# Power and Sample Size. In epigenetic epidemiology studies

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1 Power and Sample Size In epigenetic epidemiology studies

2 Overview Pros and cons Working examples Concerns for epigenetic epidemiology

3 Definition Power is the probability of detecting an effect, given that the effect is really there Or likewise, the probability of rejecting the null hypothesis when it is in fact false An example; Power of 0.8 = if we performed a study 1000 times, we would see a statistically significant difference 80% of the time

4 Why perform them Ideally: To determine the sample size required to confidently observe an anticipated effect Or, at least: To determine if there is sufficient power to detect a meaningful difference in a given sample size Required as part of a grant proposal Part of planning and designing good quality research Familiarise yourself with the data and study design Implement changes to improve the power and design

5 Limitations They are not universal but depend on; Purpose, methodology, statistical design and procedure Provide the minimum number of samples required following the best case scenario Based on statistical assumptions and data characteristics, Which if incorrect (or unknown) will lead to inaccurate estimates They are not intuitive; E.g. they may suggest a number of subjects that is inadequate for the statistical procedure Hence, power should not be the only consideration when deciding on your sample size

6 What you need to know Core elements Power Sample size Significance Effect size* These elements are all inter-related such that; If you know three you can estimate the fourth Manipulating one influences the others

7 *A note on effect size There are many ways to define and calculate effect size Difference in means Variance explained Odds ratio Standardised vs. unstandardised measures If possible use unstandardized measures Raw difference between group means Raw regression coefficients Use standardised effect sizes as a last resort Standardised difference (d): difference in means/sd Pearson s correlation coefficent (r) Cohen s recommendations Effect d r Small Medium Large

8 Deciding on levels of α and β Power (sensitivity) [1- ] Probability of finding a true effect when one does exist Type 2 error [ ]: incorrectly accepting the null hypothesis (false negative) Aim to minimise the risk of failing to detect a real effect Typical values for power are 80%, 90% and 95% Significance (p-value) [ ] Probability that an effect occurred by chance alone Type 1 error [ ]: incorrectly rejecting the null hypothesis (false positive) Aim to minimise the risk of detecting a non-real/spurious effect Typical values are 0.05, 0.01 Reducing the risk of type 1 error increased risk of type 2 error (i.e. reduced power)

9 Available Software Standard statistical packages Stata, Minitab, SPSS Sample Power, R Online web calculators Freely available software G*Power Quanto Different packages only perform specific power calculations so you will need to find one relevant to the statistical model you are planning

10 F2RL3 methylation & smoking Example of an independent two-sample t-test Breitling et al, AJHG 2011 CpG site mapping to F2RL3 was associated with smoking behaviour Average methylation in smokers was 83% compared to 95% in never smokers How many samples do we need to detect this effect? Power = 90% Significance = 0.05 Methylation characteristics: means = 83% & 95%, SD = 10% Effect size: difference in means = = 1.2] SD 10

11 G* Power 1. Select the statistical test

12 G* Power 2. Select the type of power analysis

13 G* Power 3. Input the data characteristics to determine the effect size

14 G* Power 4. Input power parameters

15 F2RL3 methylation & smoking

16 F2RL3 methylation & smoking 5. Draw plot for a range of values

17 F2RL3 methylation & smoking 5. Draw plot for a range of values

18 F2RL3 methylation & smoking 5. Draw plot for a range of values

19 F2RL3 methylation & smoking 6. Produce of table of values

20 STATA Dialog box

21 STATA 1. Select statistical test and input data characteristics

22 STATA 2. Input power parameters

23 0 Number of samples per group F2RL3 methylation & smoking > sampsi , sd1(0.10) sd2(0.10) alpha(0.05) power(.90) Estimated sample size for two-sample comparison of means Test Ho: m1 = m2, where m1 is the mean in population 1 and m2 is the mean in population 2 Assumptions: alpha = (two-sided) power = m1 =.95 m2 =.83 sd1 =.1 sd2 =.1 n2/n1 = 1.00 Sample size requirements (Power = 0.9, Alpha = 0.05) Estimated required sample sizes: n1 = 15 n2 = Absolute difference in methylation (%) between smokers and non-smokers Equal SD of 10% Equal SD of 10% Unequal SD of 15 and 20% Equal SD of 15%

