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Simplified power and sample size calculations using prevalence & magnitude of active peaks. a a a b Joke Durnez , Beatrijs Moerkerke , Ruth Seurinck and Thomas Nichols a b Department of Data Analysis, Ghent University, Belgium Department of Statistics & Warwick Manufacturing Group, University of Warwick, United Kingdom Simulations 100 full-brain datasets with smooth Gaussian noise (3 voxels) superimposed with activation (3 % BOLD change) in 4 foci (3% of total brain volume) Prevalence of activation: For small sample sizes: very conservative estimates (close to 0). Boxplot of bias of volume of activation3 ● 1.0 Estimation of π1 ● 0.2 Sample size 10 15 16 17 18 19 20 21 22 ●23 24 25 0.8 ● ● ● 0.4 ● ●● ● ●● ● ● ● ● ● ● ● ●● ● ●●●● ● ● ● ● ● ● ● ●●● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ●●●● ●● ● ● ● ●● ● ● ● ● ● ●● ●● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ●● ●● ● ● ● ● ● ● ● ●●●●● ● ● ● ●● ● ● ● ●●●● ● ● ● ●●● ● ● ● ● ●● ● ● ● ● ●● ● ● ●● ●● ● ● ● ●● ● ● ● ● ● ● ● ●●● ● ● ● ● ● ●●●●● ● ● ● ● ● ●● ●● ●● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ●● ●● ●●●● ● ● ●●● ● ● ●● ● ● ●● ● ● ● ●● ● ●● ● ●● ● ● ● ● ●● ●● ●● ●●●● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ●● ● ● ●● ●● ● ● ●● ● ● ● ● ●● ●● ● ●● ● ● ● ●● ● ● ● ● ● ●● ● ● ● ● ● ● ● ●●● ● ● ●● ● ●●● ●●● ● ●●● ●● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ●● ● ● ●●● ● ● ●● ● ● ● ●● ● ● ●●● ●●●● ● ● ● ● ●● ● ● ● ● ●● ● ● ●● ● ● ● ●● ● ● ● ● ● ● ●● ● ● ● ● ● ●● ● ● ● ● ●●●●● ●● ● ●● ● ● ● ●● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ●● ● ●●● ● ● ●●● ● ● ● ● ● ● ●● ● ● ●●●●●● ●●● ●● ● ● ● ●● ● ● ●● ● ● ● ●● ●● ● ● ● ●● ●● ● ●● ● ● ● ● ● ● ● ●● ●● ● ● ● ● ●● ● ● ● ● ● ●● ●●● ● ● ●● ●● ● ● ● ● ● ● ● ● ● ● ● ● ●●● ●●● ● ● ● ● ●●● ● ●● ● ● ● ● ● ● ● ● ●● ● ● ●●●● ● ●● ●● ● ●●● ●● ● ● ●● ●●● ● ● ● ● ● ●● ●●● ● ●●●● ●● ●● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ●●● ● ● ● ● ●●● ● ● ● ●● ●●●● ●● ● ● ● ●●● ● ●● ● ● ● ● ● ● ● ● ●● ● ● ● ●● 0.2 ● ● ● ● ● ● ● ● ● ● ●● ● ●● ● ● ● ● ●● ● ● ● ● ● ● ●● ●●● ●● ● ● ● ● ●● ●● ●●● ● ● ● ● ● ● ● ●● ● ● ●● ●●● ● ● ●● ●● ● ● ● ● ● ● ●● ● ●● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● 0.0 ● ● −0.2 ● ● ● ● −0.4 ● Bias (π^1 − π1) 0.6 ● 0.0 There is increasing concern about statistical power in neuroscience research: Low power decreases the chance of detecting a true effect Low power reduces the chance that a statistically significant result indicates a true effect (Ioannidis, 2005). ⇒ A power analysis is a critical component of any study Power analyses for fMRI are difficult: need to specify magnitude, spatial extent and location of a hypothesized effect. We present a simple way to characterize the spatial signal in a fMRI study, and a direct way to estimate power based on an existing pilot study. We must estimate (or have prior knowledge of): the volume of the brain that is activated the average effect size in activated brain regions This allows the calculation of power for given sample size, brain volume and smoothness. Estimated π^1 Introduction ● ● 0.0 0.2 0.4 0.6 0.8 10 15 16 17 18 19 20 21 22 23 24 25 1.0 ● True π1 sample size Effect size estimation: Rather conservative estimates. Boxplot of bias of estimated effect size Effect size estimation Sample size u−µ1 σ1 0.6 13 15 20 25 Sample size Department of Data analysis - H. Dunantlaan 1 - 9000 Gent 30 35 0.0 ^ Bias (δ − δ) −0.2 −0.4 ● 10 15 16 17 18 19 20 21 22 23 24 25 1.2 ● sample size 2.0 1.5 1.0 δ 2.97 3.54 3.95 4.28 µ σ=δ n 0.94 0.92 0.88 0.86 6 7 5 ● 1.0 0.6 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● Sample size 0.5 ● ● ● 15 16 17 18 19 20● 21 22 23 24 25 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 0.0 ● ● 15 16 17 18 19 20 21 22 23 24 25 15 16 17 18 19 20 21 22 23 24 25 Sample size Sample size ● References and acknowledgements Durnez, Moerkerke, Nichols (2014). NeuroImage, 84. Ioannidis (2005). PLOS Medicine, 2:6. Seurinck, de Lange, Achten, Vingerhoets (2011). Journal of Cognitive Neuroscience, 23:6. Worsley (2006). Random Field Theory in