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Cross-cultural measurement invariance testing in R in 5 simple steps

Cross-cultural measurement invariance testing in R in 5 simple steps

Here I am presenting a quick crash course to run invariance tests for cross-cultural research in R. R is a free programme and has an expanding list of awesome features that should be of interest to people doing cross-cultural work.

I am working with RStudio, but there are other options for running R. Here are the basic steps to get you started and run a cross-cultural equivalence test.

1. Set your working directory

This step is important because it will allow you to call your data file later on repeatedly without listing the whole path of where it is saved. For example, I saved the file that I am working with on my dropbox folder in a folder called ‘Stats’ that is in my ‘PDF’ folder.
I need to type this command:

setwd(“C:\\Users\\Ron\\Dropbox\\PDF\\Stats”)

Two important points:
a) for some strange reason you need double \\ to set your directory paths with windows.
b) make sure that there are no spaces in any of your file or directory paths. R does not like it and will throw a tantrum if you have a space somewhere.

2. Read your data into R

The most convenient way to read data into R is using .csv files. Any programme like SPSS or Excel will allow you to save your data as a .csv file. There are a few more things that we need to discuss about saving your data, but I will discuss this below.

Type:
justice <- read.csv(“justice.csv”, header = TRUE)

R is an object oriented language, which means we will constantly create objects by calling on functions: object <- function. This may seem weird at first, but will allow you to do lots of cool stuff in a very efficient way.

I am using a data that tested a justice scale, so I am calling my object ‘justice’.

3. Deal with missing data

If you have absolutely no missing data in your data file, skip this step. However most mortal researcher souls will have some missing data in their spreadsheet. R is very temperamental with missing data and we need to tell it what missing data is and how to deal with it. Some people (including myself) who are used to SPSS typically leave missing data as a blank cell in the spreadsheet. This will create problems. The best option is to write a little syntax command in spss to recode all blank cells to a constant number.
For example, I could write something like this in SPSS:

recode variable1 to variable12
(sysmis=9999) (else=copy) into
pj1 pj2 pj3 pj4 ij1 ij2 ij3 ij4 dj1 dj2 dj3 dj4.
Execute.

Now I can save those new variables as a .csv file and read into R using the step above.

Once you have executed step 2, you need to define these annoying 9999s as missing values.
The simplest and straightforward option is to write this short command that converts all these offending values into NA – the R form of missing data.

justice[justice==9999] <- NA

Note the square brackets and double ==. If you want to treat only a selected variable, you could write:

justice$pj1[justice$pj1==9999] <- NA

This tells R that you want only the pj1 variable in the dataframe justice to be treated in this way.

To check that all worked well, type:

summary(justice)

You should see something like this:

If all went well, now your minimum and maximum values are within the bounds of your original data and you have a row of NA’s a the bottom of each variable column.

4. Loading the analysis packages

R is a very powerful tool because it is constantly expanding. Researchers from around the world are uploading tools and packages that allow you to run fancy new stats all the time. However, the base installation of R does not include them. So we need to tell R which packages we want to use.
For measurement invariance tests, these three are particularly useful: lavaan (the key one) and semTools (important for the invariance tests).

Write this code to download and install the packages on your machine:

install.packages(c(“semTools”, “lavaan”))

Make sure you have good internet connectivity and you are not blocked by an institutional firewall. I had some problems recently trying to download R packages when accessing it from a university campus with a strong firewall.

Once all packages are downloaded, you need to call them before you can run any analyses:

library(“lavaan”)

library(“semTools”)

Important: You need to call these packages each time that you want to run some analyses, if you have restarted R or RStudio.

5. Running the analyses

R is an object oriented language, as I mentioned before. The analysis can be executed in a couple of steps. First we need to specify the model that we run by creating a new object that contains all the information. Then we tell R what to do with that model. Finally, we have various options for obtaining the results, that is the fit indices and parameter estimates as well as other diagnostic information.

