Two Way ANOVA

Two-way ANOVA

Reference:
Ferguson, G.A. & Takane, Y (1989) Statistical Analysis in Psychology and Education (Sixth Edition) London:McGraw-Hill

Analysis of Variance (ANOVA - 2+ groups) is concerned with comparing the means of two or more groups. In one way ANOVA the test simply attempts to compare group achievement on the data/mark collected. In two-way analysis of variance the groups can be subdivided using additional information ( information like gender/ethnic group).

Two-Way Analysis of Variance: . Can involve one or two factor experiments:

a) Repeated measures on one of two factors as diagram below: Marks in different subjects for each individual.

Number of scores in each ‘cell’ should be the same (can be just one). Some versions of the test can allow for missing scores (alternatively: consider interpolating the missing score a) by using mean score on that test or, b) using the mean score on all tests for that individual having taken into account different totals. Researcher would need to justify/defend, the interpolated score used.)

b) Repeated measures on one factor : where same work is marked by different teachers.
See example described below section c)

c) Repeated measures on both factors: Same individuals given scores by different teachers
on a range of samples of work

In the table below (this is real data) the work of 5 students was marked by three different markers in an FE College. Is there any difference in the standard of marking?

Note: There can be more than one score in each cell, but care must be taken for the design to fit the test, commonly the number of scores in each cell has to be the same

Student	Marker 1	Marker 2	Marker 3	Mean
1	27	26	33	28.67
2	25	23	30	26.00
3	19	21	24	21.33
4	18	22	23	21.00
5	14	17	22	17.67
Mean	20.6	21.8	26.4
F column =	18.500	F Row =	22.592
df =	2 & 8	df =	4 & 8
	Sig @ 1%		Sig @ 1%

N.B. If this data is used with one-way analysis of variance F=3.385 with 2 & 15 df. The results of which would not be significant (Intuitively the reason for this is that with the one way test no assumptions are made that the 5 scores in each group are from the same five students.).

The results of the two way analysis of variance in the above table indicates that:

the quality of the students work varies (F Row = 22.592)

the standards of marking by the markers differs. (F column = 18.5)

(Note the research design has to be fair - in this instance that the quality of the students work is not affected by the marking process, that the conditions for marking have not changed etc.)

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