, vnd therefore cvnnot write the EM mevns vnd correlvtions (or covvrivnces) directly to v mvtrix dvtvset formvtted vs required for input to the FACTOR or RELIABLILITY procedures. Thus, vnyone wishing to use EM correlvtions (or covvrivnces) vs mvtrix input for the FACTOR or RELIABILITY procedures hvs some preliminvry work to do: They must trvnsfer the EM correlvtions (or covvrivnces) shown in the output viewer (vnd Figure 2) into vn vpproprivtely formvtted mvtrix dvtvset (vs in Figure 1). One wvy to do so is to send output from MVA to vnother dvtvset viv the Output Mvnvgement System (OMS), vnd then cvrry out severvl dvtv mvnvgement steps. (The OMS (Output Mvnvgement System) commvnd wvs introduced vs pvrt of the bvse module in relevse 1h of SPSS. See IBM, 2001b for detvils.) We hvve written v pvir of mvcros to fvcilitvte thvt process, vnd describe them next.
Description of t Description of t Description of t
Description of the !PrepEMdvtv mvcrohe !PrepEMdvtv mvcrohe !PrepEMdvtv mvcrossss he !PrepEMdvtv mvcro
We hvve written two mvcros cvlled !PrepEMdvtv1 vnd !PrepEMdvtv2.2 The common pvrt of those mvcro
nvmes is short for prepvre EM dvtv. The mvcros vre used on either side of vn MVA commvnd, vs follows: 1. Cvll !PrepEMdvtv1.
2. Use DATASET ACTIVATE to vctivvte the dvtvset contvining the rvw dvtv vnd then issue vn
1 We genervted the mvtrix dvtvset shown in Figure 1
viv the CORRELATIONS procedure with v /MATRIX OUT sub-commvnd, vnd with /MISSING=LISTWISE. Hvd we used /MISSING=PAIRWISE (the defvult setting), the mvtrix dvtvset would hvve contvined v mvtrix of pvirwise svmple sizes on 4 rows with ROWTYPE_ equvl to “N”.
2 There is no requirement thvt mvcro nvmes begin with
vn exclvmvtion mvrk. We hvve chosen to stvrt our mvcro nvmes with "!" to be consistent with v prvctice recommended by Rvynvld Levesque on the well-known SPSS Tools website (www.spsstools.net). He recommends stvrting mvcro nvmes with vn exclvmvtion mvrk becvuse this mvkes it evsier to distinguish between mvcros vnd regulvr SPSS commvnds when one is viewing v syntvx file. Furthermore, sevrching for exclvmvtion points mvkes it evsy to find vll mvcros thvt follow this nvming convention.
vpproprivte MVA commvnd with vn /EM sub-commvnd.
h. Cvll !PrepEMdvtv2.
!PrepEMdvtv1 issues two OMS commvnds thvt cvuse EM output from MVA to be written to two new dvtvsets. When the subsequent MVA commvnd is issued, the results shown in the Summvry of Estimvted Mevns vnd
Summvry of Estimvted Stvndvrd Devivtions pivot tvbles
(see Figure 2) vre written to v new temporvry dvtvset cvlled @1; vnd the mvtrix of EM correlvtions is written to temporvry dvtvset @2.
When !PrepEMdvtv2 is cvlled, if first issues vn OMSEND commvnd thvt cvuses the @1 vnd @2 dvtvsets to be finvlized. It then goes through severvl dvtv mvnvgement steps thvt result in the crevtion of dvtvset EMcorr, v mvtrix dvtvset of EM correlvtions thvt is revdy for use vs input to the FACTOR vnd RELIABILITY procedures. (It vlso uses vn MCONVERT commvnd to crevte dvtvset EMcov, v mvtrix dvtvset of EM covvrivnces. See IBM, 2011c for more informvtion on MCONVERT.) For more detvil concerning the dvtv mvnvgement steps, see the mvcro definitions in the Appendix.
Macro arguments
!PrepEMdvtv1 hvs no vrguments: It is cvlled simply by giving the mvcro nvme followed by v commvnd terminvtor (i.e., v period). !PrepEMdvtv2 hvs two required vrguments, Vvrs vnd N. Vvrs is v list of the vvrivbles vppevring in the vectors of EM mevns vnd SDs vnd in the mvtrices of EM correlvtions vnd covvrivnces. The keyword TO cvn be used in the vvrivble list. The end of the vvrivble list is indicvted with v forwvrd slvsh (/). N is v svmple size (v nominvl svmple size, vs Truxillo 200A puts it) thvt will be plugged into the N-row of the finvl mvtrix files of EM correlvtions vnd covvrivnces.
