On the Relationship between ANOVA Main Effects and Average Treatment Effects.

Area of Science:

• Statistics
• Causal Inference
• Experimental Design

Background:

• Analysis of Variance (ANOVA) is a common statistical method.
• Different types of sums of squares (SS) exist for ANOVA: Type I, Type II, and Type III.
• The appropriate choice of SS type is crucial for accurate interpretation of main effects, particularly in the presence of interactions and in non-ideal experimental designs.

Purpose of the Study:

• To investigate which ANOVA sums of squares (SS) are appropriate for testing main effects.
• To determine if main effects should be considered in the presence of interactions.
• To provide guidance for applied researchers on selecting the correct SS type based on study design and model complexity.

Main Methods:

• Adopting a causal inference framework.
• Analyzing balanced, proportional, and nonorthogonal designs.
• Considering statistical models with and without interactions.
• Conducting a simulation study to assess bias in estimating average treatment effects.

Main Results:

• In balanced designs, Type I, II, and III SS all estimate the average treatment effect accurately.
• In proportional designs, Type I and II SS estimate the average treatment effect accurately, but Type III SS is biased when interactions are present.
• In nonorthogonal designs, Type I SS is always biased, and Type II and III SS are biased if interactions exist.

Conclusions:

• The selection of ANOVA sums of squares (SS) type is critical and depends on the experimental design and the presence of interactions.
• Type III SS can lead to biased estimates of average treatment effects in nonorthogonal and some proportional designs.
• Researchers should carefully consider their design and model when choosing an SS type to ensure valid statistical inference.