Sample size determination and power pdf

A comprehensive approach to sample size determination and power with applications for a variety of fields Sample Size Determination and Power features a modern introduction to the applicability of sample size determination and provides a variety of discussions on broad topics including epidemiology, microarrays, survival analysis and reliability, design of experiments, regression, and confidence intervals. The book distinctively merges applications from numerous fields such as statistics, biostatistics, the health sciences, and engineering in order to provide a complete introduction to the general statistical use of sample size determination.

Advanced topics including multivariate analysis, clinical trials, and quality improvement are addressed, and in addition, the book provides considerable guidance on available software for sample size determination. Written by a well-known author who has extensively class-tested the material, Sample Size Determination and Power: Highlights the applicability of sample size determination and provides extensive literature coverage Presents a modern, general approach to relevant software to guide sample size determination including CATD computer-aided trial design Addresses the use of sample size determination in grant proposals and provides up-to-date references for grant investigators An appealing reference book for scientific researchers in a variety of fields, such as statistics, biostatistics, the health sciences, mathematics, ecology, and geology, who use sampling and estimation methods in their work, Sample Size Determination and Power is also an ideal supplementary text for upper-level undergraduate and graduate-level courses in statistical sampling.

A comprehensive approach to sample size determination and power with applications for a variety of fields Sample Size Determination and Power features a modern introduction to the applicability of sample size determination and provides a variety of discussions on broad topics including epidemiology, microarrays, survival analysis and reliability, design of.

This book addresses sample size and power in the context of research, offering valuable insights for graduate and doctoral students as well as researchers in any discipline where data is generated to investigate research questions.

It explains how to enhance the authenticity of research by estimating the sample size and. A researcher's decision about the sample to draw in a study may have an enormous impact on the results, and it rests on numerous statistical and practical considerations that can be difficult to juggle. Computer programs help, but no single software package exists that allows researchers to determine sample size. It offers flexible guidelines for the care of these animals, and guidance on adapting these guidelines to various situations without hindering the.

This book integrates recent methodological developments for calculating the sample size and power in trials with more than one endpoint considered as multiple primary or co-primary, offering an important reference work for statisticians working in this area.

In order to ensure that the total sample size of is available at 12 weeks, the investigator needs to recruit more participants to allow for attrition. An investigator wants to compare two diet programs in children who are obese. One diet is a low fat diet, and the other is a low carbohydrate diet. The plan is to enroll children and weigh them at the start of the study. Each child will then be randomly assigned to either the low fat or the low carbohydrate diet.

Each child will follow the assigned diet for 8 weeks, at which time they will again be weighed. The number of pounds lost will be computed for each child. How many children should be recruited into the study?

To plan this study, investigators use data from a published study in adults. Suppose one such study compared the same diets in adults and involved participants in each diet group. The study reported a standard deviation in weight lost over 8 weeks on a low fat diet of 8.

These data can be used to estimate the common standard deviation in weight lost as follows:. Again, these sample sizes refer to the numbers of children with complete data. In order to ensure that the total sample size of is available at 8 weeks, the investigator needs to recruit more participants to allow for attrition. In studies where the plan is to estimate the mean difference of a continuous outcome based on matched data, the formula for determining sample size is given below:.

It is extremely important that the standard deviation of the difference scores e. In studies where the plan is to estimate the difference in proportions between two independent populations i. In order to estimate the sample size, we need approximate values of p 1 and p 2. Thus, if there is no information available to approximate p 1 and p 2 , then 0.

Similar to the situation for two independent samples and a continuous outcome at the top of this page, it may be the case that data are available on the proportion of successes in one group, usually the untreated e. If so, the known proportion can be used for both p 1 and p 2 in the formula shown above.

The formula shown above generates sample size estimates for samples of equal size. Interested readers can see Fleiss for more details. An investigator wants to estimate the impact of smoking during pregnancy on premature delivery. Normal pregnancies last approximately 40 weeks and premature deliveries are those that occur before 37 weeks. The sample sizes i. We will use that estimate for both groups in the sample size computation.

In the module on hypothesis testing for means and proportions, we introduced techniques for means, proportions, differences in means, and differences in proportions.

While each test involved details that were specific to the outcome of interest e. For example, in each test of hypothesis, there are two errors that can be committed. The first is called a Type I error and refers to the situation where we incorrectly reject H 0 when in fact it is true.

The second type of error is called a Type II error and it is defined as the probability we do not reject H 0 when it is false. In hypothesis testing, we usually focus on power, which is defined as the probability that we reject H 0 when it is false, i.

Power is the probability that a test correctly rejects a false null hypothesis. A good test is one with low probability of committing a Type I error i. Here we present formulas to determine the sample size required to ensure that a test has high power. The effect size is the difference in the parameter of interest that represents a clinically meaningful difference.

