Download Two-Way Independent ANOVA

Two-Way Independent ANOVA
Analysis of Variance (ANOVA) a common and robust statistical test that you can use
to compare the mean scores collected from different conditions or groups in an
experiment. There are many different types of ANOVA, but this tutorial will
introduce you to Two-Way Independent ANOVA.
An independent (or between-groups) test is what you use when you want to
compare the mean scores collected from different groups of participants. That is,
where different participants take part the different conditions of your study.
The term two-way simply to refers to the number of independent variables you
have; in this case, two.
You would use a Two-Way Independent ANOVA when you have the following:



one dependent variable
two independent variables
participants are only assigned to one condition for each of your IVs
This tutorial will show you how to run and interpret the output of a two-way
independent ANOVA using SPSS. To do this, let’s consider a fictional experiment
investigating the influence of mobile phone use on attention whilst driving.
Worked Example
Whilst driving, it is important to be able to reliably detect any sudden changes in
your visual environment in order to react to any potential hazards and drive safely.
To be able to do this successfully, you need to make sure you have adequate
attentional resources. We know that mobile phone use whilst driving significantly
impairs situational awareness and driving ability, as a result of drawing on
attentional resources. So imagine that we would like to explore this idea further.
There is a commonly held belief that women are better at multi-tasking than men
(although there is almost no scientific research on this topic). If this is the case, could
it be that women’s driving ability might be less affected by mobile phone use than
men's driving ability?
In this fictional study we could investigate this by looking at participants’ ability to
detect changes in a visual scene whilst driving in a driving simulator.
Half of the participants could take part in the driving task without distraction, whilst
the other half of the participants could do the task whilst simultaneously being asked
a series of scripted questions to simulate hands-free mobile phone use. An equal
number of men and women would take part in each condition.
In this example, we have:
 one dependent variable: number of changes detected
 two independent variables:
mobile use (2 levels: absent and present)
gender (2 levels: male and female)
 participants who were assigned to only one condition for each of the IVs
This is what the data collected should look like in SPSS (and can be found in the SPSS
file ‘Week 13 data.sav’):
As a general rule in SPSS, one row should contain the data provided by one
participant.
In a between-participants design, this means that we have one column for our DV
and separate columns for each of the IVs. In the IV columns, individual participants
are given a code which represents the condition that they belong to. The different
columns display the following data:

The Gender column represents our first independent variable. Codes have
been used to tell SPSS which condition each of the participants belonged to.
In this case:
o 1 = male
o 2 = female
Revisit the tutorial Adding Variables to see how this is type of coding is done.

The Mobile_Use column represents our second independent variable. This
column tells us whether or not the participant was using a mobile phone
whilst taking part in the study. Codes have been used to tell SPSS which
condition each of the participants belonged to. In this case:
o 1 = mobile phone was absent
o 2 = mobile phone was present