Let’s create the model first:
cfa.justice<-‘
pj=~pj1+pj2+pj3+pj4
ij=~ij1+ij2+ij3+ij4
dj=~dj1+dj2+dj3+dj4’

This creates the model that we can then work with. The ‘=~’ denotes that the items are loading on the latent factor. This is what it looks like:

The next set of commands sets specifies what should be done with the model. In the case of a simple CFA, we can call this function:

fit.cfa <-cfa(cfa.justice, data=justice)

To identify the model, lavaan sets the loading of the first item on each latent variable to 1. This is convenient, but may be problematic if the item is not a good indicator. An alternative strategy is to set the variance of the latent variable to 1. This can be done by adding std.lv=TRUE to the fit statement.

fit.cfa <-cfa(cfa.justice, data=justice, std.lv=TRUE)

This statement runs the analysis, but we still need to request the output.

The simplest way is to use summary again. Here is an option that prints both the fit indices and the standardized parameters.

summary(fit.cfa, fit.measures= TRUE, standardized=TRUE)

Here is a truncated and annotated version of the output:

lavaan (0.5-17) converged normally after  39 iterations
Used       Total

  Number of observations                          2518        2634

# The following shows the estimator and the Chi square stats. As can be seen, we have 51 df’s, but the model does not fit that well.

  Estimator                                                  ML

Minimum Function Test Statistic              686.657

Degrees of freedom                                   51

P-value (Chi-square)                                 0.000

Model test baseline model:
Minimum Function Test Statistic            20601.993

Degrees of freedom                                 66

P-value                                                     0.000

#The incremental fit indices are in contrast quite good. The CFI and TLI should be ideally be above .95 (or at least .90). So this model does look good.

User model versus baseline model:
Comparative Fit Index (CFI)                    0.969

Tucker-Lewis Index (TLI)                       0.960

Loglikelihood and Information Criteria:
Loglikelihood user model (H0)             -44312.277

Loglikelihood unrestricted model (H1)     -43968.949
Number of free parameters                         27

#The AIC and BIC are useful for comparing models, especially non-nested models. Not the case right now.

  Akaike (AIC)                               88678.555

Bayesian (BIC)                             88835.998

Sample-size adjusted Bayesian (BIC)        88750.212

#The RMSEA should be small. Values smaller than .08 are deemed acceptable, below .05 are good. We are doing ok-ish with this one here.

Root Mean Square Error of Approximation:
RMSEA                                          0.070

90 Percent Confidence Interval          0.066  0.075

P-value RMSEA <= 0.05                          0.000

#Another useful lack of fit index. The SRMR should be below .05 if possible. This is looking good.

Standardized Root Mean Square Residual:
SRMR                                           0.037

#Now we have a print out of all the parameter estimates, including the standardized loadings, variances and  covariances. Here it is important to check whether loadings are relatively even and strong, and that the variances and covariances are reasonable (e.g., we want to avoid very high correlations between latent variables). It is looking ok overall. Item ij4 may need some careful attention.

Parameter estimates:
Information             Expected  Standard Errors                             Standard
Estimate  Std.err  Z-value  P(>|z|)   Std.lv  Std.all

Latent variables:

pj =~    pj1   1.000                                           1.202    0.789

pj2               1.009    0.027   38.005    0.000    1.213    0.790

pj3               0.769    0.025   31.003    0.000    0.924    0.643

pj4               0.802    0.024   33.244    0.000    0.963    0.687

ij =~    ij1   1.000                                             1.256    0.879

ij2               1.064    0.014   75.039    0.000    1.335    0.957

ij3               1.015    0.015   68.090    0.000    1.274    0.910

ij4               0.808    0.022   37.412    0.000    1.014    0.644

dj =~    dj1 1.000                                              1.211    0.821

dj2               1.013    0.019   51.973    0.000    1.227    0.871

dj3               1.056    0.020   52.509    0.000    1.279    0.877

dj4               1.014    0.021   48.848    0.000    1.228    0.834
Covariances:

