Overview of this and related documents

Two alternative estimates of reliability that take into account the hierarchical structure of the inventory are McDonald’s ω_h and ω_t . These may be found in in one step using one of two functions in the psych package: the omega function for an exploratory analysis (See Figure [fig:omega.9]) or omegaSem for a confirmatory analysis using the sem package solution based upon the exploratory solution from omega. This guide explains how to do it for the non or novice user. These set of instructions are adapted from three different sets of notes that the interested reader might find helpful: A set of slides developed for a two hour short course in given in May, 2012 to the Association of Psychological Science http://personality-project.org/r/aps/aps-short.pdf as well as a short guide to http://personality-project.org/r/ for psychologists and the vignette for the psych package http://cran.r-project.org/web/packages/psych/vignettes/overview.pdf.

McDonald has proposed coefficient omega (hierarchical) (ω_h) as an estimate of the general factor saturation of a test. http://personality-project.org/revelle/publications/zinbarg.revelle.pmet.05.pdf compare McDonald’s ω_h to Cronbach’s α and Revelle’s β. They conclude that ω_h is the best estimate. (See also and http://personality-project.org/revelle/publications/revelle.zinbarg.08.pdf ).

One way to find ω_h is to do a factor analysis of the original data set, rotate the factors obliquely, factor that correlation matrix, do a Schmid-Leiman (schmid) transformation to find general factor loadings, and then find ω_h. This is done using the omega function in the psych package in . This requires installing and using both as well as the psych package .

Assuming that your data are in a file called my.data, and that you have the necessary packages installed, to find ω_h requires two lines of code:

• library(psych)

• omega(my.data)

The resulting output will be both graphical (see Figure [fig:omega.9]) and textual.

This guide helps the naive user to issue those two lines.

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[fig:omega.9]

Install R and relevant packages

To find ωh in obviously requires installing on your computer. This is very easy to do (see section [install]) and needs to be done once.

The power of is in the supplemental packages. At a minimum, you will need to install three package (psych , GPArotation and MASS . With these three packages, you will be be able to find ω_h using Exploratory Factor Analysis. If you want to find it using Confirmatory Factor Analysis, you will also need to add the sem and matrixcalc packages. For a more complete installation of a number of psychometric packages, you can install and activate a package (ctv) that installs a large set of psychometrically relevant packages. As is true for , you will need to install packages just once.

Install R for the first time

1. Download from R Cran (http://cran.r-project.org/) (see section [install])

   • Choose appropriate operating system and download compiled R

2. Install R (current version is 2.15.2)

3. Start R

4. Add useful packages (just need to do this once) (see section [installing])

   1. install.packages(c("psych","GPArotation","MASS")) #the minimum requirement

   2. or if you want to do CFA

      install.packages(c("psych","GPArotation","MASS","sem","matrixcalc"))

5. or if you want to install the psychometric task views

   1. install.packages("ctv") #this downloads the task view package

   2. library(ctv) #this activates the ctv package

   3. install.views("Psychometrics") #among others

   4. Take a 5 minute break

6. Activate the package(s) you want to use (e.g., psych)

   • library(psych) #necessary for finding omega

     psych will automatically activate the other packages it needs, as long as they are installed. Note that psych is updated roughly quarterly, the current version is 1.3.1

7. Use

Install R

[install] First go to the Comprehensive R Archive Network (CRAN) at http://cran.r-project.org: (Figure [fig:cran])

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[fig:cran]

Choose your operating system and then download and install the appropriate version

For a PC: (Figure [fig:pc])

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Download and install the appropriate version – PC

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For a Mac: download and install the appropriate version – Mac (Figure [fig:mac])

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Starting R on a PC

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Start up R and get ready to play (Mac version)

R version 2.15.0 Patched (2012-03-30 r58887) – "Easter Beagle" Copyright (C) 2012 The R Foundation for Statistical Computing ISBN 3-900051-07-0 Platform: x86₆4-apple-darwin9.8.0/x86₆4 (64-bit)

R is free software and comes with ABSOLUTELY NO WARRANTY. You are welcome to redistribute it under certain conditions. Type ’license()’ or ’licence()’ for distribution details.

