Sample Size Calculations for Analytic Studies
Transcription
Sample Size Calculations for Analytic Studies
Sample Size Calculations for Analytic Studies • Solving standard problems in Stata • Stata solutions to some non-standard problems: – Adjusting for covariates – Mediation – Clustered data – Pre-post trial designs – Fixed sample size – Categorical predictors and outcomes 1 Standard problems • Method for calculating sample size depends on – Predictor type: continuous, binary – Outcome type: continuous, binary, failure time – Effect size – α, 1- or 2-sided test, power 2 Binary predictor, continuous outcomes • Predictor prevalence 0.5 (e.g., RCTs with 1-1 allocation): – use Table 6.A. for comparing means in Designing Clinical Research (DCR) • Arbitrary predictor prevalence: – use sampsi function in Stata 3 Basic set-up using sampsi Equal allocation to groups, means of 1.4 and 1.9, SD of outcome = 2 in both groups . sampsi 1.4 1.9, sd1(2) 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 power m1 m2 sd1 sd2 n2/n1 = = = = = = = 0.0500 0.9000 1.4 1.9 2 2 1.00 (two-sided) Estimated required sample sizes: n1 = n2 = 337 337 4 A more complicated example 60% in group 2, means 1.4 and 1.6, SDs of 0.85 and 1.05, one-sided test . local r = .6/.4 . sampsi 1.4 1.6, sd1(0.85) sd2(1.05) r(‘r’) power(0.8) onesided 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 = 0.0500 (one-sided) power = 0.8000 m1 = 1.4 m2 = 1.6 sd1 = .85 sd2 = 1.05 n2/n1 = 1.50 Estimated required sample sizes: n1 = n2 = 226 339 Note: Use sampncti for large effect sizes; requires nct2 package 5 Continuous predictor, continuous outcome • If you can pose problem in terms of correlation coefficient: – use Table 6.C for in DCR – use sampsi rho in STATA • If you can pose problem in terms of slope, SDs of predictor and outcome: – use sampsi reg 6 Continuous predictor, continuous outcome, using sampsi rho Correlation of predictor and outcome = 0.3 . sampsi_rho, alt(0.3) power(0.8) Estimated sample size for Pearson Correlation Test Ho: Rho alt = Rho null, usually null Rho is 0 Assumptions: Alpha Power Null Rho Alt Rho = = = = 0.0500 0.8000 0.0000 0.3000 (two-sided) Estimated required sample size: n = 84.927811 7 Continuous predictor, continuous outcome, using sampsi reg Regression coefficient = 0.3, SD of predictor and outcome = 1 . sampsi_reg, alt(0.3) sx(1) sy(1) varmethod(sdy) power(0.8) Estimated sample size for linear regression Test Ho: slope alt = slope null, usually null slope is 0 Assumptions: Alpha Power Null Slope Alt Slope Residual sd SD of X’s SD of Y’s = = = = = = = 0.0500 0.8000 0.0000 0.3000 0.9539 1.0000 1.0000 (two-sided) Estimated required sample size: n = 82 8 Binary predictor, binary outcome • Predictor prevalence 0.5 – use Table 6.B. for comparing proportions in DCR • Arbitrary predictor prevalence: – use sampsi function in Stata 9 Binary predictor, binary outcome Exposure prevalence 2/3, outcome prevalence 20% in unexposed, 30% in exposed . sampsi 0.2 0.3, r(2) power(0.8) Estimated sample size for two-sample comparison of proportions Test Ho: p1 = p2, where p1 is the proportion in population 1 and p2 is the proportion in population 2 Assumptions: alpha power p1 p2 n2/n1 = = = = = 0.0500 0.8000 0.2000 0.3000 2.00 (two-sided) Estimated required sample sizes: n1 = n2 = 239 478 10 Continuous predictor, binary outcome • Use methods for binary predictor, continuous outcome – set r = p/(1 − p) where p is prevalence of outcome – if you know means and SDs of continuous predictor in cases and controls, use sampsi as usual – otherwise, set up using log-OR and SD of predictor 11 Continuous predictor, binary outcome Case-control study with 3 controls