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 <- 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 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] <- NA

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 look here), 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 :)

A gentle intro to cross-cultural equivalence - or how can we measure across cultures?

Psychology is the study of human behaviour and mental processes through scientific methods. The claim of psychology is often to be universal, that is applicable to all of humanity. Using scientific methods, we psychologists rely on a systematic and objective process of proposing and testing hypotheses and making predictions about the state of human nature.  Ever since the beginning of psychology as an academic discipline, the scientific quest to quantify natural occurrences to better understand and predict them in the future became one of the ultimate goals. Of course, this requires often extensive qualitative research, but ultimately the hope was and is that we can understand a behaviour or mental process so precisely that we can quantitatively measure it and also change it.

The application of such quantitative methods are now often taken for granted, even though the levels of quantification may vary. For example, we may want to select the most able person for a particular job, refer a child with learning problems to a specialist or we may wish to help a person with mental health problems to fully function in society again. Even though all these problems can be phrased in qualitative terms (a good person for the job, a child that has problems learning, a person who is not well), these are essentially quantitative problems because they always have some reference to implicit or explicit standards. A person might be BETTER qualified than another to take up a job or a person may have GREATER problems understanding concepts or material than 75% of the children of her age. Therefore, in many day-to-day situations we make implicit and intuitive quantitative statements.

If we want to make quantitative statements about a scientific concept, we run into one of the central problems in psychology. This is namely WHAT do we want to make a comparison about? Or in other words, how do we define a psychological construct so that we can measure it? A geographer, chemist or physicist is unlikely to phase the problems that psychologists have… after all, we can easily measure distances (e.g., how far is Auckland from Wellington), we have ways of dating the age of a piece of rock or we can measure the energy of particles when we collide them at the near speed of light. Psychologists on the other hand are dealing with intangible concepts that are difficult to specify. Most of you are familiar with concepts such as intelligence, attitudes, personality traits, depression or identity. However, if we were to ask you to pinpoint any of these concepts in the real world, we would be unable to do so. Our psychological terminology refer to unobserved mental constructs that we create in our community of fellow psychologists to indicate a particular set of problems, describe a particular set of behaviours or mental representations. I would argue that underlying many of these psychological terms are assumptions about relative coherence, stability, generalizability and potentially even some general biological foundations that lead to the emergence of such a syndrome. Therefore, we don’t just invent these terms on a whim, but we think that there is something meaningful to them that we think is important enough to look into and tell other people about.

Therefore, the first issue in any psychological study, even though it may not seem obvious anymore, is to clearly and unambiguously define and specify what we want to study. What is our construct or process of interest? It is at this point, that culture will throw the first curve ball at any psychologist attempting to address this question. How can we make sure that our definition or mental construct of our psychological term or process is actually valid or does have some meaning in another cultural context? How does our upbringing in a highly developed Western society influence how we think about psychological constructs? Can we assume that identity is a concept that is meaningful in a village in the lowland Amazon basin? Is our definition of depression applicable to refugees coming from Syria or Iraq? Is conscientiousness a useful term to screen out applicants for jobs in an international organization? Therefore, the first problem in any psychological study is to unambiguously define and describe the psychological process for all the populations that we are interested in. We could think of this as a mental bubble that we draw around some problem or process. Does this bubble ‘exist’ in all the different cultures that we want to include in our study? How can we find out whether this bubble is meaningful and has some value or relevance for all the local populations? We will discuss this as the question of functional equivalence.