pj ~~    ij    0.828    0.041   20.435    0.000    0.549    0.549

dj                0.817    0.040   20.187    0.000    0.562    0.562

ij ~~    dj    0.711    0.037   19.005    0.000    0.467    0.467

Variances:

pj1               0.874    0.036                      0.874    0.377

pj2               0.889    0.036                      0.889    0.377

pj3               1.213    0.039                      1.213    0.587

pj4               1.040    0.035                      1.040    0.528

ij1               0.464    0.016                      0.464    0.227

ij2               0.164    0.011                      0.164    0.084

ij3               0.336    0.013                      0.336    0.172

ij4               1.448    0.042                      1.448    0.585

dj1               0.709    0.025                      0.709    0.326

dj2               0.479    0.019                      0.479    0.242

dj3               0.489    0.020                      0.489    0.230

dj4               0.662    0.024                      0.662    0.305

pj                1.444    0.066                      1.000    1.000

ij                1.576    0.057                      1.000    1.000

dj                1.466    0.060                      1.000    1.000

However, we want to do an invariance analysis. Right now we collapsed the samples and ran an analysis across all groups. This can create problems, especially if the samples have different means (see my earlierblogpost for an explanation of this problem).

The grouping variable should be a factor, that is a string variable that has the labels. You can also use continuous variables, but then you will need to remember what each number means. In this example, I have data from three samples:

> summary(justice$nation)
Brazil        NZ         Philippines

794        1146         694

It would be informative to see whether the item loadings are similar in each group. To do this, we only need to add group=”nation” to our cfa statement.

fit.cfa.separate <-cfa(cfa.justice, data=justice, group=”nation”)

We can then print the results by using the summary statement again (remember that we have to call the new object for this analysis):

summary(fit.cfa.separate, fit.measures= TRUE, standardized=TRUE)

I am not printing the output.

Of course, this is not giving us the info that we want, namely whether the model really fits. In addition, we could ask for equal loadings, intercepts, unique variances, etc. I can’t go into details about the theory and importance of each of these parameters. I hope to find some time soon to describe this. In the meantime, have a look at this earlier article.

In R, running these analyses is really straightforward and easy. A single command line will give us all the relevant stats. Pretty amazing!!!!!

To run a full-blown invariance analysis, all you need is to type this simple command:

measurementInvariance(cfa.justice,
data=justice,
group=”nation”,
strict=TRUE)

You can write it as a single line. I just put it on separate lines to show what it actually entails. First, we call the model that we specified above, then we link it to the data that we want to analyze. After that, we specify the grouping variable (nation). The final line requests strict invariance, that is we want to get estimates for a model where loadings, intercepts and unique variances are constrained as well as a model in which we constrain the latent means to be equal. If we don’t specify the last line, we will not get the constraints in unique variances.

Here is the output, but without the strict invariance lines:

Measurement invariance tests:
Model 1: configural invariance:
chisq        df    pvalue       cfi     rmsea       bic
992.438   153.000     0.000     0.962     0.081 86921.364
Model 2: weak invariance (equal loadings):
chisq        df    pvalue       cfi     rmsea       bic
1094.436   171.000     0.000     0.958     0.080 86882.400
[Model 1 versus model 2]
delta.chisq      delta.df delta.p.value     delta.cfi
101.998        18.000         0.000         0.004
Model 3: strong invariance (equal loadings + intercepts):
chisq        df    pvalue       cfi     rmsea       bic
1253.943   189.000     0.000     0.952     0.082 86900.945
[Model 1 versus model 3]
delta.chisq      delta.df delta.p.value     delta.cfi
261.505        36.000         0.000         0.010
[Model 2 versus model 3]
delta.chisq      delta.df delta.p.value     delta.cfi
159.507        18.000         0.000         0.006
Model 4: equal loadings + intercepts + means:
chisq        df    pvalue       cfi     rmsea       bic
1467.119   195.000     0.000     0.942     0.088 87067.134
[Model 1 versus model 4]
delta.chisq      delta.df delta.p.value     delta.cfi
474.681        42.000         0.000         0.020
[Model 3 versus model 4]
delta.chisq      delta.df delta.p.value     delta.cfi
213.176         6.000         0.000         0.009

How do we make sense of this?