per case, mean of predictor 0.2 in controls, 0.4 in cases, SD of predictor 0.5 in both groups . sampsi 0.2 0.4, r(.33) sd1(0.5) 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 power m1 m2 sd1 sd2 n2/n1 = = = = = = = 0.0500 0.9000 .2 .4 .5 .5 0.33 (two-sided) Estimated required sample sizes: n1 = n2 = 265 88 12 Continuous predictor, binary outcome Cross-sectional study with outcome prevalence = 33%, OR per unit increase in predictor = 1.5, SD of predictor = 0.5: effect size equal to log(OR) × SD of predictor . local delta = log(1.5)*0.5 . sampsi 0 ‘delta’, r(0.5) sd1(1) power(0.8) 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 = 0.0500 (two-sided) power = 0.8000 m1 = 0 m2 = .202733 sd1 = 1 sd2 = 1 n2/n1 = 0.50 Estimated required sample sizes: n1 = n2 = 573 287 13 Failure time outcomes • Use stpower in STATA – stpower cox for Cox models – stpower logrank for unadjusted log-rank test – stpower exponential for trials with long accrual 14 Continuous predictor, failure time outcome Overall cumulative incidence of 15%, 10% early dropout, SD of predictor 1.5, hazard-ratio per unit increase in predictor 1.2 . stpower cox, failprob(0.15) wdprob(0.10) sd(1.5) hratio(1.2) Estimated sample size for Cox PH regression Wald test, log-hazard metric Ho: [b1, b2, ..., bp] = [0, b2, ..., bp] Input parameters: alpha = b1 = sd = power = Pr(event) = withdrawal(%) = 0.0500 0.1823 1.5000 0.8000 0.1500 10.00 (two sided) Estimated number of events and sample size: E = N = 105 778 15 Binary predictor, failure time outcome RCT, overall cumulative incidence of 15%, 10% early dropout, hazard-ratio p for treatment = 0.75, 1-1 allocation so SD of predictor = 0.5(1 − 0.5) = 0.5 (the default) . stpower cox, failprob(0.15) wdprob(0.10) hratio(0.75) Estimated sample size for Cox PH regression Wald test, log-hazard metric Ho: [b1, b2, ..., bp] = [0, b2, ..., bp] Input parameters: alpha = b1 = sd = power = Pr(event) = withdrawal(%) = 0.0500 -0.2877 0.5000 0.8000 0.1500 10.00 (two sided) Estimated number of events and sample size: E = N = 380 2811 16 Binary predictor, failure time outcome Overall cumulative incidence of 15%, 10% early dropout, prevalence of exposure 25%, hazard-ratio for exposure 1.5 . local sd = sqrt(0.25*(1-0.25)) . stpower cox, failprob(0.15) wdprob(0.10) hratio(1.50) sd(‘sd’) Estimated sample size for Cox PH regression Wald test, log-hazard metric Ho: [b1, b2, ..., bp] = [0, b2, ..., bp] Input parameters: alpha b1 sd power Pr(event) withdrawal(%) = = = = = = 0.0500 0.4055 0.4330 0.8000 0.1500 10.00 (two sided) Estimated number of events and sample size: E = N = 255 1887 17 Adjustment for covariates • Suppose that – multiple correlation of primary predictor with covariates is ρ – equivalently R2 for linear regression of primary predictor on covariates is ρ2 • Implemented in stpower using r2() option • Alternatively, compute sample size using sampsi, inflate result by 1/(1 − ρ2 ) • Hsieh recommends ρ = 0.3 → 10% inflation of N • NB: Adjusted effect size usually smaller than unadjusted 18 Binary predictor, failure time outcome Same problem as before; correlation of exposure with covariates ρ = 0.5, so R2 = 0.52 = 0.25 . local sd = sqrt(0.25*(1-0.25)) . stpower cox, failprob(0.15) wdprob(0.10) hratio(1.50) sd(‘sd’) r2(0.25) Estimated sample size for Cox PH regression Wald test, log-hazard metric Ho: [b1, b2, ..., bp] = [0, b2, ..., bp] Input parameters: alpha b1 sd power Pr(event) R2 withdrawal(%) = = = = = = = 0.0500 0.4055 0.4330 0.8000 0.1500 0.2500 10.00 (two sided) Estimated number of events and sample size: E = N = 340 2515 19 Adjustment for covariates: binary predictor, continuous outcome Prevalence of exposure 