Natural language support but running in an English locale

R is a collaborative project with many contributors. Type ’contributors()’ for more information and ’citation()’ on how to cite R or R packages in publications.

Type ’demo()’ for some demos, ’help()’ for on-line help, or ’help.start()’ for an HTML browser interface to help. Type ’q()’ to quit R.

[R.app GUI 1.43 (5920) x86₆4-apple-darwin9.8.0]

[Workspace restored from /Users/revelle/.RData] [History restored from /Users/revelle/.Rapp.history] > # > is the prompt for all commands #is for comments

Install relevant packages

[installing] Once is installed on your machine, you still need to install a few relevant “packages”. Packages are what make so powerful, for they are special sets of functions that are designed for one particular application. In the case of the psych package, that application is for doing the kind of basic data analysis and psychometric analysis that personality psychologists find particularly useful.

You may either install the minimum set of packages necessary to do the analysis using an EFA approach (recommended) or a few more packages to do both the EFA and a CFA approach. It is also possible to add many psychometrically relevant packages all at once by using the “task views” approach.

Install the minimum set

• install.packages(c("psych","GPArotation","MASS"))

Install the sem package as well

• install.packages(c("psych","GPArotation","MASS","sem","matrixcalc"))

Install all the psychometric packages from the “psychometrics” task view.

• install.packages("ctv") #this downloads the task view package

• library(ctv) #this activates the ctv package

• install.views("Psychometrics") #among others

• Take a 5 minute break

You are almost ready. But first you need to make the psych package active. You only need to do this once per session.

• library(psych) #necessary for finding omega

You are almost there. But first you need to read in your data from your data file.

Read in the data

There are of course many ways to enter data into . Reading from a local file using read.table is perhaps the most preferred. You first need to find the file and then read it. This can be done with the file.choose and read.table functions:

file.name <- file.choose() my.data <- read.table(file.name)

file.choose opens a search window on your system just like any open file command does. It doesn’t actually read the file, it just finds the file. The read command is also necessary.

Copy the data from another program using the copy and paste commands of your operating system

However, many users will enter their data in a text editor or spreadsheet program and then want to copy and paste into . This may be done by using read.table and specifying the input file as “clipboard" (PCs) or “pipe(pbpaste)" (Macs). Alternatively, the read.clipboard set of functions are perhaps more user friendly:

read.clipboard

is the base function for reading data from the clipboard.

read.clipboard.csv

for reading text that is comma delimited.

read.clipboard.tab

for reading text that is tab delimited (e.g., copied directly from an Excel file).

read.clipboard.lower

for reading input of a lower triangular matrix with or without a diagonal. The resulting object is a square matrix.

read.clipboard.upper

for reading input of an upper triangular matrix.

read.clipboard.fwf

for reading in fixed width fields (some very old data sets)

For example, given a data set copied to the clipboard from a spreadsheet, just enter the command

> my.data <- read.clipboard()

This will work if every data field has a value and even missing data are given some values (e.g., NA or -999). If the data were entered in a spreadsheet and the missing values were just empty cells, then the data should be read in as a tab delimited or by using the read.clipboard.tab function.

> my.data <- read.clipboard(sep="") #define the tab option, or > my.tab.data <- read.clipboard.tab() #just use the alternative function

For the case of data in fixed width fields (some old data sets tend to have this format), copy to the clipboard and then specify the width of each field (in the example below, the first variable is 5 columns, the second is 2 columns, the next 5 are 1 column the last 4 are 3 columns).

> my.data <- read.clipboard.fwf(widths=c(5,2,rep(1,5),rep(3,4))

Import from an SPSS or SAS file

To read data from an SPSS, SAS, or Systat file, you must use the foreign package. This might need to be installed (see [installing] for instructions for installing packages).

read.spss reads a file stored by the SPSS save or export commands.

read.spss(file, use.value.labels = TRUE, to.data.frame = FALSE,
          max.value.labels = Inf, trim.factor.names = FALSE,
          trim_values = TRUE, reencode = NA, use.missings = to.data.frame) 
          

file

Character string: the name of the file or URL to read.

use.value.labels

Convert variables with value labels into R factors with those levels?

to.data.frame

return a data frame? Defaults to FALSE, probably should be TRUE in most cases.

max.value.labels

Only variables with value labels and at most this many unique values will be converted to factors if use.value.labels  = TRUE.

trim.factor.names

Logical: trim trailing spaces from factor levels?

trim_values

logical: should values and value labels have trailing spaces ignored when matching for use.value.labels  = TRUE?

use.missings

logical: should information on user-defined missing values be used to set the corresponding values to NA?

