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EMS Table Rules



Different sources express the rules for creating an EMS Table in different ways, to include the order in which the rules are applied.  The following set of rules is based on Design and Analysis of Experiments (Montgomery) (5th Edition) Chapter 12.  These same rules are outlined in Fundamental Concepts in the Design of Experiments (Hicks) Chapter 6, but are applied in a different order.

Examples of EMS tables for GR&R and DGR&R studies are given at GRR Models and EMS Tables

The template attached at the bottom of this page might be useful when creating an EMS table.


The following rules apply to the balanced case (no missing observations) for sigma-restricted mixed models.


EMS TABLE RULES



Rule 1 - the model equation error term subscript

The error term in the model, eij..m, is written as em(ij..), where the subscript m denotes the replication subscript.  For the two-factor model, this rules implies that eijk becomes ek(ij).



Rule 2 - the model equation

In addition to an overall mean (mu) and an error term em(ij..), the model contains all the main effects and any interactions that the experimenter assumes exist.

Example - two factor model. 


Let A be a fixed effect and B a random effect.  The error term is always a random effect.



Rule 3 - classes of subscripts

For each term in the model, divide the subscripts into three classes
  • (a) live - subscripts that are present in the term and are not in parentheses
  • (b) dead - subscripts that are present in the term and are in parentheses
  • (c)  absent - subscripts that are present in the model but not in that particular term

Example - In the two factor model example, for the error term subscripts "k(ij)", the 'i' and 'j' subscripts are "dead".



Rule 4 - degrees of freedom

The number of degrees of freedom for any term in the model is the product of the number of levels associated with each dead subscript and the number of levels minus "1" associated with each live subscript.

Example:
  • For the A term, we will assume levels i = 1 to a.
  • For the B term, we will assume levels j = 1 to b.
  • For the error term (alternatively "e"), we will assume levels k = 1 to n.



Rule 5 - fixed, random, mixed effects; sigma restriction

Each term in the model has either a variance component (random effect) or a fixed factor (fixed effect) associated with it.  If an interaction contains at least one random effect, the entire interaction is considered as random.  This is known as the "restricted mixed model".

For more information on the treatment of the interaction term between a fixed and a random effect, see Variance Components (Searle) Chapter 4 (page 123).

Represent fixed effects using
Represent random effects using




Rule 6 - preparing the EMS table

To obtain the expected mean squares, prepare an EMS table.  There is a row for each model component (mean square) and a column for each subscript.
Over each subscript, write the number of levels of the factor associated with that subscript and whether the factor is fixed (F) or random (R).  Replicates (including the error term) are always considered to be random.

Example:

  levels ->
 a b n
  fixed or random ->
 FR
R
  subscript ->
 ij
k
 Ai
   
 Bj   
 ABij   
 ek(ij)   

Rule 6a - dead subscripts

In each row, write '1' if one of the dead subscripts in the row component matches the subscript in the column.

Example:

   a b n
 
 FR
R
 
 ij
k
 Ai
   
 Bj   
 ABij   
 ek(ij) 1 1 


Rule 6b - fixed effect (0) or random effect (1)

In each row, if any of the subscripts on the row component match the subscript in the column:
  • Write "0" if the column is headed by a fixed effect.
  • Write "1" if the column is headed by a random effect.
(Note:  Rule 6a takes precedence over Rule 6b.)


Example:

   a b n
 
 FR
R
 
 ij
k
 Ai
 0  
 Bj  1 
 ABij 0 1 
 ek(ij) 1 11


Rule 6c - fill in the number of levels

In the remaining empty positions, write the number of levels shown above the column heading.

Example:

   a b n
 
 FR
R
 
 ij
k
 Ai
 0b
n
 Bj a 1 n
 ABij 0 1 n
 ek(ij) 1 11


Rule 6d - finding the summed components of the expected mean square for each term in the model

To obtain the expected mean square for any model component:
  • First, cover all columns headed by live subscripts on that component.
  • Next, in each row that contains at least the same subscripts (including live and dead subscripts) as those on the component being considered, take the product of the visible numbers and multiply by the appropriate fixed or random factor from Rule 1.
  • The sum of these quantities is the expected mean square of the model component being considered.

Example:

   a b n 
 
 FR
R
 
 
 ij
k
 
 A i
 0b
n
 Bj a 1 n
 ABij 0 1 n
 ek(ij) 1 11





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