The 2-way ANOVA considers two factors than may interact. It allows to analyze the interaction directly. You can also see this as a two-factorial linear model including a coefficient for the interaction and you can use a t-test to test hypotheses about this coefficient.

The 1-way ANOVA considers only a single factor (here with 4 levels). The ANOVA is then only comparing this model to a model that considers only the grand mean. Tukey's tests are (adjusted) t-tests comparing the means of all the four groups. The interaction between the factors is not tested, because this model does not "know" that there are two factors.

Jochen Wilhelm Thanks for taking the time to respond! From my understanding of your response, it's simply the ability to test the interaction mathematically. If there was no interaction, they are kind of (I know Im being reductive here) the same. I assume the two-way ANOVA would be less frustrating for statisticians to see.

It makes me wonder if I need to brush up on my conceptual or mathematical understanding of interactions. Would the results of Tukey's test only give you a non-mathematical intuition that there was an interaction? Based on how the p-values lay across a data set analyzed by Tukey's, you could suggest there is an interaction, but you would have not done the math to actually conclude this.

But again, thank you so much. This gives me something to think about going forward with my research.

Tukey tests the differences:

1) 0X vs 10X

2) 0X vs 0Y

3) 0X vs 10Y

4) 10X vs 0Y

5) 10X vs 10Y

6) 0Y vs 10Y

The interaction would be: (6) vs (1), or, equivalently, (2) vs. (5)

So if (1) and (6) (or (2) and (5)) show recognizably different effects (not p-values), then this indicates that there might be an interaction, as a "non-mathematical intuition", as you said. Google for "interaction plot".

For a statistician it is most important that the correct questions are adressed in the analysis. Very often in experiments with designs like the one you gave, the central question is for the interaction (e.g. is the effect different between both compounds; this question cannot be answered by (5), as there is already uncertainty and possibly already some difference in (2); the same applies for (6) and (1)). The interaction must be tested specifically, and this can only be done using a model that encodes this interaction (whether you do an ANOVA F-test or a t-test of the interaction coefficient does not matter as these are equivalent). Unfortunately, in most such experiments the interaction is not specifically analyzed/tested and concluded only by "non-mathematical intuition".

Tukey's prodecure is not suited here. It makes all pair-wise comparisons, and not all are of interest (or make sense). The p-values are corrected for these unnessecary tests, so the procedure is overly conservative. Usually, in such experiments it could be important to show that the data are sufficient to demonstrate some effect (like (1) or (6), or (2) or (5)). These can be tested using the pooled standard error, using a linear model that is coding each group individually (like in a 1-way ANOVA), and these tests do not need any correction for multiple testing, because this is not a screening across several possibilities; it is adressing individually different questions.

Thanks again Jochen Wilhelm . I noticed that you have taken upon yourself to answer a lot of the stats questions on research gate for >3 tears, and usually advise to see what you are interested in.

I guess in summary:

If you are interested in an interaction, two-way ANOVA should probably be used. If you do a Tukey's, your stats will be conservative (and have a higher chance of reporting a false negative?). You will also be only comparing the means of different treatments with Tukey's.

The 2x2 table makes this scenario very interesting because the two-way ANOVA can tell you more information than it otherwise normally would in a larger table like 2x3 or 2x4 (i.e. If variable 1 [compound X] is significant, you can actually conclude whether there is a significant increase if the mean is visually higher).

If this is possibly an experiment where each treatment was a capsule of medicine (a pill), the interaction may not matter to the researcher in a clinical setting. This experiment would be less about how the compounds interact, but more about which one changes the response of interest the most. In this case, it may make sense to pursue a one-way ANOVA and compare the means with Tukey's since that is the question that is of interest (the one-way ANOVA stats would basically be pointless in this scenario).

