omega                 package:psych                 R Documentation

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_D_e_s_c_r_i_p_t_i_o_n:

     McDonald has proposed coefficient omega as an estimate of the
     general factor saturation of a test.  One way to find omega is to
     do a factor analysis of the original data set, rotate the factors
     obliquely, do a Schmid Leiman transformation, and then find omega.
     This function estimates omega as suggested by McDonald by using
     hierarchical factor analysis (following Jensen).

_U_s_a_g_e:

     omega(m, nfactors, pc = "mle",key = NULL, flip=TRUE, digits=NULL, ...)

_A_r_g_u_m_e_n_t_s:

       m: A correlation matrix or a data.frame/matrix of data 

nfactors: Number of factors believed to be group factors

      pc: pc="pa" for principal axes, pc="pc" for principal components,
          pc="mle" for maximum likelihood.

     key: a vector of +/- 1s to specify the direction of scoring of
          items.  The default is to assume all items are positively
          keyed, but if some items are reversed scored, then key should
          be specified.

    flip: If flip is TRUE, then items are automatically flipped to have
          positive correlations on the general factor. Items that have
          been reversed are shown with a - sign.

  digits: if specified, round the output to digits

     ...: Allows additional parameters to be passed through to the
          factor routines

_D_e_t_a_i_l_s:

     ``Many scales are assumed by their developers and users to be
     primarily a measure of one latent variable. When it is also
     assumed that the scale conforms to the effect indicator model of
     measurement (as is almost always the case in psychological
     assessment), it is important to support such an  interpretation
     with evidence regarding the internal structure of that scale. In
     particular, it is important to examine two related properties
     pertaining to the internal structure of such a scale. The first
     property relates to whether all the indicators forming the scale
     measure a latent variable in common. 

     The second internal structural property pertains to the proportion
     of variance in the scale scores (derived from summing or averaging
     the indicators) accounted for by this latent variable that is
     common to all the indicators (Cronbach, 1951; McDonald, 1999;
     Revelle, 1979). That is, if an effect indicator scale is primarily
     a measure of one latent variable common to all the indicators
     forming the scale, then that latent variable should account for
     the majority of the variance in the scale scores. Put differently,
     this variance ratio provides important information about the
     sampling fluctuations when estimating individuals' standing on a
     latent variable common to all the indicators arising from the
     sampling of indicators (i.e., when dealing with either Type 2 or
     Type 12 sampling, to use the terminology of Lord, 1956). That is,
     this variance proportion can be interpreted as the square of the
     correlation between the scale score and the latent variable common
     to all the indicators in the infinite universe of indicators of
     which the scale indicators are a subset. Put yet another way, this
     variance ratio is important both as reliability and a validity
     coefficient. This is a reliability issue as the larger this
     variance ratio is, the more accurately one can predict an
     individual's relative standing on the latent variable common to
     all the scale's indicators based on his or her  observed scale
     score. At the same time, this variance ratio also bears on the
     construct validity of the scale given that construct validity
     encompasses the internal structure of a scale." (Zinbarg, Yovel,
     Revelle, and McDonald, 2006).

     McDonald has proposed coefficient omega as an estimate of the
     general factor saturation of a test.  Zinbarg, Revelle, Yovel and
     Li (2005)  <URL:
     http://personality-project.org/revelle/publications/zinbarg.revelle.pmet.05.pdf>
     compare McDonald's Omega to Cronbach's alpha and Revelle's beta. 
     They conclude that omega is the best estimate. (See also Zinbarg
     et al., 2006)   

     One way to find omega is to do a factor analysis of the original
     data set, rotate the factors obliquely, do a   Schmid-Leiman
     (schmid) transformation, and then find omega.  Here we present
     code to do that.  

     Omega differs as a function of how the factors are estimated. 
     Three options are available, pc="pa"  does a principle axes factor
     analysis (factor.pa), pc="mle" uses the factanal function, and
     pc="pc" does a principal components analysis (principal).  