Model 1 is the most lenient model, no constraints are imposed on the model and separate CFA’s are estimated in each group. The CFI is pretty decent. The RMSEA is borderline.
Model 2 constraints the factor loadings to be equal. The CFI is still pretty decent, the RMSEA actually improves slightly. This is due to the fact that we have now more df’s and RMSEA punishes models with lots of free parameters. The important info comes in the line entitled Model 1 versus model 2. Here we find the difference stats. The X2 difference test is significant and we would need to reject model 2 as significant worse. However, due to the problems with the X2 difference test, many researchers treat this index with caution and examine other fit indices. One commonly examined fit index of the difference is Delta CFI, that is the difference in CFI fit from one model to the next. It should not be larger than .01. In our case, it is borderline – the delta CFI is .01.
We can then compare the other models. The next model constraints both loadings and intercepts (strong invariance). The model fit is pretty decent, we can probably assume that both loadings and intercepts are invariant across these three groups.
In contrast, constraining the latent means shows some larger problems. The latent means are likely to be different.

6. Further statistics

In this particular case, the model fits pretty well. However, often we run into problems. If there is misfit, we either trim the parameter (drop parameters or variables from the model) or we can add parameters. To see which parameters would be useful to add, we can request modification indices.
This can be done using this command:
mi <- modificationIndices(fit.cfa)
mi

The second line (mi) will print the modification indices. It gives you the expected drop in X2 as well as what the parameter estimates would be like if they were freed.

If we want to print only those modification indices above a certain threshold, let’s say 10, we could add the following line:
mi<-modificationIndices(fit.cfa)
subset (mi, mi>10)
mi

This will give us modification indices for the overall model. If we want to see modification indices for any of the constrained models, we can request them after estimating the respective model.

For example, if we want to see the modification indices after constraining the loadings to be similar, we can run the following line:

metric <-cfa(cfa.justice,
data=justice,
group=”nation”,
group.equal=c(“loadings”))
mi.metric<-modificationIndices(metric)
mi.metric

This will now give us the modification indices for this particular model.

There are more options for running constrained models. For example, this line gives the scalar invariant model:

scalar <-cfa(cfa1,
data=justice,
group=”nation”,
group.equal=c(“loadings”, “intercepts”))

As you can see, these models are replicating the models implied in the overall analysis that we got with the measurementInvariance command above.

Summary

I hope I have convinced you that measurement invariance in R using lavaan and semTools is a piece of cake. It is an awesome resource, allows you to run lots of models in no time whatsoever and of course it is free!!!!! Once you get into R, you can do even more fancy stuff and run everything from simple stats to complex SEM and ML models in a single programme.

More info on lavaan can be found here (including a pdf tutorial).

I am still in the process of learning how to navigate this awesome programme. If you have some suggestions for simplifying any of the steps or if you spot some mistakes or have any other suggestions… please get in touch and let me know 🙂
If there are some issues that are unclear or confusing, let me know too and I will try and clarify!
Look forward to hearing from you and hope you find this useful!

Cross-cultural structural invariance testing: How to run the procrustean factor rotation magic in R

Cross-cultural structural invariance testing: How to run the procrustean factor rotation magic in R

It has been a while since I last posted some stats related material. Today I am getting back to this amazing topic and focus on how we can compare factor structures across cultural samples. I have done this previously with SPSS. Today I am focusing on R, which is way cooler.

In cross-cultural psychology, we often use factor analysis (or principal component analysis) to examine the factor structure of an instrument. But how can we tell whether the factors that we find are comparable? And how similar are they to each other? In order to do this, we need to make the factor structures maximally comparable with each other and then get an overall estimate of factor similarity. This is what Procrustean Rotation and indices such as Tucker’s Phi are all about.