My experiment is similar to directly above

I am really only interested whether there will be a higher rate of false positives, and if the mean of response changes between these 3 treatments specifically (see attachment for a visual):

X) is the mean of 0X:0Y different from 10X:0Y? We hypothesize 10X will enhance an effect

Y) is the mean of 10X:10Y different from 10X:0Y? We want to know if 10Y reduces the effect of 10X.

Z) is the mean of 0X:0Y different from 0X:10Y? We hypothesize not.

In the attached visual, I can tell that:

1) there would be a significant interaction - there is a difference in the differences of 0Y,0X and 10Y,0X compared to the differences of 0Y,10X and 10Y,10X (not sure how to notate this)

2) The main effect of Y is significant because there is a difference in means between 0Y conditions and 10Y conditions

3) The main effect of X is significant because there is a difference in means between 0X conditions and 10X conditions

In this specific 2x2 table structure, because the mean of 10X conditions are higher than the mean of 0X conditions, I can look at the means and say 10X increased the response. Similarly, because the mean of the 10Y conditions are lower than the 0Y conditions, I can look at the means and say that 10Y lowered the response. Are these 3 parameters always sufficient enough to conclude that 10X alone increases (or decreases) a response in a 2x2 table? I think it's true, but I do not think this conclusion is very intuitive based on how (I think) the math works. I think (im not a statistician) I often see scientists misinterpret what interactions compare (e.g. in my data set I have seen interactions be described as the difference between the 0X:10Y condition vs the 10X:0Y condition exclusively, which to me sounds like a misinterpretation because this description breaks when you increase the table size).

If the table was any larger (e.g. 2x3, 2x4, 2x5) because I added more concentrations of Y, and the data got complicated (in the real world with this type of concentration data it shouldn't but lets say it did), I could not confidently say whether the response increased or decreased in any given condition. Just that the main effect of Y was significant, or the main effect of X was significant. Thus, would require some sort of post-hoc test (Tukey's) to determine where the differences are.

Taken together, if I am only interested in the differences in the mean of X,Y,Z - I really am only interested in Tukey's post-hoc test, arn't I?

One thing I can't figure out - which this may actually be the answer to my original question - why does my software use the DFd value to calculate Tukey's post-hoc test. I thought post-hoc tests would be independent of ANOVAs. In my randomized-block, this difference in degrees of freedom results in different p-values. Interesting. I will have to look into this.

"Are these 3 parameters always sufficient enough to conclude that 10X alone increases (or decreases) a response in a 2x2 table?"

To conclude that there is a positive effect of X, given Y=0 ("10X alone increases the response") it is sufficient to compare 10X,0Y to 0X,0Y.

The three tests you do are NOT helpful to judge the interaction of X and Y. In your particular case the figure does show a strong interaction, and this is likely also statistically significant, but you have not found/identified/demonstrated it by the tests you supposed. Imaginge the mean for X0,Y0 was a bit higher, and the mean for X0,Y10 was a bit lower (so that there was some negative effect of Y, given X=0). It's not clear then if the negative effect of Y, given X=10, is stronger than the negative effect of Y, given X=0. Note that this has nothing to do with the fact whether or not any of the simple tests is significant. It does not matter if X0,y= vs. X0,Y10 is significant or not. You have to consider the uncertainty in the estimated effects. If you want to make the point, that the effect of X is decreased in the presence of Y, you must test the effect of Y, given Y=0 vs. the effect of X, given Y=10 - in other words: you must test the interaction of X and Y.

"Taken together, if I am only interested in the differences in the mean of X,Y,Z - I really am only interested in Tukey's post-hoc test, arn't I?"

No. First of all, I don't think that these tests answer the relevant question, i.e.: is the effect of X decreased in the presence of Y (as I tried to explain above). But even if we assume that you are really not interested in the interaction, then These teste are not the complete Tuley contrasts (2 are missing), and it's not a screening, so you scarifice power when using Tukey's procedure.

I can't help with the last question, as I don't really understand what you mean, what software you use and how this software works.