An example of reading from an SPSS file and then describing the data set to make it looks ok.

> library(foreign) > datafilename <- "http://personality-project.org/r/datasets/finkel.sav" > eli <- read.spss(datafilename,to.data.frame=TRUE, use.value.labels=FALSE)

> describe(eli,skew=FALSE)

var n mean sd median trimmed mad min max range se USER* 1 69 35.00 20.06 35 35.00 25.20 1 69 68 2.42 HAPPY 2 69 5.71 1.04 6 5.82 0.00 2 7 5 0.13 SOULMATE 3 69 5.09 1.80 5 5.32 1.48 1 7 6 0.22 ENJOYDEX 4 68 6.47 1.01 7 6.70 0.00 2 7 5 0.12 UPSET 5 69 0.41 0.49 0 0.39 0.00 0 1 1 0.06

Some simple descriptive statistics before you start

Although you probably want to jump right in and find ω, you should first make sure that your data are reasonable. Use the describe function to get some basic descriptive statistics. This next example takes advantage of a built in data set.

my.data <- sat.act #built in example – replace with your data describe(my.data)

var n mean sd median trimmed mad min max range skew kurtosis se gender 1 700 1.65 0.48 2 1.68 0.00 1 2 1 -0.61 -1.62 0.02 education 2 700 3.16 1.43 3 3.31 1.48 0 5 5 -0.68 -0.07 0.05 age 3 700 25.59 9.50 22 23.86 5.93 13 65 52 1.64 2.42 0.36 ACT 4 700 28.55 4.82 29 28.84 4.45 3 36 33 -0.66 0.53 0.18 SATV 5 700 612.23 112.90 620 619.45 118.61 200 800 600 -0.64 0.33 4.27 SATQ 6 687 610.22 115.64 620 617.25 118.61 200 800 600 -0.59 -0.02 4.41

There are, of course, all kinds of things you could do with your data at this point, but read about them in the vignette for the psych package http://cran.r-project.org/web/packages/psych/vignettes/overview.pdf.

Using the omega function to find ω

Two alternative estimates of reliability that take into account the hierarchical structure of the inventory are McDonald’s ω_h and ω_t. These may be found using the omega function for an exploratory analysis (See Figure [fig:omega.9]) or omegaSem for a confirmatory analysis using the sem based upon the exploratory solution from omega.

Background on the ω statistics

has proposed coefficient omega (hierarchical) (ω_h) as an estimate of the general factor saturation of a test. http://personality-project.org/revelle/publications/zinbarg.revelle.pmet.05.pdf compare McDonald’s ω_h to Cronbach’s α and Revelle’s β. They conclude that ω_h is the best estimate. (See also and http://personality-project.org/revelle/publications/revelle.zinbarg.08.pdf ).

One way to find ω_h is to do a factor analysis of the original data set, rotate the factors obliquely, factor that correlation matrix, do a Schmid-Leiman (schmid) transformation to find general factor loadings, and then find ω_h.

ω_h differs slightly as a function of how the factors are estimated. Three options are available, the default will do a minimum residual factor analysis, fm=“pa" does a principal axes factor analysis (factor.pa), and fm=“mle" provides a maximum likelihood solution.

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. Arbitrarily reversing items this way can overestimate the general factor. (See the example with a simulated circumplex).

β, an alternative to ω, is defined as the worst split half reliability. It can be estimated by using iclust (Item Cluster analysis: a hierarchical clustering algorithm). For a very complimentary review of why the iclust algorithm is useful in scale construction, see .

The omega function uses exploratory factor analysis to estimate the ω_h coefficient. It is important to remember that “A recommendation that should be heeded, regardless of the method chosen to estimate ω_h, is to always examine the pattern of the estimated general factor loadings prior to estimating ω_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 ω_h estimates occasionally produced in the simulations reported here." .

