Term 2 L5: Factorial ANOVA

Question 1

What is a factor?

Answer 1

A factor is a categorical (nominal or ordinal) independent variable.

Question 2

Factorial designs are named according to _________ and __________

Answer 2

the number of factors and
the number of categories in the factors.

In our case, we have a:
two-way 2×3 factorial design

so factors
2x3 because the first factor has 2 levels and the second has 3

Question 3

The number of groups in factorial design is equal too

Answer 3

product (multiplication) of the numbers of factor categories.
For example,

2×3 factorial design, the number of groups
is: 2×3 = 6.
The same is the case with more than two factors:
A 2×4×5 factorial design has 3 factors, with the numbers of
categories 2, 4, and 5, respectively. Hence, the number of
groups is 2×4×5 = 40.

Question 4

What do descriptive stats give?

why use factorial ANOVA? (2 reasons)

DS, SE, SP tttnh-tagma=itpop
(whether the observed group difference are likely to have come from a population where the observer effects don’t exist)

PLUS

interesting to test H0 of effects of ____ _______and their _________

most elegant procedure =

Answer 4

descriptive results give an indication of an
interaction effect in our sample

However, it is possible that these differences are due to sampling error.

We need a statistical procedure that allows us to test whether the observed group differences are likely to have come from a population where the observed effects do not exist (the null
hypothesis that all group means are equal in the population).

In addition, it is interesting to test null hypotheses concerning the effects F1 F2 , and their interaction.

The most elegant procedure Factorial analysis of variance.

Question 5

What does a Factorial analysis do?

partitions the total variation of the dependent variable into various sources

Answer 5

It’s a statistical model is to account for (“explain”) the variation of a dependent variable through a set of independent factors.

It partitions the total variation of the dependent variable into various sources:
• the variation due to factor I (here: Sex of Patient)
• the variation due to factor II (here: Treatment)
• the variation due to the interaction of factors I and II
• “Error”: residual variation (within-group variation) that
cannot be explained by any of the independent variables that we have included in the model.

This logic extends to a case where we have more than two factors. For example, with three factors I, II, and III, we could consider three two-way interactions (III, IIII, IIIII) and also a three-way interaction (III*III)

Question 6

What are the assumptions for FA?

Answer 6

same assumptions as one-way ANOVA:
• Independence of observations
——– (random assignment to groups, except with respect to a ‘natural’ independent variable, such as sex or age group, which cannot be manipulated experimentally)

• Equal variances across all groups;

• Normal sampling distribution of the mean within
each group.

Question 7

How can you tell how well the model fits the data?

PES, R2

Answer 7

To get an indication look at the proportion of the variance in the dependent variable (e.g. weight change) that is explained by the independent variables (eg. Sex, treatment)

This proportion is given by R2 ,
here, R2 = 0.452. (The overall effect size R2 = η2 , where η2 is “Partial Eta Squared”).

This tells us that 45.2% of the variance of Weight Change is explained by the statistical model.

Question 8

what is the mathmatical formula for R2 and how does it map onto the SPSS output?

Answer 8

𝑅2 = 𝑆𝑆𝐶𝑜𝑟𝑟𝑒𝑐𝑡𝑒𝑑𝑀𝑜𝑑𝑒𝑙/ 𝑆𝑆𝐶𝑜𝑟𝑟𝑒𝑐𝑡𝑒𝑑𝑇𝑜𝑡𝑎l

Question 9

what figures do we need from the SPSS output to analyse the H0 in a FA?

F=(,)=, P=

Answer 9

F, Df,Df error, p

The test statistic is F(5,24)=3.961. The associated p-value (p=0.009) tells us that
the model prediction is statistically significant at α=0.05.
So, we reject the null hypothesis and conclude that not all group means are the
same.

Question 10

what figures do we need from the SPSS output to analyse the H0 (that all group means are the same) in a FA?

F=(,)=, P=

Answer 10

F, Df,Df error, p

The test statistic is F(5,24)=3.961. The associated p-value (p=0.009) tells us that
the model prediction is statistically significant at α=0.05.
So, we reject the null hypothesis and conclude that not all group means are the
same.

Question 11

How do you report the main effects of a FA?

Answer 11

A main effect is the effect of one factor ignoring all of the other factors.

Here, treatment accounts for 22.6% of the variance of “Weight Change”, an effect that is statistically significant: F(2,24)=3.507, p=0.046.

