Statistical Analysis and Modeling in R

Determining the Number of Factors

A critical first step is parallel analysis. This technique compares the eigenvalues of your observed data against those of a randomly generated dataset of the same size. Factors are retained only if their eigenvalues exceed those produced by the random data.

Extracting and Rotating Factors

Once the number of factors is determined, the researcher must choose an extraction method (such as Principal Axis Factoring, fm = "pa") and a rotation method. An oblimin rotation is often preferred when researchers assume that the underlying factors are correlated rather than orthogonal.

The output of a factor analysis includes:

• Standardized Loadings: These represent the correlation between an item and a factor. High loadings indicate that an item is a good indicator of that construct.
• Item Complexity: This refers to how much an item loads onto multiple factors. A "complex" item may be difficult to interpret because it does not clearly belong to a single dimension.

Linear Modeling and Regression

Linear models allow us to predict a dependent variable based on one or more independent variables.

Simple and Multiple Regression

In a simple linear model, we might predict "Trust" based on "Years of Education". However, to account for more complexity, we use multiple regression, adding variables like "Political Orientation" to see if they improve the model.

When evaluating these models, we look at:

1. Multiple R-squared ($R^2$): This value represents the proportion of variance in the dependent variable that is explained by the independent variables.
2. Residual Standard Error: This indicates the average amount that the observed values deviate from the regression line.

Mediation and Interaction Effects

Advanced modeling often explores how variables relate through indirect paths or how they influence each other.

• Mediation: This occurs when an independent variable affects a dependent variable through an intermediate "mediator" variable.
• Interaction Effects: An interaction occurs when the effect of one independent variable on the outcome depends on the level of another variable. For example, the impact of education on happiness might differ depending on whether a person is married or divorced.

Advanced ANOVA and Group Comparisons

When the independent variables are categorical (e.g., Marital Status or Sex), we utilize Analysis of Variance (ANOVA). Using the afex package, researchers can perform complex designs, such as a two-way ANOVA, to test for main effects and interaction effects between two categorical factors.

The output provides an Anova Table (Type III tests), which helps determine if the differences in means between groups are statistically significant. To make these results interpretable, the sjPlot package can be used to generate predicted plots, visualizing how the different groups compare across the model's dimensions.