You may ask: Why do we need rotations which such weird Greek mythological names (if you wonder about the history of the name, look up the  mighty evil rogue Procrustes  on google)? The problem is that simply speaking any factor rotation is arbitrary and there are infinite possible solutions that can be mathematically fitted to any factor structure. Which means that there is a good chance that sample specific fluctuations will make factors look quite different. Apparently dissimilar factor structures might be more similar than we think; procrustean rotation is necessary to judge how similar they are.

Hence, I will cover the magic of how to do this in R, a free and awesome statistics program. Assuming that you are new to R, I will cover the basics of how to set your path and get your data in. If you know what you are doing, you can skip forward to the latter section.

Step 1. Set your working directory

You need to set a working directory. This step is important because it will allow you to call your data file later on repeatedly without listing the whole path of where it is saved. For example, I saved the file that I am working with on my USB drive.

I need to type this command:

setwd(“F:\\”)

If I had saved all the data on my dropbox folder in a folder called ‘Stats’ that is in my ‘PDF’ folder, then I would need to type this command:

setwd(“C:\\Users\\Ron\\Dropbox\\PDF\\Stats”)

Two important points:

a) for some strange reason you need double \\ to set your directory paths with windows. You could also use / instead of \\ (e.g., setwd(“C:/Users/Ron/Dropbox/PDF/Stats”)).  This is just to confuse you… But R is still awesome.

b) make sure that there are no spaces in any of your file or directory paths. R does not like it and will throw a tantrum if you have a space somewhere.

Step 2. Read your data into R

The most convenient way to read data into R is using .csv files. Any programme like SPSS or Excel will allow you to save your data as a .csv file.

You need to type:

ocb=read.csv(“ocb_efa.csv”, header=TRUE)

R is an object oriented language, which means we will constantly create objects by calling on functions: object

I am using a data that tested an organizational citizenship behavior scale, so I am calling my object that contains the data ‘ocb’. Just as a bit of background, I am using data from Fischer and Smith (2006). They measured self-reported work behaviour in British and East German samples, which they called extra-role behaviour. Extra-role behavior is pretty much the same as citizenship behaviour, voluntary and discretationary behaviour that goes beyond what is expected of employees, but helps the larger organization to survive and prosper. These items were supposed to measure a more passive component (factor 1) and a more proactive component (factor 2). We will need this info on the expected factors below…

The command header=TRUE (or you could make it sure and just type T) tells R that the variable names are included.

Step 3. Preparing your data (dealing with missing data, checking your data, etc.)

R does not like missing data. We will need to define which values are missing. I previously coded all missing data as -999 in SPSS or EXCEl. Now I have to declare that these annoying -999s should be treated as missing values.

If you type:

summary(ocb)

You will see that the minimum value is -999. The simplest and straightforward option is to define the missing values is to write this short command that converts all these offending values into NA – the R form of missing data.

ocb[ocb==-999]<-NA

Note the square brackets and double ==. If you want to treat only a selected variable, you could write:

ocb$ocb1[ocb$ocb1==-999]

This tells R that you want only the the first variable in the dataframe ocb to be treated in this way.

To check that all worked well, type:

summary(ocb)

You should see something like this:

If all went well, now your minimum and maximum values are within the bounds of your original data and you have a row of NA’s a the bottom of each variable column.

As you can see, we have a variable called country with 1’s and 2’s. This is not that useful, because last time I checked, these are not good names for countries and might be a bit confusing.

The best option is to convert this variable in what is called a factor in R (don’t confuse it with factor analysis). Basically, it becomes a dummy variable and we can give it labels. In my case, I have data from British and German employees, so I am using UK and German as labels.

