Bios 6649: Clinical Trials - Statistical Design and Monitoring

Transcription

1 Bios 6649: Clinical Trials - Statistical Design and Monitoring Spring Semester 2015 John M. Kittelson Department of Biostatistics & nformatics Colorado School of Public Health University of Colorado Denver c 2015 John M. Kittelson, PhD Bios pg 1 (i) (Review) Distinction between sample and parameter (ii) (Review) Steps in scientific/statistical inference (iii) Sampling distribution (vs distribution of the data) Bios pg 2

2 We use statistics calculated in sample to make inference about unknown quantities in parameter : nferential Question: µ A = µ B Underlying Population: µ A : Hypothetical mean, treatment A µ B : Hypothetical mean, treatment B Sample Treatment A Statistics: X A Treatment B X B Bios pg 3 We have discussed key elements in structuring inference: 1. dentify primary outcome 2. Choose probability model 3. Choose functional of interest 4. Choose contrast for comparing groups 5. Specify ( hypotheses) 6. Select statistical standard for evidence 7. nference about the underlying population Sample vs parameter : The first 4 elements structure outcome (both sample and parameter ) The 5th element provides the map between parameter and the clinical standards for benefit, harm, or equivalence. The 6th and 7th elements are calculated in sample and provide inference about parameter. Bios pg 4

3 Sampling distribution vs distribution of data (Normal data) Distribution of data Sampling distribution of average Bios pg 5 Sampling distribution vs distribution of data (Log-normal data) Distribution of data Sampling distribution of average Bios pg 6

4 Sampling distribution vs distribution of data (mixture data) Distribution of data Sampling distribution of average Bios pg 7 (i) (Review) Four elements of (ii) (Review) Statistical standards for scientific evidence (iii) Frequentist versus Bayesian approaches Bios pg 8

5 (i) Four elements of We use ˆ (observed trial result) to estimate the true underlying value 1. Point estimate: ˆ is the best" estimate of. 2. nterval estimate: Values of that are consistent with the trial results. 3. Expression of uncertainty (p-value): To what degree is a particular hypothesis (the null" hypothesis) consistent with the observed trial results? 4. Decision: Based on the above measures, what decision should be reached about the use of a new therapy? Bios pg 9 The scientific objective is to identify the hypotheses that have (or have not) been ruled out by the trial s results. Let U( ref ˆ obs ) represent a statistical measure of the consistency between the trial s result ˆ = ˆ obs and the hypothesis = ref. Reject the hypothesis = ref when U( ref ˆ obs ) is small; specifically when: U( ref ˆ obs ) < 2 = 0.05 is the common (universal?) standard Bios pg 10

6 Usual frequentist standard By usual frequentist criteria U( ref ˆ obs ) is equal to the smaller of: P(ˆ ˆ obs = ref ) P(ˆ apple ˆ obs = ref ) The 95% confidence interval represents the range of ref that cannot be rejected according to this standard. Bios pg 11 Usual Bayesian standard n Bayesian constructions U( ref ˆ obs ) can be defined as the smaller of: P( ref ˆ = ˆ obs ) P( apple ref ˆ = ˆ obs ) The 95% credible interval is commonly calculated as the range of ref that cannot be rejected according to this standard. (The above is perhaps the most common approach to Bayesian in. Other approaches are also used.) Bios pg 12

7 Frequentist versus Bayes approaches: Practically, both are scientifically relevant: The strength of the trial s results under some hypothesis (frequentist) Evidence that overpowers one s prior beliefs (Bayes). Theoretical foundations: Joint probability : P(, ˆ ) Frequentists use the likelihood": P( ˆ ) Bayesians use a posterior distribution P( ˆ ). Bios pg 13 Frequentist versus Bayes approaches (con t): Bayes and frequentist approaches are related through the likelhood by Bayes rule: P( ˆ ) = R P(ˆ )P( ) P(ˆ )dp( ) where P( ) denotes a prior" distribution for the true treatment effects. The prior distribution is subjective Choosing a different prior gives a different answer. (The questions answered are different, but both are scientifically relevant.) Currently, statisticians are not doing a good job of objectively explaining the overlap and differences to scientists. Sometimes one approach is often presented as right and the other as wrong. The distinction is lost on scientific investigators. Bios pg 14

8 (i) (Review) Central limit theorem (ii) (Review) mplications for frequentist inference (iii) mplications for Bayesian inference Bios pg 15 (i) Central limit theorem: For sample sizes that are not too small, most treatment effect statistics (ˆ ) are normally distributed regardless of the distribution of the data. (ii) mplications for frequentist inference: Assumptions about the distribution of the data are not necessary in order to assure accurate probability statements (C, p-values). U( ref ˆ obs ) is safely computed using the normal distribution since ˆ N (, V ) by CLT. Bios pg 16

9 (iii) mplications for Bayesian inference: With large sample sizes the likelihood approaches a normal distribution. Regular priors with low statistical information are eventually overwhelmed by the data. These principles often mean that the posterior distribution is normal. Thus (for regular prior distributions), U( ref ˆ obs ) can often be computed using the normal distribution. Bios pg 17