1-Way ANOVA

We previously looked to see who was heavier, males or females, from our Body Composition.omv data. We found that females were significantly heavier than males. However, now we would like to know if the course the student is taking has an impact on their weight (remember this is not cause an effect, just correlation!)

We have a new data file to use here, so open up Course Sport Weight VO2max.omv. Our data file has 4 separate groups in the Course variable, one per degree course, and we are going to compare Weight across the 4 courses.

Before running the test, we need to check that the data is parametric. We know that the observations are independent, and the dependent variable (Weight) is ratio. We just need to confirm that the data within each group is normally distributed and whether or not we satisfy the condition of sphericity.

You should already know how to find some basic information on this data, so run some Descriptives and the Shapiro-Wilk for the 4 different courses.

We’ll look at homogeneity of variance next. We can’t do this the same way as previous weeks, using the t-test menu (try it, and see what error message you get). Instead, click Analyses, then ANOVA, and One-Way ANOVA.

These menus should be familiar to you now, so move Weight into the Dependent variables box, and Course into the Grouping variable box. Check the Homogeneity of variances tick box under assumptions and check the Levene’s test results table in the Results viewer.

In the same way as for the t-test, we are looking to see whether the Levene Statistic is < 0.05, i.e. is the variance significantly different between each group. Here the p value is 0.439, so we can be confident that the variances are not significantly different.

This tells us which parametric ANOVA to run - so under Variances now, we can tick Assume equal.

Turn on Descriptives too. Finally, we need to decide what post-hoc tests we will do - that is, the ANOVA will tell us there is a difference between our 4 groups, and the post-hoc tests will tell us exactly where that difference is.

Under Post-Hoc Tests, select Tukey. This is the Tukey Honest Significant Difference, or Tukey HSD, test. It will compare each group against every other group, but make adjustments behind-the-scenes to minimise the family-wise error rate from running multiple tests at once. Turn on all the options under Statistics that Jamovi offers here, too.

The output form the ANOVA is then presented in the Results viewer. Look at the One-Way ANOVA table. This is where we find our test statistic. In the t-test, this was the t statistic, but in the ANOVA it is the F statistic.

Here, F = 3.144, and the significance is p = 0.034. As the p-value is < 0.05, we take this as a significant result. This does not tell us where the difference occurs, only that there is an effect somewhere. For this particular test it just tells us that the means of the weights of the students in each of the courses is different, so at least one of the groups has a greater/lesser mean weight than the other groups.

Let’s look at the post-hoc table now. This is presented with all the groups on both the rows and columns of the table - the cell where the two cross is the post-hoc comparison results for that particular row against that particular column. To avoid repetition, Jamovi only shows unique combinations.

As we turned on the option to flag significant comparisons, we first just need to look for any cells marked with an asterisk. Look along the first row - the first cell we can see with an asterisk is for Sports Therapy v. Sport Psychology Coaching and Physical Education (p = 0.032). We can see the mean difference between the two groups is 18.5 kg, telling us that Sports Therapy is significantly lighter than Sports Psyc (at the 5% level).

Finally, we need to write up the results from our ANOVA:

There was a significant difference in the weight of the students across the London Metropolitan Sports courses F(3, 45) = 3.144, p = 0.034. Post hoc comparisons using Tukey’s HSD indicated that the weight of students taking Sports Therapy are significantly lighter (M = 70.24, SD = 18.99 kg) than students studying Sport Psychology, Coaching and Physical Education (M = 88.75, SD = 12.80 kg) p = 0.032.

Where previously we included against the t-statistic the degrees of freedom, which was calculated as number of cases -1, in the ANOVA we show the number of groups -1, followed by number of cases -1. You can get these values from the ANOVA table.

Now you try: As well as making a record of the course each student is on, we have also recorded the sport the student takes part in to see if there is any link between a student’s sport and their weight. Run a One-Way ANOVA with a Tukey HSD post-hoc comparison again in the Course Sport Weight VO2max.omv data file.

Kruskal-Wallis test

While the ANOVA is fairly robust to violations of its assumptions (e.g. when there is heterogeneity of variance of similar size groups), there is still a fully non-parametric version (the Kruskal-Wallis test) that can be used if we find that our data really fails to meet the assumptions required for a regular 1-way ANOVA, such as normality.

