Correct way to build composite variables

[Diogo Borges Provete]

Hi, 

I'm fairly new to SEM and not sure how to use some of the advanced techniques, such as latent and composite variables. 

I have 7 macroclimate variables in my dataset that I want to summarise in a statistical composite variable, called climate. But after reading Jim Grace's papers and online documentation of the piecewiseSEM R package, I am not sure what is the correct way to build such a variable, because it influences more than one response variable in my path diagram, for example: 

mod1 <- lm(bird.rich  ~ BIO03+BIO04+BIO07+BIO11+BIO15, data=full_data)

lm(bird.fdisp  ~ BIO03+BIO04+BIO07+BIO11+BIO15, data=full_data) 

lm(bird.dr  ~ BIO03+BIO04+BIO07+BIO11+BIO15, data=full_data)

lm(bird.mpd  ~ BIO03+BIO04+BIO07+BIO11+BIO15, data=full_data)

My question is, should I build just one composite variable, say using the first model, something like this:

# get loadings 

beta_x1 <- summary(mod1)$coefficients[2, 1] 

beta_x2 <- summary(mod1)$coefficients[3, 1] 

beta_x3 <-  summary(mod1)$coefficients[4, 1] 

beta_x4 <-  summary(mod1)$coefficients[5, 1] 

beta_x5 <-  summary(mod1)$coefficients[6, 1] 

# compute factor scores

composite <- beta_x1 * BIO3 + beta_x2 * BIO4 + beta_x3 * BIO7 +beta_x4 * BIO11+ beta_x5 * BIO15

or make separate variables for each response?

To complicate things even more, I have one SEM for each vertebrate group...

Thank you in advance,

Diogo

[Keith Markus]

Diogo,

Funny timing, I am covering this topic in my SEM class this evening.  :-)

First, I will follow the terminology of Bollen and Bauldry (2011, Psychological Methods) in distinguishing formative measurement models from composite models.  In a formative measurement model, it is assumed that one has only a sample of the possible causes of the latent variable, and thus the latent variable has a non-zero residual variance representing the omitted causes.  In contrast, a composite model fixes the disturbance variance to zero because the composite is defined as the weighted sum of the observed variables that cause it.  As such, it seems to me that a composite should never have any causes other than its index variables -- the items that measure it, they are not properly indicators because they do not indicate as reflective items do -- and that composites should never have correlated disturbances.  In other words, the designation of a latent variable as a composite places severe restrictions on what other parameters makes sense in the model without undermining the variables status as a composite.

You could model each vertibrate group as a separate group in a multi-group model.

Regarding your main question, there is no one unique composite of a set of indices.  There are a plenum of such composites.  Moreover, unlike a formative construct, a composite is not a latent variable.  We may use the tools for latent variables to represent a composite.  However, because the disturbance is zero, the composite variable is a fully determinate function of the indices, in this sense, it is an observed variable even if you do not compute it in your data set.

One possibility is that a single composite "mediates" the effects of the indices on the outcomes.  In that case, you can model all the outcomes as effects of the same composite.

Another possibility is that one linear composite "mediates" the effects on one subset of outcomes, and another linear composite mediates the effects on another subset of outcomes.  In this case, a model with a single composite will not adequately reproduce the covariance matrix.  You need to include more composites in your model to account for these different patterns of causal effects.

However, you are on much better footing in doing this with a composite than with a formative construct.  With a formative construct, adding another latent variable has implications for the interpretation of the model:  implying that some further attribute exists in nature and has causal effects.  With a composite, on the other hand, there is no such existential import to adding another latent variable to the model.  The composite is just a convenient way to summarize a highly structured set of causal effects of the indices.  It does not require that the composite correspond to an attribute that exists in nature above and beyond a simply mathematical transformation of the attributes represented by the indices.  If you add an extra composite, you are just relaxing some constraints on the patter of causal effects.

I put the word 'mediates' in quotes earlier to allow for this possibility that the composite itself may just represents a series of constraints on the effects of the indices on the outcomes without having any distinct existence or unique causal efficacy on its own.  It is formally a mediation model, but semantically the direct effects of the composite may not correspond to anything in the system being modeled, just a convenient way to impose certain constraints on other effects.

Keith

------------------------
Keith A. Markus
John Jay College of Criminal Justice, CUNY
http://jjcweb.jjay.cuny.edu/kmarkus
Frontiers of Test Validity Theory: Measurement, Causation and Meaning.
http://www.routledge.com/books/details/9781841692203/