     For ability items, it is typically the case that all items will
     have positive loadings on the general factor.  However, for
     non-cognitive items it is frequently the case that some items are
     to be scored positively, and some negatively.  Although probably
     better to specify which directions the items are to be scored by
     specifying a key vector, if flip =TRUE (the default), items will
     be reversed so that they have positive loadings on the general
     factor.  The keys are reported so that scores can be found using
     the 'score.items' function.

     Output from omega can be shown using the  'omega.graph' function.

     Beta, an alternative to omega, is defined as the worst split half
     reliability.  It can be estimated by using ICLUST (a hierarchical
     clustering algorithm originally developed for main frames and
     written in Fortran and that is now available in R.  (For a very
     complimentary review of why the ICLUST algorithm is useful in
     scale construction, see Cooksey and Soutar, 2005). 

     The 'omega' function uses exploratory factor analysis to estimate
     the omega_h coefficient.  It is important to remember that  ``A
     recommendation that should be heeded, regardless of the method
     chosen to estimate omega_h, is to always examine the pattern of
     the estimated general factor loadings prior to estimating omega_h.
     Such an examination constitutes an informal test of the assumption
     that there is a latent variable common to all of the scale's
     indicators that can be conducted even in the context of EFA. If
     the loadings were salient for only a relatively small subset of
     the indicators, this would suggest that there is no true general
     factor underlying the covariance matrix. Just such an informal
     assumption test would have afforded a great deal of protection
     against the possibility of misinterpreting the misleading omega_h
     estimates occasionally produced in the simulations reported here."
     (Zinbarg et al., 2006, p 137).

     A simple demonstration of the problem of an omega estimate
     reflecting just one of two group factors can be found in the last
     example.

_V_a_l_u_e:

  alpha : Cronbach's alpha

 schmid : The Schmid Leiman transformed factor matrix and associated
          matrices

schmid$sl: The g factor loadings as well as the residualized factors

schmid$orthog: Varimax rotated solution of the original factors

schmid$oblique: The oblimin transformed factors

schmid$fcor: the correlation matrix of the oblique factors

schid$gloading: The loadings on the higher order, g, factor of the
          oblimin factors

     key: A vector of -1 or 1 showing which direction the items were
          scored.

_N_o_t_e:

     Requires the GPArotation package

_A_u_t_h_o_r(_s):

     <URL: http://personality-project.org/revelle.html> 
      Maintainer: William Revelle    revelle@northwestern.edu           

_R_e_f_e_r_e_n_c_e_s:

     <URL: http://personality-project.org/r/r.omega.html> 

     Revelle, W. (1979).  Hierarchical cluster analysis and the
     internal structure of tests. Multivariate Behavioral Research, 14,
     57-74. (<URL:
     http://personality-project.org/revelle/publications/iclust.pdf>)

     Zinbarg, R.E., Revelle, W., Yovel, I., & Li. W.  (2005).
     Cronbach's Alpha, Revelle's Beta, McDonald's Omega: Their
     relations with each and two alternative conceptualizations of
     reliability. Psychometrika. 70, 123-133.  <URL:
     http://personality-project.org/revelle/publications/zinbarg.revelle.pmet.05.pdf>

     Zinbarg, R., Yovel, I., Revelle, W. & McDonald, R. (2006). 
     Estimating generalizability to a universe of indicators that all
     have one attribute in common:  A comparison of estimators for
     omega.  Applied Psychological Measurement, 30, 121-144. DOI:
     10.1177/0146621605278814 <URL:
     http://apm.sagepub.com/cgi/reprint/30/2/121>

_S_e_e _A_l_s_o:

     'omega.graph' 'ICLUST', 'ICLUST.graph', 'VSS', 'schmid ',
     'make.hierarchical '

_E_x_a_m_p_l_e_s:

     ## Not run: 
     test.data <- Harman74.cor$cov
     my.omega <- omega(test.data,3)       
     print(my.omega,digits=2)
     #

     ## End(Not run)
     #create 9 variables with a hierarchical structure
     jen.data <- make.hierarchical()
     #with correlations of
     jen.data
     #find omega 
     jen.omega <- omega(jen.data,digits=2)
     jen.omega

     #create 8 items with a two factor solution, showing the use of the flip option
     sim2 <- item.sim(8)
     omega(sim2)   #an example of misidentification-- remember to look at the loadings matrices.