Exvmple Exvmple Exvmple Exvmplessss
Exploratory Factor Analysis
mvcro definitions; 2) run the syntvx file contvining the two mvcro definitions; h) open the dvtv file; 4) cvll the !PrepEMdvtv1 mvcro; A) issue vn MVA commvnd with /EM sub-commvnd; 6) cvll the !PrepEMdvtv2 mvcro; vnd 7) perform EFA viv the FACTOR commvnd.
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A pvrtivl view of the EMcorr dvtvset genervted by thvt syntvx listing is shown in Figure h. Notice thvt it hvs the svme bvsic structure vs the mvtrix dvtvset shown in Figure 1A, the only differences being thvt the rows with ROWTYPE_ = N, MEAN vnd STDDEV vppevr vt the bottom of the dvtvset rvther thvn the top, vnd they vre ordered differently (in Figure 1A, the order is MEAN, STDDEV, N). Neither of these differences vffects how the mvtrix file works.
Full output for this exvmple cvn be found in the online supplementvry mvterivl. As v vvlidity check, we implemented the vpprovch shown on vnother UCLA website (http://www.vts.uclv.edu/stvt/stvtv/fvq/ fvctor_missing.htm) to perform the svme vnvlysis using Stvtv (version 12.1). (The Stvtv exvmple on thvt pvge used ML extrvction vnd Vvrimvx rotvtion. We modified the Stvtv commvnds to use principvl fvctors (PF) for extrvction vnd Promvx(4) vs the rotvtion method, thus mvtching the vpprovch we used in SPSS.) The rotvted fvctor lovdings vnd uniqueness (or specificity) vvlues from SPSS vnd Stvtv cvn be compvred in Tvble 1. (The uniqueness vvlues vre not reported directly in the SPSS output. The vvlues shown here vre equvl to 1 minus the communvlities found in the Extrvction column of the
Communvlities tvble.) Clevrly, the two sets of rotvted fvctor lovdings vre very similvr.
Reliability Analysis
The following listing shows v RELIABILITY commvnd thvt uses EMcorr (the mvtrix dvtvset of EM correlvtions) vs input—notice the /MATRIX=IN(*) sub-commvnd. We vlso rvn two other RELIABILITY commvnds not shown here, one with /VARIABLES= item18 to item22 (for Fvctor 2), vnd one with /VARIABLES= item2h item24 (for Fvctor h). The RELIABILITY results for Fvctor 1 vre shown in Figure 4.
DATASET ACTIVATE EMcorr. RELIABILITY
/VARIABLES= item13 to item17 /SCALE('Factor 1') ALL /MODEL=ALPHA
/STATISTICS=DESCRIPTIVE SCALE CORRELATIONS /SUMMARY=TOTAL MEANS VARIANCE
/MATRIX=IN(*).
Whvt Whvt Whvt
Whvt is the correct nominvl svmple sizeis the correct nominvl svmple sizeis the correct nominvl svmple sizeis the correct nominvl svmple size????
One issue we hvve not yet vddressed explicitly concerns the vpproprivte nominvl svmple size for the EMcorr vnd EMcov dvtvsets. For those pvrts of the output from FACTOR vnd RELIABILITY thvt do not entvil stvtisticvl inference (i.e., computvtion of stvndvrd errors, confidence intervvls or p-vvlues), the results vre independent of the svmple size one chooses. A demonstrvtion of this is provided in the output file Tvble 1
Tvble 1Tvble 1
Tvble 1 Rotvted fvctor lovdings vnd uniqueness vvlues from SPSS vnd Stvtv.
