I was assigned the task to present how to calculate the size of a population when you're trying to set up study of any sort, either a cohort study, a case control study, a randomized trial. We're going to try to simplify this as much as possible. The idea behind obtaining an adequate number of a sample size is that we can't really test the whole population at hand. So typically, we want to obtain a certain number of a population, and from that population, we're going to try to test the hypothesis. And from that, we're going to reach a conclusion that hopefully we can generalize the population. The problem that we face is efficiency, and efficiency is knowing how many people that we need to actually study so we don't waste money and resources. And the worst thing to do when you're trying to do a study is retrospectively say, I didn't have enough patients involved in a study, so what can I do to fish and try to come up with anything from doing a trial? So we have a trade-off with how much money we have and how much we need to spend per sample. And like Dr. Haynes was alluding to with the crash trial, $100 per patient, but they studied 2,000 patients in order to reach a conclusion, and there was an extra 8,000 that they did not analyze in time before getting to that conclusion. So it's important to know how much money you have and how much resources you're going to spend on each trial to obtain your adequate conclusion. So sample size is determined by multiple factors. Significance level is very important, and I'm going to try to query the residents to see whether they think the effect of sample size. For example, Carl, what do you think sample size and significance level are related? Is it a positive relationship or a negative relationship? Can you explain a little bit more? As your sample size increases, the significance of your results increase because they're more generalizable to a population, you've reduced the random error. So can you say it in reverse? So if you're going to look at an alpha of 1 or 0.1 versus an alpha of 0.001, what would the effect of sample size be? Would you need less or more people involved in a trial? So with an increased significance level or with a decreased significance level, you'd need more people involved in your trial. Okay. How about Michael? What do you think about the power? How is that related to the sample size? Can you repeat the question? So if you have a power, for example, of 80% and a power of 95%, how would that relate to your sample size? Would that increase the number needed to obtain that power or decrease that number to obtain a higher power? So you're saying, you're talking about beta? Or you're saying like your power, like I don't understand the question. So power is one minus beta. So if you're increasing the power of your study, would you need more or less patients? Typically you would need more patients to increase the power of the study. Okay. How about the expected prevalence of factors of interest? So Nick, can you explain that and see if that relates to higher or lower number of patients involved? Sure. So if the factor of interest is more common, then you need less patients in your study. And if it's rarer, then you need more patients. Okay. Nina, can you elaborate on test efficiency? Would you try to explain what does it mean? For example, to have efficient test in order, or an inefficient test in order to obtain lower or higher number of patient samples. Need to select a test that's going to be able to simplify your data. So if you have multiple variables, you'll need to select a test that's going to corroborate or include all of those variables, or you need to split up the variables and be able to compare with a simpler test. So an example is pregnancy test. It's very, very highly sensitive for pregnancy compared to gene chip analysis and trying to predict gene anomalies. Well, you need to do a triplicate analysis of a test in order to reach some form of sensitivity. So you need a lot of tests to be done when you do that. And the design of the study also is very important. So you need to take into account your primary and secondary outcome while doing the calculation of the samples. You also need to look ahead and learn from previous studies. So you have to look at the literature and see what are the factors that are involved in the attrition rate. For example, people are lost to follow-up. So you'll see, based on experience from previous trials, how many people were lost to follow-up. So you have to factor it in. Lack of compliance, response to surveys, different people in different locations, response and the calculated ahead-of-time methods in order to obtain the adequate number of samples that you need. So going back to the two-by-two table and looking at the factors involved in obtaining a power of a sample or a number of samples. So the type one error and the beta are the most important values that will dictate most of the power or most of the sample size that is required for a certain power in a study. So Christopher, can you tell me what does beta really mean when you're looking at the two-by-two table? Can you give me a real-life example about a beta error? I think it's the... So if you have a beta of 80%, then you're 80% sure you're not making a type two error. So a type two error, I have, for example, an HIV test and I do the test on a patient. And the test comes back as negative, but the patient is definitely HIV positive. This is what a beta two error or a beta error means or type two error. So you're missing out on a test. The test is giving you a false result from reality. So it's a false negative result. And from that, you can obtain the power of that test. And from a one minus beta, you can get the power of the study that you can plug into the formula in order to obtain your sample size. So how to increase the power of your study? You can increase the sample size. You increase the desired difference. So if you're looking at a very small difference between two populations, you need a very large amount of patients to detect that difference. And if you increase the significance level by changing the confidence interval, and I'm going to show a couple of examples to reiterate what Dr. Barker was saying about confidence interval and the value of the N or the sample size that decreases or changes the confidence interval that will change ultimately your power involved in the study. So consider these three examples. You have three different populations with three different Ns. Each N is a multiple of 10, so 10, 100, and 1,000. In all three studies, your exposure group incidence is