To my understanding, Factorial ANOVA seems most appropriate to use when there are 2 independent variables/factors, e.g., one which is a type of population with 2 levels (clinical and sub-clinical populations) and the other independent variable/factor e.g., type of intervention (MBCT, ACT, DBT, MBSR), and one dependent variable (improved mental health). Not entirely sure if I should use an one-way ANOVA or Factorial ANOVA. Seems like the Factorial ANOVA has more often been used to compare multiple groups and interventions and their relationships. Please correct me if I'm wrong, Factorial ANOVA is used when there are two or more independent variables/factors at several different levels (in this case would be a 4x2, otherwise stick with the basic one way ANOVA (one independent variable).
Any other type of analysis that would be appropriate to use in this case?
You can do a multi-factorial ANOVA only if you have multiple (=2 or more) independent (experimental/explanatory/predictor) variables (what are all factors for sure; if these were all numeric variables, we would not talk about ANOVA but about multiple regression, and if it was a mix of factros and numerical variables it would be called a general linear model).
You must do multi-factorial ANOVA if you are interested in interactions (which is often the most relevant scientific question but not recognized by scientists).
If you are not interested in interactions, you can always do a one-factorial ANOVA by coding the independent factors as one single factor of all possoblelevel-combinations (a 4x2 experiment, for instance, would be seen as a 1x8 experiment). This is technically as valid as the multi-factorial ANOVA (this is where I kindly disagree with Jos Feys), but it does not allow you to neatly test interactions (which would be the main purpose of the multi-factorial analysis).
PS: one-way = one-factorial and multi-way = multi-factorial. I orefer "factorial" over "way" as I think this term is more spot-on.
If you have very strong/sound reasons not to expect an interaction between the 2 factors, you can stick to basic one-way ANOVA. (The example you give seems to suggest a multilevel/ hierarchical regression. Your subjects seem to be nested within clinical or sub-clinical level, in which they are not independent from each other.)
You can do a multi-factorial ANOVA only if you have multiple (=2 or more) independent (experimental/explanatory/predictor) variables (what are all factors for sure; if these were all numeric variables, we would not talk about ANOVA but about multiple regression, and if it was a mix of factros and numerical variables it would be called a general linear model).
You must do multi-factorial ANOVA if you are interested in interactions (which is often the most relevant scientific question but not recognized by scientists).
If you are not interested in interactions, you can always do a one-factorial ANOVA by coding the independent factors as one single factor of all possoblelevel-combinations (a 4x2 experiment, for instance, would be seen as a 1x8 experiment). This is technically as valid as the multi-factorial ANOVA (this is where I kindly disagree with Jos Feys), but it does not allow you to neatly test interactions (which would be the main purpose of the multi-factorial analysis).
PS: one-way = one-factorial and multi-way = multi-factorial. I orefer "factorial" over "way" as I think this term is more spot-on.
From the way the scenario is described, it sounds like you want to use a (multi-) factorial approach as you are interested in the interaction of the Group and the Intervention, right? Like you want to know if these interventions behave the same in the Clinical group as in the Sub-clinical group. For example, look at the following plot. Here, we can see that the effect of Tutor types is not the same for Males as it is for Females. https://ademos.people.uic.edu/Chapter13_files/figure-html/unnamed-chunk-16-1.png
I also wanted to add ---- in response to a sub-topic running through this thread ---- that, in general, if data are analyzed as a one-way anova, the interaction effect can still be teased out using contrasts.
(Maybe search for "orthogonal contrasts", "planned contrasts", or "single degree-of-freedom contrasts").
This is useful, for example, if you had a design which is a 2 x 2 factorial plus a control treatment. Let's say you are growing corn and one treatment variable is nitrogen rate (100 kg/ha and 200 kg/ha) and the other treatment variable is phosphorus rate (25 kg/ha and 50 kg/ha), but you also have a control treatment that receives no fertilizer. You can analyze this design as a one way anova with 5 treatments (control, 100/25, 100/50, 200/25, and 200/50). The contrast (0, 1, 1, -1, -1) tests the effect of N (100 vs 200). The contrast (0, 1, -1, 1, -1) tests the effect of P (25 vs 50). And the contrast (0, 1, -1, -1, 1) tests the interaction of N and P.
You could do other planned tests such as (4, -1, -1, -1, -1) to compare control to all other treatments.
Notice with this last contrast that we just created a super-category of groups called "all other treatments". This is useful if, say, we are testing some aspect of several wines, and wanted to compare red wines to wine wines.
As the number of levels of the factors gets larger, this gets more complicated, but can still be carried out.
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