Still using Course Sport Weight VO2max.omv, we have the same students, but we want to measure how ‘fit’ they actually are by analysing their VO2max values, and seeing if different Sports are represented by different VO2max values.

• H₀: There is no difference in the VO2max values of students who take part in different sport
• H₁: There is a difference in the VO2max values of students who take part in different sports

Before we run the Kruskal-Wallis test, we need to be sure that the parametric assumptions have been violated, otherwise we might as well use the 1-way ANOVA. Look at those now.

Let’s run the Kruskal-Wallis. Click Analyses, ANOVA, then One-Way ANOVA (Kruskal-Wallis). As before, move VO2max into Dependent variables, and Sport into Grouping variable. We won’t worry about effect size, but tick DSCF pairwise comparisons.

The test statistic (χ², pronounced chi-squared) here is 36.248, and the significance is p < .001. So we can clearly see that there is a significant difference between the type of sports taken part in and the VO2max of the participants, but as with the 1-way ANOVA it does not tell us which group(s) is/are significantly different to the others. For that, we should look at the pairwise comparisons table below.

This is laid out slightly differently than the Tukey HSD table from before, but does a similar thing - shows us the comparison of each group with every other group. Look at the p column for our significance values. We don’t get the mean differences in this table, so go back to our Descriptives to get a sense of which group has the greater VO2max in each comparison.

Lots of the groups are significantly different from each other! Only Football v. Rugby, and Hockey v. Triathlon are not significantly different.

Let’s write up our results:

There was a significant difference between the sport the students take part in and their VO2max results, χ²(3) = 36.248, p < 0.001. Follow up testing was performed with Dwass-Steel-Critchlow-Fligner pairwise comparisons. It appears that VO2max is significantly different between Football and Hockey (W = 5.483, p < 0.001), Football and Triathlon (W = 5.247, p = 0.001), Rugby and Hockey (W = 6.360, p < 0.001), and Rugby and Triathlon (W = 6.043, p < 0.001).

Repeated measures 1-way ANOVA

This is the partner measure of the 1 way independent measures ANOVA. Where the 1-way independent ANOVA examined several groups for one particular measure or result, this time we will be looking at 1 group for several (repeated) results. For instance, we might want to examine 1 group under several conditions (e.g. 5mg caffeine, 10mg caffeine and then 20 mg caffeine intake) and the effect on sprint performance (50m sprint time). For this test, I took all the Sport and Dance Therapists, and performed a double blind randomised placebo-controlled trial.

Open up the Caffeine.omv data file. We’ll assume our data is parametric (feel free to test it yourself if you’re curious).

Click Analyses, then ANOVA, and Repeated Measures ANOVA.

This is probably the most different of all the menu options we’ve seen so far. Here, we need to tell Jamovi the names of all our levels of independent variable (Caffeine intake) first - so in the top box marked Repeated Measures Factors, give our factor a name (Caffeine) and enter 4 labels - 0mg, 5mg, 10mg, 20mg.

Then, drag each variable - Sprint_0, Sprint_5, etc. - into the Repeated Measures Cells box (where it says ‘drag variable here’) in the same order that we just labelled the factors above. Ignore the ‘Between Subject Factors’ and ‘Covariates’ boxes.

Underneath, tick Partial η² and enter Sprint Time as the Dependent Variable Label.

Then, under Assumptions, tick Sphericity tests and make sure all three Sphericity corrections are also ticked. Under Post Hoc Tests, move Caffeine over into the testing box. Let’s change things up for our post-hoc method and tick Bonferroni rather than using Tukey again.

Finally, under Estimate marginal means, move Caffine into the Term 1 box and tick Marginal means plots and Marginal means tables.

This all produces a lot of information in the Results viewer, so we will work through just the important bits.

Under the Estimated Marginal Means - Caffeine table, we can find some basic descriptives for our ANOVA. These are both useful to check that your variables are entered in the right order. It already looks like the greater the amount of caffeine ingested, the quicker the 50m sprint performance, but we are unable to say yet whether this is statistically significant.

Next, find the Test of Sphericity table. As with other pre-test tests, we want this test to be non-significant, to show that Sphericity has not been violated. Our p here is greater than 0.05, so we can assume that sphericity has not been violated, and can assume that the variances of the differences between levels are equal.