Similar to the margin of error in confidence interval applications, the effect size is determined based on clinical or practical criteria and not statistical criteria. The concept of statistical power can be difficult to grasp. Before presenting the formulas to determine the sample sizes required to ensure high power in a test, we will first discuss power from a conceptual point of view. We compute the sample mean and then must decide whether the sample mean provides evidence to support the alternative hypothesis or not.

This is done by computing a test statistic and comparing the test statistic to an appropriate critical value. However, it is also possible to select a sample whose mean is much larger or much smaller than When we run tests of hypotheses, we usually standardize the data e. To facilitate interpretation, we will continue this discussion with as opposed to Z. The rejection region is shown in the tails of the figure below.

This concept was discussed in the module on Hypothesis Testing. Now, suppose that the alternative hypothesis, H 1 , is true i. 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. If the true mean is 94, then the alternative hypothesis is true. The critical value The upper critical value would be The effect size is the difference in the parameter of interest e.

The figure below shows the same components for the situation where the mean under the alternative hypothesis is Notice that there is much higher power when there is a larger difference between the mean under H 0 as compared to H 1 i.

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 Notice also in this case that there is little overlap in the distributions under the null and alternative hypotheses. If a sample mean of 97 or higher is observed it is very unlikely that it came from a distribution whose mean is The inputs for the sample size formulas include the desired power, the level of significance and the effect size.

The effect size is selected to represent a clinically meaningful or practically important difference in the parameter of interest, as we will illustrate. The formulas we present below produce the minimum sample size to ensure that the test of hypothesis will have a specified probability of rejecting the null hypothesis when it is false i. In planning studies, investigators again must account for attrition or loss to follow-up. The formulas shown below produce the number of participants needed with complete data, and we will illustrate how attrition is addressed in planning studies.

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:. The formula for determining sample size to ensure that the test has a specified power is given below:. ES is the effect size , defined as follows:. Similar to the issue we faced when planning studies to estimate confidence intervals, it can sometimes be difficult to estimate the standard deviation.

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.

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. A cross-sectional study is planned to assess the mean fasting blood glucose levels in people who drink at least two cups of coffee per day. The mean fasting blood glucose level in people free of diabetes is reported as The effect size represents the meaningful difference in the population mean - here 95 versus , or 0.

In the planned study, participants will be asked to fast overnight and to provide a blood sample for analysis of glucose levels. Therefore, a total of 35 participants will be enrolled in the study to ensure that 31 are available for analysis see below. 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:.

The formula for determining the sample size to ensure that the test has a specified power is given below:. The numerator of the effect size, the absolute value of the difference in proportions p 1 -p 0 , again represents what is considered a clinically meaningful or practically important difference in proportions. A medical device manufacturer produces implantable stents. How many stents must be evaluated? Do the computation yourself, before looking at the answer.

This book is designed to help users quickly learn the primary features of the PSS application, a point-and-click interface for power analysis and sample size determination.

The book can be used as a text in a senior-level or a graduate course on sample size methodology. Annotated list of tables in appendixSupplemental problems at the end of book.

Addressing the overarching theme of sample size determination for correlated outcomes, this book provides a useful resource for biostatisticians, clinical investigators, epidemiologists, and social scientists whose research involves trials This text describes the following available approaches for estimating sample size in social work research and discusses their strengths and weaknesses: power analysis; heuristics or rules-of-thumb; confidence intervals; computer-intensive Skip to content.

Advanced topics including multivariate analysis, clinical trials, and quality improvement are addressed, and in addition, the book provides considerable guidance on available software for sample size determination.

Written by a well-known author who has extensively class-tested the material, Sample Size Determination and Power: Highlights the applicability of sample size determination and provides extensive literature coverage Presents a modern, general approach to relevant software to guide sample size determination including CATD computer-aided trial design Addresses the use of sample size determination in grant proposals and provides up-to-date references for grant investigators An appealing reference book for scientific researchers in a variety of fields, such as statistics, biostatistics, the health sciences, mathematics, ecology, and geology, who use sampling and estimation methods in their work, Sample Size Determination and Power is also an ideal supplementary text for upper-level undergraduate and graduate-level courses in statistical sampling.

Computer programs help, but no single software package exists that allows researchers to determine sample size across all statistical procedures. In concise, example-rich chapters, Dattalo covers sample-size determination using power analysis, confidence intervals, computer-intensive strategies, and ethical or cost considerations, as well as techniques for advanced and emerging statistical strategies such as structural equation modeling, multilevel analysis, repeated measures MANOVA and repeated measures ANOVA.

He also offers strategies for mitigating pressures to increase sample size when doing so may not be feasible. Whether as an introduction to the process for students or as a refresher for experienced researchers, this practical guide is a perfect overview of a crucial but often overlooked step in empirical social work research. The determination of sample size and the evaluation of power are fundamental and critical elements in the design of clinical trials. If the sample size is too small, important effects may go unnoticed; if the sample size is too large, it represents a waste of resources and unethically puts more participants at risk than necessary.