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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.
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.
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.
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>.
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 GenderMobile_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 ( ηp^2 ) 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 , ηp^2 = Partial Eta Squared
...along with a sentence, explaining what you have found. For example:
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
