You always can do anything. If you want to publish the results, you just should be able to convince the editors/reviewers that the experiment and the data analysis was sensible and that your conclusions seems to be supported by the data you have.

For most studies, either the sample size is given or the (interesting, relevant) effect size is given. If I am interested in demonstrating some relevant effect with a desired precision, then I can calculate the required sample size (depending on the kind of analysis and the expected variance). In basic research - which is often exploratory! - we usually don't have any opinion about the size of a "relevant effect" (any effect size is interesting), nor do we often have much knowledge about the expected variance. Additionally, we often do not know a-priori an apporpriate model (which co-variables to include? what to adust for? ...). All we want is an as-good-as-possible guess about the effects we see in the data we have, to come up with a model that best represents our data and does not contradict things we already know well from other studies. And this is the way to go when the sample size is given. All you can do is to see what effect sizes your data will show, and with what precision you can estimate these effects.

You always can do anything. If you want to publish the results, you just should be able to convince the editors/reviewers that the experiment and the data analysis was sensible and that your conclusions seems to be supported by the data you have.

For most studies, either the sample size is given or the (interesting, relevant) effect size is given. If I am interested in demonstrating some relevant effect with a desired precision, then I can calculate the required sample size (depending on the kind of analysis and the expected variance). In basic research - which is often exploratory! - we usually don't have any opinion about the size of a "relevant effect" (any effect size is interesting), nor do we often have much knowledge about the expected variance. Additionally, we often do not know a-priori an apporpriate model (which co-variables to include? what to adust for? ...). All we want is an as-good-as-possible guess about the effects we see in the data we have, to come up with a model that best represents our data and does not contradict things we already know well from other studies. And this is the way to go when the sample size is given. All you can do is to see what effect sizes your data will show, and with what precision you can estimate these effects.

https://class.coursera.org/dataanalysis-001/human_grading/view/courses/294/assessments/4/selfGradingSets/174/selfGradings/0

Jochen - thanks. I agree with most of what you say, but there certainly are assumptions of logistic regression that i wouldn't want to violate by using a smaller than optimum sample size. also, i want to convince people to give me money to do the study, and there just might be statistician on the review committee!

Roberts- thank you, that link looks pretty interesting.

Assumptions define what you know *a pirori* about the data. There is no way for the data to violate your assumptions. However, assumptions may be inappropriate, you may miss to consider some aspects of the data, its variability, important covariates and so on. Actually, this has nothing to do with the specific data or sample size, but with what all you can know (and what you cannot know) about the data before seeing it.

After having the data, you can perform an analysis, fit your model, and then you can check if the residuals can still tell you some important structure in your data that your model did not consider. This is an exploratory analysis.

If you are going for a confirmatory analysis, then the model is already specified and appropriate and you know well what to expect from your data, so there is no question for the sample size because of the assumptions (only because of the power).

The problem of small sample size can affect the properties of Maximum Likelihood estimators (like the logistic). Perhaps this paper can provide you with some indications: http://www.polmeth.wustl.edu/media/Paper/hart99.pdf.

Lavanya, according to this paper (http://ageconsearch.umn.edu/bitstream/103771/2/Bergtold%20et%20al.%20Logit%20Bias%20Paper.pdf) sample size should not be a main concern for logistic regression.

There are also these notes (http://courses.washington.edu/b536/Archive/handouts/Lecture11.pdf) where it is explained the use of exact logistic regression as an alternative estimator to the logistic in presence of (very) small sample sizes

I guess much would depend on a) how randomized this experiment is, b) how similar are the treated and the controls wrto a number of obvious intervening factors (such as age, sex, genetic propensity to the desease). The more treatment is assigned randomly and the more groups are pretty close wrto those characteristics the smaller the sample you need to achieve a given level of accuracy of your estimates. Hope this helps.

Thank you everyone for very interesting and helpful answers!

Http://www.dartmouth.edu/~eugened/power-samplesize.php

Lavanya, how did you study work out?

Virginia - sadly the study never took off.

Dear Lavanya,

an alternative is to apply exact logistic regression, which is used when the sample size is too small for a regular logistic regression. See the link below for more information.

http://www.ats.ucla.edu/stat/stata/dae/exlogit.htm 

Best, Everton

R users may use the package "elrm"

https://cran.r-project.org/web/packages/elrm/index.html

http://www.statpower.net/Content/312/Handout/Hsieh(1989).pdf

eq 4) (single regressor) and 5) (multiple regressors).

These give you n (sample size) for a given precision level (desired type I and type II error) of the estimated beta coefficient(s).