Quasi-F Ratios

I. Declaring a factor fixed or random has important consequences.

A. Two random factors in one model

1. Repeated measures with an additional random factor

a) Example: Judges rate behavior of subjects in different conditions

(S^R X T^F X J^R)

2. Random factor nested within fixed factor

a) Example: (Clark, 1973) Words of different types rated by subjects (W^R(C^F) X S^R)

II. Quasi-F Ratio (cf. Winer, 1972; Satterthwaite, 1946)

A. General Principle: construct F ratio so that all of the components of the E(MS) for the term of interest appear in the denominator and the numerator except for the variance due to the factor of interest.

B. Example: S^R X T^F X J^R

Source E(MS) Error

1. T jns²_T + ns²_JT +js²_ST +s²_JST ?

2. J nts²_J + ts²_JS 6

3. S jts²_S + ts²_JS 6

4. JT ns²_JT + s²_JST 7

5. ST js²_ST + s²_JST 7

6. JS ts²_JS

7. JST s²_JST

1. Quasi-F for T

MS_T + MS_JST

F"=

MS_JT + MS_ST

2. F" is not distributed as c² but will approximate F when df are adjusted:

(MS_T + MS_JST)²

df_numerator=

(MS_T²/df_T) + (MS_JST²/df_JST)

(MS_JT+MS_ST)²

df_denominator=

(MS_JT²/df_JT) + (MS_ST²/df_ST)

C. Can also use F', not as common. May yield negative estimated mean square.

1. Example: S^R X T^F X J^R

MS_T

F'=

MS_JT + MS_ST - MS_JST