Non-Recursive Models (Feedback Loops) in SEM

A non-recursive model is a structural model where there is a feedback loop between two con- structs. A recursive model is what we would consider a structural model where there are no reciprocal influences between constructs, or specifically, no feedback loops exist. The problem with feedback loops is you are creating an infinite loop in estimating the parameters between the feedback variables. Let’s look at another example using the construct of Customer Delight to help understand the complexities of non-recursive modeling. In this example, Adaptive Behav- ior of the employee influences customers’ feelings of delight but also a second construct called “Gratitude”. Gratitude is the customers’ feelings of appreciation or thankfulness to the employee. Customer Delight then influences the variable “Trust in the retailer”, or “Trust” for short, which is the degree to which a customer believes a retailer has their best interest in mind.The construct of “Gratitude” influences a construct called “Confidence in the Retailer”. Confidence in a Retailer is the belief in the ability of the retailer to perform a service.The constructs of Trust and Confi- dence in the Retailer have a reciprocal relationship where the more a customer trusts the retailer, the more confidence they have in them; and vice versa, the more confidence a customer has in the retailer, the more trust they will place with them. See Figure 10.33.

Figure 10.33 Structural Model With a Feedback Loop

The problem with non-recursive models is they tend to be under-identified. With a feedback loop included in the model, you need to make sure you have enough predictor variables included for each part of the feedback loop. In the example provided, both Trust and Confidence in the Retailer have a unique predictor. We could not have a relationship from Customer Delight to Confidence in the Retailer without the model being under- identified. The dependent variables in the feedback loop need to have what is called an “instrumental” variable. An instrumental variable is where a predictor will have a relation- ship with only one side of the feedback loop. Thus, the predictor will be “instrumental” in understanding the other side of the feedback loop. In our example, Gratitude is the instrumental variable for the construct of Trust, and Customer Delight is the instrumental variable for Confidence in the Retailer. If Customer Delight had a relationship to Trust and Confidence in the Retailer, then the construct of Confidence in the Retailer would not have an instrumental variable anymore. You would need an additional variable having a relationship only with Trust to have an instrumental variable for the Confidence in the Retailer construct. With most non-recursive models, you want to make sure you are pro- viding enough predictor variables for each side of the feedback loop to be identified. Often you will find that only one half of the loop will be significant; the influence is not strong enough to close the loop.This is usually because you do not have enough unique predictors for each side of the feedback loop.

With non-recursive conceptualizations, model fit can be especially problematic if you are using a path model.You are better off to use a full structural model with measurement items and error terms included in order to help explain the structural relationships and feedback loop. Lastly, you also need to determine the stability of your parameter estimates in the feedback loop. Since you are creating an infinite loop with two regression weights, you need to determine if those weights are considered “stable”; or, put another way, the linear dependency in the feedback loop will provide a clearly defined relationship. AMOS will calculate a Stability Fit Index to help with this. If the stability index falls between −1 and +1, regression weights are considered stable. Saying that, let’s look at the example in AMOS.

You can see that a full structural model is created and a feedback loop is drawn between the constructs of Trust and Confidence with the Retailer. One important component with a feedback loop is that the error terms for the unobservable constructs in the feedback loop need to be correlated. Since both constructs are influencing each other, it’s advisable to cor- relate the error terms.

Figure 10.34 Trust and Confidence Feedback Loop Modeled in AMOS

Figure 10.35 Estimate Output Showing Regression Weights for Direct Relationships and Feedback Loop

After initially finding a significant feedback loop, we need to check the stability of the relation- ships in the feedback loop.When AMOS detects that a non-recursive model is being analyzed, it will calculate the Stability index in the output. On the left-hand side, there is a tab called “Notes for Group/Model” that will show you the Stability index between Trust and Confidence in the Retailer. If you have multiple feedback loops, it will show you the index for all the proposed loops.

Figure 10.36  Stability Index for Trust and Confidence Variables

Next, let’s check the model fit statistics.The results show that the model even with a feed- back loop has an appropriate fit to the data. Thus, we can have confidence in our results with the inclusion of a feedback loop due to the significant regression weights, an acceptable stabil- ity index, and appropriate model fit.

