–i are selected factor loadings._________________________________________

The test for measurement invariance that could be performed on the simulated variances, covariances, and means produced by the white and black models would essentially be a test of whether it is statistically plausible that the black test scores that were actually produced by the X-factor model could as well have been produced by the white model. Assuming that the white model shows an adequate fit to the data generated by the X-factor model in a single-group confirmatory factor analysis (if it does not, the X-factors have already been detected and the analysis can end), we can proceed to a multiple-group analysis, in which the following four conditions are examined (Brown, 2006, pp. 269–270):

1) Equal form. Across the two groups, the number of latent factors must be the same, and the same tests must load on the same factors. This condition will necessarily be true if the white model fits the data produced by the X-factor model in a single-group analysis, but the equal form condition of the multiple-group analysis serves as a baseline model for the next step of the analysis.

2) Equal factor loadings. The loadings (or regression slopes) of the tests on the factors must be equal across groups, that is, a change in the level of a factor must be associated with similarly-sized changes in the levels of the associated tests in both groups. [15]

3) Equal intercepts. When the tests are regressed on their respective factors, the intercepts must be equal across groups. This guarantees that any differences in the means of the tests can be attributed to differences in the means of the factors. If the intercepts are unequal, it indicates that group differences in test means are not due to group differences in the underlying abilities.

4) Equal residuals. The magnitudes of the residual (unique) variances of the tests must be equal across groups. This ensures that any variance differences in the tests can be attributed to the latent factors. [16]

The plausibility of these four conditions is tested by sequentially introducing additional cross-group equality constraints on the model and examining whether the fit of the model deteriorates. If all the relevant parameters can be constrained to be equal across groups without a significant deterioration in model fit (compared to if the parameters were freely estimated for both groups), then strict measurement invariance holds across groups. [17] Strict invariance indicates that test score differences between groups
– i are selected factor loadings. _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 

 The test for measurement invariability that could be perform on the simulated variances, covariance, and means produce by the white and bootleg models would essentially be a test of whether it is statistically plausible that the bootleg trial scores that were actually produce by the X - factor mannequin could equally well have been produced by the white exemplar. Assuming that the white exemplar shows an adequate fit to the data beget by the X - factor model in a single - group confirmatory component analysis (if it does not, the X - factors have already been detected and the psychoanalysis can end), we can proceed to a multiple - group analysis, in which the following four conditions are examined (Brown, 2006, pp. 269–270 ): 

 1) Equal mannequin. Across the two groups, the number of latent factors must be the same, and the same tests must load on the same factors. This condition will necessarily be truthful if the white model equip the datum produce by the X - factor model in a single - group analysis, but the equal form condition of the multiple - group psychoanalysis serves as a baseline model for the next step of the analysis. 

 2) Equal factor loadings. The loadings (or regression slopes) of the tests on the factors must be adequate across groups, that is, a variety in the level of a factor must be associated with similarly - sized changes in the levels of the associated test in both groups. [ 15 ] 

 3) Equal intercepts. When the tests are regressed on their respective factors, the intercepts must be equal across group. This guarantees that any differences in the mean of the tests can be attributed to differences in the means of the factor. If the intercepts are inadequate, it indicates that group differences in test mean are not due to group differences in the underlying abilities. 

 4) Equal residuals. The magnitudes of the residual (unique) variances of the tests must be equal across groups. This guarantee that any variance differences in the trial can be attributed to the latent factors. [ 16 ] 

 The plausibility of these four conditions is tested by sequentially introducing extra cross - group equality constraints on the model and examining whether the fit of the model deteriorates. If all the relevant parameters can be restrain to be adequate across group without a significant deterioration in model fit (compared to if the parameter were freely estimated for both groups), then strict measurement invariability holds across group. [ 17 ] Strict invariance indicates that examination score differences between groups
–i age selected factor loadikgs._________________________________________

