C-reactive protein, the metabolic syndrome and prediction of cardiovascular events in the Framingham Offspring Study. If the true mean is 94, then the alternative hypothesis is true. The standard deviation of the outcome variable measured in patients assigned to the placebo, control or unexposed group can be used to plan a future trial, as illustrated below. In studies where the plan is to estimate the proportion of successes in a dichotomous outcome variable (yes/no) in a single population, the formula for determining sample size is: where Z is the value from the standard normal distribution reflecting the confidence level that will be used (e.g., Z = 1.96 for 95%) and E is the desired margin of error. An investigator wants to estimate the mean birth weight of infants born full term (approximately 40 weeks gestation) to mothers who are 19 years of age and under. The range of p is 0 to 1, and therefore the range of p(1-p) is 0 to 1. The probability of a Type II error is denoted β, and β = P(Do not Reject H0 | H0 is false), i.e., the probability of not rejecting the null hypothesis if the null hypothesis were true. Nevertheless, the study was stopped after an interim analysis. Power and Sample Size Determination Bret Hanlon and Bret Larget Department of Statistics University of Wisconsin|Madison November 3{8, 2011 Power 1 / 31 Experimental Design To this point in the semester, we have largely focused on methods to analyze the data that we have with little regard to the decisions on how to gather the data. This tutorial shows how to determine the optimal sample size. The effect size is the difference in the parameter of interest that represents a clinically meaningful difference. If they anticipate a 10% attrition rate, the investigators should enroll 556 participants. In our test, we selected α = 0.05 and reject H0 if the observed sample mean exceeds 93.92 (focusing on the upper tail of the rejection region for now). In fact, it is the objective of the current study to estimate the prevalence in Boston. We conduct a study and generate a 95% confidence interval as follows 125 + 40 pounds, or 85 to 165 pounds. These data can be used to estimate the common standard deviation in weight lost as follows: We now use this value and the other inputs to compute the sample sizes: Samples of size n1=56 and n2=56 will ensure that the 95% confidence interval for the difference in weight lost between diets will have a margin of error of no more than 3 pounds. 3 Enter the size of population (e.g. How many women 19 years of age and under must be enrolled in the study to ensure that a 95% confidence interval estimate of the mean birth weight of their infants has a margin of error not exceeding 100 grams? The challenge becomes the desired sample size to meet this 80% power. In studies where the plan is to perform a test of hypothesis comparing the mean of a continuous outcome variable in a single population to a known mean, the hypotheses of interest are: H0: μ = μ 0 and H1: μ ≠ μ 0 where μ 0 is the known mean (e.g., a historical control). A 95% confidence interval will be estimated to quantify the difference in mean HDL levels between patients taking the new drug as compared to placebo. Power is defined as 1- β = P(Reject H0 | H0 is false) and is shown in the figure as the area under the rightmost curve (H1) to the right of the vertical line (where we reject H0 ). A two sided test will be used with a 5% level of significance. If data are available on variability of the outcome in each comparison group, then Sp can be computed and used to generate the sample sizes. If a study is planned where different numbers of patients will be assigned or different numbers of patients will comprise the comparison groups, then alternative formulas can be used. Here we shed light on some methods and tools for sample size determination. 2005; 142: 393-402. Buschman NA, Foster G, Vickers P. Adolescent girls and their babies: achieving optimal birth weight. If the process produces more than 15% defective stents, then corrective action must be taken. Here we present formulas to determine the sample size required to ensure that a test has high power. When we run tests of hypotheses, we usually standardize the data (e.g., convert to Z or t) and the critical values are appropriate values from the probability distribution used in the test. This is the first choice you need to make in the interface. A statistical test is much more likely to reject the null hypothesis in favor of the alternative if the true mean is 98 than if the true mean is 94. The numerator of the effect size, the absolute value of the difference in proportions |p1-p0|, again represents what is considered a clinically meaningful or practically important difference in proportions. If the process produces more than 15% defective stents, then corrective action must be taken. Therefore, the manufacturer wants the test to have 90% power to detect a difference in proportions of this magnitude. In order to compute the effect size, an estimate of the variability in systolic blood pressures is needed. The investigators feel that a 30% increase in flu among those who used the athletic facility regularly would be clinically meaningful. Child: Care, Health and Development. When we use the sample size formula above (or one of the other formulas that we will present in the sections that follow), we are planning a study to estimate the unknown mean of a particular outcome variable in a population. The figure below shows the same components for the situation where the mean under the alternative hypothesis is 98. stical power: (a) the significance level (α), (b) the magnitude or size of the treatment effect (effect size), and (c) the sample size (n). However, it is more often the case that data on the variability of the outcome are available from only one group, often the untreated (e.g., placebo control) or