40%, Effect size 0.25, ρ = 0.3 . local r = 0.4/0.6 . sampsi 0 0.25, sd1(1) r(‘r’) power(0.8) 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 = 0.0500 (two-sided) power = 0.8000 m1 = 0 m2 = .25 sd1 = 1 sd2 = 1 n2/n1 = 0.67 Estimated required sample sizes: n1 = 314 n2 = 210 . dis round((314+210)/(1-0.3^2)) 576 20 Sample size to show mediation • To show mediation, we need to show that 1. primary predictor associated with mediator 2. mediator independently associated with outcome, adjusting for primary predictor 3. coefficient for primary predictor changes when mediator added to the regression model • If 1 holds, then 2 implies 3 • Determine sample size needed to show 2: – mediator independently predicts outcome, adjusting for primary predictor and covariates 21 Mediation example Cox model, cumulative incidence 25%, early dropout 15%, adjusted HR per SD increase in continuous mediator 1.2, correlation of mediator w/ primary predictor, covariates 0.2 . stpower cox, failprob(0.25) wdprob(0.15) hratio(1.2) sd(1) r2(0.04) Estimated sample size for Cox PH regression Wald test, log-hazard metric Ho: [b1, b2, ..., bp] = [0, b2, ..., bp] Input parameters: alpha b1 sd power Pr(event) R2 withdrawal(%) = = = = = = = 0.0500 0.1823 1.0000 0.8000 0.2500 0.0400 15.00 (two sided) Estimated number of events and sample size: E = N = 246 1158 22 Cluster randomized trials • Clusters of average size nc are randomized 1-1 to treatment or control • Compute number of patients for trial ignoring clustering • Inflate N by “design effect”: 1 + ρ(nc − 1), where ρ is the within-cluster correlation • sampclus implements this after sampsi (but not stpower) 23 Cluster randomized trials Binary outcome, incidence 30% in control, 20% in treatment, 1-1 allocation of clusters, nc = 25, ρ = 0.02 . sampsi 0.3 0.2 Estimated sample size for two-sample comparison of proportions Test Ho: p1 = p2, where p1 is the proportion in population 1 and p2 is the proportion in population 2 Assumptions: alpha power p1 p2 n2/n1 = = = = = 0.0500 0.9000 0.3000 0.2000 1.00 (two-sided) Estimated required sample sizes: n1 = n2 = 412 412 . dis round(412*(1+0.02*24)) 610 24 Cluster randomized trials . sampclus, obsclus(25) rho(0.02) Sample Size Adjusted for Cluster Design n1 (uncorrected) = 412 n2 (uncorrected) = 412 Intraclass correlation = .02 Average obs. per cluster = 25 Minimum number of clusters = 49 Estimated sample size per group: n1 (corrected) = 610 n2 (corrected) = 610 25 Randomized trials, within-cluster randomization • Patients are randomized 1-1 to treatment or control within clusters • Design effect is 1 − ρ, does not depend on nc • Compute sample size ignoring clustering, multiply result by 1 − ρ 26 Clustered data: complex surveys • Design effects vary by predictor and outcome, can be less than 1 • Simple rules do not apply • Use design effect to adjust sample size calculated assuming independence, if you can get a reasonable estimate 27 RCTs measuring change in a continuous outcome • Outcome measured at baseline and follow-up • No expected difference in means at baseline (why?) • Analysis options – analyze follow-up outcome, ignoring baseline – analyze change scores – analyze follow-up outcome, adjusting for baseline (ANCOVA) • Use sampsi 28 RCT measuring change in continuous outcome 1-1 allocation, equal baseline means, effect size at follow-up 0.5 SD, correlation of baseline and follow-up outcome 0.3 . sampsi 0 0.5, sd1(1) pre(1) post(1) r01(.3) Estimated sample size for two samples with repeated measures Assumptions: alpha = 0.0500 (two-sided) power = 0.9000 m1 = 0 m2 = .5 sd1 = 1 sd2 = 1 n2/n1 = 1.00 number of follow-up measurements = 1 number of baseline