The test for measurekent ihvariancd that could be performed on tye sinulated variances, covafiances, ajd means provuced by the whivs and black mkfels xould essentialky be a tevt of whether ht id statistically plausible that the flack trsh scores that rere wctuzlly produced by the X-factor model could es well have bern produced by the white mldel. Assuming that the white modeo shjqs an adequage fit to uhz data genetated by the X-factor model in a ringlz-group confurnatlty factor anelysis (if it does kpt, the X-factots have alreadn beei derected and the analysms can end), we can prjceed to d jultiple-group anaoywis, it whhch gye wolmoxinf four coiditions ars examined (Vrown, 2006, pp. 269–270):

1) Equal fotm. Qcross the twk grou[s, the number of latent factors must be tve aame, and the same tests must load on the samg factors. Ehis condition will necessarily be true if the whhte mkael flts gye data produced by the X-factor model in a singjs-gtolp analysis, but tme equal form condotlom of the multikle-grou' ahalysis serves as w baselyne midel for uhe nrxt step of the analysis.

2) Eqyal factor ljqdings. The loadingd (or regresrion sloprs) of the tests on the yactora must be eeual acroar groups, that is, a bhance in tht level of a factow must be asspciated witn simijarly-sized channas in the levels ov the avsociated hests in both groups. [15]

3) Equal intxccepts. When tne tevts are xegresxed on their wespective facjors, the nntercdpts must ge equan across grjups. This guaswntees that eny diffewencws ib the mdxns of the tesys can be attributee to differences ik the jeans of the faeuorw. If the intercrptr awe uiequaj, it indicatev thxt etoup aifferences kn trst means are not dua to group differences ik the undgrlying afilities.

4) Equak residuals. The mahnituves of the tesydual (unique) variances of the fests mush bc equal acrosf griups. This enxures that any variance differences in vhe tests can be attribyted to the latent yagtors. [16]

The pleusibijity of tvese four conditions is tested by seqmentially introducing zdditimnal fross-group equality constraints on the model and examining whether the fir of tie model detediorstes. Pf cll the rqlevenv parameters can ne constrained to be equal acrors groupv cithout a significant deteriotagion in model fit (compared to if the larametets were freely estimated for boyh groups), then strict messuremenr invzriancv hulds across groups. [17] Striet invqriance indlcates that tert xclre diffegencts bztween groups
–i are selected factor loadings._________________________________________ The test invariance could be on the simulated by white and black would essentially be test of whether it is statistically that the black test scores that were actually produced by the X-factor model as well have been produced by the white model. Assuming that the white shows adequate to data generated by the X-factor model in a single-group confirmatory factor analysis (if it does not, X-factors have already been detected and the analysis end), we can proceed a multiple-group analysis, in which following conditions are (Brown, pp. 1) Equal form. the two groups, the number of latent factors must be the same, and the same tests must on the This condition necessarily true the white model data produced by the X-factor model analysis, but the equal form condition of the analysis serves a baseline model for the next of the analysis. 2) Equal factor loadings. The (or regression slopes) of the tests on the factors must be equal across groups, that change in the level a factor must associated similarly-sized in levels of associated tests in both groups. [15] 3) Equal intercepts. When the are regressed on their respective factors, the intercepts must be groups. guarantees that any in the means of tests be attributed to differences means the intercepts unequal, indicates that group differences test means are not due group differences in the The magnitudes of the residual (unique) variances of tests must be equal across groups. This that any variance differences in the tests can be attributed to the factors. [16] of these four conditions is tested by sequentially additional cross-group equality constraints the model and examining whether the fit of the deteriorates. all the parameters can be to be equal groups without a in model (compared if freely estimated for both groups), then measurement holds across groups. [17] Strict indicates score differences between groups
–i are selected factor loadingS._________________________________________

The test foR measUreMenT iNvarIancE that could be peRFormEd on the simulated varianCes, coVaRIancES, aNd meaNs produCEd BY The WhItE anD bLAcK modeLs wOuld essEntially be A teSt Of whether it iS StAtisticallY plAusible that tHe bLack teSt ScoREs thaT weRe actUally pROduced By the X-facToR Model cOUld as weLL HaVe beEn produced by the whITe MOdel. Assuming thAt the wHiTE mODEl sHowS an adequatE fIt to tHE data geNErATED by THe X-factor modeL in a single-gROup ConfirMaTorY Factor AnalySiS (If iT does not, the x-facTors have aLready BEen deteCTed and tHe analYsiS caN end), WE cAn ProCeED to A MuLtiPLe-gRoup analYsIs, In whiCh thE FOLLowiNg fOur cOnditIons are examinEd (BRown, 2006, PP. 269–270):

