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Sample size and statistical power ดร. อรพิน กฤษณเกรียงไกร ดร. อรพิน กฤษณเกรียงไกรคณะสาธารณสุขศาสตร์มหาวิทยาลัยนเรศวร

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Going from population to sample Populations, parameters & taking a census Populations, parameters & taking a census Samples, statistics, and Samples, statistics, and Getting a sample from a population Getting a sample from a population –Random sampling process Simple random selection of subjects from population Simple random selection of subjects from population Stratified random sampling Stratified random sampling Cluster/multistage sampling Cluster/multistage sampling –Non-random sampling process Convenience sampling Convenience sampling Snowball sampling Snowball sampling

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Hypothesis testing Begin with assumption of “no difference”, and when that is untenable, conclude a difference Begin with assumption of “no difference”, and when that is untenable, conclude a difference –Ho: µ = 100 H a : µ ≠ 100 Determine standards of rareness, α Determine standards of rareness, α Assuming Ho, what is µ & σ of sampling distribution? Assuming Ho, what is µ & σ of sampling distribution? Assuming Ho, how rare is observed measure? Assuming Ho, how rare is observed measure? Compare rareness of observed measure to α Compare rareness of observed measure to α –If observation is rare, conclude Ho is false. –If observation is not rare, conclude Ho cannot be rejected.

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Hypothesis testing & α The normal distribution extends to infinity in both directions. The normal distribution extends to infinity in both directions. We choose our level at which results are “not normal” We choose our level at which results are “not normal” This level, α, expresses how “rare” something has to be to claim something is different This level, α, expresses how “rare” something has to be to claim something is different But because the normal distribution extends to infinity, what we claim as different might not be…and we make a Type 1 error, Pr (Type 1 Error) = α But because the normal distribution extends to infinity, what we claim as different might not be…and we make a Type 1 error, Pr (Type 1 Error) = α

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Hypothesis testing and errors Ho is True Ho is False You decide: Reject Ho Type 1 Error Probabilit y = α Correct decision Probabilit y = 1-β You decide: Fail to reject Ho Correct decision Probabilit y = 1-α Type II Error Probabilit y = β

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Hypothesis testing and errors You set alpha You set alpha A type II error is when you fail to reject the null hypothesis, but you should have rejected it A type II error is when you fail to reject the null hypothesis, but you should have rejected it Both errors always exist when you test Both errors always exist when you test As you increase alpha you increase beta and vice versa. As you increase alpha you increase beta and vice versa.

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Visualizing α and β Pr (Type 1 Error) = Area under Ho to right of A and to left of B Pr (Type 1 Error) = Area under Ho to right of A and to left of B Pr (Type 2 Error) = Area under Ha, to right of B Pr (Type 2 Error) = Area under Ha, to right of B But we don’t know Ha But we don’t know Ha σxσx σxσx BA µ µ Ha Ho

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Hypotheses: one & two tailed Ho: µ = 8.0 Ha: µ ≠ 8.0 Prob > lzl Large (>0.05) Small (<0.05) Ho: µ < 8.0 Ha: µ > 8.0 Prob > z Large (>0.05) Small (<0.05) Ho: µ > 8.0 Ha: µ < 8.0 Prob < z Large (>0.05) Small (<0.05) µ µ µ

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ความคาดเคลื่อน 2 ประเภทที่ อาจเกิดขึ้นในการ ตั้งสมมติฐาน : Type I error: Reject H 0 when it is true. Type I error: Reject H 0 when it is true. Type II error: Fail to reject H 0 when it is false. Type II error: Fail to reject H 0 when it is false. เราสามารถควบคุมความน่าจะเป็นของ Type I error ได้โดยการกำหนด ให้ น้อยๆ เช่น 0.05 ในการควบคุม Type II error เราสามารถ ทำได้โดยการเพิ่มขนาดของกลุ่มตัวอย่าง นิยาม : 1- = power ของการทดสอบ สมมติฐาน ซึ่งเป็นความน่าจะเป็นใน การปฎิเสธ H 0 เมื่อ H 0 ไม่ถูกต้อง

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There are two types of error we can commit in hypothesis testing: Type I error: Reject H 0 when it is true. Type I error: Reject H 0 when it is true. Type II error: Fail to reject H 0 when it is false. Type II error: Fail to reject H 0 when it is false. We control the probability of committing a type I error by choosing a small value for , e.g. = 0.05 Question: How can we control , the probability of committing a type II error? Answer: By controlling the sample size. As the sample size increases the probability of committing a type II error decreases.

