Homogeneity of Variance > Hartley’s Test / Fmax
What is The Fmax Test?
The Fmax test (also called Hartley’s Fmax) is a test for homogeneity of variance. In other words, the spread (variance) of your data should be similar across groups or levels. Compared to Levene’s test, Hartley’s test is fairly simple to figure out by hand.
An assumption of the Fmax test is that there are an equal number of participants in each group.
Steps
Example question: Run Hartley’s Fmax test for two conditions X and Y. Each condition (level) has 10 participants with variances of X = 12 and Y = 4.
Step 1: Find the variances for each group/level/condition. If you’re running an ANOVA test, the s^{2} values (i.e. the variances) will be shown in the output. Otherwise, you can calculate by hand (or use our variance and standard deviation calculator).
For this example, our variances are 12 and 4.
Step 2: Divide the larger variance by the smaller variance to find the F_{max} ratio:
Note: If the ratio is close to 1, you can stop here, as your data shows homogeneity of variance.
In this example, we have 12/4 = 3. This ratio isn’t close to 1, so go to the next step.
Step 3: Look up the number of levels and the degrees of freedom (number of items in each level minus 1) in the F_{max} table.
For 9 degrees of freedom (n – 1 = 10 – 1 = 9) and a k of 2 (k is the number of levels/groups), the table shows an F_{max} value of 4.04.
Step 4: Compare your calculated value from Step 2 to the table value from Step 3.
- If your calculated value from Step 2 is smaller than the table value, the variance is homogeneous.
- If your calculated value from Step 2 is larger than the table value, the variance is not homogeneous.
Our calculated value of 3 is smaller than the table value of 4.04, so our variance is homogeneous.
Hartley’s F_{max} Table
Degrees of freedom: Subtract 1 from your sample size to find the degrees of freedom. For example, if you have 10 items in your sample, then df = 10 – 1 = 9.
k: is the number of levels/groups or conditions.
DF |
K (Number of levels/groups/treatments) |
||||||||||
2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |
2 | 39.0 | 87.5 | 142 | 202 | 266 | 333 | 403 | 475 | 550 | 626 | 704 |
3 | 15.4 | 27.8 | 39.2 | 50.7 | 62.0 | 72.9 | 83.5 | 93.9 | 104 | 114 | 124 |
4 | 9.6 | 15.5 | 20.6 | 25.2 | 29.5 | 33.6 | 37.5 | 41.1 | 44.6 | 48.0 | 51.4 |
5 | 7.2 | 10.8 | 13.7 | 16.3 | 18.7 | 20.8 | 22.9 | 24.7 | 26.5 | 28.2 | 29.9 |
6 | 5.82 | 8.38 | 10.4 | 12.1 | 13.7 | 15.0 | 16.3 | 17.5 | 18.6 | 19.7 | 20.7 |
7 | 4.99 | 6.94 | 8.44 | 9.70 | 10.8 | 11.8 | 12.7 | 13.5 | 14.3 | 15.1 | 15.8 |
8 | 4.43 | 6.00 | 7.18 | 8.12 | 9.03 | 9.78 | 10.5 | 11.1 | 11.7 | 12.2 | 12.7 |
9 | 4.03 | 5.34 | 6.31 | 7.11 | 7.80 | 8.41 | 8.95 | 9.45 | 9.91 | 10.3 | 10.7 |
10 | 3.72 | 4.85 | 5.67 | 6.34 | 6.92 | 7.42 | 7.87 | 8.28 | 8.66 | 9.01 | 9.34 |
12 | 3.28 | 4.16 | 4.75 | 5.30 | 5.72 | 6.09 | 6.42 | 6.72 | 7.00 | 7.25 | 7.43 |
15 | 2.86 | 3.54 | 4.01 | 4.37 | 4.68 | 4.95 | 5.19 | 5.40 | 5.59 | 5.77 | 5.95 |
20 | 2.46 | 2.95 | 3.29 | 3.54 | 3.76 | 3.94 | 4.10 | 4.24 | 4.37 | 4.49 | 4.59 |
30 | 2.07 | 2.40 | 2.61 | 2.78 | 2.91 | 3.02 | 3.12 | 3.21 | 3.29 | 3.36 | 3.39 |
60 | 1.67 | 1.85 | 1.96 | 2.04 | 2.11 | 2.17 | 2.22 | 2.26 | 2.30 | 2.33 | 2.36 |
If you prefer an online interactive environment to learn R and statistics, this free R Tutorial by Datacamp is a great way to get started. If you're are somewhat comfortable with R and are interested in going deeper into Statistics, try this Statistics with R track.
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in your Fmax table the CV value is wrong for 2 treatments and df=7; should be 4.99.
Thanks for spotting that. It’s now fixed.
what if the Df is 29 and K = 2.
29 is not presented in the table. please help me. thank you
You could extrapolate. 29 is very close to 30.