Preparing to Present Your Summary of a Two-WayFactorial
ANOVA
The factorial ANOVA for you example may have two or more factors
(classification variables). Professor Karl wants you to focus in on only two of the
factors. Here I shall simply refer to the two factors as “A” and “B.”
You should prepare an A x B table of cell means and calculate the
marginal means. There may or not be such a table in your article. You may
have to estimate the cell means from a plot. If the statistical presentation in your
article is so poor that you are unable to do this, find another article.
When I call on you, you should be prepared to, with little delay, explain
what A and B are, what the scores are, and whether the main effect of A, the
main effect of B, and/or the interaction are statistically significant. You should be
prepared to give supporting statistics – F, degrees of freedom, and p. If reported,
you should share estimates of magnitude of effect for each of these three effects.
Then you should use the table of cell means to describe each of the significant
effects.
Be prepared to criticize the statistical presentation, if it fell short of
adequate. Were estimates of effect size provided? Were they accompanied by
confidence intervals? If the interaction was significant, were simple main effects
analyzed?
Past Examples of Independent Samples Factorial ANOVA
Published Example of Factorial ANOVA

López-Aumatell, R., Martínez-Membrives, E., Vicens-Costa, E., Cañete,
T., Blázquez, G., Mont-Cardona, C., & ... Fernández-Teruel, A. (2011).
Effects of environmental and physiological covariates on sex differences in
unconditioned and conditioned anxiety and fear in a large sample of
genetically heterogeneous (N/Nih-HS) rats. Behavioral and Brain
Functions, 7(48).
This journal is an “Open-Access” journal. Such journals are published
strictly online. The authors must pay a fee to have their manuscript published.
The fee may be small ($200) or large ($2,000 or more). The publisher has
relatively few expenses – just need to create the pdf and post it on a web server,
so the temptation is for all sorts of scam artists to set up shop publishing bogus
open access journals. I get several solicitations a week from such scammers,
inviting me to publish my work for a fee. While there are likely exceptions, one
should generally expect that articles published in open access journals are not of
high quality. The article summarized here is no exception to that rule.
The authors tested rats in an elevated zero- maze (ZM). The maze,
similar to that described by Shepherd et al [20] comprised an annular platform
(105 cm diameter; 10 cm width) made of black opaque plywood and 65 cm
above the ground level. It had two open sections (quadrants) and two enclosed
ones (with walls 40 cm height). The subject was placed in an enclosed section
facing the wall. The apparatus was situated in a black testing room, dimly
illuminated (approximately 50 lux at the level of the apparatus) with red
fluorescent light, and the behaviour was videotaped and measured outside
the(ZM-LAT), time spent in the open sections (ZM-T), number of entries in the
open sections (ZM-E), number of stretched attend postures (ZM-SAP) and
number of defecation boluses (ZM-D) were measured for 5 minutes
There were 5 levels of rat color: white, brown, black, brown spotted and
black spotted.
The interaction between color and sex was significant, F(4, ?) = 2.4, p <
.05.
Notice that the authors did not report the degrees of freedom, did not give
exact p values, and gave no strength of effect estimates. Each analysis was
based on 284 or fewer rats. Total degrees of freedom were 283 or fewer. For
each effect, the denominator degrees of freedom were less than 283. The
authors reported that the F of 2.4 was significant. The critical value of F on 4 and
200 degrees of freedom is 2.42; on 4 and 500 degrees of freedom it is 2.39. The
p is not much less than .05.
You can see the interaction in the plot above. Black bars are for
male rats, white for female rats. For every color, the male rats were more
fearful (entered fewer open sections in the maze) than the female rats, but
the magnitude of the sex difference varied across colors. That is, the
interaction is monotonic.
The authors reported having used the Duncan range test to make
multiple comparisons. I have not heard of this test in many years. It
became obsolete many years ago.

Bente, G., Feist, A., & Elder, S. (1996). Person perception effects of
computer-simulated male and female head movement. Journal of
Nonverbal Behavior, 20, 213-228.

The participants were student observers recruited from an introductory
course for psychology majors at the University of Cologne. Sixteen
female and 16 male undergraduate psychology students with a mean age
of 27.9 (SD = 7.7) took part in the study.

Observers were shown computer-simulated sequences of male and
female head movement and were asked to rate their impressions of the
computer actors on the screen. The independent factors were (1) sex of
observer and (2) sex of movement origin. The dependent variable
measured impressions of the computer actors on the screen.
Measurement of the observers' impressions of the computer-simulated
actors was based on a 27-item, seven-point, bipolar, adjective checklist
recently developed for use in media research. Were there 27 dependent
variables or were the 27 items used to compute some smaller number of
scale scores?

This was an experimental study because the independent variables were
manipulated (the sex of movement origin was switched in different
conditions). Quasi-experimental actually. One IV was manipulated
(gender of origin), but the other was not (gender of the observer).

The analysis showed male behavior in general was perceived as more
active and mobile, whether observed by men or women. The gender of
observer was significant only on rating the material exciting or boring- men
found it more boring.
Author’s statements:
The sex of observer factor revealed significant results for only one
item: boring or exciting, F(1, 28) = 5.67, p < .05. The male observers rated
the stimulus material as more boring than the female observers did (M = 0.34, SD = 0.88 for male observers; M = -0.01, SD = 0.78 for female
observers).
The effects of the sex of movement origin were the most important with
respect to our research questions. The data show that male head
movement was rated as significantly more active, F(1, 27) = 60.64, p <
.001, more mobile, F(1, 27) = 70.71, p < .001, more open, F(1, 28) = 14.24,
p < .001, and more compassionate, F(1, 28) = 20.80, p < .001, than female
head movement.
There were no significant two-way interactions between sex of observer
and sex of movement origin.
Karl’s Comments
It looks like the authors actually did 27 two-way ANOVAs. With three tests
of significance in each ANOVA, that is 3 x 27 = 81 tests. I assume the
usual .05 criterion of significance was employed, especially since one
effect is reported as significant at “p < .05” -- note that the authors fail to
give exact p values. If the truth were that there are no effects of any of the
variables studied here, one would expect in 81 tests to obtain .05(81) = 4
significant results, each of which would be a Type I error. The authors
report 5 significant effects, an outcome that would not be very surprising to
a person who believed that the variables studied do, in fact, have no
effects at all in the population.