Sample size calculation for hierarchical 2x2 full factorial trial

[Marco Riguzzi]

Dear Community,

I need to calculate the sample size (number of clusters and average number of observations per cluster) needed for a hierarchical 2x2 full factorial trial. Trial centers will be randomized to the four conditions (base, base+A, base+B, base+A+B). In each center, the primary outcome (metric, may be skewed) will be measured for multiple individuals. The effect of A and the effect of B on the outcome as well as their interaction will be of interest.

I intend to use a mixed linear effects model if the distribution of residuals permits (I hope that I will not have to resort to a generalized model or another alternative, there is a chance though I suppose). I will make assumptions on ICC (expected to be rather low, somwhere between 0.02 and 0.04) and cluster size variation. Alpha=0.05, power=0.8.

I am familiar with power calculation for cluster-randomized trials with a hierarchical data structure that are analyzed by a mixed linear effects model where a single treatment effect is of interest, but not for factorial trials. I am also familiar to a limited degree with power calculation for 2x2 full factorial trials, but not with hierarchical data structure.

Your help is greatly appreciated!

Marco

[Joseph Coveney]

I am familiar with power calculation for cluster-randomized trials with a hierarchical data structure that are analyzed by a mixed linear effects model where a single treatment effect is of interest, but not for factorial trials.

Well, it's argued here—with follow-up here and here—that the interaction is the weakest among the effects estimated in factorial models.

So, if your interest lies in the interaction term, then use the method that you're familiar with to set the sample sizes (number of clusters and per-cluster size) to achieve the desired power for that contrast.

. . . power=0.8.

I take it that this isn't a so-called human subjects study; I was under the impression that IRBs / ECs typically demand a power of at least 0.9.

[Clyde Schechter]

I was under the impression that IRBs / ECs typically demand a power of at least 0.9.

That hasn't been my experience, neither as a researcher submitting to an IRB, nor when I was an IRB member. 0.8 is generally accepted.

[Joseph Coveney]

I was under a mistaken impression then. Thanks for the correction, Clyde.

[Marco Riguzzi]

Thank you very much Joseph and Clyde for answering!

The trial is going to be about family members of patients who die in intensive care or survive after being at high risk of death or permanent impairement.

So, if your interest lies in the interaction term, then use the method that you're familiar with to set the sample sizes (number of clusters and per-cluster size) to achieve the desired power for that contrast.

Thank you for the material, Joseph. As it turns out, I am not familiar enough with the method to do the sample size calculation for the interaction term. I have done the calculation for a hierarchical cluster-randomized two-arm trial (using R, package clusterpower). It yields 8 clusters per condition for the two-arm trial (total of 16 clusters) with 50 individuals per cluster (total of 800 individuals). Question 1): How do I get from there to the sample size (number of clusters and per-cluster size) required for the interaction term of the 2x2 factorial?

The material you provided explains why, for an interaction effect half the size of one of the main effects (+A, +B), 16 times the sample size is required. I need to understand this: I was advised by a colleague to solve my task by first answering the question for sample size as if there were only one factor (so a two-arm trial), which I have done as described in the previous paragraph, with the only difference between that and the 2x2 factorial being that the latter has 2 fewer df for error. Question 2): Is this correct and how does this relate to the material you generously provided?

I surely must be missing something important here, so thank you for your patience.

Your help is truly very much appreciated!

[Marco Riguzzi]

I cannot edit my previous post any more. Is the solution to Question 1) 16 times as many clusters as needed for the two-arm trial in total (2x8x16=256) with the same number of individuals per cluster (50), with each of the four conditions (base, base+A, base+B, base+A+B) getting one fourth of the 256 clusters (64)? (Given that the interaction is expected to be half the size of one of the two main effects.)

And, if the interaction was expected to have the same size as the main effects, would we then need 4 times (instead of 16 times) as many clusters as for the two-arm trial, and if the interaction was twice the size of the main effects would we need the same number of clusters as for the two-arm trial?

Or would the clusters not be evenly distributed across the four conditions so that "base+A+B" would get more than the other conditions?

Thanks again!

[Roman Mostazir]

Not sure if this will help but there is a R-package H2X2Factorial for calculating sample size for hierachical 2X2 design.

[Marco Riguzzi]

Thank you, Roman.

The algorithm works with a parameter pi_z that is the proportion of individuals randomized to the individual-level treatment. In my trial, I will not have an individual-level treatment, only two treatments at the cluster level and their interaction. How can I model this with the H2X2FFactorial package?

[Roman Mostazir]

I am sorry that I don't know. I have not used that H2X2FFactorial package myself and am really in stringent time condition to decipher it further and to assess whether it is suitable for your question. It was an on the fly aid to you to ensure you know a relevant package exists.

However, I also am not being able to connect your research hypotheses and the prospective analytical model. You probably will be better off by clearly explaining what is it you are going to study, the hypotheses being tested and the analytical model.

You said in #1 that you intend to use linear mixed effect model, in #5 you said the study is about the family members of patients who died or survived and in #8 you said that you don't have individual level treatment. If #8 is true i.e., only families receive tratment-A or B, I don't see how number of individuals is important here. You probably just need a sample size calcualtion for number of families with family level difference (effect size) regardless the number of people in there and correct for the design effect (icc). Given the families are a random sample, the findings then is generalisable to the wider population whether treatment-A or B is better for the families. It is of course the obvious case that the number of family members will vary between families but that is generalisable given randomisation at family level. Further, when you select a family randomly, you cannot help with the number of members in that family as you cannot icrease or decrease the size of the family. Therefore, if you discard a randomly selected family just because of it's small number of family members, your research will be liable for selection bias.


I may be completely wrong in my understanding of your study but as I said, explaining clearly what you intend to do (randomisation, treatment, hypotheses, your outcome of the study and the analytical model) might give you a better answer from the respected members of this forum.

[Clyde Schechter]

I don't see how number of individuals is important here.

O.P. has given a very incomplete description of his problem. But if the outcome is measured at the individual level, even though the intervention is delivered at the family level, then, yes, the number of individuals is important. Designs with group-level intervention and individual-level outcome assessment are pretty common in health services research, and somewhat common in epidemiology as well.

[Marco Riguzzi]

Thank you for your reply, Roman. I probably did not describe my study in sufficient detail. In any case, I think I have found the solution in: Ahn, C., Heo, M., and Zhang, S. 2015. Sample Size Calculations for Clustered and Longitudinal Outcomes in Clinical Research. CRC Press. New York. Thanks again for the suggestion which may come in handy in the future.

[Roman Mostazir]

O.P. has given a very incomplete description of his problem. But if the outcome is measured at the individual level, even though the intervention is delivered at the family level, then, yes, the number of individuals is important. Designs with group-level intervention and individual-level outcome assessment are pretty common in health services research, and somewhat common in epidemiology as well.

Totally agree Clyde and as you have indicated too, the number of individuals are important only if the outcome varies at individual level too. The confusion arose from #8 where Marco said that the treatment is not at individual level which might be translated as family being the unit of analysis where outcome will not vary within individuals in family. The lack of description nature of the problem created this confusion and it is another good example of why we request new forum members to read the FAQ section before posting stressing on the point of clarity of questions. However, I am glad that Marco has found a solution.