unexposed group. In planning studies, we want to determine the sample size needed to ensure that the margin of error is sufficiently small to be informative. In order to evaluate the properties of the screening test (e.g., the sensitivity and specificity), each pregnant woman will be asked to provide a blood sample and in addition to undergo an amniocentesis. Again, here we are planning a study to generate a 95% confidence interval for the difference in unknown proportions, and the formula to estimate the sample sizes needed requires p1 and p2. In hypothesis testing, we usually focus on power, which is defined as the probability that we reject H0 when it is false, i.e., power = 1- β = P(Reject H0 | H0 is false). A medical device manufacturer produces implantable stents. The 3 remaining patients received a second infusion with feces from a different donor, with resolution in 2 patients. If the null hypothesis is true, it is possible to observe any sample mean shown in the figure below; all are possible under H0: μ = 90. The amniocentesis is included as the gold standard and the plan is to compare the results of the screening test to the results of the amniocentesis. A critical component in study design is the determination of the appropriate sample size. The critical values for a two-sided test with α=0.05 are 86.06 and 93.92 (these values correspond to -1.96 and 1.96, respectively, on the Z scale), so the decision rule is as follows: Reject H0 if   < 86.06 or if   > 93.92. The sample size computation is not an application of statistical inference and therefore it is reasonable to use an appropriate estimate for the standard deviation. Each child will then be randomly assigned to either the low fat or the low carbohydrate diet. The effect size is selected to represent a clinically meaningful or practically important difference in the parameter of interest, as we will illustrate. Rejection Region for Test H0: μ = 90 versus H1: μ ≠ 90 at α =0.05. From the Epi Info™ main page, select StatCalc. In planning the study, the investigator must consider the fact that some women may deliver prematurely. From the figure above we can see what happens to β and power if we increase α. An investigator is planning a clinical trial to evaluate the efficacy of a new drug designed to reduce systolic blood pressure. Sample size estimates for hypothesis testing are often based on achieving 80% or 90% power. Provide examples demonstrating how the margin of error, effect size and variability of the outcome affect sample size computations. However, it is more often the case that data on the variability of the outcome are available from only one group, usually the untreated (e.g., placebo control) or unexposed group. 4 Enter the expected frequency (an estimate of the true prevalence, e.g.80% ± your minimum standard). The procedure to determine sample size depends on the proposed design characteristics including the nature of the outcome of interest in the study. • The larger the sample size, the higher will be the degree of accuracy, but this is limited by the availability of resources. The sample sizes (i.e., numbers of women who smoked and did not smoke during pregnancy) can be computed using the formula shown above. To solve for n, we must input "Z," "σ," and "E.". In order to ensure that the total sample size of 500 is available at 12 weeks, the investigator needs to recruit more participants to allow for attrition. Each will be asked to rate the severity of the pain they experience with their next migraine before any treatment is administered. To determine the required sample size to achieve the desired study power, or to determine the expected power obtainable with a proposed sample size, one must specify the difference that is to be detected. Sample Size to Conduct Test of Hypothesis. (Yuk!) Suppose that the investigators thought a sample of size 5,000 would be reasonable from a practical point of view. For example, if α=0.05, then 1- α/2 = 0.975 and Z=1.960. If the investigator believes that this is a reasonable estimate of prevalence 2 years later, it can be used to plan the next study. Lenth, R. V. (2001), ``Some Practical Guidelines for Effective Sample Size Determination,'' The American Statistician, 55, 187-193. Within each study, the difference between the treatment group and the control group is the sample estimate of the effect size.Did either study obtain significant results? This calculator allows you to evaluate the properties of different statistical designs when planning an experiment (trial, test) utilizing a Null-Hypothesis Statistical Test to make inferences. Illness from C. difficile most commonly affects older adults in hospitals or in long term care facilities and typically occurs after use of antibiotic medications. In studies where the plan is to perform a test of hypothesis comparing the proportion of successes in a dichotomous outcome variable in a single population to a known proportion, the hypotheses of interest are: where p0 is the known proportion (e.g., a historical control). N (number to enroll) * (% following protocol) = desired sample size. Again, these sample sizes refer to the numbers of children with complete data. It is customary to calculate sample size based on power (Adcock, 1997). The investigator would like the margin of error to be no more than 3 units. An investigator hypothesizes that in people free of diabetes, fasting blood glucose, a risk factor for coronary heart disease, is higher in those who drink at least 2 cups of coffee per day. The following example demonstrates how to calculate a sample size for a cohort or cross-sectional study. The investigator must enroll 258 participants to be randomly assigned to receive either the new drug or placebo. by feces infusion versus antibiotic therapy. However, in metabolic phenotyping, there is currently no accepted approach for these tasks, in large