My friend and I have arguments about some statistical assumptions. He thinks that if he has 4 constructs, and every construct has 3 variables, for example. To test a mediation analysis, he can take the mean average of these variables, compute it, name the average means, and start the mediation test. My argument is that he has to obtain the latent variable through load factors and make a path structural equation model. is both methodologies right? which is better?
The resolution to your argument is an empirical issue. If there is relatively little variation in then loadings, then adding them will be adequate because it assigns equal weights to each variable.
FYI equal weights is the assumption with coefficient alpha, so a high alpha (say .80 and above) would suggest that simply add the variables would be adequate. Also, averaging simply divides by a constant, so it just makes the results more interpretable.
The manifest aggregate (average, summary) score will likely have less than perfect reliability (i.e., Rel is typically < 1, measurement error variance in the aggregate > 0), unless you have many components/items to aggregate (in which case we sometimes see Rel > .95 or so). When the aggregate score has less than perfect reliability, bias in the manifest path coefficients and their standard errors may result. Therefore, the SEM (latent variable) approach is generally preferable even when the components are essentially tau-equivalent/have equal loadings because the SEM approach accounts for unreliability (measurement error) in the components and estimates the paths and mediated effect(s) among latent variables that are by definition free of measurement error. That being said, there are also drawbacks of latent variable approaches. For example, the results may be incorrect/biased if you cannot find a truly well-fitting model. However, in that case the aggregate/sum score approach would be incorrect also.
This is an example where you are taking responses to three variables and creating one summary variable. Consider two very different approaches. The first, let's call this blind to reality, does not consider what the variables are, what the summary measure is supposed to measure, nor how the summary measure is to be used. The second wisely considers the situation, what the summary variable is, and how it is to be measured. Call this the wise approach. Either of these may be appropriate, but in different situations. Consider a financial expenditure and health survey. Suppose the three variables are: how often you buy Granny Smith apples, how often you buy Fuji apples, and how often you buy Honeycrisp apples (three popular varieties in the US). The summary variable is supposed to be "how often you buy any of these three" and it will be used to help gauge how much people spend on different food categories. I assume I am like most people and have a favorite (GS), and that is what I buy. At least for apple eaters I expect the correlations of the three questions to be negative (but I don't know). The blind to reality approach would base the summary variable on these correlations. The analyst might even reverse score one the variables so things "fit right." I don't know what the correlations would be so I can't say if this would squeeze through whatever checks there might be. The wise approach would just add these together. Of course there are situations where the factor analysis approach would be better, and Christian Geiser and David L Morgan can describe examples of these. My main point is that the answer to how to combine variables is: it depends, and overall the thinking approach is at least worth considering.
Christian Geiser I disagree about adding "bias" because all of the models you mentioned (most notably tau equivalence) assume random errors. So, reduced reliability my "attenuate" (degrade) your ability to detect a significant effect, but it will not bias the estimate.
When the estimate (zero-order correlation coefficient or path/regression coefficient) is underestimated (too small, attenuated), would you not say it is biased? Also, in more complex path models, measurement error may lead to over- or underestimation of effects. The original question was about mediated (indirect) effects. When the path coefficients are over- or underestimated, the mediated effects (which are given as the product of path coefficients) may be biased also.
This is funny, because just recently I simulated data in lavaan to compare the difference between the correlations of latent variables and the correlations of mean variables from the manifest variables (not mediation specifically). My simulations were not too systematic or exhaustive, but the overall pattern showed that the correlations between the latent variables were higher and closer to the population values than correlations of the mean variables, even if the simulated model was tau equivalent, i.e. all loadings were identical. (I did not find a situation where the coefficients were overestimated by the mean variables, as mentioned by Christian, but my simulations were not systematic as already said)
Since in mediation one is interested in the product of paths, any underestimation is detrimental, therefore, I would say (unless you have specific situations as described by Daniel Wright) that using latent variables should be more useful. Just my two cents.
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