Sex accounts for 1% of the variance of Weight Change, which is not statistically significant: F(1,24)=0.231, p=0.635.

Question 12

How do you report the interaction of a FA?

Answer 12

The interaction of Treatment by Sex is statistically significant:

F(2,24)=6.280, p=0.006.
It accounts for 34.4% of the variance of Weight Change.

Question 13

What must you consider when reporting ME and I in a FA?

Answer 13

In general, for a factorial ANOVA: if an interaction effect is present, it is not meaningful to interpret the main effects of the variables involved in the interaction, on their own.

In the example study…
• The main effect for Sex is not significant
• However, it would be wrong to say that the sex of the patient is unimportant in determining treatment outcome
• Since there is a sex*treatment interaction, we have to consider the effect of treatment separately for each sex

Question 14

what indicates the effect size in FA?

Answer 14

partial eta sqaured

Partial eta-squared is one estimate of the effect of a factor or interaction, after having “partialed out” effects of other factors or interactions in the model.

Partial η𝑇𝑟𝑒𝑎𝑡 ∗𝑆𝑒𝑥 2 = 𝑆𝑆𝑇𝑟𝑒𝑎𝑡∗𝑆𝑒𝑥/ (𝑆𝑆𝑇𝑟𝑒𝑎𝑡∗𝑆𝑒𝑥 + 𝑆𝑆𝐸𝑟𝑟𝑜r)

Question 15

Things to remember about partial eta squared

which values will it lie between?

can the individual PES values for F1,f2 and I be more that he total effect size (R2)?

What can it be used to compare?

Answer 15

Partial eta squared will be a number between 0 and 1, where:

0 indicates that a factor or interaction does not explain any
variation in the outcome variable,

1 indicates that a factor or interaction explains all the variation in the outcome variable that remains after variation explained by other factors/interactions has been “partialed” out.

Note that the partial eta squared values of Treatment (0.226), Sex (0.010), and Treatment*Sex (0.344) sum to more than the total effect size (0.452). Equally, it is possible that the sum of the values of partial eta squared is larger than 1.

Partial eta squared values can be used to compare the effects of factors and interactions. So, we can say that in this experiment, Treatment*Sex has the largest effect size, Treatment the second largest, and Sex the smallest.

Question 16

How to calculate degrees of freedom

Answer 16

Corrected Model df = Number of groups – 1

F1 : Main effect: (Number of levels of this factor – 1)
F2: Main effect: (Number of levels of this factor – 1)
Error: (Sample size – number of groups)

Interaction (no. of levels of factor “F1”−1)
× (no. of levels of factor “F2”−1)

Question 17

See slides for examples of interaction w and w/o interactions

Answer 17

general rule if it cross or is looking like it has done or will cross then interaction if the lines are the same distance apart each time it won’t

Question 18

How to report in writing a FA

Answer 18

The aim of this study was to assess the effectiveness of two treatments for anorexia nervosa in girls and boys. The participants were fifteen girls and fifteen boys, who were randomly allocated to one of three experimental
conditions: Treatment A, Treatment B, and a Waiting List (no treatment), forming a 2×3 factorial design.

We conducted all statistical tests at the 5%
significance level. The dependent variable was the weight change of each patient after X weeks
of treatment.

The main effect of treatment was statistically significant (F2,24= 3.507, p=0.046). As expected, the group of patients who received no treatment did not experience weight change on average. There was no significant main effect of Sex (F1,24= 0.231, p=0.635).
There was a statistically significant Sex by Treatment interaction (F2,24= 6.280, p=0.006).

(Remember to provide an interpretation of the interaction effect)!!!!!!!: (below)
As the plot of “Estimated Marginal Means” shows, Therapy A resulted in a substantial weight gain for boys, but only a small gain for girls. Conversely,
Therapy B resulted in a substantial weight gain for girls, but a small gain for

Question 19

Homogentiy of varience for FA

if the variaences are not equal does this mean we can’t carry out FA?

Answer 19

The variances (and hence the standard deviations) of the six groups are
clearly not the same in this sample. 

Despite the observed differences in variance between the six groups, the Levene’s test statistics is not significant. So, we can assume that variances are equal “in the population” (in this case, this would mean
that boys and girls have the same variance, and that the three different treatments do not differ in their effect on the variance).

The observed differences in variance between groups are likely to be due to sampling error (random variation)