24 Power F2RL3 methylation & smoking > sampsi , sd1(0.15) sd2(0.20) alpha(0.05) n(100) Estimated power for two-sample comparison of means Test Ho: m1 = m2, where m1 is the mean in population 1 and m2 is the mean in population 2 Assumptions: alpha = (two-sided) m1 =.9 m2 =.83 sd1 =.15 sd2 =.2 n2/n1 = 1.00 Power achieved (n = 100, Alpha = 0.05) Estimated power: power = Absolute difference in methylation (%) between smokers and non-smokers Equal SD of 10% Equal SD of 15% Unequal SD of 15 and 20%

25 Other statistical tests G* Power Correlations & regressions (univariate, multiple variate, logistic) Means (one, two, many groups, un/paired, non-parametric) Proportions (one, two groups, un/paired) Variances (one, two groups) STATA sampsi (one, two groups, un/paired, means, proportions) fpower (one-way anova) powerreg (regression)

26 Challenges Non-normality of DNA methylation data Can try transform the data Popular transformations don t always work They make interpretation of results more difficult Transformations that modify the data too much can actually lose more power Can categorise the data Requires more samples given less power Can perform non-parametric tests Few programs perform non-parametric power calculations Those that do still assume the data is normally distributed

27 Challenges Non-normality of DNA methylation data there is minimal power loss associated with the non-parametric tests even when the data are distributed normally, while the power gains of these tests when normality is violated are substantial (Kitchen, Am J Ophthalmol. 2009) If the data is normal, the Mann-Whitney test has been estimated to be ~0.96 times as powerful as the t-test If the data isn t normal, the Mann-Whitney test is more powerful than the t-test [Therefore, if you have enough power for a t-test, you'll have enough power for a Mann-Whitney.]

28 Challenges Lack of prior knowledge Public databases Detailing spectrum of genome wide DNA methylation distributions across multiple populations, ages, tissues and cells are not yet available Literature Specific sample & tissue populations may be different Don't always give the relevant data characteristics Pilot data Not always possible [Small sample sizes in pilot data can be misleading]

29 Challenges Multiple testing Alpha inflation = the more tests you perform, the more likely you are to see a false-positive effect (Type 1 error) P-value ( ) = probability that an effect occurred by chance P-value of 0.05 = 5% of all tests performed (or 1 in 20) Could perform alpha adjustments Bonferroni correction [0.05/n o of tests] Genome-wide significance is estimated at p = 10-6 and 10-8 for GWAS Overly stringent Limited data regarding correlation across genome-wide CpG sites

30 Best practice Perform a range of power calculations covering a range of scenarios ABSOLUTE CHANGE IN METHYLATION POWER FOR P=1.0x10-8 TWO GROUP COMPARISON (N=280 VS. 990) SD of methylation = 8% 3% % % % SD of methylation = 10% 4% % % SD of methylation = 12% 5% % Blurb: There is more than 66% power to detect an absolute change in methylation of 5% between the two groups (n=280 vs990), assuming the variance in methylation is 12% or less, at the p x10-8 significance level. (Power calculations were performed in STATA and are based on standard unpaired t- tests, which assume normality of data and equal variances between groups.)

32 References Jacob Cohen Cohen, Statistical Power Analysis for the Behavioral Sciences,1988 Russell V. Lenth Lenth, Some practical guidelines for effective sample-size determination. Am. Stat. 55: , 2001 Kitchen, Non-parametric versus parametric tests of location in biomedical research. Am J Ophthalmol. 147(4): , 2009 Breitling et al. Tobacco-smoking-related differential DNA methylation: 27K discovery and replication. Am J Hum Genet. 88(4):450-7, 2011 G*Power

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Tests of Differences: two related samples What are paired data? Frequently data from ecological work take the form of paired (matched, related) samples Before and after samples at a specific site (or individual)

### Contents 1. Contents

Contents 1 Contents 3 K-sample Methods 2 3.1 Setup............................ 2 3.2 Classic Method Based on Normality Assumption..... 3 3.3 Permutation F -test.................... 5 3.4 Kruskal-Wallis

### SPSS Explore procedure

SPSS Explore procedure One useful function in SPSS is the Explore procedure, which will produce histograms, boxplots, stem-and-leaf plots and extensive descriptive statistics. To run the Explore procedure,

### PASS Sample Size Software. Linear Regression

Chapter 855 Introduction Linear regression is a commonly used procedure in statistical analysis. One of the main objectives in linear regression analysis is to test hypotheses about the slope (sometimes

### Power Analysis and Sample size estimation

Power Analysis and Sample size estimation The Power Analysis implements the techniques of statistical power analysis, sample size estimation, and advanced techniques for confidence interval estimation.