Is there a way to do Tukey's HSD test for the combined results of a one-way ANOVA and a two-way ANOVA? Or should both be followed by separate tables?

Question

Tested the inhibitory effect of 2 essential oils on bacteria at 5 different concentrations. Thus, the first independent variable is the type of oil used and the second is the concentrations. A two-way ANOVA with replication (since I had 10 trials per conc.) in that case would be appropriate according to my knowledge.

After that, I combined the oils in binary combination (5 different ratios) to see their inhibitory effect. If I am assuming correctly, that is a third independent variable and in that case, a one-way ANOVA would be appropriate.

Edit: For my dependent variable, I measured the inhibition zone produced. For the individual effect, I had the following percent concentrations 10%, 30%, 50%, and so on increasing by 20% each time (don't want to fully write all the concentrations so I don't get flagged for plagiarism. Which means when I combined them the 5 ratios were 10:90, 30:70, and so on.

Ultimately, I am trying to deduce whether the individual effect is better than the combined effect.

Edit: This was my initial question — How would I do Tukey's HSD test now? I have access to SPSS. Should I have one table of Tukey's HSD test for the individual effect and one for the combined?

While waiting for an answer, I continued with reading on the internet and found that I can't even do Tukey's HSD test because I only have 2 types of oil and to actually do it, it had to be 3 or more? Is that true, can someone elaborate? Do I still carry on with the one-way ANOVA for the combined effect? If so, what would I do for the individual effect? So I can then determine whether the combined is greater than the individual effect.

Answer 1

Tukey's HSD is a test for multiple populations with a hypothesis that their means are all the same. This is applicable to your situation, you have samples from 10 populations or even 15 when you include the mixed ratio's. Tukey's HSD can also already be applied directly without first performing ANOVA.

• How can I get a significant overall ANOVA but no significant pairwise differences with Tukey's procedure?
• The theory behind Tukey's HSD test.

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In your case, a problem with the use of Tukey's HSD test is possibly the interpretation. Your 15 groups are not completely unrelated but observations of groups where there are two continuous regressor variables changed. Possibly you may do something like fitting a curve or specific relationship to the data. For example, when you observe a linear pattern like

here it might be that the Tukey's HSD only considers concentration 1 and concentration 4 as different. Also, due to the 10 different pairwise comparisons that it considers, it strongly corrects for multiple comparisons while that might not be neccesary.

There are several ways to consider relationships, but with only 5 concentrations a quadratic relationship should be able to cover many possibilities. Then you can do ANOVA on the 2 parameters of the regression instead of the 4 parameters when fitting each additional group seperately.

Another problematic case, what is the interpretation in a situation below, when the Tukey's HSD test considers concentration 1 and concentration 2 different but none of the others?

Probably you are gonna ignore it or give it not much weight. That indicates that Tukey's HSD test is not using the best alternative hypothesis and is not gonna be very powerful.

Answer 2

That you only have 2 different types of oil doesn't prevent you from performing a Tukey HSD test. That's a test on all pairwise comparisons of means. You have 5 different combinations of those oils in your second set of tests, for 5 potentially different mean values (and 10 pairwise comparisons).

I don't know, however, that a Tukey test on those 5 means will answer what seems to be your fundamental question:

Ultimately, I am trying to deduce whether the individual effect is better than the combined effect.

The way that you phrased that, I don't think that you care about whether the effect of combined 30% Oil A and 70% Oil B is different, say, from that of combined 70% Oil A and 30% Oil B. But that's the type of thing you would be evaluating with the Tukey test on the 5 means from the combination treatments.

It seems that the comparisons you need to make are things like: is the effect of combined 30% Oil A and 70% Oil B different from the sum of their individual effects at those concentrations? You do seem to have data that could inform that type of comparison, if your understanding of the subject matter indicates that such sums of different individual treatment outcomes represent valid measures.* If that's what you primarily need and those sums are valid measures, then there are only 5 comparisons of primary interest. You could simply evaluate all 15 single/double tests in a single model (you don't need to do a formal ANOVA) and restrict the multiple comparisons to the 5 of primary interest. The fewer multiple comparisons you need to adjust for, the higher your power to detect a true effect.

You should know that the best ways to evaluate synergistic and antagonistic effects between treatments, as in your study, are still a matter of some dispute. See for example "A critical evaluation of methods to interpret drug combinations," Scientific Reports 10: 5144 (2020). What sometimes seems to be very simple at first ends up depending on hidden assumptions that aren't always met in practice.

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*You might be able to evaluate that in your own data, if you model outcome as a continuous function of concentration within each of the oils. For example, is the predicted effect at 60% Oil A (which you don't have, but could interpolate) the same as what you would expect from the sum of 10% and 50% individually, double the effect of 30% individually, and equal to the difference between 70% and 10% individually?