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Is it necessary in structural equation modeling (SEM) to incorporate all potential independent variables that could affect the dependent variable? Or is it acceptable to examine the influence of only a select few independent variables on the dependent variable?

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In addition to what Jeremy Miles wrote, we typically include an error (residual) term in a structural model for each dependent (endogenous) variable, thereby acknowledging that we do not know (or haven't measured/don't have access to) all possible predictors and/or causes of our dependent variables in the model. Of course, not including certain relevant independent variables can lead to bias in the estimated regression (path) coefficients for those independent variables that are included in the model.

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    $\begingroup$ Regression has an error term too. You just don't show it or so anything with it. (I would even argue that it's latent.) $\endgroup$ Commented Sep 13, 2023 at 15:26
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Multiple regression is an example of structural equation modeling. It's often easier to think about what you would do in multiple regression, because that's true in SEM.

So, it is as necessary to incorporate all predictor variables in SEM as it is in multiple regression.

Which is to say that it depends on the question that you are asking, and the conclusions that you want to draw. But 'all potential independent [predictor?] variables' includes everything in the universe, so no.

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  • $\begingroup$ Thank you for your response. My current project involves investigating the influence of various inventory management issues on consumer shopping behavior. As we know, there are various other factors in addition to inventory management issues that can impact consumer shopping behavior. Therefore, I am trying to understand whether it's appropriate to utilize SEM solely to model various aspects of inventory management and their impact on consumer shopping behavior, while not including other factors that might impact the dependent variable. $\endgroup$
    – Marjaan
    Commented Sep 12, 2023 at 18:35
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    $\begingroup$ McElreath has cautioned against putting as many potential predictors as possible into scientific models. It is an example of what he calls "causal salad". See this talk for an introduction. $\endgroup$
    – Galen
    Commented Sep 12, 2023 at 20:18
  • $\begingroup$ Yeah, you should think hard about your predictors - Willem Saris wrote years ago (I completely forget where) that if your R^2 isn't 0.9 or above you can't interpret your regression, because maybe you forgot an important predictor. I don't necessarily agree, but it's an interesting thought (IMHO). $\endgroup$ Commented Sep 13, 2023 at 0:00
  • $\begingroup$ It seems me that you conflate SEMs with regression. I suggest to don't this and embrace another perspective. Read here: stats.stackexchange.com/questions/63417/… $\endgroup$
    – markowitz
    Commented Sep 13, 2023 at 7:51
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    $\begingroup$ SEMs are a form of regression. They're historically motivated by causal thinking, but they are just statistical models by themselves. If you don't put causal assumptions in, then you won't get causal inferences out. $\endgroup$
    – Galen
    Commented Sep 13, 2023 at 14:52
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I think choosing a "correct" SEM model boils down to these important questions:

  • Does my model make sense? Often I see cross-lagged models that sometimes become giant cob webs with no real interpretability and it makes me question what the end goal of such an analysis is. The important part is that your model has some actual utility outside of AMOS or lavaan. As some others have alluded to already, designing an interpretable design using something like DAGs helps narrow down what's actually important to your question and which are not.
  • Can my data actually support this? It is well known that a large part of SEM is fitting an implied relationship to the actual data, and this is contingent on the data actually behaving in the way you are considering. Loading up a ton of predictors without considering how important they are can lead directly to poor fit models and issues with model convergence, and there are a variety of reasons this happens which are directly relevant to which predictors you include.
  • Even if my data supports this, is it necessary? Related to concerns already noted here about whether or not you are overfitting your paths, another issue is whether or not you are wasting energy on including as many paths as possible. If for example we know that $X1$, $X2$, and $X3$ have substantial loadings onto a latent factor $X$, but then we have an $X4:X30$ variables which are statistically significant but have overall weak loadings, we may not really need to include them in a model given they don't explain much anyway.
  • Are there better models than mine? I think its not always great to think of your specified model in isolation, but rather one of an infinite continuum of models that are possible. This is sometimes called the issue of alternative or equivalent models. Related to my earlier point about DAGs, you should consider constructing other models to see what the weak points of your model are, what the strong points are, and what may alternatively explain the phenomena you are after. This will help solve which predictors are ultimately important.

If these points could be summarized into one sentence, I would say that your goal is to "explain phenomena in a way which is best explained by the data."

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Basically, yes you can consider adding all the variables that influence output variable(s).

Most assumptions in that regard are no different than linear regression: too many inputs will lead to a Neyman-Scott kind of bias. Additionally, you need to be sure of their causal relationship. Many sort of "kitchen sink" models fail to consider which "inputs" might actually be colliders or mediators, which shouldn't be included as an effect.

The advantage of SEM is that, unlike OLS, you can specify this kind of structure in the modeling and avoid bias and perhaps even boost efficiency. It's just quite a bit harder to extract the correct inference.

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You're asking because you're examining how much one variable can explain the other variable--that is, to find the regression coefficient, I assume. Given that different variables can "steal" the weight from different variables, and that this does not represent a change in the actual data, but a distinct interpretation depending on the variables for the same data, then yes, it can be a problem: if you use an incomplete model, you're going to end up with results that could have different interpretations compared to those of a complete model.

As others have suggested in comments directly to your question, this means you need to question the causal relationship between the variables. As they have also pointed out, this is no different than a regression, except that you're gonna specify certain residual characteristics depending on whether you're analyzing latent or observed variables.

Keeping with our usual unscrupulous regression examples, imagine that you create a SEM model that explains surfing accidents on one day by the number of soda cans found on the beach the next day. The model is likely to return some level of explanation. However, once you add the number of

No statistical model is able to tell anything about causality. Some are better than others at helping you formulate a theory about it, but it's the theory that has to do the heavy lifting. So no one can expect the SEM model to be explaining causality. As long as you explain what you're trying to show with the model and the downsides to doing it that way, using an incomplete model is not a big problem. Most often, this intermediate step towards a complete model is worth it.

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