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

Linda Graefe1, Sonja Hahn2, Axel Mayer3

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Summary

The choice of ANOVA sums of squares (SS) impacts the accuracy of estimating average treatment effects. Type III SS can be biased in proportional and nonorthogonal designs, especially with interactions.

Keywords:
ANOVAAverage causal effectanalysis of varianceaverage treatment effectmain effectsums of squares

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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.