Next is the main ANOVA output. Find the Within Subjects effect table (ignore the between subjects table completely).

In the Within Subjects table, all we are concerned with in this test is the first row of data, the Caffeine records, and we read the line with ‘None’ under the Sphericity Correction column.

F (the repeated measures test statistic) is 23.563, and p < .001, showing that there is a significant difference between the groups (this confirms what we thought we saw in the means). But still, we do not know yet exactly which level of caffeine is significantly different from the others.

For that, let’s jump down to the Post-Hoc Comparisons table. This reads across the rows (not rows x columns like the independent ANOVA post-hocs).

We want to compare based on the control group (0 mg), so we can see that the time taken for the 50m sprint is significantly less when ingesting either 5 mg (p < 0.001), 10 mg (p < 0.001) or 20 mg (p < 0.001) of caffeine compared to no caffeine

What is interesting though is that while 10 mg and 20 mg are significantly better than no caffeine at all, they are not significantly better than just 5 mg (5 mg compared to 10 mg, p = 1.000 and 5 mg compared to 20 mg, p = 0.376). 10 and 20 mg are also not significantly different to each other (p = 1.0).

If this is a little complex to get your head around, then if we bring in the graph of this from the estimated marginal means section, it will hopefully become clearer:

From the graph it is clear to see that the 3 caffeine groups (5 mg, 10 mg, and 20 mg) are much lower than the no caffeine group (0 mg), but the three caffeine groups are very close to each other.

We can now report what we have found:

The results show that the sprint performance was significantly affected by the ingestion of caffeine F(3, 18) = 25.563, p < 0.001. Post hoc tests using the Bonferroni correction revealed that 5 mg of caffeine significantly improved performance times (p < 0.001), but any increase above that was not significantly different to the 5 mg ingestion (5-10 mg: p = 1.00, 5-20 mg: p = 0.376, 10-20mg: p = 1.00).

Friedman’s ANOVA

Friedman’s ANOVA is the non-parametric version of the Repeated Measures ANOVA.

Similar to the Kruskal-Wallis 1-way ANOVA (for independent groups), the Friedman’s test is used to detect differences in treatments within one group across multiple test attempts. For instance, for the Kruskal-Wallis we could look at the perceived fitness across each of the courses, but how could we analyse the perceived fitness of the overall cohort at different time points? This is where Friedman’s ANOVA can be used.

Similarly while the Mann-Whitney U was used to compare two groups based on their perceived fitness, the Friedman will allow for multiple groups.

In this example we will use the PerceivedFitness measure once more, but this time from a new data file with some extra data. Load the FitPerception.omv data file.

In this we have asked our group for their perceived fitness level (a nominal data type), but this time they were asked at the start of their first, second, and final year of their degree course, presented as FitPercept1, FitPercept2, and FitPercept3 respectively. We now want to know if their perception of their fitness level changed over the period of their degree.

Click Analyses, ANOVA, and Repeated Measures ANOVA (Friedman).

This menu is similar to the parametric repeated measures test from from, but a bit simpler due to being non-parametric. Here, all we need to do is move all 3 variables into the Measures box, and tick all the options underneath.

We can see from the Descriptives table and plot that there appears to be a drop in perceived fitness level each year (perhaps as sports science knowledge increases?).

From the Friedman results table, we see the χ² (chi-square) test statistic (22.531), the test statistic’s degrees of freedom (3 groups, so 3-1 = 2 degrees of freedom), and the significance, p < 0.001. This shows that the level of study does affect the perception of fitness.

Finally, we can look at the pairwise comparisons results to see how each year differs from each other.

Here, every comparison differs significantly from each other, indicating that fitness perception has decreased a significant amount each year of the degree course.

Let’s write up our findings:

The perceived fitness level of the students changed significantly over the 3 years of study, (χ² = 22.531, p < .005). Durbin-Conover tests were used to follow up on this finding. It appeared that perceived fitness level significantly dropped between year 1 and year 2 of study (T = 3.124, p = 0.002),between year 2 and year 3 of study (T = 2.187, p = 0.031), and between year 1 and year 3 of study (T = 5.311, p < 0.001).