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Power = 1- , It is the probability of rejecting H 0 when it is false, i.e. the probability of detecting a true alternative hypothesis. Assume it is well established that participants in a 6 months smoking cessation program reduce their daily number of cigarettes by 10, on average. The program reduce their daily number of cigarettes by 10, on average. The program coordinators would like to know whether increasing the number of session per week will lead to a significantly greater average reduction in the number of cigarettes smoked per day. They choose a random sample of 5 new participants and increase the number of sessions from 2 to 3 week. They find that the average reduction in the sample is 15 with a standard deviation of 10.

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H 0 : The increase is not significant A one sample t-test yields a p-value greater than 0.1. We fail to reject H 0 and conclude that the increased reduction of cigarettes smoked per day is not significant. This seems strange, since the increase appears to be important. Question: Was the sample size large enough to detect the increase as significant? Did we have enough power to detect the alternative hypothesis? Answer: Probably not.

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Note: Power depends on a many factors including the -level and the magnitude of the effect we would like to detect. What magnitude is important depends on biological considerations or experience. It is not a statistical question. Note: The sample size is generally chosen such that the power is greater than 80%. Why don’t we try to get more than 80% power? Many studies have limited budgets and the sample size must be kept as small as possible. 80% is considered a reasonable compromise.

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Problems with sample size/ power calculations: and level are completely arbitrary and level are completely arbitrary The magnitude of the effect we would like to detect is arbitrary The magnitude of the effect we would like to detect is arbitrary To determine sample size or power we need an estimate of the standard deviation (if we are estimating a mean), or of the disease rate in the absence of exposure prevalence in the absence of disease) and the relative size of the compared groups (if we are estimating a RR or OR). These estimates are guess work or come from small pilot studies and are often inaccurate. To determine sample size or power we need an estimate of the standard deviation (if we are estimating a mean), or of the disease rate in the absence of exposure prevalence in the absence of disease) and the relative size of the compared groups (if we are estimating a RR or OR). These estimates are guess work or come from small pilot studies and are often inaccurate. The sample size is often predetermined by the availability of eligible study subjects rather than statistical formulas. The sample size is often predetermined by the availability of eligible study subjects rather than statistical formulas. Note: Even power curves (plots of power vs. sample size) generally don’t alleviate these problems.

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Avoid hypothesis testing when possible and use confidence intervals instead. Even though the level of confidence is arbitrary, confidence intervals are preferable because they provide us with an estimate of the effect and a measure of the precision of the estimate. Even if the sample size is small, confidence intervals still provide us with a lot of information. solution

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Reference ไพบูลย์ โล่ห์สุนทร ระบาดวิทยา ภาควิชาเวช ศาสตร์ป้องกัน คณะแพทยศาสตร์ จุฬาลงกรณ์ มหาวิทยาลัย 2540 ไพบูลย์ โล่ห์สุนทร ระบาดวิทยา ภาควิชาเวช ศาสตร์ป้องกัน คณะแพทยศาสตร์ จุฬาลงกรณ์ มหาวิทยาลัย 2540 Annette Bachand, Introduction to Epidemiology: Colorado State University, Department of Environmental Health Annette Bachand, Introduction to Epidemiology: Colorado State University, Department of Environmental Health Leslie Gross Portney and Mary P. Watkins (2000). Foundations of Clinical Research: Applications to Practice. Prentice-Hall, Inc. New Jersey, USA Leslie Gross Portney and Mary P. Watkins (2000). Foundations of Clinical Research: Applications to Practice. Prentice-Hall, Inc. New Jersey, USA

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Hypothesis Testing State the hypotheses. Formulate an analysis plan. Analyze sample data. Interpret the results.

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