A brief introduction to randomisation methods Peter T. Donnan Professor of Epidemiology and Biostatistics

Treatment Allocation Methods Overview Fixed Methods: Simple randomisation Stratification Minimisation Adaptive Methods: Urn randomisation Biased Coin Play-the-winner

Randomised Controlled Trial Parallel-group Randomised Controlled Trial Eligible subjects RANDOMISED Intervention Control

RANDOMISED CONTROLLED TRIAL (RCT) Random allocation to intervention or control so likely balance of all factors affecting outcome Hence any difference in outcome ‘caused’ by the intervention

Treatment Allocation Methods Randomisation is main allocation method in scientific experiments First proposed by Fisher (1935) ‘The Design of Experiments’ Two Properties : Unbiased allocation Balances covariates, known and unknown

Treatment Allocation Methods Properties required for unbiased and efficient treatment comparison: 1. Equal distribution of known covariates 2. Equal distribution of unknown, or unmeasured, covariates 3. Balanced group size Random allocation is the best means of ensuring equal distribution of unknown covariates

Random Allocation Methods Simple Randomisation – 0,1 computer generated list: Coin toss pr (A) = 0.5 Least predictable method – can have long runs of same treatment Risk of covariate imbalance especially with short sequences i.e. small trials

PROCESS OF RANDOMISATION Example - 5-digit Random numbers below 75792, 80169, 94071, 67970, 91577, 84334 03778, 58563, 29068, 90047, 54870, 23327 With two treatments can be converted to: A B B A B A A B A B A B Where last digit even = A and odd = B

RANDOMISATION Power of RCT is RANDOMISATION Facilitates blind objective unbiased assessment of outcome – removes selection bias But note not necessarily what the patient wants Nor what the physician prefers In a sense trial patients acting altruistically

PROCESS OF RANDOMISATION Usually generate random numbers (statistical software or spreadsheet) Note that to be GCP–compliance requires: Record of seed used to generate the random numbers so that list is replicable Record of patient allocation

OUTCOME OF RANDOMISATION Often just a randomised list ABBAAABB…. To ensure treatment balance use randomised blocks e.g. size 4 ABBA ABAB BAAB BABA Electronic 24 hr telephone randomisation may be necessary or web-based Usually provided by a trials unit

Example of Parallel-group RCT Sebag-Montefiore et al Lancet 2009; 373: 811-820 Trial of short course preoperative radiotherapy vs. initial surgery with selective postoperative chemotherapy for operable rectal cancer (n=1350) Reduction of 61% of risk of local recurrence with preoperative radiotherapy HR = 0.39 (95% CI 0.27, 0.58) Consistent evidence that short course preoperative radiotherapy is effective treatment for operable rectal cancer

Restricted Allocation Methods Restricted methods improve balance but more predictable: Permuted blocks – e.g BAAB ABAB …. Stratification Minimisation Biased coin Urn randomisation Optimal biased coin

Permuted Blocks BAABABBA AABABBBA BBAAABAB Guarantees balance in group size, at end of each block • Predictable, especially if block size known • Predictability depends on block size n • Randomly vary block size • Start at random point in first block • Put details in protocol? No! n p 2 0.75 10 0.65 20 0.62

Stratified Randomisation Randomise separately within each strata e.g. centre • Each randomisation list should use restricted methods e.g. permuted blocks • Ensures balance of known prognostic factors • Limited to two or three factors, strata multiply • ICH-E9 recommends stratification by centre • Group small centres

Stratified Parallel-group Randomised Controlled Trial Eligible subjects Stratum Mild Moderate Severe RANDOMISED Active Control Active Control Active Control

Minimisation Balances a number of known prognostic factors Deterministic case of Pocock-Simon (1975) method Start with simple randomisation and after, say n=56 we end up with …… ARM A ARM B Smoker 9 7 Non-smoker 17 23 Male 13 16 Female 14 Total 26 30

Minimisation  

Minimisation Next patient is female smoker then GA = 1 x | 10-7 | + 1 x | 13-14 | = 4 GB = 1 x | 9-8 | + 1 x |12-15 | = 4 ARM A ARM B Smoker 9 7 Non-smoker 17 23 Male 14 16 Female 12 Total 26 30 Hence imbalance is equal so patient is allocated to treatment by chance pr=0.5

