Hello Seyyed,

If you are asking, could n = 2 subgroups (each having some number of individuals nested within the subgroup) be nested within each of two major independent groups, in a given design, the answer is "Yes."

If you are asking, is one-way ANOVA suitable for handling such a data set, the answer is, unfortunately, "No." You have a doubly-nested design (cases nested within subgroups, which are subsequently nested within groups; sometimes called a hierarchical linear model).

You could accurately estimate the main effect of independent group differences via one-way anova or independent t-test by (a) computing subgroup mean scores, and (b) using these (4 in all) mean scores as the data points for comparing the two independent groups. However, note that effective sample size is 4 cases, so statistical power will be very low unless the degree of difference across groups is very large. (This is the case as well in the doubly nested design: two sublevels is not an optimal number for evaluating differences.)

Good luck with your work.

A one-way anova may be appropriate, depending. There is a traditional approach in agronomy. Let's say you have four fertilizer treatments: Fertilizer A, Fertilizer B, Fertilizer C, and Fertilizer D. It would be natural to analyze this design as a one-way design. However, let's say that Fertilizer A and Fertilizer B are organic fertilizers and Fertilizer C and Fertilizer D are chemical fertilizers. If you analyze the experiment with a one-way ANOVA, you can then test the difference between organic and chemical fertilizers with a single degree-of-freedom contrast. And you can also test the difference within organic fertilizers and then within chemical fertilizers. ... Typically the contrasts are orthogonal... Depending on the philosophy, there may be a requirement that you can only conduct as many contrasts as will "use up" the degrees-of-freedom in the original anova. ... I have an example here in R about wine, with the caveat that I wrote it: https://rcompanion.org/rcompanion/h_01.html . I know these types of contrasts are also easy to conduct in SAS, I think using PROC GLM. It's been a while.

Dear David Morse & Sal Mangiafico Thanks for your constructive responses and nice statistical explanations.

My research design is similar to the fertilizers one mentioned by Salvatore.

Dear David Morse

Which statistical test is the most suitable to compare the mean scores of four subgroups nested within two main independent groups if we do not consider one-way ANOVA as the appropriate one?

Group 1=2 subgroups

Group 2= 2 subgroups

Hello Seyyed,

If your goal is to compare 4 mean scores (2 each in group 1 and group 2), then you're pretty much out of luck; there really isn't a good alternative test.

Some might suggest a Mann-Whitney test, but with n1 = n2 = 2, there's simply no assortment of values that would yield a statistically significant result at typical alpha levels. The lower bound of sample sizes in order to identify a difference at the .05 level is n1 = n2 = 3. Again, it's the power issue.

An exact test (making no distributional assumptions) won't work either, for the same reason.

I think the best you can do is try anova or independent t for this situation.

If, on the other hand, you wish to compare two subgroups, one with the other, you may certainly do that (assuming a decent number of cases per subgroup). Independent t or Mann-Whitney or exact/resampling tests could be used.

Finally, here's a question for you. Which hypothesis is most important for you to test:

1. Group 1 results = Group 2 results;

2. Subgroup 1 = Subgroup 2 results; and Subgroup 3 = Subgroup 4 results; or

3. Subgroup 1 = Subgroup 2 = Subgroup 3 = Subgroup 4 results?

If #1, and you're willing to consider all responses within a group as independent observations, then Independent t OR Mann-Whitney U OR exact/resampling test. If not, then you have the n1 = n2 = 2 issue, discussed above.

If #2, and you're willing to make same assumption as for #1, then same methods as for #1 (but, there would be two tests).

If #3, and you're willing to make the same assumption as for #1, then ANOVA, or Kruskal-Wallis, OR exact/resampling test. Note that this test carries the information associated with hypothesis #1 as well.

Good luck with your work.

Seyyed Morteza Hashemi Toroujeni , the appropriate analysis may depend on your design. ... In my fertilizer example, there is no nesting of treatments in reality. It's just a conceptual grouping, where we wish to compare between or within some treatments because we are interested in such comparisons. ... However, there are also cases where treatments are nested in reality. A common design is the split plot design. You might look up this design and analysis. Often this kind of design would be analyzed with a hierarchical, or mixed-effects, model. If you can get a copy of Douglas Montgomery, Design and Analysis of Experiments, there is a chapter on Nested and Split-Plot Designs that may be helpful. But you should be able to find information on split plot designs online. ... At this point, it's not clear to me what your design is.

David Morse & Sal Mangiafico thanks for your great explanations