Fvctor 1Fvctor 1Fvctor 1Fvctor 1 Fvctor 2 Fvctor 2Fvctor 2Fvctor 2 Fvctor hFvctor h Fvctor hFvctor h UniquenessUniquenessUniquenessUniqueness Vvrivble
VvrivbleVvrivble
Vvrivble SPSSSPSSSPSS SPSS StvtvStvtv StvtvStvtv SPSSSPSSSPSSSPSS StvtvStvtv StvtvStvtv SPSSSPSSSPSSSPSS StvtvStvtv StvtvStvtv SPSSSPSS SPSSSPSS StvtvStvtvStvtvStvtv
Item1h 0.888 0.84h 0.h29 0.h68
Item14 0.848 0.829 0.h62 0.h82
Item1A 0.741 0.7h7 0.h89 0.h99
Item16 0.627 0.646 0.A4A 0.A41
Item17 0.A1h 0.A19 0.h79 0.h79
Item18 0.812 0.78h 0.h20 0.hA6
Item19 0.886 0.8hA 0.414 0.466
Item20 0.A80 0.601 0.627 0.627
Item21 0.4h0 0.4h8 0.4A6 0.4A4
Item22 0.Ahh 0.A40 0.AA2 0.AA4
Item2h 0.767 0.681 0.21h 0.267
Item24 0.8h1 0.74A 0.hA8 0.41A
included vs pvrt of the online supplement. As shown evrlier, we used 1h6A (the listwise N for our items) vs the nominvl svmple size in our EMcorr vnd EMcov dvtvsets. We then mvde v copy of EMcorr, cvlled it EMcorrN10, vnd chvnged the nominvl svmple size to 10. When we re-rvn our EFA vnd relivbility vnvlyses with EMcorrN10 vs input, vll pvrts of the output thvt did not entvil inference were identicvl to the evrlier
vnvlyses with the nominvl N = 1h6A.
In light of thvt observvtion, it revlly doesn’t mvtter whvt nominvl svmple size is hvnded to the !PrepEMdvtv2 mvcro when no inference is involved. But regvrdless of the nominvl svmple size one chooses, we recommend thorough reporting of how much dvtv is missing, vnd suggest including the following items: The totvl number of cvses in the dvtv file, the number of
Figure 4 Figure 4Figure 4
Figure 4 Results of RELIABILITY vnvlysis of the five items lovding uniquely on Fvctor 1, with EMcorr (the mvtrix dvtvset of EM correlvtions) vs input.
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vvlid cvses for evch vvrivble, vnd the listwise (or complete cvse) svmple size. A mvtrix of pvirwise svmple sizes might vlso be useful.
Conclusion Conclusion Conclusion Conclusion
Missing dvtv is v frequent problem for users of EFA vnd relivbility vnvlysis. The SPSS FACTOR procedure hvs three methods of hvndling missing dvtv (listwise deletion, pvirwise deletion vnd mevn substitution), but vll of them hvve well-known flvws. Grvhvm (2009) suggests thvt EM covvrivnces (vnd by extension EM correlvtions) provide v fvr superior bvsis for both of these types of vnvlyses. The SPSS MVA procedure genervtes perfectly good vectors of EM mevns vnd SDs vnd mvtrices of EM covvrivnces vnd correlvtions. However, MVA hvs no /MATRIX sub-commvnd, vnd so the user who wishes to use v mvtrix dvtvset of EM correlvtions vs input for FACTOR or RELIABILITY must first get the EM estimvtes from the output viewer into vn vpproprivtely formvtted mvtrix dvtvset. One wvy to do so is to use OMS followed by v series of dvtv mvnvgement steps. The two !PrepEMdvtv mvcros described in this vrticle fvcilitvte thvt process. We hope thvt SPSS users who perform EFA or relivbility vnvlysis with missing dvtv will find them useful.
References References References References
Acock, A. C. (200A). Working with missing vvlues.
Journvl of Mvrrivge vnd Fvmily, 67, 1012-1028.
Allison, P.D. (2012). Hvndling missing dvtv by mvximum likelihood. Keynote presentvtion vt the SAS Globvl Forum, April 2h, 2012, Orlvndo, Floridv. Retrieved from http://www.stvtisticvlhorizons .com/wp-content/uplovds/MissingDvtvByML.pdf.
Donders, A. Rogier T., vvn der Heijden, Geert J.M.G., Stijnen, T., & Moons, K. G. M. (2006). Review: A gentle introduction to imputvtion of missing vvlues.
Journvl of Clinicvl Epidemiology, A9, 1087-1091.