Minimisation What happens if give more weight to smoking (2)? GA = 2 x | 10-7 | + 1 x | 13-14 | = 7 GB = 2 x | 9-8 | + 1 x |12-15 | = 5 ARM A ARM B Smoker 9 7 Non-smoker 17 23 Male 14 16 Female 12 Total 26 30 Hence patient is deterministically allocated to B

Minimisation Dynamic – uses allocations and patient characteristics Deterministic – predictability high in theory, low in practice? Uses categorical covariates Does require more complex programming But ensures balance on known factors TCTU system incorporates minimisation ICH-E9 recommends a random element be added pr (0.7-0.8)

Treatment- or Response-Adaptive Randomisation Biased Coin Randomisation Urn randomisation Play-The-Winner

Biased Coin Adaptive technique, which is a modified version of flipping a coin Start as simple randomisation, with an unbiased (fair) randomisation process (pr = 0.5) If group size becomes unequal then the probability of treatment allocation changes to FAVOUR THE SMALLER GROUP Alter prob. so if arm A has nA < nB Then pr(A) > 0.5 If arms are balanced then use equal Prob i.e. 0.5

Biased Coin • Absolute difference generally used e.g. if group size differs by >3, use ratio of 2:1 to favour smaller group • `Big stick’ randomisation, force next patient into smaller group • Improves balance but can become predictable

Urn Randomisation More flexible Adaptive method is Urn Randomisation Probability of treatment assignment depends on the magnitude of imbalance Start with two coloured balls in an urn • Sample with replacement • Add extra ball of opposite colour to the one selected each time

Urn Randomisation Draw Allocation 0 1 1 1 2 1 Etc…..

Urn Randomisation Improves balance at start of trial • simple randomisation as trial progresses • Effect depends on no. balls at start and no. balls added e.g. start with 10 red, 10 blue for less effect, add more balls each time for greater effect • Useful for smaller trials In larger trials urn randomisation eventually behaves like complete simple randomisaton

Other Adaptive Methods – Play-The-Winner If one treatment is clearly inferior over time, many patients getting the weaker drug Zelen (1969) suggested classify each patient’s outcome as ‘success’ or ‘failure’ Starts as simple randomisation If patient ‘success’ then allocate same treatment to next patient. If patient ‘failure’ then allocate different treatment to next patient

Play-The-Winner Prob of patient’s allocation is unknown at start and depends on prob of ‘success’ Trial stops when fixed pre-specified number of ‘failures’ observed or predetermined sample size is reached Benefit to the patient is more patients get ‘successful’ treatment

Play-The-Winner Drawbacks: Balance not an aim of method and imbalance can occur High susceptibility to selection bias Recent outcomes determine subsequent allocation and researcher can guess what next assignment will be Randomisation prob. changes over time So time trend in outcome confounds treatment effect and biases its estimate Can magnify early random differences

Optimal Biased Coin Dynamic, uses allocations and patient characteristics • Coin is biased so that next allocation minimises treatment effect variance , based on OLS regression model • Continuous covariates • Continuous outcome • Non-deterministic • Complex, matrix computation Atkinson A.C, Statistics in Medicine, 1982

Summary For large trials differences in methods have little effect so stick to simple randomisation Methods described mainly for smaller trials Permuted –block randomisation prone to selection bias – keeping these unknown reduces this potential How much detail in protocol? Omit size of blocks in protocol Urn randomisation – use instead of blocks? Urn randomisation reduces selection bias, and protects against high imbalance better than simple randomisation with small samples

Summary Optimal biased coin – an alternative to minimisation but difficult to program – rarely used Adaptive methods more complex to program and difficult to implement Stratification factors – No evidence base for stratifying by many factors Always stratify by centre? Probably So fixed randomisation methods generally preferable Treatment allocation methods in clinical trials: a review. Leslie A. Kalish and Colin B. Begg. Stats in Med, Vol.4, 129-144 (1985)

Remember Primary goal of randomisation is to guarantee independence between treatment assignment and outcome Any other goals are secondary (covariate balance, equal group sizes,…)

TCTU randomisation system (TRuST) provides simple, stratified and minimisation methods of randomisation