You can type:

ocb$country<-factor(ocb$country,   #specifies the variable to be recoded

levels = c(1,2), #specifies the numeric values

labels = c(“UK”, “German”)) #specifies the labels assigned to each numeric value

If you wonder, the # allows me to add annotations to each command line, that tell me (and you) what is going on, but R is ignoring these sections.
If you type, summary(ocb) again, you should now see that there 130 responses from the UK and 184 from Germany.
There is one more thing we need to do. In our analyses, we want to compare the factor analysis results of the two samples. Therefore, we need to create two data sets for each sample that include only the variables that we need for our factor analysis. This can be achieved with the subset command, which creates a new object with only the data that we need for each analysis. At the same time, we can also use this command to select only the relevant variables for our factor analysis.
To create the UK data set, you can type:
ocb.UK<-subset(ocb, #creates a new data frame using the original ocb data frame
               country==”UK”, #this is the variable that is used for subsetting, note the double ==
               select=c(2:10)) #we only need the continuous variables which were in column 2 to 10
To see whether it worked, type:
summary(ocb.UK) #check that it worked
nrow(ocb.UK) #check that it worked, this command will give you the number of rows
Then repeat the procedure to create the German data set:
ocb.German<-subset(ocb,
                   country==”German”,
                   select=c(-1)) #if you wonder, this is an alternative way of selecting the variables, by dropping the first column which had the country dummy factor
To check, you know the drill (summary or nrow).

Step 4. Installing and loading the analysis packages for your analysis

R is a very powerful tool because it is constantly expanding. Researchers from around the world are uploading tools and packages that allow you to run fancy new stats all the time. However, the base installation of R does not include them. So we need to tell R which packages we want to use.
For the type of measurement invariance tests that I am talking about today, we will need these two: psych (written by William Revelle, an amazing package, check out some of the awesome stuff can do with this package here) and GPArotation.
Write this code to download and install the packages on your machine:
install.packages(c(“psych”, “GPArotation”))
Make sure you have good internet connectivity and you are not blocked by an institutional firewall. I had some problems recently trying to download R packages when accessing it from a university campus with a strong firewall.
Once all packages are downloaded, you need to call them before you can run any analyses:
library(“psych”)
library(“GPArotation”)
Important: You need to call these packages each time that you want to run some analyses, if you have restarted R or RStudio. Now we should be ready to start our analyses.

Step 5. Run the analysis in each sample

I have used the name factor analysis so far. Technically, I am going to use principal component analysis (PCA). There is a lot of debate whether factor analysis or principal component analysis are better… I touched upon this in class, but will not repeat it here. Let’s just stick with PCA for the time being and be happy. I will also continue to use the term ‘factors’, even though this is factually incorrect (they are principal components) and I am likely to burn in statistical hell. I am happy to brave this risk…
To run the PCA, we need to type a short command line. Let’s break it down. pca_2f.uk is the name that I gave the new object that R will create. The name is pretty much up to you, I called it pca (because I am running a PCA) with 2 factors (hence 2f) based on the British data (voila, this is what uk stands for). The command ‘principal’ tells R what to do:  run a principal component analysis. After the open brackets, I first specify the data object (ocb.UK), then how many factors I want to extract (nfactors=2), followed by the type of rotation (I decided to go with varimax rotation, which is a form of orthogonal rotation that assumes independence of factors). So this is what I write:
pca_2f.uk<-principal(ocb.UK,
                     nfactors=2,

                     rotate=”varimax”)

If you run it, nothing will happen. We just created an object that contains the PCA results. To actually see it, we can either call all the output by typing:

pca_2f.uk

Or we could sort the factor loadings by size and suppress small factor loadings (for example, factor loadings smaller than .3). To get this, write:

print.psych(pca_2f.uk, cut=0.3, sort = T)

Now you should see some output like this:

As you can see, the first item loaded on both factors. However, overall there seems to be a pretty neat two-factor structure.

Now you need to do the same thing for the German data set. This is not rocket science and I hope you would have come up with the same code like this:

pca_2f.german<-principal(ocb.German,
                         nfactors=2,
                         rotate=”varimax”)
print.psych(pca_2f.german, cut=0.3, sort = T)
The output looks like this:

The first item loads much more clearly on factor 2 in this German data set compared to the British data set. But what can say about this difference? We can’t really compare to the two factor results, because there might be arbitrary changes due to sample fluctuations or other funny jazz (this is a highly technical term).  Now we get to the crux of this whole issue, because we need to do Procrustean rotation. Procrustean rotation (have you looked up Procrustes yet?) does what the name says, it rotates and fits one solution to the other, making them directly comparable.

