Hi,

Sample size basically depends on the statistical test you will apply for comparison, for sample size calculation in case of sensitivity and specificity you can read

PMID: 15195324

PMID: 9249923

Hi Fahmi,

The confidence interval surrounding your statistical test results will tell you how much you can generalise your results to the whole population as it will tell you the likely range that the result will take in the total population with, e.g. (if a 95% confidence interval), 95% confidence. The larger your sample, the narrower your confidence interval and the more accurate your results are (the more powerful your statistical tests of the null hypothesis will be). A sample size of 32 is quite small so I imagine your confidence intervals (for sensitivity, specificity, predictors of mortality etc.) will be quite large, meaning that you won't be that confident of what the true population values are (assuming your sample is representative of the population).

Hope this helps,

Tim

statistical tests related to sensitivity and especificity depends also on the incidence/prevalence of the interest variable on the sample.

Researches have some kind of fixation on sample size. Although size matters, because size will determine the confidence intervals (as stated above), it's even more important to ensure that sample was random! A biased sample has no salvation...

Hi Fahmi, just to get to one part of your question: the p-value is about how likely your sample is to come from a population where whatever it is you are testing (let's say a nonzero difference between two means, or a nonzero correlation between two variables) does not exist. So if you get a p-value of less than 0,05, it tells you there is less than 5 chances out of a hundred that your sample came from a population where that wasn't true. That's true whether your sample was 10 or 10,000. Your p-value is already based on the sample size. So if it is a t-test, for example, it took a higher value of t in a sample of 32 to get a p-value of 0,05 than it would have if your sample was 300.

You asked if you can determine predictors of mortality from this sample. You didn't say how. If you are thinking of using multiple regression, you won't get too far because you don't have a lot of degrees of freedom. Your model will only accommodate a couple of, maybe three, predictors, and you would have to be careful that the predictors themselves or a combination of them aren't just identifying individual cases or maybe two or three specific cases.

Finally, with a small sample, you have very low stastical power. That is an ability to find something that really is there in the population. Your chances of finding something that is in the population aren't very good unless it is somethng very big. This is also called Type II error. You can learn more by Googling "statistical power" and "effect size." Bob

I also meant to say that the size of the population rarely has anything to do with it, even though there are ways to correct statistics when you approach the size of the population. How can I say such a thing? Well let's say you are studying first grade students in San Francisco and you administer a test to every first grader in the city.. Some would think you got the whole population. But are you really just studying the first graders that are there in the 2012-2013 school year? Probably not, because you are probably trying to learn something more general about improving the schools, the curriculum, the teachers that is gong to be important over time. Or you might be trying to learn some larger lessons about first graders in general. It's pretty unlikely you are trying to learn about just those first graders. By the time you analyze the data, those kids will probably have moved on and not even be first graders any more. The point is we almost never think about the population. Somebody might say to me, we have 10,000 people at our University, what percentage of our students to we need to survey. That 10,000 is irrelevant. I do my statistical power analysis based on the comparisions I might want to make in the data or the sort of confidence intervals I want. My calculatons are going to be the same whether the population was 10,000 or 100,000 It's possible my sample might get so large that I could actually apply the corrections to my statistics, but honestly, I have never used them.

Regarding mortality, A question I have in response is: How are you planning to measure this? Are you interested in whether patients die within a certain amount of time or are you interested in following them until they die? One of these is a categorical variable the other is a time to event variable. More importantly,they are appropriate in different settings. Is this disease completely curable? or is it something that a patient will die with or from long term? It is important to think about these things in order to choose the coreect test. Once you have the correct test, then you can move to consideraations of precision and power etc. In your case it appears that you have only one group. Are you categorizing your adenosine deaminase results into a positive/negative like variable? I guess the last question that comes to mind is: How big would the estimate of the relation (OR, slope parameter, event rate, etc.) between the predfictor and mortality have to be in order to be "medically significant"?

The point of "generalized" is not how large your sample size is, but how you get your sample. If you did not randomly sample form the population, generalization to population is almost impossible.

I think your sample size is small and it can not be represent to whole population if you have not collected the sample randomly.

Chih-long Yen is correct, if u must refer to the larger population from your findings you must be extremely cautious in your interpretations. secondly as already mentioned sample size is a secondary concern in this matter.

Generalizability and sample size are associated in that with small sample size you are more prone to have missed a relevant subgroup of your target population. If an omitted group has a different response to the condition or treatment, your estimate carries a bias. Obviously, that has nothing to do with confidence limits. (A 95% confidence limit tells you that if you repeat the same experiment a 100 times, your estimate will fall into the respective interval 95 times).

In addition, simple random sampling is just one way to ensure representativeness of your sample. If you have an idea of how your target population is composed, you could also go for a stratified random sampling scheme.

Another issue with small sample sizes that has not been mentioned is that classical statistical tests are asymptotic. They make certain assumptions that are only reached in infinity. In particular, this is the case with confidence limits. There is a solution to that however: use resampling techniques, such as the bootstrapping. Or you take the Bayesian route via MCMC sampling.

Martin, although your answer is right, let me add one important commentary. When you say that random sampling is just one way to ensure representativeness and that stratified sampling could be used too, I believe that you are trying to say that SIMPLE random sampling is just one way and that stratified RANDOM sampling could be used too. Don't you?

I am saying that because stratified non-random sampling is the quota sampling, which is not a probabilistic sampling method and can bias the results...