If we are confident that there is some value to this mental bubble of ours (let’s say, depression, personality or identity) and that the terms are meaningful in two or more cultures, then we need to find good indicators for it. In psychological terms, this is called operationalization. How can we empirically say that one person has more of this latent category quality that we just created with our mental bubble compared to another person? What would be a good indicator to tell us that one person is better for a job compared to another person or that one person is a better learner than another, who in turn may need some help? Here again, culture will throw lots of beautiful little challenges at us. We need to find indicators that are meaningful and relevant in each cultural context, but obviously we would still need to be able to compare the results across contexts. Therefore, we can’t have indicators that are relevant and meaningful in each context, but cannot be compared across cultures. We want to aim for some level of comparability. For example, is staying late at your desk a good indicator of being conscientious? Or could it be seen as being disorganized and incompetent? What if people are unfamiliar with office jobs? Is the number of items that you circled the temple this morning before going to work a better indicator of your conscientiousness? Is the ability to track animals over long distances and varied terrain a good indicator of concentration?  Or should we give people lots of d’s and b’s and p’s and q’s and then ask them to count how many p and q’s were together in each line? Should we measure intelligence by asking people to name as many types of medicinal plans for diarrhoea? Or give them complex questions about history and philosophy? This problem of identifying good measurement indicators will be called structural equivalence. Obviously, how we define and how we operationalize a construct is very much dependent on each other. For this reason, some researchers lump the two terms together as construct equivalence. For reasons that we will discuss later, I prefer to keep them separate.

So, we now have a mental bubble and we have a number of indicators that give us some clue about the latent bubble. However, we don’t actually know how good each of these indicators is in representing that latent bubble. We need to find a way to show us how well each indicator works in each of our cultures. In other words, is the same indicator better in capturing a key aspect of our construct in one culture compared to another? For example, is going to parties and having lots of friends a good indicator of extraversion? Is having many wives a good indicator of social status in all cultures? Is staying late at work to finish a good indicator in all cultures for high conscientiousness? This problems is called metric equivalence. It is the question about the relative strength of the indicator-latent variable relationship. In technical terms, we are concerned with the equivalence of factor loadings or item slopes in classic test theory or the item discriminability in item response theory.

Finally, we may be convinced that our indicators work equally well in all contexts. Each questionnaire or test items is really giving us a good and reliable insight into the construct. But there may be still problems. Some items, even though they have the same relationship with the latent construct in all cultures, may still be a bit more difficult or easier in one context compared to another.  If I would ask you to name the capital of Benin, most of you would probably struggle finding the correct answer. Benin is a country that is quite far from our thoughts and most of us will never set foot in this place or may not have heard about it in the media. However, if I would ask you about the capital city of one of your neighbouring countries, you would probably quite easily be able to name it. Therefore, asking about the capital of Benin would be easier for somebody living in Togo or Nigeria compared to somebody living in NZ or Denmark. This is the issue of full score or scalar equivalence. Technically, we would look at the invariance of item intercepts (in a multi-group CFA) or the differential item difficulty (in IRT).

In summary, measuring psychological attributes or processes across cultural contexts is quite difficult. I gave some relatively superficial and easy examples to make this a relatively non-technical and easy intro to the problem. We need to define our construct – draw our mental bubble around what we want to study. The first step in any cultural study then is to make sure that this construct or mental bubble is meaningful and functional in all cultures that we want to study. Once we think this is the case, we need to find good indicators that are observable and give us some insight into the position or state of an individual in relation to our mental bubble. We then need to discuss whether the indicators are equally good in all contexts or whether some are better in telling us something about a person or process in one cultural context compared to another. Finally, we need to find out whether all indicators are equally easy or difficult. Only once we have fulfilled this last criterion can we actually make any comparisons between individuals or groups across cultures. This is a tough task and unfortunately, most studies that you will see in the literature do fall well short of it. But this is the challenge that we really need to meet in order to develop a meaningful and universal psychological science. 

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 earlier blogpost 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!

A crisis in cultural psychology? Lack of replications, bias & publication pressures

Social psychology is facing an existential crisis. Ype Poortinga and I took the opportunity to examine how cross-cultural psychology fares in comparison. What is the background? A collective drive for presenting novel, sexy and sensational findings has propelled social psychology into a minefield of public mistrust and claims of being a pseudoscience. The list of sins in the eyes of the public are long: Central methods at the core of the discipline such as priming have been challenged, the drive to find significant differences has led to a neglect of the meaningfulness of psychological findings, publication pressures opened the doors for unscientific data massaging and most notoriously, glamorous stars of the discipline have been found to fabricate their data. There has rarely been a month since the now infamous Staples affair, when the field was not in the spotlight of public and internal scrutiny. This series of events has led to some agonizing soul-searching among psychologists.