measurements = 1 correlation between baseline & follow-up = 0.300 29 RCT measuring change in continuous outcome Method: POST relative efficiency = adjustment to sd = adjusted sd1 = 1.000 1.000 1.000 Estimated required sample sizes: n1 = 85 n2 = 85 Method: CHANGE relative efficiency = adjustment to sd = adjusted sd1 = 0.714 1.183 1.183 Estimated required sample sizes: n1 = 118 n2 = 118 Method: ANCOVA relative efficiency = adjustment to sd = adjusted sd1 = 1.099 0.954 0.954 Estimated required sample sizes: n1 = 77 n2 = 77 30 Same design, but with pre-post correlation of 0.7 Method: POST relative efficiency = adjustment to sd = adjusted sd1 = 1.000 1.000 1.000 Estimated required sample sizes: n1 = 85 n2 = 85 Method: CHANGE relative efficiency = adjustment to sd = adjusted sd1 = 1.667 0.775 0.775 Estimated required sample sizes: n1 = 51 n2 = 51 Method: ANCOVA relative efficiency = adjustment to sd = adjusted sd1 = 1.961 0.714 0.714 Estimated required sample sizes: n1 = 43 n2 = 43 31 If sample size is fixed • In secondary analyses, sample size usually a done deal • sampsi can compute power for fixed N and effect size • stpower can compute either power or minimum detectable effects when other inputs are specified • In grants, power for a well-motivated effect size more convincing than minimum detectable effects 32 Power for fixed sample and effect sizes Cox model, N=2515, 15% overall cumulative incidence, 10% dropout, 25% exposed, HR for exposure 1.5 . local sd = sqrt(0.25*(1-0.25)) . local n = round(2515*0.9) . stpower cox, failprob(0.15) hratio(1.3(0.1)1.6) sd(‘sd’) r2(0.25) n(‘n’) Estimated power for Cox PH regression Wald test, log-hazard metric Ho: [b1, b2, ..., bp] = [0, b2, ..., bp] +------------------------------------------------------------------------+ | Power N E B1 SD Alpha* Pr(E) R2 | |------------------------------------------------------------------------| | .441616 2264 340 .262364 .433013 .05 .15 .25 | | .64254 2264 340 .336472 .433013 .05 .15 .25 | | .800117 2264 340 .405465 .433013 .05 .15 .25 | | .901134 2264 340 .470004 .433013 .05 .15 .25 | +------------------------------------------------------------------------+ 33 Minimum detectable hazard ratios Same set-up as last slide . local sd = sqrt(0.25*(1-0.25)) . local n = round(2515*0.9) . stpower cox, failprob(0.15) sd(‘sd’) r2(0.25) n(‘n’) power(0.9) hr Estimated hazard ratio for Cox PH regression Wald test, hazard metric Ho: [b1, b2, ..., bp] = [0, b2, ..., bp] Input parameters: alpha = sd = N = power = Pr(event) = R2 = 0.0500 0.4330 2264 0.9000 0.1500 0.2500 (two sided) Estimated number of events and hazard ratio: E = hratio = 340 0.6256 . dis 1/.6256 1.5984655 34 Categorical predictors and outcomes • Categorical predictors: – compute Ns for pairwise differences with reference group (multiple comparisons) – for overall effect, use fpower or simpower functions in STATA http://www.ats.ucla.edu/stat/stata/dae/fpower.htm • Categorical outcomes – nominal outcomes: compute Ns for pairwise differences with reference group – ordinal outcomes: Whitehead paper gives methods available for proportional odds model, but n/a in Stata 35 Summary • Stata sampsi and stpower commands can do a lot, including making tables • stpower can account for covariate adjustment; with sampsi inflate sample size by 1/(1 − ρ2 ) • Handle mediation like a confounding problem • sampclus can inflate sample size for cluster-randomized trials with continuous or binary endpoint; for Cox model, inflate sample size by design effect 1 + ρ(nc − 1) • sampsi can handle simple pre-post designs • Downloadable Stata packages (and your biostat mentors) can deal with more complicated problems 36