Although ω_h is uniquely defined only for cases where 3 or more subfactors are extracted, it is sometimes desired to have a two factor solution. By default this is done by forcing the schmid extraction to treat the two subfactors as having equal loadings.

There are three possible options for this condition: setting the general factor loadings between the two lower order factors to be “equal" which will be the $\sqrt{r_{ab}}$ where r_ab is the oblique correlation between the factors) or to “first" or “second" in which case the general factor is equated with either the first or second group factor. A message is issued suggesting that the model is not really well defined. This solution discussed in Zinbarg et al., 2007. To do this in omega, add the option=“first" or option=“second" to the call.

Although obviously not meaningful for a 1 factor solution, it is of course possible to find the sum of the loadings on the first (and only) factor, square them, and compare them to the overall matrix variance. This is done, with appropriate complaints.

In addition to ω_h, another of McDonald’s coefficients is ω_t. This is an estimate of the total reliability of a test.

McDonald’s ω_t, which is similar to Guttman’s λ₆, (see guttman) uses the estimates of uniqueness u² from factor analysis to find e_j². This is based on a decomposition of the variance of a test score, V_x into four parts: that due to a general factor, g⃗, that due to a set of group factors, f⃗, (factors common to some but not all of the items), specific factors, s⃗ unique to each item, and e⃗, random error. (Because specific variance can not be distinguished from random error unless the test is given at least twice, some combine these both into error).

Letting $\vec{x} = \vec{cg} + \vec{Af} + \vec {Ds} + \vec{e} $ then the communality of item$_j$, based upon general as well as group factors, h_j² = c_j² + ∑ f_ij² and the unique variance for the item u_j² = σ_j²(1 − h_j²) may be used to estimate the test reliability. That is, if h_j² is the communality of item$_j$, based upon general as well as group factors, then for standardized items, e_j² = 1 − h_j² and
$\omega_t = \frac{\vec{1}\vec{cc'}\vec{1} + \vec{1}\vec{AA'}\vec{1}'}{V_x} = 1 - \frac{\sum(1-h_j^2)}{V_x} = 1 - \frac{\sum u^2}{V_x}$
Because h_j² ≥ r_smc², ω_t ≥ λ₆.

It is important to distinguish here between the two ω coefficients of McDonald, 1978 and Equation 6.20a of McDonald, 1999, ω_t and ω_h. While the former is based upon the sum of squared loadings on all the factors, the latter is based upon the sum of the squared loadings on the general factor.
$\omega_h = \frac{ \vec{1}\vec{cc'}\vec{1}}{V_x}$

Another estimate reported is the omega for an infinite length test with a structure similar to the observed test. This is found by
$\omega_{\inf} = \frac{ \vec{1}\vec{cc'}\vec{1}}{\vec{1}\vec{cc'}\vec{1} + \vec{1}\vec{AA'}\vec{1}'}$

It can be shown In the case of simulated variables, that the amount of variance attributable to a general factor (ω_h) is quite large, and the reliability of the set of items is somewhat greater than that estimated by α or λ₆.

Using the omega function

Just call it. For the next example, we find ω for a data set from Thurstone. To find it for your data, replace Thurstone with my.data.

> omega(Thurstone)

Omega Call: omega(m = Thurstone) Alpha: 0.89 G.6: 0.91 Omega Hierarchical: 0.74 Omega H asymptotic: 0.79 Omega Total 0.93

Schmid Leiman Factor loadings greater than 0.2 g F1* F2* F3* h2 u2 p2 Sentences 0.71 0.57 0.82 0.18 0.61 Vocabulary 0.73 0.55 0.84 0.16 0.63 Sent.Completion 0.68 0.52 0.73 0.27 0.63 First.Letters 0.65 0.56 0.73 0.27 0.57 4.Letter.Words 0.62 0.49 0.63 0.37 0.61 Suffixes 0.56 0.41 0.50 0.50 0.63 Letter.Series 0.59 0.61 0.72 0.28 0.48 Pedigrees 0.58 0.23 0.34 0.50 0.50 0.66 Letter.Group 0.54 0.46 0.53 0.47 0.56