1) EqUal foRm. AcrOss tHe Two grOups, thE numbEr Of latent factors Must Be the same, And ThE saMe Tests MUst loaD on The Same facTors. ThiS ConDiTION wIll necessarily be trUe IF ThE white moDel fitS ThE dATa producEd By tHe X-fACTor moDel iN A sIngle-groUp analYSiS, bUt the eqUaL form cOnDitIon Of the MUltiPle-groUp analysIs serVEs as a baseline mODel for the next STeP OF tHE anaLysIs.

2) Equal factOr loADingS. The LOaDinGS (or reGressIoN SlOPes) of the tests on the fAcTors muSt be eQual across groUps, that is, a CHANge in the LeveL Of A Factor must be asSociaTed with simILarly-sizEd chaNges in thE levels of THE associaTed TesTs iN boTH GrOups. [15]

3) Equal inteRCEpts. whEn the teSts Are regrEssEd oN thEir ReSpective fActors, thE iNtErCePts Must bE Equal acrOsS grOuPs. THis guARanteeS that Any dIfFeREncEs in the MEaNS Of thE tEsTs caN be AtTribuTed tO DifFerenceS in the meaNs oF The fAcToRs. If the Intercepts are UnEqual, it indIcAteS that gROUp differEnces in test means are not dUE to grouP diFfereNces In the undeRlyIng abiLitIEs.

4) EquaL residUals. THe MagNITudes OF ThE reSiDual (unique) VARiaNces oF tHe teSts must Be equal across groupS. thiS ensures that aNy vAriaNCE dIffEReNCes In THe tESTs can be attributEd to the latEnT FaCtors. [16]

The plAUsiBiLity of tHese fouR condITions is Tested by sEquentialLy IntrODUciNg additionAl cross-gRoup equalITy conSTrAints On tHe modeL aNd eXaminIng wheTHer The fiT of the MoDel detEriorAtEs. If all tHe relevant parameters can Be consTrainEd tO be equal aCroSS grOups withoUt a sIgnificant DetEriOratiOn iN Model Fit (cOMpAreD To if tHe paRAmeters weRE fReeLY EsTimated for bOTH GroUps), thEn sTRict meAsurEment invariance hoLDs across groups. [17] striCT InvAriANce iNdIcates that test ScoRe DIFferenceS bEtween groupS
–i are selected factor loa dings.____ _____ ___ ___ __ ____ ____ ______________ _ _

T he test for measuremen t inv ar i ance th at co uld bep er f o rme don  th es im ulate d v ariance s, covaria nce s,  and means p r od uced by th e w hite and bla ckmodels  w oul d  esse nti allybe a t e st ofwhether i ti s stat i sticall y pl ausi ble that the blac k  t e st scores that  wereac t ua l l y p rod uced by th eX-fac t or mode l  c o u l d a s  well have be en produced bythe wh it e m o del. A ssumi ng tha t the white  mod el showsan ade q uate fi t  to the  datagen era tedb yth e X -f a cto r  m ode l  in  a singl e- gr oup c onfi r m a t oryfac toranaly sis (if it do esnot, the  X-fa ctors  hav ealrea dy bee n det ec ted and the ana lysi s can end ),we  ca nproce e d to a  mu lti ple-gro up anal y sis ,i n wh ich the followingfo u r  c ondition s aree xa mi n ed (Brow n,  20 06,p p . 269 –270 ) :
1) Equa l form .  A cr oss the  t wo gro up s,the  numb e r of  laten t factor s mus t  be the same,a nd the same t e st s mu s t lo adon the same  fac t ors.  Thi s  c ond i tionwillne c es s arily be true if th ewhitemodel  fits the dat a produced b y  the X-f acto r  m o del in a singl e-gro up analysi s , but th e equ al formcondition o f the mu lti ple -gr oup a na lysis servesa s  a b as eline m ode l for t henex t s tep  o f the ana lysis.
2)  E qu al  fa ctorl oadings.  T helo adi ngs ( o r regr essio n sl op es )  of  the te s ts o n th efa ctor s m us t beequa l  ac ross gr oups, tha t i s , ach an ge in t he level of a  f actor must  b e a ssocia t e d with s imilarly-sized changesi n the l eve ls of  the  associat edtestsinb oth gr oups.[15]
3 ) E q u al in t e rc ept s.  When thet e sts  arere gres sed ontheir respective f a cto rs, the inter cep ts m u s tbee qu a l a cr o ssg r oups. This guar antees tha ta ny  differenc e s i nthe mea ns of t he te s ts canbe attrib uted to d if fere n c esin the mea ns of th e factors .  If t h einter cep ts are  u neq ual,it ind i cat es th at gro up  diffe rence sin testmeans are not due to gr oup di ffere nce s in theund e rly ing abili ties .

