Tips and Tricks on EMS

• If something is randomly assigned.
• If something is nested.
• If something crosses it.

Example 1. A two-factor, fixed-effects design

• Subjects are randomly assigned to method and
• Subjects are nested in gender.
• Error term is s:MG (includes both fixed factors M and G)
• Add s:MG to both main effects and to the 2-way interaction: M, G, and MxG
• Graphically depict the design: see below.

Example 2: A three-factor, fixed-effects design

• Subjects are randomly assigned to method (M) and
• Subjects are nested in gender (G) and
• Subjects are also nested in school (E)
• Error term is s:MGE (includes all 3 fixed factors)
• Add s:MGE to all three main (fixed) effects, all three 2-way interactions, and the 3-way interaction.
• Graphically depict the design: Same as Example 3 below, but the EMSs are different.

Example 3: A three-factor, mixed-effects design

• School (E) becomes a random factor (rename it to e)
• The EMS for each term starts out just like Example 2 with s:MGe added to all terms.
• But now you add new information to the EMS:

• Main Effects: Here are the EMSs
  • M changes: School crosses method Mxe so M becomes M + Mxe + s:MGe
  • G changes: School crosses gender Gxe so G becomes G + Gxe + s:MGe
  • e does not change: School cannot cross itself (no exe) so e remains e + s:MGe

• 2-Way Interaction Terms: Here are the EMSs
  • Mxe remains Mxe + s:MGe
  • Gxe remains Gxe + s:MGe
  • MxG changes: Method and Gender both cross with school so you add the 3-way interaction term and it becomes MxG + MxGxe + s:MGe

• 3-Way Interaction Term: Here is the EMS
  • The 3-way interaction remains: MxGxe + s:MGe

  Graphically depict the design: see below. The design is the same, whether the effects are fixed or mixed.

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