Figure 10.37 Model Fit Statistics of Feedback Loop Model

The example provided had just one feedback loop, but more complex models can have multiple feedback loops in a single model. Obviously, these models can be extremely com- plicated and often problematic in identification and stability. If you have multiple feedback loops in a model and one of the loops is considered “unstable”, then the model as a whole is considered unstable even if the second loop has an acceptable stability index. Ultimately, many examples of reciprocal relationships can be modeled in SEM; the key is to make sure you have enough predictor variables to help explain the feedback loop.

Addressing Missing Data in AMOS

One of the nice perks of using AMOS is it uses a full information maximum likelihood method (FIML) to handle missing data. This method allows model parameters and standard errors to be estimated directly from the available data. With this method, the likelihood function is computed separately for variables that are not missing data, and a separate function is run for variables that are missing data. The two likelihood functions are then maximized together to find an estimate. Note this function is not an imputation but a method of computing maximum likelihood estimates of parameters.

If you have missing data, to use the full information maximum likelihood approach you need to go to the Analysis Properties  and select “Estimate means and intercepts” under the “Estimation” tab at the top.This will allow AMOS to run your analysis via a FIML method.

Figure 10.38 Using the Full Information Maximum Likelihood Method for Missing Data

Note that if you don’t check the estimate means and intercepts option and have missing data, AMOS will prompt you with the following error message.

Figure 10.39 Error Message in AMOS if Missing Data Is Present

Once the estimate means and intercept option is chosen, AMOS will run the analysis for a measurement model or structural model and provide regression estimates for all estimated parameters.

One of the assumptions of SEM is that you have a complete data set that is not missing data. The FIML method will initially run your analysis with missing values, but if you want more advanced analysis options like bootstrapping or modification indices, AMOS will not perform these functions with missing data. In essence, you cannot run more complex functions with missing data that is addressed by FIML. This will require you to delete or impute the miss- ing values. In Chapter 2, I talk about how to impute missing values in different ways (series mean and linear interpolation), but now we need to address imputation through regression imputation.

If you want to address imputing missing data via regression in AMOS, you will need to use the AMOS data imputation function . This is located under the Analyze tab at the top menu in AMOS. After clicking this function, a pop-up window will appear. You want to make sure “Regression imputation” is chosen in the pop-up window. AMOS wants you to rename your data file if an imputation is performed. If you do not specifically state a new file name, AMOS will alter the name to your existing file name and put “imputed” on the end of the name. See Figure 10.40.

Figure 10.40 AMOS Imputation and Renaming of Data File

Once you have labeled your new data set and hit the “Impute” button at the bottom, AMOS will replace all missing values in all variables.

A word of caution needs to be given about using the AMOS imputation function. If you design a model in theAMOS graphics window with unobservable and observable constructs, and then decide to impute your data set, AMOS imputation will try to impute the values for the missing observable variables, but it will impute a score for the latent/unobservable variables as well.AMOS will create an imputed column which is the labeled name of your unobservable construct.You have two options to address this: (1) you can go into the new imputed file and delete all columns that are unobserv- able names; or (2) you can change the names of all the unobservables in your AMOS model. If you don’t address this problem,AMOS will think your unobservable construct is an observable and will give you an error message. Obviously, a better option is to impute the data before you conceptualize your model with unobservable constructs. Lastly, to run your conceptual model, you will now need to read in the new file name and data that is imputed.AMOS does not automatically do this after an imputation.You need to make sure that AMOS is using the imputed data going forward.

The AMOS software can also use a Bayesian imputation to perform “multiple imputation”. Multiple imputation is where the missing data is estimated multiple times.The Bayesian analy- sis estimates the missing value of the data multiple times based on the total variables of the study. This will produce numerous different completed data sets that include the different imputation of the missing data. The individual data sets are examined and then, ultimately, combined to form a single imputed data set. This method is labor intensive but has shown to be a valid method of handling missing data (Allison 2003).

Source: Thakkar, J.J. (2020). “Procedural Steps in Structural Equation Modelling”. In: Structural Equation Modelling. Studies in Systems, Decision and Control, vol 285. Springer, Singapore.