Addressing methodological vulnerabilities in research on behavior and culture

Ype Poortinga and myself used the opportunity of the 22nd International conference of Cross-Cultural Psychology (organized by IACCP) to critically examine how cross-cultural psychology as a sister discipline of social psychology is faring. We assembled an A-list of leading cross-cultural psychologists and former editors of the flagship journal for research on culture and psychology (Journal of Cross-Cultural Psychology). Our instructions were simple: we requested them to critically evaluate the methods of our field and comment on ways how our field may move forward. Ype and I also provided a summary of our own concerns about the state of the field. The session was exceptionally well attended and the panel managed to create a lively debate and exchange of views with each other and the audience. This was particularly remarkable given the technical challenges, the double booking of the room and the incredible heat, lack of seats and oxygen in the late afternoon (it felt like a 2 hour sauna session). I have received quite a few requests for our slides, so I am summarizing some key points from our introductory presentation, the talks by Peter Smith, Johnny Fontaine and David Matsumoto as well as discussion that followed the presentations. I will also outline some ideas of the next steps that we are considering taking.

Poortinga and Fischer: Why questionable null-hypotheses and convergent search for evidence erode research on behavior and culture

Null hypothesis significance testing is the modus operandi for conducting research in psychology overall. At the same time, it has come under increasing pressure and scrutiny. Some quotes from some recent papers illustrate the various problems with the state of psychology:

Ioannides (2005): “[A] research finding is less likely to be true when … when effect sizes are smaller; when there is … lesser preselection of tested relationships; … greater flexibility in designs, definitions, outcomes, and analytical modes; … and when more teams are involved in a scientific field in chase of statistical significance”
Vul et al. (2009) report on “voodoo correlations” in fMRI: “We show how … nonindependent analysis [of voxels] inflates correlations while yielding reassuring-looking scattergrams”
Simmons et al. (2011) on “false-positive psychology”: “… flexibility in data collection, analysis, and reporting dramatically increases actual false-positive rates. In many cases, a researcher is more likely to falsely find evidence that an effect exists than to correctly find evidence that it does not”.

The application of the experimental research paradigm with an emphasis on null-hypothesis significance testing is particular problematic in cross-cultural psychology, because some of the basic assumptions of experimental design are violated by default:

a) There is no random assignment of respondents to conditions and

b) The experimenter has little control over conditions and ambient events.



This figure shows these problems in a nice way and clearly highlights that cross-cultural studies do not even meet the conditions for good quasi-experimental designs and have significant shortcomings. 

Further challenging current experimental practices, Simmons, Nelson and Simonsohn (2011) eloquently exposed the problems of researcher degrees of freedom and the impact of quite innocent appearing research practices on significance levels. They demonstrated how a logically impossible hypothesis (listening to songs about age will decrease the age of listeners) can be empirically supported. Applied to the topic of their investigation, they discovered the psychological equivalent of the proverbial fountain of youth by using questionable research practices. The following figure shows the outcomes of their simulation study and the impact of four researcher degrees of freedom on significance levels. We highlighted the relevance of these conditions for cross-cultural research. 

First, assuming per definition that culture is a shared meaning system, any two cultural variables will be correlated to a significant degree. The very nature of the phenomena under investigation makes finding significant differences more likely. This non-independence is well recognized and the negative impact on significance testing is well recognized in methods circles but not well-understood in general cross-cultural research circles.

Second, a researcher may add 10 more observations or cases to the study if a first examination did not reveal any significant differences. This is probably a more common practice in cultural priming studies, but may be less of an issue in comparative survey studies.

The third questionable practice is controlling for third variables, especially if their impact is not theoretical grounded. In their case study, Simmons et al. used gender as an example, but in cross-cultural psychology it is often GDP at the country level or some demographic variables at the individual level that is entered as a covariate. This is a double-bind of cross-cultural psychology, on one hand we need to control for other variables that may explain any differences between samples, on the other hand, these simulations demonstrated that such practices have a sizable impact on significance levels.

The last questionable practice is to drop (or not) one of the conditions. The equivalent in cross-cultural psychology is to omit samples that may not fit the expected pattern (outlier removal). Talking to other researchers, this seems a common practice.