The Changes column displays how many changes to the scene each
participant successfully detected whist in the driving simulator. This is our
dependent variable.
As mentioned at the beginning of this tutorial, the Two-Way Independent ANOVA
compares the scores of different groups across two variables.
In this case, we want to compare the change detection scores for the two mobile
phone usage conditions, the two genders and we want to look at the interaction
between these variables.
To start the analysis, begin by CLICKING on the Analyze menu, select the General
Linear Model option, and then the Univariate… sub-option.
The “Univariate ANOVA” box should now appear.
This is where we tell SPSS what our independent and dependent variables are. We
can begin the analysis by defining our IVs.
Note that Gender of Participant is already selected.
CLICK on the arrow to the left of the Fixed Factors box to add this variable to the
analysis.
Using the same method, we also need to select the second IV (Mobile_Use) and add
it to the analysis.
Now, we need to tell SPSS what the dependent variable is. Select the variable
Changes Detected and move it across to the Dependent Variable box...
Now that our variables have been defined, we are almost ready to run the ANOVA.
But before we do, we need to ask SPSS to produce some other information for us, to
help us understand our data.
First, we need to tell SPSS which descriptive and inferential statistics
we want it to produce. CLICK on the Options button to do this.
This opens the Univariate Options box. To produce information for the different
variables and conditions, highlight all of the factor names in the box, as is shown
here. When doing this yourself, if you simultaneously hold down the Shift key you
can click on and highlight all of the variable options in one go. To move the variables
across to the Display Means for box, CLICK the arrow to the left of it.
In the bottom half of the
dialogue box, there are a
number of tick box options
that you can select to get
more information about the
data in your output.
In this example we are just
going to select three.
1. Descriptive statistics, so SPSS will produce some means and standard
deviations for us.
2. Estimates of effect size, as we need to report this when writing up our
analysis.
3. Homogeneity Tests. This is because an assumption of the independent
ANOVA is that the groups you are comparing have a similar dispersion of
scores. This assumption is called homogeneity of variance. If it is not met,
the ANOVA may not be reliable and this must be taken into account when
interpreting your results. As such, this option allows you to test the
assumption of homogeneity (or equality) or variance.
Once these options have been selected, click on Continue to return
to the main dialog box.
Almost there… but before we can run the analysis, we need to tell SPSS to produce
some more things for our output.
There are a number of other buttons we could select here. If any of our IVs had
more than two levels (or conditions) we would need to select the Post-Hoc tests
button now. But as both Gender and Mobile_Use only have two (male vs female;
absent vs present), we can skip this option on this occasion.
Visit the tutorial on One-Way Independent ANOVA to see more about the Post Hoc
options available in SPSS.
So, next we want to tell SPSS to create a graph of our data for us. This will help us
interpret any interaction there might be between the two Independent Variables.
We can do this by CLICKING on the Plots button.
Here we are going to tell SPSS what
type of graph we want. As both
variables have the same amount of
levels, it doesn’t really matter which
order you put them into the graph here
(although it is usually better to put the
IV with the most levels on the
horizontal axis).
As such, move Gender across to the
Horizontal Axis box and Mobile_Use
over to the Separate Lines option, by
CLICKING on the arrows to the left of
the relevant options.
Once these options have been selected,
this graph needs to be added to the
Plots box.
To do this, CLICK on the Add button.
Once the graph has been added, click
on Continue to return to the main
dialog box.
We are now ready to run the analysis!
CLICK OK to continue
You can now view the results in the output window:
But what does this show you? Let’s look at the output tables one at a time.
Between-Subjects Factors
The first box in your output is just here to remind you what values you have assigned
the different levels of your variables, and what they mean. You may find it useful to
refer back to this when interpreting your output.
From looking at the box you should be able to see that for the first factor, Gender,
there are two levels where: 1 = Male 2 = Female
For Mobile_Use, the numbers represent whether or not participants used a mobile
phone whilst undertaking the detection task: 1 = Absent and 2 = Present.
Descriptive Statistics
This table gives you your descriptive statistics. It’s good to look at and report these,
as they can give you an important insight into the pattern of your data.
From this table we can see the average (mean) change detection scores for each of
the possible variable conditions. You can also see the variation in the data (i.e.
spread of scores) for the different groups from the standard deviation.
From this table we can see that on average, fewer changes were detected in the
mobile present condition (mean = 15.00) than in the mobile absent condition
(mean = 20.80).
We can also see that the overall memory scores for the males and females were
relatively similar (means = 17.80 and 18.00 respectively).
This table also shows the breakdown of change detection scores for the males and
females for each of the mobile phone conditions. Both genders appear to perform
worse in the mobile phone condition, than in the mobile absent condition.
While this suggests that there is no interaction between the variables, we need to
look at the inferential statistics to confirm this.
Levene’s Test of Equality or Error Variances
As mentioned earlier, an assumption of ANOVA is that the groups you are comparing
have a similar dispersion of scores (otherwise known as homogeneity of variance).
The Levene’s Statistic tells us whether or not this is the case.
If the test is significant this indicates that there are statistically significant differences
in the way in the data are dispersed, suggesting that the assumption of homogeneity
has not been met.
As we want the variances to be similar, we are looking for a non-significant result
here. In this example, the Sig. column tells us that this is what we have found,
as p = .345, which is greater than 0.05. So we can say:

Levene’s test confirmed that the assumption of homogeneity of variance has
been met, F(3,76) = 1.12, p>.05
Tests of Between-Subjects Effects
This is the most important table in the output. This is where we get our inferential
statistics for the Analysis of Variance (ANOVA). The key columns you need to
interpret your analysis are:

df stands for degrees of freedom. Degrees of freedom are crucial in
calculating statistical significance, so you need to report them. We use them
to represent the size of the sample, or samples used in the test. Don’t worry
too much about the stats involved in this though, as SPSS automatically
controls the calculations for you.
With Independent ANOVA, you need to report the df values for all of your
variables and interactions. In this case, you would need to know the dfs in
the rows labelled Gender, Mobile_Use and Gender*Mobile_Use. In
addition, you also need to report the residual error df, which can be found in
the Error row.

F stands for F-Ratio. This is the test value calculated by the Independent
ANOVA, you need to report the F values for all of your variables and
interactions, in this case: Gender, Mobile_Use and Gender*Mobile_Use.
It is calculated by dividing the mean squares for each variable or interaction
by the error mean squares. Essentially, this is the systematic variance (i.e.
the variation in your data that can be explained by your experimental
manipulation) divided by the unexpected, unsystematic variance. If you’re
looking for a significant effect, then you want there to be more systematic
variance than unsystematic (error) variance. The larger your F-Ratio the
more likely it is your effect will be significant.

Sig stands for Significance Level. This column gives you the probability that
the results could have occurred by chance, if the null hypothesis were true.
The convention is that the p-value should be smaller than 0.05 for the F-ratio
to be significant. If this is the case (i.e. p < 0.05) we reject the null
hypothesis, inferring that the results didn’t occur by chance (or as the result
of sampling error) but are instead due to the effect of the independent
variable. However, if the p-value is larger than 0.05, then we have to retain
the null hypothesis; that there is no difference between the groups.