Friston, Ashburner, Kiebel, Nichols and Penny, Statistical Parametric Mapping. Elsevier, London This work was carried out using the STEVIN Supercomputer Infrastructure at Ghent University, funded by Ghent University, the Flemish Supercomputer Center (VSC), the Hercules Foundation and the Flemish Government â department EWI. 0.6 0.4 Power to detect effect 0.8 1.0 10 ● 0.0 π1 = 0.7 π1 = 0.6 π1 = 0.5 π1 = 0.4 π1 = 0.32 π1 = 0.2 π1 = 0.1 current study 0.2 1.0 0.8 ● ● ● Power with varying sample size and effect size 0.6 0.4 0.2 0.0 Power to detect effect ● ● ● 0.5 ● ● peak heights Power with varying sample size and π1 ● ● ● ● ● ● Expected peak height under activation µ1 = δ = 4.04 √ Effect size µ/σ = δ/ n = 1.13 Step 3: estimate power for given sample size, estimated peak height under activation and prevalence of activation (π1) ● Bias on power estimation ● ● 0.8 6 ● ● 4 True power ● 0.2 peak p−values 5 ● Power calculations: Good estimates for realistic values of power with FWER control at the 5% significance level. 0.4 4 ● peak heights Estimated power 3 1.1 3 0.4 0.2 2 ● ● ● 0.0 2 ● 1.0 ● ● ● ● Estimated power from pilot study (n=15) 0.0 0.8 ● ● ● 10 15 20 25 0.6 Density 0.5 0.6 ● ● ● ● 1.0 1.0 2.0 1.5 1.0 Density 0.0 0.4 ● 1.0 ● 0.2 ● ● ● ● ● Sample size ● 0.0 ● ● ● ● Average estimated distribution over all simulations for different sample sizes Estimated alternative distribution Estimated null distribution Estimated total distribution 0.8 2.5 Estimated null distribution Estimated alternative distribution ● ● ● ● ● ● ● 0.0 π1 = 0.31 / δ = 4.04 / Power = 0.22 3.0 π1 = 0.31 0.9 ● Alternative distribution: Constant effect size (µ/σ) over different sample sizes to be used in power analyses. Density Data from Seurinck et al. (2011). The original study was analyzed voxelwise with FWER-control at the 5% significance level. We reanalyzed the data peakwise with FDR-control at the 5% significance level. Sample size: 13 Step 1: compute prevalence of activation Step 2: estimate truncated distributions Distribution of peak heights 0.8 ● ● ● True effect size in activated peaks (δ) µ1 and σ1 can be estimated using maximum likelihood, where µ1 is the expected peak height in activated regions. Power can be estimated for a given thresholdt as P (T > t|Ha) with T the T -statistic of the peak. Distribution of 48 peak p−values 0.7 Example 0.2 1.1 0.6 0.7 0.8 0.9 1.0 ● −0.5 1−Φ 10 15 16 17 18 19 20 21 22 23 ● 24 25 −1.0 f (x|π0, µ1, σ1, u) = (1 − π1)u exp(−u(x − u)) + π1 x−µ1 σ1 ●● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ●● ● ● ●● ● ● ● ● ● ●● ● ●● ●● ● ● ●● ● ● ● ● ●● ● ●● ● ●●● ● ● ● ● ● ● ●● ● ● ● ● ●●●● ● ● ● ● ● ● ● ● ● ● ●● ● ● ●● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●●●● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ●● ● ●● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ●● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ●● ●● ● ● ● ● ● ●● ● ● ●● ● ● ● ● ● ● ● ●● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●●●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ●● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ●● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ●● ●●● ● ●● ● ● ● ● ●●●● ● ●● ● ● ● ● ● ● ●● ● ● ●● ● ●● ● ●● ● ● ● ●● ● ● ●● ●● ● ●● ●●●● ●● ● ● ● ●● ● ● ● ● ● ● ● ●●● ● ● ● ●● ● ● ● ● ●● ●● ● ● ● ● ● ● ● ● ● ● ●● ● ● ●● ● ● ● ●● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ●● ● ● Estimated power − True power 1 σ1 ϕ ^ Estimated effect size in activated peaks(δ) We start from peak p-values in a group level analysis. We estimate π1, proportion of peak p-values thare are non-null, as described in Durnez, Moerkerke & Nichols (2014). We assume that the null distribution of peak values is an exponential distribution (Worsley, 2007). We assume that the alternative distribution of peak values is a truncated normal distribution (truncated at excursion threshold u). Therefore, the distribution of peak values can be written as a mixture: 0.4 1.2 Methods ● ● 40 10 13 15 20 25 30 δ = 0.7 δ = 0.8 δ = 0.9 δ=1 δ = 1.12 δ = 1.2 δ = 1.3 current study 35 40 Sample size [email protected] da.ugent.be