These individual practices individually increase the likelihood of finding significant results only in a relatively minor way, but the combination of these practices will lead to substantively inflated ratios of significance results: a significant result at the magical .05 significance level is 60% more likely if you combine all four of these questionable practices! Based on conversations with colleagues and observations of publication trends, these practices are common in cross-cultural psychology. This now means that we probably need to question a good number of empirical findings published!

A further issue is that the null hypothesis of no difference is likely to be rejected if there is a difference on any third variable that is related to the dependent variable. In such instances, there is a high rate of Type 1 errors (false positive results). One pressing issue is method bias. In questionnaire studies, response biases such as acquiescence or yes-saying are particularly salient.

The next figure shows the probability of finding a significant result as a function of sample size and the size of the bias. The various lines show the various levels of bias in terms of the standard deviation. If the bias effect is small (e.g., 1/16th of the standard deviation on the DV), increasing sample sizes are not increasing the probability of finding a significant effect by much. However, when bias approaches a .25 of the standard deviation, the probability of finding a significant effect in a sample of 100 participants approaches 60%. You may argue that ¼ of a standard deviation is large. However, it is not an unrealistic scenario given the prevalence and extent of response styles in questionnaire research – see for example our earlier research showing that response styles produce bigger effect sizes than 1/3 of theoretically important research studies.



These two simulation studies suggest that cross-cultural differences might be spurious and driven by method effects. In addition, our field seems to be driven by differences and appears to pay unduly emphasis on differences, without questioning their validity. The next figure shows the emphasis on differences and the lack of studies hypothesizing and finding similarities. This graph is adapted from a review by Brouwers and others, published in JCCP in 2004. As can be seen, the majority of studies expect differences only (N=55) and only 25 studies expected differences and similarities. At the same time, 57 studies found both. Most importantly, given laws of probability, we should also have studies that expect and report only similarities. Brouwers and colleagues did not find a single study that either hypothesized or reported similarities only. Where are these studies?




The points raised so far should not be understood as challenging the experimental methods underlying comparative research. We would urge our colleagues to critically question some of our designs and analytic procedures. In the larger experimental literature, a number of strategies have been proposed, including:

- stricter designs (larger n, Button et al., 2013) for more power
- stricter analysis (p < . 005, Johnson, 2013)
- prevention of experimenter bias (O. Klein et al., 2012)
- more transparency (e.g., pre-registration of hypotheses)
-replication across multiple researchers and labs (R. A. Klein et al., 2014)

We see it as a good sign that replication studies have achieved new status. For example, an earlier attempt of our lab to replicate the culture-level value structure by Schwartz using data from the Rokeach Value Survey faced some real uphill battle in getting it published. The saving grace to get it published seemed to be the appearance of a new value type that was not evident in the earlier Schwartz circle (a replication of this new value type is still outstanding). The new emphasis on replication in my opinion is a major achievement. The first findings of this new wave of replications are coming in. For example, the following graph shows the replication success of a number of studies in the ‘many labs’ replication report. Some of the older studies hold up well to scrutiny, but many of the newer findings, in particular priming studies are not replicable.

What is also noteworthy is that in the original dissemination of these findings, the lack of cross-cultural differences in the patterns was emphasized. Some commentators were quick to jump on that and suggested that careful and experimentally strict replications will do away with cross-cultural differences. We may want to challenge such an assumptions, but these comments clearly demonstrate that we as cross-cultural and cultural psychologists need to engage with the replication debate. We cannot sit back and pretend that the replication crisis does not affect us!

Replications vary along an underlying dimension, with exact replications being at one end and conceptual replications forming the opposing end. The conventional experimental wisdom is to prioritize exact replications or to stick as closely as possible to the original designs (close replications) with large samples sizes to have high power to detect effects. Of course, we know that exact replication in a cross-cultural context is problematic due to the different cultural conditions of participants.

However, an even more important point for us is that the presence of bias (e.g., response styles, speed-accuracy trade-offs) challenges the validity of exact or close replications. A replication of a biased study is a replication of a biased study.