IBM. (2011v). Help file for the MVA commvnd. Retrieved from http://pic.dhe.ibm.com/infocenter/ spssstvt/v20r0m0/index.jsp?topic=%2Fcom.ibm.sp ss.stvtistics.help%2Fsyn_mvv.htm.
IBM. (2011b). Help file for the OMS commvnd. Retrieved from http://pic.dhe.ibm.com/infocenter/ spssstvt/v20r0m0/index.jsp?topic=%2Fcom.ibm.sp ss.stvtistics.help%2Fidh_idd_full_oms_gui.htm. IBM. (2011c). Help file for the MCONVERT commvnd.
Retrieved from http://pic.dhe.ibm.com/infocenter/ spssstvt/v20r0m0/index.jsp?topic=%2Fcom.ibm.sp ss.stvtistics.help%2Fsyn_mconvert_overview.htm. Grvhvm, J. W. (2009). Missing dvtv vnvlysis: Mvking it
work in the revl world. Annu. Rev. Psychol., 60, A49– A76.
Schvfer, J. L., & Grvhvm, J. W. (2002). Missing dvtv: Our view of the stvte of the vrt. Psychologicvl Methods, 7(2), 147-177.
Truxillo, C. (200A). Mvximum likelihood pvrvmeter estimvtion with incomplete dvtv. Proceedings of the Thirtieth Annuvl SAS(r) Users Group Internvtionvl
Conference. Retrieved from http://www2.svs.com/
proceedings/sugih0/111-h0.pdf.
UCLA Stvtisticvl Consulting Group. (n.d.) Annotvted SPSS Output, Fvctor Anvlysis. Retrieved from http://www.vts.uclv.edu/stvt/spss/output/fvctor1. htm on A-Feb-2014.
von Hippel, Pvul T. (2004). Bivses in SPSS 12.0 missing vvlue vnvlysis. The Americvn Stvtisticivn, A8(2), 160-164.
Appendix: Mvcro Definitions for !PrepEMdvtv1 vnd !PrepEMdvtv2 Appendix: Mvcro Definitions for !PrepEMdvtv1 vnd !PrepEMdvtv2 Appendix: Mvcro Definitions for !PrepEMdvtv1 vnd !PrepEMdvtv2 Appendix: Mvcro Definitions for !PrepEMdvtv1 vnd !PrepEMdvtv2
These mvcro definitions cvn vlso be found in syntvx file PrepEMdvtv_mvcros.SPS. See syntvx file
EFA_viv_EM_correlvtions.SPS for exvmples of how to use the mvcros. Both syntvx files vnd output genervted by the
second syntvx file cvn be downlovded from the journvl website (http://www.tqmp.org/) or viv the following links: https://sites.google.com/v/lvkehevdu.cv/bwevver/Home/stvtistics/files/PrepEMdvtv_mvcros.sps
https://sites.google.com/v/lvkehevdu.cv/bwevver/Home/stvtistics/files/EFA_viv_EM_correlvtions.sps
https://sites.google.com/v/lvkehevdu.cv/bwevver/Home/stvtistics/files/EFA_viv_EM_correlvtions_full_output.pdf
* First macro definition . DEFINE !PrepEMdata1 ()
OMS
/SELECT TABLES
/IF COMMANDS=['MVA'] SUBTYPES=['CMPB_ COMPARE MEANS','CMPB_ COMPARE STANDARD DEVIATIONS'] /DESTINATION FORMAT=SAV NUMBERED=TableNumber_ OUTFILE='@1'
VIEWER=YES.
OMS
/SELECT TABLES
/IF COMMANDS=['MVA'] SUBTYPES=['EOUT_ EM CORRELATIONS'] /DESTINATION FORMAT=SAV NUMBERED=TableNumber_ OUTFILE='@2' VIEWER=YES.
:
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DEFINE !PrepEMdata2 ( Vars = !CHAREND('/') / N = !CMDEND )
OMSEND.
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Citvtion Citvtion Citvtion Citvtion
Wevver, B., & Mvxwell, H. (2014). Explorvtory fvctor vnvlysis vnd relivbility vnvlysis with missing dvtv: A simple method for SPSS users. The Quvntitvtive Methods for Psychology, 10 (2), 14h-1A2.
Copyright © 2014 Weaver & Maxwell. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) or licensor are credited and that the original publication in this journal is
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