Sandro, you are perfectly right.

For the results to be validated on the public at large the ample size must be significantly large: In view of financial constraints many studies can ill afford to do so especially those from developing countries

Just a comment to Robert Brennan's answer. The p-value is actually not the probability of the nullhypothesis -- or that your sample came from a popualtion where the effect of interest is not present. It is the probability of getting at least as extreme results as the observed one, in a situation where the nullhypothesis really is true. So it is the probability of data in a specified scenario, rather than the probability of the hypothesis.

A low p-value indicates that something rare has happened, given that the nullhypothesis is true. The setting is turned on its head, and the conclusion made is that the nullhypothesis is rejected (and some alternative hypothesis is consulted). This also explains why we don't interpret p-values quantitatively, but have to decide whether we accept or reject the nullhypothesis. Bayesians can calcuate probabilities for different hypotheses, but they first have to assign prior probabilties to all of them.

THERE IS NO SIMPLE ANSWER. You must balance the risks associated with Type 1 and 2. Read a good probability sampling book or an elementary book that goes into this problem. I wrote one when I was youngerbut it is now out of print.

30 samples and more can be used for statistical analysis. We should also keep in mind that results of such studies that include small sample size must be interpreted cautiously.

In public interest and not influenced by personal prejudice such results should be not over interpreted...generalising..

The sample size rule of thirty only refers to many elementary textbooks which allow readers that significance level read from the standard normal table approximate the values in Student-T tables.Minimum sample sample sizes are determined by alpha , beta probabilities and estimated stand deviation of the population. Please read an elementary textbook on sample size determination.

Tweite misstated the meaning of the 95% confidence interval. Probably because English is his send languade. I do not given defintions in German because of the same reason.

The 95% interval state that if the process was repeated 100 time, 95 such interval would approximate the parameter correctly. The parameter is a constant and the contrustucted confidence intervals vary from sample to sample. Tweite committed an elementary mistake but I am sure he will rectify quickly.

Dr. Khan,

There is no minimal depending on the type of research design. For instance if you are working with qualitative or case studies then the popoulation may be 5 people for a case study. However, especially if you are not an expert right away at statistics. You should try to have a large sample size. This will help you meet the internal and external validity of the study. The larger your sample size in any study follows the law of the Central Limit Theorum. Look this up and become familiar with it. Even if you have a really serious skewness or kurtosis problem, if you increase the sample size you will most always end of with a normal curve as that is the law.

However, to speak in regards to learning about the importance of population size, it is very important. You want to have a demographically balanced population and then look at how the results may be applicable to different populations and situations as this is also a factor. However, to be succinct, you should especially when there is uncertainty increase that sample size and Mr. Central Limit Theorem.

I like your questions. Questions are the way you will learn about what to do. The knowledge of statistics is a life long study. It is fascinating. Increasing programming learning SAS and then increasing statistical interpretation will help you design better research studies. You as the lead researcher need to know your statistics. It is not imperative, it will add to your credibility as a researcher. I love statistics and I learned so much from SAS. This is the most important bit of advice that is relevant to all Professors and students worldwide. SAS will give you FREE instruction and training for all Professors whom utilize SAS and their students whom are actual research competent. I get eleven courses worth $11000.00 free at sas.com all the time. I get video lessons, applications and data sets. There is also a forum there and they will answer and help you grow. Statistics from a programmer is like giving the researcher a fish to eat. Statistics training for the researcher is giving the researcher to knowledge of "how to fish." This is very significant.

if the population size is just 32, you better do the census NOT sampling...

Minimum or optimum sample size is related to your risk of incurring  Type ! and Type 2  errors, the assumed standard deviation of the population and the tolerance level. Read a book of statistics of sampling and you will never ask this question again. The issue was solved many years ago.

Excellent question!

Other answers (Flom, Dunn) already mentions that

1) for large n (sample size), many distributions can be approximated to normal. [Courtesy: Central Limit Theorems]

2) the approximation becomes better with larger n, given other things remaining the same.

However, if you are wondering why 30? Why not 25 or 40 or something else?

Note that actual answer will depend:

1) how much error are you going to tolerate

2) size of the effect

3) exact distribution family (like t vs chi-square vs gamma)

[ search for keyword like "level and power of statistical test" to know more ]

Given all these, how did the thumb rule of 30 become part of the folklore?

According to the story, I heard from my statistics professor, this is as follows:

Before computers became ubiquitous, statistical tables were used to get  the cutoffs points for different tests at some popular significance levels ( like 1%, 2%, 5%) (They are used even now, I’ve used many times in college)

This means that there will be a page in the book with cutoff for t distribution at those levels, with different degrees of freedom (n) (with one row for each n)

From the publishers' point of view, it’s convenient that each distribution takes a page, like one for t, one for Chi-square, one for normal, and so on)

This limits the maximum n you can go to, and after, that you are supposed to approximate with normal. 

For common page sizes and font sizes, somehow 30 appeared to be this limit, and became so popular that "above 30 it is normal" was repeated enough to become part of the folklore.

Caveat: for n random samples sample mean is approximately t distributed with (n-1) degrees of freedom. This means sample size should be at least 31.

But thumb rules being thumb rules, these kind of details are never paid attention to (the cutoff of 30 is so crude that this doesn’t matter anyway).

Hi Fahmi,

Generally, for any inferential statistic, a sample size of less than 500 may not be adequate.