part due to the unknown nature of the expected effect. In sample size computations, investigators often use a value for the standard deviation from a previous study or a study performed in a different but comparable population. The effect size is the difference in the parameter of interest (e.g., μ) that represents a clinically meaningful difference. Feuer EJ, Wun LM. Statistical power is the most commonly used metric for sample size determination. For you computations, use a two-sided test with a 5% level of significance. This is a situation where investigators might decide that a sample of this size is not feasible. A 95% confidence interval will be estimated to quantify the difference in weight lost between the two diets and the investigator would like the margin of error to be no more than 3 pounds. The rejection region is shown in the tails of the figure below. Sample size for case-control studies is dependent upon prevalence of exposure, not the rate of outcome. The figure below shows the distributions of the sample mean under the null and alternative hypotheses.The values of the sample mean are shown along the horizontal axis. A medical device manufacturer produces implantable stents. Because we have no information on the proportion of freshmen who smoke, we use 0.5 to estimate the sample size as follows: In order to ensure that the 95% confidence interval estimate of the proportion of freshmen who smoke is within 5% of the true proportion, a sample of size 385 is needed. How precisely can we estimate the prevalence with a sample of size n=5,000? Pain will be recorded on a scale of 1-100 with higher scores indicative of more severe pain. Circulation. For example, suppose we want to estimate the mean weight of female college students. In order to estimate the sample size that would be needed, the investigators assumed that the feces infusion would be successful 90% of the time, and antibiotic therapy would be successful in 60% of cases. Really? Systolic blood pressures will be measured in each participant after 12 weeks on the assigned treatment. Again, these sample sizes refer to the numbers of participants with complete data. The sample size is then calculated so that inferences and decisions about the parameter can be correctly made. Hoenig, John M. and Heisey, Dennis M. (2001), ``The Abuse of Power: The Pervasive Fallacy of Power Calculations for Data Analysis,'' The American Statistician, 55 , … The Z 1-β values for these popular scenarios are given below: For 80% power Z 0.80 = 0.84 For 90% power Z 0.90 =1.282 The mean birth weight of infants born full-term to mothers 20 years of age and older is 3,510 grams with a standard deviation of 385 grams. Based on data reported from diet trials in adults, the investigator expects that 20% of all children will not complete the study. However, the investigators hypothesized a 10% attrition rate (in both groups), and to ensure a total sample size of 232 they need to allow for attrition. Studies that have either an inadequate number of participants or an excessively large number of participants are both wasteful in terms of participant and investigator time, resources to conduct the assessments, analytic efforts and so on. Sample size determination is an integral part of any well-designed scientific study. This leaves: Now divide both sides by "E" and cancel out "E" from the numerator and denominator on the left side. We will use that estimate for both groups in the sample size computation. The effect size represents the meaningful difference in the population mean - here 95 versus 100, or 0.51 standard deviation units different. When planning a clinical trial to investigate a new drug or procedure, data are often available from other trials that may have involved a placebo or an active control group (i.e., a standard medication or treatment given for the condition under study). To be informative, an investigator might want the margin of error to be no more than 5 or 10 pounds (meaning that the 95% confidence interval would have a width (lower limit to upper limit) of 10 or 20 pounds). We evaluated diffuse optical spectroscopic imaging (DOSI), an experimental noninvasive imaging technique that may be capable of assessing changes in mammographic density. An investigator wants to estimate the prevalence of breast cancer among women who are between 40 and 45 years of age living in Boston. The formula for determining the sample sizes to ensure that the test has a specified power is: where ni is the sample size required in each group (i=1,2), α is the selected level of significance and Z 1-α /2 is the value from the standard normal distribution holding 1- α /2 below it, and 1- β is the selected power and Z 1-β is the value from the standard normal distribution holding 1- β below it. Each student will be asked if they used the athletic facility regularly over the past 6 months and whether or not they had the flu. Samples of size n1=232 and n2= 232 will ensure that the test of hypothesis will have 80% power to detect a 5 unit difference in mean systolic blood pressures in patients receiving the new drug as compared to patients receiving the placebo. When planning a clinical trial to investigate a new drug or procedure, data are often available from other trials that involved a placebo or an active control group (i.e., a standard medication or treatment given for the condition under study). This is done by computing a test statistic and comparing the test statistic to an appropriate critical value. Samples of size n1=33 and n2=33 will ensure that the test of hypothesis will have 80% power to detect this difference in the proportions of patients who are cured of C. diff. An alternative is to conduct a matched case-control study rather than the above unmatched design. The mean birth weight of infants born full-term to mothers 20 years of age and older is 3,510 grams with a standard deviation of 385 grams. Suppose one such study compared the same diets in adults and involved 100 participants in each diet group. Antibiotic therapy sometimes diminishes the normal flora in the colon to the point that C. difficile flourishes and causes infection with symptoms ranging from diarrhea to life-threatening inflammation of the colon. Suppose that the screening test is based on analysis of a blood sample taken from women early in pregnancy. (Do the computation yourself, before looking at the answer.). Statistical Methods for Rates and Proportions. is a bacterial species that can be found in the colon of humans, although its numbers are kept in check by other normal flora in the colon. 