See these links, please:

https://statisticsbyjim.com/regression/interpret-r-squared-regression/

https://www.researchgate.net/post/How-does-one-calculate-effect-size-if-only-regression-coefficient-is-given

https://www.statisticssolutions.com/free-resources/directory-of-statistical-analyses/effect-size/

https://statisticsbyjim.com/regression/low-r-squared-regression/

https://www.spss-tutorials.com/effect-size/

https://stats.stackexchange.com/questions/15749/how-to-report-effect-size-measures-r-and-r-squared-and-what-is-a-non-technical-e

https://www.researchgate.net/post/Low-R-squared-values-in-multiple-regression-analysis

Hello Camille,

The target effect size, if expressed as R-squared, should represent the minimum amount of variance explained by the multiple linear regression model that you believe is sufficiently noteworthy to detect, if it exists in the population.

That value can and may be different from what a prior study has reported.

As you no doubt recognize, the smaller the target ES, the larger the requisite sample size for a given statistical method (all other things equal).

By tying your ES to what the previous study reported, you're telling the rest of humanity that you believe a R-squared value of 0.7% is large enough to be noteworthy. In the social/behavioral science realm, I doubt that a lot of folks would endorse that perspective. But that's just my opinion; yours may differ.

Now, let's address the framework.

This sounds like:

1. A 4 (between: condition) x 2 (between: age) x 2 (within: news type) anova. The key issues are: (a) which hypothesis/hypotheses are of greatest interest to you; and (b) what is a defensible target ES for each one that you identify? Once you have this, evaluate g*power for each hypothesis. Of the resultant values for requisite N, go with the one that is largest.

or, equivalently:

2. An OLS regression model containing the same information, dummy coded (1 df for age level and for news type as main effects, 3 df for condition as a main effect. Again, you'll need to decide what hypothesis is most important (is it overall explanatory power of the model? Is it whether the condition x news type interaction is significant?). Overall (full model) explanatory power may be evaluated using g*power's first regression option. The additional explanatory power of a model with vs. a model without a specific interaction may be evaluated using g*power's second regression option (akin to a stage-wise or "hierarchical" regression).

Good luck with your work.

As David already pointed out, the Rsquared found in the original study is very small in the field of social sciences and pschology, so presumably the authors mixed up "statistical significance" and "practical relevance". Since in real data, there will never be an absolute nil effect, but always a small deviation and hence, a sample size large enough will flag this as "significant".

Therefore, when calculating YOUR sample size, you should thing about the Smallest Effect Size Of Interest (SESOI, cf. Anvari & Lakens, 2021). So which is the smallest difference between your groups/conditions that would matter in the real world, and NOT what have others found, yet, since this could be a fluke or due to overpowered studies (N>700 for simple ANOVAs seems quite large). For example, your reported interaction of b=0.0054, what does it mean? On what scale? Is this difference important? How large is the typical distribution on that scale (since the standard error for the significance test is scaled by the sample size and does not represent the original scale).

Anvari, F., & Lakens, D. (2021). Using anchor-based methods to determine the smallest effect size of interest. Journal of Experimental Social Psychology, 96, 104159.

Thank you so much for all these answers! I am indeed aware of the potential practical non-significance of the effect I am trying to replicate. That said, the author is quite well known in his field, so I wonder if I have not simply missed something when trying to explain my case, because all of this would suggest that giving importance to such a study is somewhat aberrant (rightly so if one only looks at the R squared found...) yet the study has already been replicated many times in the US. Also, I am unfortunately constrained by my limited resources (I think I will not be able to go beyond 400 participants maximum...) and whether we hope to find an effect or not, the study is going to be conducted no matter what.

But I still wanted to be as rigorous as possible in my written report. On this point, the most important hypothesis is indeed to see if there is a significant interaction between condition and type of news (namely we want to know if a certain type of nudging, corresponding to the Treatment conditions, can improve the quality of the information successively shared by the participants; and only secondarily we will look at whether this effect differs according to the age group considered for a more developmental perspective). All the studies on this "accuracy nudge to reduce spreading of fake news online" paradigm having been conducted by the same author, who got enough attention to start conducting even field studies on social networks, I confess that I am not aware of an effect that I myself would consider desirable, and that all the researchers I met advised me to calculate my sample size from their data, which are the following (attached file). I will of course continue to try on my own, and am only passing it on to you in case you have time to look at it, but this reframing has already been very useful to me - thanks again!

Hello again Camille,

I'm intrigued that the guidance you've gotten thus far from others (not in this thread) has been to emulate the (apparently gigantic) sample sizes used in other studies, without pausing to consider the utility of embracing such a small target effect size.

No matter, here are a couple of methods and considerations that you might adapt for setting your own threshold, whether for a main effect, an interaction, or some degree of variance accounted for:

1. Invite a small group of persons who do or are likely to understand the domain. Ask them something like: How much of a change in this (your target) DV would you consider non-chance and meaningful, if it occurred due to some variable or treatment? (You should make the language more specific to your DV and IV(s), of course.) Work until you reach either a consensus (as in the Delphi technique), or go with something like the median of their estimates.

2. Think of Jacob Cohen's metric for differences: Standard deviations (not standard errors, which Rainer correctly points out are driven both by variation and sample size; you want a metric that is independent of N). How much of a SD difference would convince you that something's going on (and is not just random perturbation)?

3. Perspective and context matter: Small changes/differences may be useful to agencies/organizations if applied to numerous cases (such as saving a few cents per piece in manufacturing), but may be negligible to individuals.

Put another way: Consider a medical intervention that increases the odds of survival (over 5 years) by 3%, but costs $100,000. Is that ES (change in survival odds) noteworthy? Now, consider an intervention that increases the odds of survival by 3%, but costs $1. Is that noteworthy? I suspect many folks would give different answers to the two scenarios. So, context matters.

As this moment is, apparently, early doors for this line of research, you may not have the luxury of a broadly-vetted ES threshold. The best you can do, especially if you're working against an upper bound constraint on sample size, is select a target ES value that makes sense to you, and explain the rationale you used.

Good luck with your project.