One-Way Independent ANOVA Analysis of Variance (ANOVA) is 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 One-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 in the different conditions of your study. The term one-way simply to refers to the number of independent variables you have; in this case, one. A one-way independent ANOVA is similar to an independent t-test, in that they are both used to analyse data from a between participants design. However, while the t-test limits you to situations where you only have one independent variable with two levels, an ANOVA can be used when you have more than two conditions. You would use this test when you have the following: one dependent variable one independent variable (with 3 or more conditions) each participant takes part in only one condition This tutorial will show you how to run and interpret the output of a one-way independent ANOVA using SPSS. One-Way Independent ANOVA in SPSS This tutorial uses a subset of real data collected by the Enduring Love project. This research sought to establish the factors that affect the longevity of people s relationships, and investigated how enduring relationships are experienced by couples at different generational points in their life course. The data was collected using a questionnaire, which included a measure of how happy participants were with their relationship/partner. In this example we are only interested in one aspect of this project: whether people in different age groups experience their relationships differently. In this example we have: one independent variable: age group one dependent variable: happiness score The IV had three levels: young, middle-aged and older adults
This is what the data looks like in SPSS. It can also be found in the SPSS file: Week 9 EL subdata.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 (Happiness_Score) and a separate column for our IV (Age). For the IV, participants are given a code which indicates the experimental condition that they belong to. In this example, the different columns display the following data: ID : This column contains the ID number assigned to the participants. We use these numbers as identifiers instead of participant names, as this allows us to collect data while keeping the participants anonymous. Age : The Age column represents our independent variable. Codes have been used to tell SPSS which condition each of the participants belonged to. In this case: 1 = Young adults (under 35 years of age) 2 = Middle-aged adults (aged 35-55) 3 = Older adults (aged over 55) Revisit Tutorial 3: Adding Variables to see how this is done. Happiness_Score : This column displays each participant's happiness score. As part of this study, participants filled out a questionnaire which included a measure of how happy participants were with their relationship/partner. Their score on this measure is our DV and ranges from 1 to 5, with higher scores representing greater happiness with their partner and relationship.
As mentioned at the beginning of this tutorial, the One-Way Independent ANOVA compares the scores of different groups on a certain variable. In this case, we want to compare the happiness scores for the three different age groups. To start the analysis, begin by CLICKING on the Analyze menu, select the General Linear Models, and then the Univariate sub-option. Note: you can also run a One-Way Independent ANOVA through the Compare Means menu. But as you will use the General Linear Models option more often, this tutorial focuses on this method. The Univariate dialog box should now appear. This is where we tell SPSS what our independent and dependent variables are. CLICK on the blue arrows to move Overall Happiness Score into the Dependent Variable box and Participant Age into the Fixed Factors box, as seen here. Now that both variables have been added, 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 better understand our data.
First, it s often a good idea to illustrate your data by producing a graph. To do this, CLICK on the Plots button on the right of the Univariate dialog box. This opens the Profile Plots box. The variable Age should already be selected for you. CLICK on the arrow to the left of the Horizontal Axis box to add this variable to the plot. Then CLICK Add. Once it is added to the Plots window (as below), CLICK Continue. Next, we need to ask SPSS to run some Post-Hoc tests. CLICK on the Post Hoc option in the Univariate dialog box to do this. This opens up the Post Hoc dialog box. Post Hoc tests essentially conduct multiple comparisons between all of the possible pair combinations of your conditions to find out exactly where any significant differences lie. Age should already be selected for you. CLICK on the blue arrow to add this variable to the Post Hoc Tests for: box. There are two sections within this box: The first is for tests that we can use when variances are equal (i.e. when the assumption of homogeneity of variance has been met). There are a number of different options you could choose from here. In this case, SELECT Bonferroni.
The second section, at the bottom of the Post Hoc Multiple Comparisons dialog box is for tests that we would use when the variances of the different conditions cannot be assumed to be equal (i.e. when the assumption of homogeneity of variance has been violated). In this section, SELECT the Games-Howell option. Once you have selected all of your options, CLICK on Continue to close this dialog box do this. Finally, we need to tell SPSS exactly what information we want it to produce as part of the ANOVA. CLICK on the Options button in the Univariate dialog box to This opens the Options dialog box: To produce information for the independent variable and its conditions, SELECT Age in the Factors box and move it across to the Display Means for box by CLICKING on the blue arrow. In the bottom half of the dialog 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. We want SPSS to produce some descriptive statistics for us (like means and standard deviations), so CLICK on this option. 2. It is good practice to report the effect size of your analysis in your write up, so SELECT this option too 3. Finally, 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. To test the assumption of homogeneity (or equality) or variance, CLICK on the Homogeneity Tests option. CLICK on Continue when your box looks like that shown above.
