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Consequences of Measurement Error Application

Researchers in marketing and the social and behavioral sciences often work with theories that posit relationships between concepts. In Amos, the concepts are latent variables that are measured by one or more indicators using structural equation modeling (SEM). The measures are often imperfect and contain measurement error. Consequences of measurement error include:

In practice, researchers use reliable multiple indicators to try to overcome these problems. The following example illustrates how multiple indicators can help remedy measurement error.

No provision for measurement error

In a study of coupon usage, Richard Bagozzi considers a model in which attitudes affect behavior both directly and by way of intentions. He compares models in which single and multiple indicators are used.

The model with no provision for measurement error appears in the following figure.

This figure shows a model with no provision for measurement error.

This path diagram illustrates the use of a single indicator to represent each latent variable:

This model makes no provision for measurement error. The indicators are treated as perfect measures of their respective latent variables.

Amos produces the following regression estimates.

The table shows regression estimates.
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Note that these coefficients are un-standardized.

Amos produces the following variance estimates.

The table shows variance estimates produced by Amos.
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Finally, Amos produces the following estimates of R-square.

The table shows estimates for R-square.
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Note that to estimate the model you must set certain regression weights to 1 and certain variances to 0. These values appear in the Amos input model graphic shown above.

Here is Amos graphic output for the above model.

The image shows an output graph for the model above.

Measurement error taken into account in attitudes

This model appears in the following figure.

This graph shows the model with measurement errors taken into account.

In this model, there are now three indicators of attitudes, namely, X1, X2, and X3. To make the model identified, you must fix the scale of attitudes. You can either set the variance of attitudes to a constant, or you can constrain one of the regression weights from attitudes to its indicators to 1. We did the latter in this instance.

Amos produces the following regression weights for this model.

The table shows regression weights for the model.
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Note that the estimated impact of attitudes on intentions has increased by 64 percent from 0.76 to 1.25, and that the estimated impact of attitudes on behavior has more than doubled from 0.24 to 0.56. The impact of intentions on behavior has decreased slightly, from 0.36 to 0.28.

Amos produces the following variance estimates for this model.

The table shows variance estimates for the model provided by Amos.
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Finally, Amos produces the following R-squared values.

The table shows R-squared values for the model.
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The R-squared for intentions has increased from 0.252 to 0.369 while that for behavior has increased from 0.393 to 0.445. The squared multiple correlations for X1, X2, and X3 are 0.542, 0.603, and 0.825 respectively. Note that due to measurement error these values are not 1. Based upon these values, you might conclude that X3 is the most reliable indicator of attitudes.

Here is the estimated model in an Amos graphic.

The graph shows the estimated model.

Measurement error taken into account in both attitudes and intentions

This model appears in the following figure.

The graph shows the model with measurement error taken into account in both Attitudes and Intentions.

This model augments the previous model by adding a second indicator (Y2) for intentions. To make the model identified, you must specify a constraint: either fix the variance of intentions or set the regression weight for one of intentions' indicators to 1. Here, we did the latter.

Amos produces the following regression weights for this model.

The table shows regression weights for the model produced by Amos.
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Regarding the structural part of the model, the regression coefficients differ slightly from the second model above.

Amos produces the following variances for this model.

The table shows variances produced for this model.
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Finally, Amos produces the following R-square values.

The table shows R-square values for the model.
Click to View Larger Image

The R-square for intentions improved from 0.369 to 0.460, while that for behavior improved slightly from 0.445 to 0.450.

Here is the model output in an Amos graphic.

The graph shows model output produced by Amos.

Summary

The structural equation modeling framework found in Amos enables you to use multiple indicators of your latent variables. The example above shows some of the benefits from SEM, including:

References

Bagozzi, R.P. 1996. Structural Equation Models in Marketing Research: Basic Principles, in Principles of Marketing Research. Richard Bagozzi (ed.), Cambridge: Blackwell Publishers.

Bollen, K. 1989. Structural Equations with Latent Variable. New York: John Wiley and Sons.