the 10 patients per independent variable is a rough estimate for even being able to do the analysis (to avoid over-fitting) however your sample size calculation needs to consider your ability to statistically discover a difference in your data material. You would need to know what is the expected difference in incidence of the hypotension outcome between your paracetamol treated and non-treated patient groups. If this is uncommon you will need a larger patient population. 

If this is really uncommon then prospective data collection will be a challenge. Also are you doing a trial (meaning actively giving or not giving paracetamol to patients in a randomised fashion, AKA an RCT?) or are you doing a observational study, where you do not influence the clinicians choice to treat or not treat. Normally when you need to do an adjusted analysis like you outline it is because you are observing real life data and the two populations are different. If the data are observational, and if the outcome is rare you might want to look into doing a case-control study rather than a cohort of patients. 

A final word is that normally trials (RCTs) are not good tools to look at many safety issues since rare outcomes demand very large trials, while retrospective (or prospective collection if you set up the data collection in many wards/hospitals) observational studies are better powered to at least discover these safety issues and potentially reveal information about what factors are associated.

Hi Vini Bains,

The research you are working in sounds quite well and your question is quite interesting.

Sample size is probably one of the hottest topics when facing the design of a research. In general, we could use the famous statement “the more, the best”. That is not false, the more cases you have, more precise are your estimations. The ideal case is when you can study the whole population. Unfortunately, studying the entire population is not practical in most research settings (it is expensive, time consuming or a waste of resources). As a result, we need to select a “small” set of the interest population to try to generalize from sample estimates to population parameters. However, having a very big sample size does not guarantee your inferences from sample to population are completely valid. In my opinion, “representativeness” is also a key thing to keep in mind when estimating sample size (and when sampling data).

Let me give you an example. Imagine we are interested in finding a risk factor to suffer from heart attack for a given population (i.e., a country). Imagine someone offers you the possibility to have a sample of 10,000 (n1) and someone else offers you a sample of size 5000 (n2). Which sample would you choose? I suppose you would prefer “the more” because it is “the best” and you will decide to have a sample of 50,000. But imagine the n2 sample size comes from the country capital (a very big urban and industrial zone) whereas the n1 sample is a random sample taking into account the whole population of the country and having blocked for controlled for confusing variables. Which sample would you prefer? It seems to me that in this situation size is less important that representativeness. In my opinion, n2 could be biased.

Anyway, I come back to the main issue in your question. Logistic regression is one of these techniques requiring relatively big sample sizes to obtain unbiased estimators. For example, some epidemiologic studies require about 20,000 cases to reliably find risk factors related with diseases. Although you could use exact logistic regression to overcome some problems related with sample size (i.e., Mehta & Patel, 1995) my recommendations below are more related with classical binary logistic regression. If you decided to work with exact logistic regression you could use LogXact (Cytel Software, www.cytel.com) or the “elrm” package for R (https://cran.r-project.org/web/packages/elrm/vignettes/elrm.pdf).

You can find some helpful tables in Hsieh (1989) to allow you to decide on sample size for your research. The recommendation from your statistician is not bad (10 cases per independent variable) but I would suggest to use another criteria based on “events per variable”. The term “events per variable” (in binary logistic regression) refers to the number of positive cases in your outcome (dependent) variable. A positive case, for example, is an ill person in an epidemiologic study searching for a risk factor. Positive cases are normally (this is also my recommendation) coded with 1 and negative cases are coded with 0. Some studies suggest that the critical factor to decide sample size is the rate between the number of positive cases in your sample and the number of independent variables. More specifically, they suggest that certain reduction in error estimation and increase in statistical power (to cite just two benefits) is reached when the event per variable is about 10 or higher. In other words, the key is not the number of cases per variable but the number of positive cases per variable.

Good luck with the research and do not forget the thing about representativeness.

Best wishes,

Jorge.

References

Hsieh, F. Y.  (1989).  Sample size tables for logistic regression. Statistics in Medicine, 8, 795–802.

Mehta, C. R. & Patel, N. R. (1995). Exact logistic regression: theory and examples. Statistic in Medicine, 14, 2143–2160.