about 60% or 0.6, and the control group is 40%. And if you calculate the two-by-two table, for each one, the relative risk is going to be 1.5. Houshal, can you tell me how I got the relative risk of 1.5? What did I calculate? To calculate the relative risk, you did A over A plus B, so 3 over 8, and then you divided that by the no, so 2 over 7. Excellent. Perfect. So in every table, you'll see that the relative risk is the same. It's 1.5 because the numbers don't change, they become multiples of 10. But the confidence interval for the first one is between 0.4 and 5.4, so pretty wide confidence interval, and the significance value has not reached significance. And if you multiply that number by 10, you start to see how tight the confidence interval becomes from 0.4 to 5. It's basically shrunk by half. And if you do that for 1,000, you have a significant change in the confidence interval, and you are much more confident about your result of relative risk of 1.5. So again, to summarize, for the three different values of multiples of 10, you can increase your confidence interval and increase the power and the value of power beyond that. So switching gears, I'm trying to calculate some sample sizes for different clinical scenarios. You have to keep in mind before you start the trial is to figure out what you are testing. For example, is it qualitative data? Does the person have GBM or not, or is it length of survival of GBM with months or days? And that will dictate the type of formula that you need to use and what type of analysis that you need to use in order to calculate your sample size. So these are four simple formulas for four simple scenarios for qualitative and quantitative data. So I'm going to give four different examples just to reiterate these problems. So for the first problem, we have a population with a certain standard deviation of 46. And you wanted to see if you can calculate a sample size that will determine a sample error of four. So the difference basically between samples is four. And you want to be 99% confident in your decision. So if you take the formula from the Z number of 99%, you will get a 2.58 from the table of Z. And you already have the standard deviation and the calculated difference. You can pretty much calculate the number needed for your sample size of 880. And if you decrease that confidence interval, you can see that your number is going to be cut back significantly because you're going to go down to a Z of 1.96 if you're using 95%. And 95 versus 98 will make a huge leap in your sample size. Z is calculated. You get certain tables with Z numbers based on your confidence interval that you need. So for 95%, it's 1.96. For 99 percentile, it's 2.56. These are pre-calculated tables that are present in appendices in the end of statistics books. The second problem that I want to discuss was a study to determine the effect of two drugs on length of survival in GBM. The previous studies was one drug had a standard deviation of 8 months, and the other one had a standard deviation of 12. The calculated number that is required is for a 95% confidence interval, and you wanted a difference of 3 months between these two groups to obtain the sample size. So from the previous formula for qualitative data, and by calculating the ANOVA, analysis of variance of these two variables at the 95% interval, you can calculate the F, and you already have the two standard deviation and a difference. So it immediately pops up as N of 243 between the two arms of that group. The third example is for qualitative data. It's not quantitative data, so it's ratios that you're going to depend on. And in this example, you're looking at the proportion of anemia in lymphoma patient, and it's typically the proportion is about 30 percent or 0.3. And if you want to compute the sample size at a confidence of 95th percentile, looking for a 4% difference, going back to the formula, it's the same thing. For 95%, the Z is 1.96, and you can calculate from the proportion and the difference required, and you can get the total number required in that study. So if we want to bump it up into a higher level of confidence at 99th percentile, you can see it. You switch 1.96 and then add 2.56, and that will pretty much increase the sample size at least by three times. And finally, if you have two proportions of patients and you're trying to look at the numbers needed between these two groups, for example, diabetes versus, you know, two patient populations, diabetes or not diabetes, and the incidence of hypertension, which would be higher in diabetes, about 70 versus 40, and you want to calculate the sample size in order to determine a difference of about 4% in the incidence of hypertension, the formula is a little bit different in this situation. You'll take the average of the frequency or the proportion between these two groups, and if you don't know the previous proportion, you can select an average of 0.5 of frequency as an out for you if you don't know the other side of the population. And from that, you can calculate the N, and in each group, you will need 2,400 patients. So these are four examples of trying to calculate sample size in each population, and depending on the type of variable that you're using, qualitative or quantitative, or one-sided t-test or two-sided t-test, or if it's an ANOVA type of trial or a MANOVA type of trial. So ahead of time, you have to ask all these questions in order to put all this together in order to calculate your sample size for trying to decrease the amount of effort you put into a trial and to come up with certain answers to your questions. So finally, I just want to share with you guys a really nice program that makes life much easier. Instead of going through the mental exercise of calculating patients, based on your preconceived trial and what you think you're dealing with, what type of data, you can just plug in the numbers that you need. For example, if an effect size of one between two populations, and you're looking at an alpha of 0.5 and a power of eight, you can do, if it's a one-sided tail, for example, you can immediately get the numbers needed for a sample size, and you can change based on a t-test, an f-test, an exact test, or a chi-square test based on what type of population you're having. So it's a very simple software to try to get your total number of patients that you need in a trial without going through the pains of trying to figure this out with a mathematical formula. If anybody wants that program, I'm more than happy to share it. It's called G-Star Power.

sample size

study

population

hypothesis

generalization