Partial Eta Squared. While the p-value can tell you whether the difference
between conditions is statistically significant, partial eta squared (ηp2) gives
you an idea of how different your groups are. In other words, it tells you
about the magnitude of your effect. As such, we refer to this as a measure of
effect size.
To determine how much of an effect your IV has had on the DV, you can use
the following cut-offs to interpret your results:
o 0.14 or more are large effects
o 0.06 or more are medium effects
o 0.01 or more are small effects
How do we write up our ANOVA results?
So we know which columns we need to look at, but how do we write this up and
what numbers do we need to use?
To make life easier for you, SPSS groups your analysis into rows: one row for each
variable and interaction. You can then report each one separately using this
formula:
F (IV df, error df) = F-Ratio, p = Sig, ηp2 = Partial Eta Squared
...along with a sentence, explaining what you have found. For example:

There was no significant effect of Gender on participants' change detection
scores (F(1,76) =.11, p > .05, ηp2 = .001).

There was a significant main effect of Mobile_Use on participants’ change
detection scores (F(1,76) = 87.89, p < .001, ηp2 = .54).

There was no significant interaction between Gender and Mobile_Use
(F(1,76) = .24, p > .05, ηp2 = .003).
But the question is, what does this all mean? We know we have a significant effect of
mobile use, but no interaction or effect of gender... but we need to explain what this
actually means in English!! To do that, we need to refer back to our Descriptive
Statistics to see what is happening in each of our different conditions.
Estimated Marginal Means
These boxes show you the breakdown of the means for the different levels of your
IVs, and all of the different conditions. For an Independent ANOVA, the mean values
are the same as those displayed the Descriptive Statistics box at the beginning of
your output.
These boxes also give you the Standard Error rather than the Standard Deviation.
Sometimes you need to report this… but not in this example.
As a quick reminder of what the means tells us, we can see that for our IV Gender,
the detection scores were similar for both males and females.
Looking at the means for the Mobile_Use conditions, we can see that the change
detection scores were worse for the mobile present condition.
Finally, the interaction table shows us that the difference in performance between
mobile absent and present conditions was similar for both males and females.
Profile Plots
This is the final part of the output. The Means Plot graph can often be useful in
helping you to visualise your interaction and it is a good idea to include it in your
write up.
The flat blue line indicates that males and females performed similarly in the mobile
absent condition.
The flat green line also suggests similar performance by the two genders in the
mobile present condition.
You should also look at the gap between the different points on the graph. In this
case, the distance between the green (mobile present) and blue (mobile absent)
lines is similar for both males and females.
What have we learned?
When interpreting and reporting our results, we need to report both the descriptive
and inferential statistics.
We know from our ANOVA table, that the effect of mobile use on change detection
scores was significant, and that performance was best in the mobile absent
condition.
We know that there was no significant main effect of gender, as performance was
very similar for males and females.
We also know that there was no significant interaction between our two IVs.
This lends support to the hypothesis that:

Mobile use can deteriorate change detection abilities
However, there was no effect of gender overall; and there is no evidence that
women were less effected by mobile use than men.
How do we write up our results?
When writing up the findings for a Two-Way ANOVA in APA format, you need to
include all of the relevant information covered by the previous slides. To do this, you
need to answer all of the following questions that are relevant to your study (we can
skip points 2 and 4 for this example, as they do not apply):
 What were the inferential statistics for your IVs and interaction
o i.e. what were the 3 main ANOVA results
 For significant IVs with more than 2 levels, where did the significant
difference(s) lie
o i.e. what were the results of any post hoc tests you ran
 What was the pattern and direction of these differences
o i.e. what were the means and descriptive statistics for your conditions
 If the interaction term was significant, what does it mean in English
o i.e. describe what was happening in the different conditions
It is usually helpful to the reader of your results if you include a table of the means
and standard deviations for all of the conditions in your results. If your interaction is
significant, it may also help to include a graph of your results. You can edit your
graph by double clicking on it in the output in SPSS.
How do we write up our results?
For this example, you might end up writing a results section that looks a bit like this:
The results of the Two Way Independent ANOVA showed that there was no
significant main effect of Gender (F(1,76) =.11, p > 0.05, ηp2 = .001) with males
(mean = 17.80; SD = 4.22) and females (mean = 18.00; SD = 3.80) producing similar
change detection scores.
In contrast, there was a significant main effect of Mobile_Use on memory scores
(F(2,18) = 36.255, p < 0.001). Participants in the mobile present condition
performed significantly worse (mean = 15.00, SD = 2.84) than those in the mobile
absent condition (mean = 20.80, SD = 2.63).
In addition, no significant interaction was found between the two varaibles
(F(1,76) = 0.24, p > 0.05, ηp2 = .003).
These findings suggest that mobile phone use may impair change detection ability,
and that this impairment is similar for both genders.
This brings us to the end of this tutorial. Why not download the data file from this
tutorial and see if you can run the analysis yourself? Practice makes perfect!