In addition, if we have two samples and we define one sample as belonging to X culture and the other sample as belonging to a Y culture (this could be anything: collectivistic vs individualistic; independent vs interdependent self-construals, honour vs dignity; holistic vs analytic thinking), then any difference on whatever variable will be statistically related to the presumed X-Y difference. Therefore, replications in cross-cultural psychology need to be positioned towards the conceptual replication end and require additional methodological safeguards.

We suggest that cross-cultural replications need to:

-ensure validity of procedures in local context

-empirical checks on the postulated antecedent (what theoretical process is likely to drive these expected patterns and to empirically test these theoretical processes)

-manipulation checks (including a “no-difference” condition, on what variable or set of variables do we NOT expect a difference)

-control on likely alternative explanations (e.g., response styles).


In summary of the points so far, cross-cultural psychology suffers from many of the same shortcomings that have created the crisis in social psychology. A somewhat humorous account borrowing from Dante’s version of hell is provided by this cartoon (by the Neurosceptic, published in Perspectives in Psychological Science). Our research culture that emphasizes differences instead of similarities leads a state of limbo, overselling and post-hoc story-telling. Our narrow orientation towards ghost in the machine variables (such as collectivism, self-construals and values) lead to overselling (everything needs to be explicable by single dimensions, typically of personal relations or self-construals), post-hoc story telling and p-value fishing. From personal experience publishing cross-cultural research, nearly any difference can with a bit of theoretical creativity be related back to individualism-collectivism, self-construals or any of the other fashionable constructs these days. These biases in orientation and the researcher practices and researcher degrees of freedom then lead to p-value fishing and creative outlier utilization. Of course, the absence of no-difference studies suggest a significant file drawer problem.

Our suggestions are therefore:

Better designs (including efforts to reduce bias, testing of alternative theoretical processes, etc.)

Planned replications

Depositing hypotheses and methods in a public archive prior to data collection

Peter Smith: To understand cultural variation let’s sample cultural variations

Peter Smith and colleagues suggested a rather straightforward approach for addressing some of the concerns. Their recommendation was to go beyond two culture comparisons and to sample cultural variation more broadly, e.g., by studying multiple Asian and non-Asian samples that are typically lumped together as collectivist, interdependent, holistic, etc. In addition, Peter and colleagues included more diverse instruments capturing conceptually similar constructs to examine variability in intended constructs across a broader range of instruments. Peter presented some preliminary data that supported the usefulness of this approach. However, he also acknowledged that the current study has some important limitations, including studying students, not having enough samples yet to properly examine effects (e.g., though multi-level modeling) and a high demand on participants (e.g., completing long sets of questionnaires).

Johnny Fontaine: A plea for domain representation

Johnny presented a more technical account of domain representation that examined the meaning of constructs across a larger number of languages and cultural contexts. Using examples from the emotion domain, he showed that we can avoid confusion and biases in meaning through the use of sophisticated non-metric statistical methods in combination with elaborate designs that allow separating situational and personal characteristics. His approach demands a theoretical analysis of possibly important variables that need to be incorporated into the research design. Johnny really got the methods guns blazing in his presentation and I have to admit that the heat of the room by that time had fried my brain. As a consequence, I was not able to follow all the intricate steps in the procedure and not having a seat did not allow me to take good notes (but the graphs looked very convincing). He is working on a manuscript detailing the procedures and I am certainly looking forward to reading it when it is ready.

David Matsumoto: Random thoughts about methodological vulnerabilities in research on behavior and culture.

David broadened the symposium by focusing on the broader research climate in culture and psychology. Most people in this overheating room will have appreciated his first demand: before he started talking he requested everyone to stand up from their seats. Beyond bringing some oxygen into our brains, this also then became a beautiful point of reference for his short and sharp presentation. Here are his three main arguments (my paraphrasing):

Point 1: Study behavior

Point 2: Respect the literature

Point 3: The current pressures on young academics makes following recommendations 1 and 2 challenging

The first point is obvious – our discipline confuses self/other/peer-reports of behavior for behavior. I do not have hard stats here, but from memory – I cannot remember a single cross-cultural social psyc study in the last year or two in JCCP that studied actual behavior. He pointed out that everyone had stood up when he asked at the beginning of his presentation – a success rate of 100%. In contrast, when asking people whether they would stand up in a seminar room when asked by the presenter (e.g., on a scale from 1-7), there would have been significant variability and the mean would definitely been lower than 100%. Drawing upon his own research on emotion display, he argued that triangulation of research method is necessary.