42 43. For example, if 5% of the women are expected to delivery prematurely (i.e., 95% will deliver full term), then 60 women must be enrolled to ensure that 57 deliver full term. A sample of size n=869 will ensure that a two-sided test with α =0.05 has 90% power to detect a 5% difference in the proportion of patients with a history of cardiovascular disease who have an elevated LDL cholesterol level. It is important to note that this is not a statistical issue, but a clinical or a practical one. ES is the effect size, defined as: where | μ 1 - μ 2 | is the absolute value of the difference in means between the two groups expected under the alternative hypothesis, H1. A two sided test will be used with a 5% level of significance. Fleiss JL. The formula above gives the number of participants needed with complete data to ensure that the margin of error in the confidence interval does not exceed E. We will illustrate how attrition is addressed in planning studies through examples in the following sections. The procedure is most useful for setting up Phase II control charts, i.e., control charts designed to monitor real-time performance of a process once standard operating conditions have been … The sample size is an important feature of any empirical study in which the goal is to make inferences about a population from a sample. CONCLUSIONS • Sample size determination is one of the most essential components of every research Study. The application will show three different sample size estimates according to three different statistical calculations. How many children should be recruited into the study? The Z 1-β values for these popular scenarios are given below: ES is the effect size, defined as follows: where μ 0 is the mean under H0, μ 1 is the mean under H1 and σ is the standard deviation of the outcome of interest. Of these 3 factors, only the sample size can be manipulated by the investigator because the significance level is usually selected before the study, and the effect size is determined by the effectiveness of the treatment. This procedure is designed to help determine the appropriate sample size and parameters for common control charts. Studies that are much larger than they need to be to answer the research questions are also wasteful. The Cohort or Cross-Sectional window opens. During the manufacturing process, approximately 10% of the stents are deemed to be defective. Wechsler H, Lee JE, Kuo M, Lee H. College Binge Drinking in the 1990s:A Continuing Problem Results of the Harvard School of Public Health 1999 College Health, 2000; 48: 199-210. In statistical hypothesis terms, power is the probability of rejecting the null hypothesis when it … If so, the known proportion can be used for both p1 and p2 in the formula shown above. Boston, MA: Duxbury Press, 1982. Normal pregnancies last approximately 40 weeks and premature deliveries are those that occur before 37 weeks. The plan is to enroll participants and to randomly assign them to receive either the new drug or a placebo. In studies where the plan is to perform a test of hypothesis comparing the means of a continuous outcome variable in two independent populations, the hypotheses of interest are: where μ 1 and μ 2 are the means in the two comparison populations. Rutter MK, Meigs JB, Sullivan LM, D'Agostino RB, Wilson PW. Ramachandran V, Sullivan LM, Wilson PW, Sempos CT, Sundstrom J, Kannel WB, Levy D, D'Agostino RB. In participants who attended the seventh examination of the Offspring Study and were not on treatment for high cholesterol, the standard deviation of HDL cholesterol is 17.1. Top An investigator wants to plan a clinical trial to evaluate the efficacy of a new drug designed to increase HDL cholesterol (the "good" cholesterol). Conclusion. [Note: We always round up; the sample size formulas always generate the minimum number of subjects needed to ensure the specified precision.] However, the estimate must be realistic. This concept was discussed in the module on Hypothesis Testing. The values of p1 and p2 that maximize the sample size are p1=p2=0.5. This value can be used to plan the trial. Therefore, a sample of size n=31 will ensure that a two-sided test with α =0.05 has 80% power to detect a 5 mg/dL difference in mean fasting blood glucose levels. The estimated effects in both studies can represent either a real effect or random sample error. While a better test is one with higher power, it is not advisable to increase α as a means to increase power. Because we purposely select a small value for α , we control the probability of committing a Type I error. Power calculations tell us how many patients are required in order to avoid a type … Note that β and power are related to α, the variability of the outcome and the effect size. One case will be matched to one control. In planning studies, investigators again must account for attrition or loss to follow-up. The formula for determining the sample size to ensure that the test has a specified power is given below: where α is the selected level of significance and Z 1-α /2 is the value from the standard normal distribution holding 1- α/2 below it.
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