20241222 - troubleshooting improper solutions

08-Structural-Equation-Modeling.Rmd

@@ -150,6 +150,7 @@ For example, the model you fit should depend on how you conceptualize each const
 There are important practical issues to consider with both [reflective](#reflectiveConstruct) and [formative](#formativeConstruct) models.\index{construct!reflective}\index{construct!formative}
 An important practical issue is model identification—adding enough constraints so that there is only one, best answer.\index{structural equation modeling!model identification}
 The model is identified when each of the estimated parameters has a unique solution.\index{structural equation modeling!model identification}
+For ensuring the model is identifiable, see the criteria for identification of the measurement and structural model [here](https://davidakenny.net/cm/identify_formal.htm) (archived at https://perma.cc/5C9E-LBWM).\index{factor analysis}\index{factor analysis!identifiability}
 
 Degrees of freedom in a SEM model is the number of known values minus the number of estimated parameters.\index{structural equation modeling!degrees of freedom}
 The number of known values in a SEM model is the number of variances and covariances in the variance-covariance matrix of the manifest (observed) variables in addition to the number of means (i.e., the number of manifest variables), which can be calculated as: $\frac{m(m + 1)}{2} + m$, where $m = \text{the number of manifest variables}$.\index{structural equation modeling!degrees of freedom}

15-Factor-Analysis-PCA.Rmd

@@ -595,6 +595,33 @@ Maximum likelihood [factor analysis](#factorAnalysis) can make use of all availa
 Mplus, which is often used for [SEM](#sem) and [factor analysis](#factorAnalysis), will notify you if you are removing many participants in [CFA](#cfa)/[EFA](#efa).\index{factor analysis!decisions}\index{factor analysis}\index{structural equation modeling}\index{factor analysis!confirmatory}\index{factor analysis!exploratory}
 The `lavaan` package [@R-lavaan] in `R` also notifies you if you are removing participants in [CFA](#efa-cfa)/[SEM](#sem) models.\index{factor analysis!decisions}\index{factor analysis}\index{structural equation modeling}\index{factor analysis!exploratory}
 
+### Troubleshooting Improper Solutions {#troubleshooting-factorAnalysis}
+
+A [PCA](#pca) will always converge; however, a [factor analysis](#factorAnalysis) model may not converge.\index{principal component analysis}\index{factor analysis}
+If a [factor analysis](#factorAnalysis) model does not converge, you will need to modify the model to get it to converge.\index{factor analysis}
+Model identification is described [here](#modelIdentificationTypes-sem).\index{factor analysis}\index{factor analysis!identifiability}
+
+In addition, a [factor analysis](#factorAnalysis) model may converge but be improper.\index{factor analysis}\index{factor analysis!improper solution}
+For example, if the [factor analysis](#factorAnalysis) model has a negative residual variance or correlation above 1, the model is improper.\index{factor analysis}\index{factor analysis!improper solution}
+A negative residual variance is also called a Heywood case.\index{factor analysis}\index{factor analysis!improper solution}
+Problematic parameters such as a negative residual variance or correlation above 1 can lead to problems with other parameters in the model.\index{factor analysis}\index{factor analysis!improper solution}
+It is thus preferable to avoid or address improper solutions.\index{factor analysis}\index{factor analysis!improper solution}
+A negative residual variance or correlation above 1 could be caused by a variety of things such as a misspecified model, too many factors, too few indicators per factor, or too few participants (relative to the complexity of the model).\index{factor analysis}\index{factor analysis!improper solution}
+
+Here are various ways you may troubleshoot improper solutions in [factor analysis](#factorAnalysis):\index{factor analysis}\index{factor analysis!improper solution}\index{factor analysis!troubleshooting}
+
+- Modify the model to be a represent a truer representation of the data-generating process\index{factor analysis}\index{factor analysis!improper solution}\index{factor analysis!troubleshooting}
+- Simplify the model\index{factor analysis}\index{factor analysis!improper solution}\index{factor analysis!troubleshooting}
+  - Estimate fewer paths/parameters\index{factor analysis}\index{factor analysis!improper solution}\index{factor analysis!troubleshooting}
+  - Reduce the number of factors estimated/extracted\index{factor analysis}\index{factor analysis!improper solution}\index{factor analysis!troubleshooting}
+- Increase the sample size\index{factor analysis}\index{factor analysis!improper solution}\index{factor analysis!troubleshooting}
+- Increase the number of indicators per factor\index{factor analysis}\index{factor analysis!improper solution}\index{factor analysis!troubleshooting}
+- Examine the data for outliers\index{factor analysis}\index{factor analysis!improper solution}\index{factor analysis!troubleshooting}
+- Drop the offending item that has a negative residual variance (if theoretically justifiable because the item may be poorly performing)\index{factor analysis}\index{factor analysis!improper solution}\index{factor analysis!troubleshooting}
+- Use different starting values\index{factor analysis}\index{factor analysis!improper solution}\index{factor analysis!troubleshooting}
+- Test whether the negative residual variance is significantly different from zero; if not, can consider fixing it to zero (not ideal)\index{factor analysis}\index{factor analysis!improper solution}\index{factor analysis!troubleshooting}
+- Examine the data for possible outliers\index{factor analysis}\index{factor analysis!improper solution}\index{factor analysis!troubleshooting}
+
 ## Getting Started {#gettingStarted-factorAnalysis}
 
 ### Load Libraries {#loadLibraries-factorAnalysis}