Before we get there, take a deep breath and have a look at this picture…

Feeling more relaxed and calmer now? Let’s move on to the real stuff!

Step 5. Run the Procrustean rotation

For those of you who have done the procrustean rotation stuff in SPSS (for a reminder, have a lookhere), you might have braced yourself for a massive typing exercise with lots of random error messages and annoying missing commas, semi-colons and winged brackets. Fear not – R is making it much easier.
To run the actual procrustean rotation, we need to type one little command line. To break it down again, we create a new object that contains our rotated factor loadings. I called it ‘pca2.uk.rotated’. We tell R what to do (run a Target Rotation… hence, called ‘TargetQ’), specify what factor loadings we want to rotate and what we want it to rotate it to – our target. I used the German sample as the target. This is a pretty arbitrary choice, but I decided to use it because a) the German sample is larger and b) the German sample had a slightly cleaner initial structure.
Here is the command:
pca2.uk.rotated<-TargetQ(pca_2f.uk$loadings, Target=list(pca_2f.german$loadings))
If we now call the object (just type the name of the object), we should see something like this:
The first first item still does show up as loading on both factors, but the loading on the first factor is somewhat reduced. We could now start a bit of a tea leaf reading exercise and look at all the little changes that have happened after rotation. This can be informative and if you have your own data sets, this is probably a good thing to do. Yet, these impressions do not allow us to get a sense of how statistically similar the two factor solutions are. Do these differences matter?
Hence, the final step for today… We need to calculate the overall similarity.

Step 6. Compute Factor Congruence Coefficients

There are a number of different ways to calculate factor congruence or factor similarity. The most common one is Tucker’s Phi. You can read up more about it in a chapter that I have written together with Johnny Fontaine. Send me a message if you want a copy.
To get Tucker’s Phi, we again have to write a single command line. The command is simple: ‘factor.congruence’ and all we need to specify is which loadings from what analyses we want to analyze. In our case, we want to compare the original German factor loadings with the procrustean rotated British loadings. Hence, we write:
factor.congruence(pca2.uk.rotated$loadings,pca_2f.german$loadings)
We will see a 2 x 2 matrix, which has Tucker’s Phi on the diagonal. As you should see, the similarity for factor 1 is .94 and for factor 2 is .97. If you compare it with the standards that we discuss in the book chapter, this is pretty good similarity. The small changes that we see across the two samples do not matter that much.
If you want another indicator, we could compute the correlation between the two factor structures. This again is relatively straightforward. Without creating a new object, we could just type (note that we use the same structure as for the factor.congruence statement):
cor(pca2.uk.rotated$loadings,pca_2f.german$loadings)
The correlation matrix shows us on the diagonal that the correlation for factor 1 is .87 and for factor 2 is .93. Therefore, the correlation coefficient suggests that factor 2 is pretty similar. However, factor 1 is not doing that great. Maybe item 1 is a big dodgy after all.
As we discuss in the chapter, it can be useful to compare the different indices. If they agree – you are sweat and you can happily go your way comparing the factor structures. If they diverge (as they do a wee bit in this case), you may want to explore further. In our case, it might make sense to remove the first item and redo the analyses. If we do this and re-run all the steps after excluding ocb1 (see the subsetting command at step 3), we will find the two structures are now beautifully similar. Nearly like identical twins… Who would have thought that of ze Germans and ze Brits…
I hope you have enjoyed this little excursion into R and procrustean rotation. I am a big fan of the capabilities of R and what you can do with it for cross-cultural analyses. I hope I got you inspired too.
Any questions or comments, please get in touch and comment 🙂
Now… rotate and relax 🙂