If your data sample is skewed to the point that a certain group is underrepresented, you might want to use a weighted score where underrepresented groups will have a stronger weight in the analysis.An example of this might be where 80% of your sample is female and you need to let the males in the sample have a stronger weight because they are underrepresented. Other examples of this can be where your sample is heavily skewed to one race and you want to include weights to the other races in order to represent a certain population.Weighting a score can be done for lots of reasons, but ultimately it is done so that your sample represents a specified group or population. Let’s use our structural model example again that Adaptive Behavior influences Customer Delight which influences Positive Word of Mouth. In our data collection, 80% of the respond- ents were Caucasian and 20% were African American. If we know that the customers of the retailer who were surveyed has a demographic makeup of 70% Caucasian and 30% African American, then our sample is underrepresenting the African American group, and we might want to give their responses more weight in the analysis.

If we look at the specifics of our data file (Figure 10.1), there is a column labeled “Race” where a “1” denotes a customer is Caucasian and a “2” represents a customer that is African American. Next, you need to determine the weight that needs to be applied across the data.To calculate the weight, you can use the following formula:

**% of the population/% of the sample in the data**

** **If we know that 30% of our customers are African American and 20% of our sample is African American, then the weighting for this group is:

30 / 20 = 1.50

For the Caucasian group who are overrepresented (with 80% of the sample), the weight of their scores is:

70 / 80 = 0.875

Weightings that are greater than 1 mean that those respondents are given more weight or importance in the analysis.Weightings less than 1 mean that the respondents’ scores are given less emphasis in the analysis. After finding the weighting for each race, you need to form a new column in your SPSS data, and you can call it “weight”. For all the respondents, who are classified as a “1” in Race, you will include a weight of 0.875. For all respondents with a 2 in the Race field, you will include a weight of 1.50. See Figure 10.1.

*Figure* *10.1* Scores Weighted in SPSS

After establishing the weights for each group, you need to apply the weights to the data. SPSS has a helpful function that will apply a weight to a whole row (or respondent). In the “Data” menu option at the top of SPSS, there is a function at the bottom called “Weight Cases”. After selecting that option, you will see a pop-up menu.This menu will ask what column you want to use as the weight for the data. I labeled the column simply “Weight”. After selecting the “Weight” variable, hit the OK button.The weight will now be applied to all of the data. See Table 10.1 to view an example with the weight function on and off in SPSS.

*Figure 10.2* Weight Cases Pop-Up Window in SPSS

Please note that SPSS will continue to run subsequent analyses with the weighted scores until you tell SPSS to stop weighting the scores. To turn off the weightings, just go back into the “Weight Cases” pop-up menu, select “Do not weight cases”, and hit OK.

After applying the weight in SPSS and saving the data file, you would think that it would be a simple process of pulling in these revised scores to AMOS, but you would be wrong. If you have a weighted score saved in the SPSS data and try to use it in AMOS, you will get an error message.This error message states that AMOS will not recognize the weights and AMOS will give each case a weighing of 1 (which means no weighting at all). To use a weighted score, we have to use a work-around.

*Figure 10.3* AMOS Error Message When Trying to Use a Weighted Score Data File

Let’s go back to the SPSS file where our data is located. To work around this problem, we are going to get the covariance matrix of the weighted scores. We will also get the weighted means and standard deviations for each indicator or construct. Once we get this information,

we will create a new SPSS file that will set up the adjusted data as a covariance matrix that we can use as input in AMOS. If you are examining a path model, this is a pretty easy process. If you are testing a full structural model, then it gets more laborious because you need to form a covariance matrix that includes every indicator in the model.

For instance, let’s use our three construct model that we have used previously (Adaptive Behav-ior → Customer Delight → PositiveWOM) and set this up as a weighted score.After applying the weight in SPSS, we could use the correlation function in SPSS to get the means, standard devia- tions, and covariance matrix, but the results will be in a very unfriendly format. Another option presents the results in a little more manageable form: the reliability analysis option in SPSS.

In this function, we would include all the indicators for each of the three variables.We are not concerned with the reliability across all three constructs; we are just going to choose some options in this analysis to give us the means, standard deviation, and covariance matrix. To do this, we go to the “Analyze” function at the top menu in SPSS, then “Scale”, and then we select “Reliability Analysis”.

*Figure* *10.4* Using the Reliability Analysis Function to get a Covariance Matrix

*Figure* *10.5* Statistic Option in Reliability Analysis

To see the full output of this analysis, see Figure 10.6. Once we have this information, we can set up a separate file that will be used as the input for AMOS.To see how to use a covariance matrix as input in AMOS, refer to page 133. Obviously, this would have been easier if we could have just brought in the weighted scores directly from SPSS, but this alternative will give us the same results. It is a little more labor intensive to set up the weighted covariance input file, but you can still perform your analysis with weighted scores through this alternative way.

*Figure* *10.6* Reliability Analysis Output Using Weighted Scores

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.

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