With eigenvalues of: g F1* F2* F3* 3.58 0.96 0.74 0.71

general/max 3.71 max/min = 1.35 mean percent general = 0.6 with sd = 0.05 and cv of 0.09

The degrees of freedom are 12 and the fit is 0.01

The root mean square of the residuals is 0 The df corrected root mean square of the residuals is 0.01

Compare this with the adequacy of just a general factor and no group factors The degrees of freedom for just the general factor are 27 and the fit is 1.48

The root mean square of the residuals is 0.1 The df corrected root mean square of the residuals is 0.16

Measures of factor score adequacy g F1* F2* F3* Correlation of scores with factors 0.86 0.73 0.72 0.75 Multiple R square of scores with factors 0.74 0.54 0.52 0.56 Minimum correlation of factor score estimates 0.49 0.08 0.03 0.11 >

Estimating ω_h using Confirmatory Factor Analysis

The omegaSem function will do an exploratory analysis and then take the highest loading items on each factor and do a confirmatory factor analysis using the sem package. These results can produce slightly different estimates of ω_h, primarily because cross loadings are modeled as part of the general factor.

> omegaSem(r9,n.obs=500)

Call: omegaSem(m = r9, n.obs = 500) Omega Call: omega(m = m, nfactors = nfactors, fm = fm, key = key, flip = flip, digits = digits, title = title, sl = sl, labels = labels, plot = plot, n.obs = n.obs, rotate = rotate, Phi = Phi, option = option) Alpha: 0.75 G.6: 0.74 Omega Hierarchical: 0.66 Omega H asymptotic: 0.84 Omega Total 0.78

Schmid Leiman Factor loadings greater than 0.2 g F1* F2* F3* h2 u2 p2 V1 0.70 0.53 0.47 0.93 V2 0.70 0.52 0.48 0.94 V3 0.54 0.32 0.68 0.91 V4 0.53 0.46 0.50 0.50 0.57 V5 0.44 0.44 0.39 0.61 0.50 V6 0.40 0.32 0.26 0.74 0.59 V7 0.31 0.31 0.21 0.79 0.48 V8 0.34 0.44 0.30 0.70 0.37 V9 0.24 0.36 0.19 0.81 0.32

With eigenvalues of: g F1* F2* F3* 2.18 0.52 0.08 0.44

general/max 4.21 max/min = 6.17 mean percent general = 0.62 with sd = 0.24 and cv of 0.39

The degrees of freedom are 12 and the fit is 0.03 The number of observations was 500 with Chi Square = 14.23 with prob < 0.29 The root mean square of the residuals is 0.01 The df corrected root mean square of the residuals is 0.03 RMSEA index = 0.02 and the 90 BIC = -60.35

Compare this with the adequacy of just a general factor and no group factors The degrees of freedom for just the general factor are 27 and the fit is 0.21 The number of observations was 500 with Chi Square = 103.64 with prob < 6.4e-11 The root mean square of the residuals is 0.05 The df corrected root mean square of the residuals is 0.08

RMSEA index = 0.076 and the 90 BIC = -64.15

Measures of factor score adequacy g F1* F2* F3* Correlation of scores with factors 0.86 0.63 0.25 0.59 Multiple R square of scores with factors 0.74 0.39 0.06 0.35 Minimum correlation of factor score estimates 0.48 -0.21 -0.88 -0.30

Omega Hierarchical from a confirmatory model using sem = 0.68 Omega Total from a confirmatory model using sem = 0.78 With loadings of g F1* F2* F3* h2 u2 V1 0.73 0.54 0.46 V2 0.68 0.29 0.54 0.46 V3 0.51 0.22 0.31 0.69 V4 0.54 0.47 0.51 0.49 V5 0.45 0.42 0.38 0.62 V6 0.39 0.31 0.25 0.75 V7 0.34 0.34 0.23 0.77 V8 0.36 0.39 0.28 0.72 V9 0.26 0.33 0.18 0.82

With eigenvalues of: g F1* F2* F3* 2.21 0.49 0.14 0.38