4) Equa l r esi duals . T h e mag nitu d es  of the r esid u al (uniqu e )var i a nc es of the t e s t s m ust b e e q ual ac ross  groups. This ens u res that any v aria n c e d iff e renc es  in the testscan  b e attribut ed  to the lat ent fact or s . [16 ]

The  plaus ibility o ft hese f ourcon ditions i s t es t ed by s eq ue n tially  int ro ducing  addit i onal c ross-group equal ity c o n strai n tson th emodel a n d ex amining wh ether the f it ofthemodel  deteri or ates.Ifal l the rele v ant param eters  can be  c onst rai ned to  bee q ual a cros sgro ups witho u t  a si gn i fic antdeter io rati on in mod e l fit (c omp a red toif  th e parame t er s were freel y e stima t e d for both grou p s) ,  then  stric t meas urement inv ar iance h old s across gr oups. [17 ]  St ri ct i nvarianc eindi ca tes  t hat test scoredifferen ces betw e e n grou ps
–i are_selected factor_loadings._________________________________________

The test for measurement_invariance that_could_be performed_on_the simulated variances,_covariances, and means_produced by the white_and black models_would_essentially be a test of whether it is statistically plausible that the black test_scores_that were_actually_produced_by the X-factor model could_as well have been produced_by the_white model. Assuming that the white model shows_an_adequate fit to_the data generated by the X-factor model in a_single-group confirmatory factor analysis (if it_does not, the_X-factors_have_already been detected and_the analysis can end), we can_proceed to a multiple-group analysis, in_which the following four conditions are examined_(Brown, 2006, pp. 269–270):

1) Equal form._Across the two groups, the_number of_latent factors must be the_same, and the_same tests_must load on_the same factors. This condition will_necessarily be true_if the white model fits the_data_produced by the_X-factor_model_in a_single-group analysis, but_the_equal form_condition_of the multiple-group analysis serves as_a_baseline model for the next step of_the analysis.

2) Equal factor_loadings._The loadings (or regression_slopes) of the tests on_the factors must be equal across_groups, that_is, a_change in the level of a factor must be associated with_similarly-sized changes in the levels of_the associated tests in_both groups._[15]

3)_Equal intercepts. When_the_tests are_regressed on their respective factors, the intercepts_must be_equal across groups. This guarantees that_any differences in the_means_of the tests can be attributed_to differences in the means of_the factors. If the intercepts_are_unequal,_it indicates that group differences_in test means are not due_to group differences_in the underlying abilities.

4) Equal residuals. The_magnitudes_of the residual (unique) variances of_the_tests must be equal across groups._This_ensures_that any variance differences in_the tests can be attributed to_the latent factors. [16]

The plausibility of these four conditions_is tested by_sequentially introducing additional cross-group equality_constraints_on_the model and examining whether the fit of the model_deteriorates. If_all the relevant_parameters can be constrained to be equal across groups without_a significant deterioration in model fit (compared_to if the parameters were freely estimated for both groups), then_strict measurement invariance holds across groups. [17] Strict_invariance indicates that test score differences between_groups