We are now ready to run the analysis! CLICK OK in the Univariate dialog box to continue. You can now view the results in the Output window. Here, SPSS produces all of the statistics that you asked for. There is quite a lot of output for Analysis of Variance, but don t worry - this tutorial will talk you through the output you need, box by box.
Between-Subjects Factors This box is just here to remind you what values you have assigned the different levels of your variable, 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 there are three levels for your Participant Age variable, where the number represents the age group the participants fell into: 1 = 16-34 years 2 = 35-54 years 3 = 55 and above Descriptive Statistics The next table in the output gives you your descriptive statistics. It s good to look at and report these, as they can give you an initial insight into the pattern of your data. From this table we can see that on average, participants in the youngest age group had the highest happiness scores (mean = 4.23), then the oldest age group (mean = 4.17), followed by the middle group (mean = 3.79). We can also see the spread of scores for the different groups from the standard deviation. But how should we interpret the difference between the means? To find out whether this observed pattern is significant, we need to look at the inferential statistics. Test of Homogeneity of Variances As mentioned earlier, an assumption of ANOVA is that the groups you are comparing have a similar dispersion of scores, or variance. The Levene s Statistic tests this. If the test is significant, this means that the group variances are likely to be significantly different, suggesting that the assumption of homogeneity of variance has not been met. As such, we are looking for a non-significant result here. In this example, this is what we have found, as p =.07, 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(2,297) = 2.75, p>.05
Tests of Between-Subjects Effect 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 two of the df values. In this case, you would need to know the df in the row representing your IV (i.e. in the row labelled Age). 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 your variable, which can be found in the Age row. It is calculated by dividing the mean squares for the variable by its 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 (η p 2 ) gives you an idea of how different your samples 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? To report the statistics from an ANOVA, you can use the formula: F (IV df, error df) = F-Ratio, p = Sig, η p 2 = Partial Eta Squared...along with a sentence, explaining what you have found. For example: There was a significant main effect of Age on participants' happiness scores (F(2,297) = 7.34, p =.001, η p 2 =.05). But what does this effect mean? To explain this we need to refer back to our Descriptive Statistics to see what is happening in each of our different conditions. Estimated Marginal Means This box shows you the breakdown of the means for the different levels of your IV. For an Independent ANOVA, the mean values are the same as those displayed the Descriptive Statistics box at the beginning of your output. This box also gives you the Standard Error rather than the Standard Deviation for your different conditions. 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 the youngest and oldest groups have similar happiness scores, both of which are higher than the middle-aged group.
Multiple Comparisons The Multiple Comparisons table essentially does what the title implies. It carries out multiple comparisons between every possible combination of pairs for your conditions. Depending on whether or not your assumption of homogeneity of variance has been met, you would either use the Bonferroni or the Game Howell test. In this case, as Levene s test revealed that the assumption has been met, we can use the Bonferroni post hoc tests: the top half of the table. Reading this table is actually very simple. In the first column we can see the list of our three conditions. For each of the conditions, we can then simply read across each of the rows to see which two groups are being compared. For example, the third row in the table is comparing the middle and younger age groups: To find out whether these comparisons are significant or not, we need to look at Sig column (and the asterisks in the Mean Difference column). Remember, the p-value needs to be less
than.05 to be significant. Here we can see that while there is no significant difference between the older and younger groups, there is a significant difference between: the younger and middle-aged group the older and middle-aged group Profile Plots This is the final part of your output. The Means Plot graph can often be useful in helping you to visualise your results and it is a good idea to include it in your write up. In this case, from the graph we can see that while the happiness scores are relatively similar for the younger and older age groups the middle-aged group scored lower than both of them. The trick now is to put all of the information from your output together to make a results section that is sensible and meaningful!
How do we write up our results? When writing up the findings from your analysis in APA format, you need to include ALL of the relevant information from the output. What were the inferential statistics for your IV? o i.e. what was the ANOVA result If your finding was significant, where do the significant differences lie? o i.e. what were the results of your post hoc tests What was the pattern and direction of these differences? o i.e. what were the means and descriptive statistics for your conditions For this example, you might end up writing a results section that looks a bit like this: The results of the one way independent ANOVA showed that there was a significant main effect of Age on Happiness Scores (F(2,297) = 7.34, p =.001, η p 2 =.05). Bonferroni post hoc tests showed that while the middle-aged group (mean = 3.79, SD =.98) scored significantly lower than both the younger (mean = 4.23, SD =.80; p =.001) and older groups (mean = 4.17, SD =.85; p =.007), the difference between the older and younger groups did not reach significance (p>.05). These findings support the hypothesis that age has a significant effect on perceived happiness in relationships. It is usually helpful to both you and the reader of your results if you include a table of the means and standard deviations for all of the conditions in your results. It can also help to include a graph of your results. You can alter your graph by double clicking on it in the output in SPSS. This brings us to the end of the tutorial. Why not download the dataset used in this tutorial and see if you can produce the same output on your own. Remember, practice makes perfect!