Dear Vini. Sorry to say that, but in your case I'd strongly advise you to work together with an experiented statistician and have him do the power analysis. There are many pitfalls in this area and it's easy to produce numbers that do not relate to reality at all. (E.g.you have to consider the distribution of your variables on either side and so on; one can perhaps tell more after having seen the results of your pilot study). 10 observations per factor in logReg may well be too few even when only considering validity of the model (leave alone power).

Nonetheless: for making the 1st steps w.r.t power analysis and sample size calculation I'd recommend to have a look at GPower (http://gpower.hhu.de/), which has a nice GUI and comes with a tutorial to the topic.

Other than that: all major statistics packages have now procedures for samplesize calculation included. And in particular for clinical trials there are quite some specialized packages available. Major downside of the latter is that they are costly, as FDA, EMEA certification/approval etc. costs lots of money and time.

Thank you everyone for the sound statistical advice! and so FAST too! it is much appreciated!

Kristian, yes we are currently doing an pilot RCT, my first, enrollment is currently slow as it is brain injury and liver failure season in our ICU but I hope (weird thing to hope) it will pick up once the flu season hits. the trial is listed on clinicaltrials.gov (NCT02280239). And yes, I think I have to agree an observational study may be a more cost effective choice to explore confounding variables. Thanks for the suggestion. The research to date has been mostly observational and as such the incidence (20-30% ish depending on the study) is probably an over-estimate. (hopefully this RCT can offer a less biased finding)

Jorge, Thanks for the detailed response, and the references, which I have already added to my collection. And your elaboration of sorting out sample size as per "events per variable" found particularly helpful. It makes a lot more sense and is more nuanced than the "10 per variable" advice I got from one statistician who I think was a bit too busy to attend to my neophyte questions.

Josef, Don't be sorry. You are absolutely right. I totally need my statistician back! Everyone is on vacation right now and I needed a draft done to pass around to those with more knowledge than I before anyone got back. This was a shot in the dark, a what could I figure out in the interim kinda thing.

Thanks everyone for starting me on a good path and giving food for thought... And as I will be on this path for a while, if anyone else has anything to add, please do!

Vini

Hello Vini.  As Kristian noted, one thing you need to bear in mind is the issue of over-fitting your model (i.e., having more explanatory variables than can be supported by your sample size).  In linear models, the rule of 10 is often used (i.e., you must have at least 10 observations per model degree-of-freedom).  Not everyone agrees that is a good rule of thumb for linear models (see first link below), and it certainly is not a good rule of thumb for binary logistic regression.  For logistic regression, the rule of thumb needs to be expressed in terms of events-per-variable, not observations-per-variable, where an event is defined as the less frequent of the two outcome variable categories (e.g., event = death, assuming fewer patients died than survived).  In his nice article (2nd link below), Mike Babyak ends up recommending at least 15 events-per-variable (IIRC) to avoid over-fitting logistic regression models. 

Again, as Kristian noted, this is a separate issue from whether you have enough power to detect an effect size of some size that you must specify.  Your statistician probably has some software to deal with that issue. 

HTH.

http://www.angelfire.com/wv/bwhomedir/notes/linreg_rule_of_thumb.txt

http://www.cs.vu.nl/~eliens/sg/local/theory/overfitting.pdf

Vini,

I am going to recommend a 1998 article published in "Statistics in Medicine" in by F. Y. HSIEH1*, DANIEL A. BLOCH2 AND MICHAEL D. LARSEN3 for your review on that very subject.

Cheers,

Bob Yaffee

I think Tabachnik and Fidell (2019) 7th edition is an excellent and definitive and contemporary resource to deal with your question. https://www.pearson.com/us/higher-education/program/Tabachnick-Using-Multivariate-Statistics-7th-Edition/PGM2458367.html?tab=contents. 10 cases per predictor is a traditional approach, I suggest go for the new one. Many now recommend 20 cases per predictor in any event but this approach is also kind of outdated. Tabachnik and Fidell have what you need and are a credible source.