The second point highlights the emphasis of getting to know more about previous research. In the current research environment, researchers need to present novel result and theory. There is no incentive for (or penalty for not) reading older research that may have been conducted 10 or 40 years ago. Journal editors are keen to get citations to recent papers to increase the journal’s Impact Factor. Yet, this leads to impoverished and non-cumulative research.

The last point highlights the constraints that young researchers pre-tenure are facing: more publications in less time. Studies of behavior are time consuming and therefore are less appealing. Reading relevant literature in one’s field or neighboring disciplines is also detracting from writing articles and funding applications. David emphasized that IACCP has a richer intellectual tradition than many mainstream researchers who have discovered culture and now publish in high-impact journals.

Some of the discussion points

The discussion turned repeatedly on a number of points. I will try and summarize some of the key ones that stood out for me.

Representativeness of samples: One key concern that came up repeatedly was that studying students is not appropriate for making claims about cultural processes. Students are not good representatives of the larger population.

Studying nations: One early comment that drew spontaneous applause from the audience was that psychology has failed in studying culture. Instead, psychologists are studying nations. Yet, nations are highly diverse and consist potentially of many subcultures. Various other commentators picked up similar themes throughout the discussion. One issue that is related here was the relative emphasis on between-country/culture differences and the lack of attention to within-country/culture differences. Both Geert Hofstede and Shalom Schwartz were in the audience, but they remained silent – it would have been nice to hear their responses to some of these comments (and both have done some interesting work that would have been informative in this debate).

Lack of strong theory: Peter Richerson argued that psychologists lack strong theory and recommended looking to neighboring disciplines such as biology for inspiration. David Matsumoto defended psychology in his response, suggesting that psychology has some good theories. But he also added that we need truly exploratory work that can understand phenomena on their own terms. My thought on this is that we have not enough strong theory (in a philosophy of science perspective) and that exploratory research with attention to various alternative explanations may bring us closer to developments of stronger theories of culture (e.g., by including the possibilities of no differences, attention to alternative processes beyond the usual suspects in current psychological thinking on culture).

Validity of findings: One point that occurred in various disguises in a number of comments was the importance of validity of findings in the local context. Amina Abubakar was the first to get this point across in the debate: To what extent can cultural psychology and cross-cultural research as a method of choice yield insights into the minds and behaviours of people in a specific context? How applicable and relevant is cross-cultural research for people around the world? This is a major question and needs some serious contemplation as we face a rapidly changing world and need to collectively respond to multiple pressing challenges (e.g., increasing intergroup conflict, climate change, decreasing natural resources).

Next steps

An immediate opportunity following this debate arose the next day after the round-table discussion. Ype challenged the assembly that methods issues need more attention and in response Walt Lonner as the founding editor of JCCP suggested a methods oriented special issue for JCCP. We had a discussion during the coffee break and he invited us to write a proposal for a special issue. Any thoughts for topics and contributors for such a special issue addressing the methods challenges are much welcome (please flick me an email or respond below – I would love to hear from you).

Looking at some other associations (APS comes to mind here), we could adopt some of their criteria for publication – there have been some interesting suggestions and changes in policies recently. Even JPSP now publishes replications (hooray!!!!!!)!

Overall, I think that the overall change in research climate is promising. There has never been a more positive time to discuss how we collectively do research, there is much promise of change in the air and I strongly believe that collectively we can make a positive change. Without this conviction, we would not have had the symposium and such a large crowd keen to brave tropical temperatures and horrible conditions in the late afternoon to debate a topic so passionately. I felt humbled by this enthusiasm of the audience and the positive comments that we received over the next couple of days. I look forward to continuing this debate and hearing your opinions and suggestions!