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Statistics Notes Treatment allocation by minimisation Douglas G Altman, J Martin Bland In almost all controlled trials treatments are allocated by randomisation. Blocking and stratification can be used to ensure balance between groups in size and patient characteristics. 1 But stratified randomisation using several variables is not effective in small trials. The only widely acceptable alternative approach is minimisation, 2 3 a method of ensuring excellent balance between groups for several prognostic factors, even in small samples. With minimisation the treatment allocated to the next participant enrolled in the trial depends (wholly or partly) on the characteristics of those participants already enrolled. The aim is that each allocation should minimise the imbalance across multiple factors. Table 1 shows some baseline characteristics in a controlled trial comparing two types of counselling in relation to dietary intake. 4 Minimisation was used for the four variables shown, and the two groups were clearly very similar in all of these variables. Such good balance for important prognostic variables helps the credibility of the comparisons. How is it achieved? Minimisation is based on a different principle from randomisation. The first participant is allocated a treatment at random. For each subsequent participant we determine which treatment would lead to better balance between the groups in the variables of interest. The dietary behaviour trial used minimisation based on the four variables in table 1. Suppose that after 40 patients had entered this trial the numbers in each subgroup in each treatment group were as shown in table 2. (Note that two or more categories need to be constructed for continuous variables.) The next enrolled participant is a black woman aged 52, who is a non-smoker. If we were to allocate her to behavioural counselling, the imbalance would be increased in sex distribution (12+1 v 11 women), in age (7+1 v 5 aged > 50), and in smoking (14+1 v 12 non-smoking) and decreased in ethnicity (4+1 v 5 black). We formalise this by summing over the four variables the numbers of participants with the same characteristics as this new recruit already in the trial: Behavioural: 12 (sex) +7 (age) +4 (ethnicity) +14 (smoking) = 37 Nutrition: 11+5+5+12=33 Imbalance is minimised by allocating this person to the group with the smaller total (or at random if the totals are the same). Allocation to behavioural counsel- Table 1 Baseline characteristics in two groups 4 Behavioural counselling Nutrition counselling Women 82 84 Mean (SD) age (years) Ethnicity: 43.3 (13.8) 43.2 (14.0) White 94 96 Black 37 32 Asian 3 5 Current smokers 47 44 BMJ VOLUME 330 9 APRIL 2005 bmj.com Downloaded from bmj.com on 28 February 2008 Table 2 Hypothetical distribution of baseline characteristics after 40 patients had been enrolled in the trial Behavioural counselling (n=20) Nutrition counselling (n=20) Women 12 11 Age >50 Ethnicity: 7 5 White 15 15 Black 4 5 Asian 1 0 Current smokers 6 8 ling would increase the imbalance; allocation to nutrition would decrease it. At this point there are two options. The chosen treatment could simply be taken as the one with the lower score; or we could introduce a random element. We use weighted randomisation so that there is a high chance (eg 80%) of each participant getting the treatment that minimises the imbalance. The use of a random element will slightly worsen the overall imbalance between the groups, but balance will be much better for the chosen variables than with simple randomisation. A random element also makes the allocation more unpredictable, although minimisation is a secure allocation system when used by an independent person. After the treatment is determined for the current participant the numbers in each group are updated and the process repeated for each subsequent participant. If at any time the totals for the two groups are the same, then the choice should be made using simple randomisation. The method extends to trials of more than two treatments. Minimisation is a valid alternative to ordinary randomisation, 2 3 5 and has the advantage, especially in small trials, that there will be only minor differences between groups in those variables used in the allocation process. Such balance is especially desirable where there are strong prognostic factors and modest treatment effects, such as oncology. Minimisation is best performed with the aid of software—for example, minim, a free program. 6 Its use makes trialists think about prognostic factors at the outset and helps ensure adherence to the protocol as a trial progresses. 7 1 Altman DG, Bland JM. How to randomise. BMJ 1999;319:703-4. 2 Treasure T, MacRae KD. Minimisation: the platinum standard for trials. BMJ 1998;317:362-3. 3 Scott NW, McPherson GC, Ramsay CR, Campbell MK. The method of minimization for allocation to clinical trials: a review. Control Clin Trials 2002;23:662-74. 4 Steptoe A, Perkins-Porras L, McKay C, Rink E, Hilton S, Cappuccio FP. Behavioural counselling to increase consumption of fruit and vegetables in low income adults: randomised trial. BMJ 2003;326:855-8. 5 Buyse M, McEntegart D. Achieving balance in clinical trials: an unbalanced view from the European regulators. Applied Clin Trials 2004;13:36-40. 6 Evans S, Royston P, Day S. Minim: allocation by minimisation in clinical trials. http://www-users.york.ac.uk/zmb55/guide/minim.htm. (accessed 24 October 2004). 7 Day S. Commentary: Treatment allocation by the method of minimisation. BMJ 1999;319:947-8. Education and debate Cancer Research UK Medical Statistics Group, Centre for Statistics in Medicine, Oxford OX3 7LF Douglas G Altman professor of statistics in medicine Department of Health Sciences, University of York, York YO10 5DD J Martin Bland professor of health statistics Correspondence to: D Altman doug.altman@ cancer.org.uk BMJ 2005;330:843 843
6 The Cardiac Arrhythmia Suppression Trial I. Preliminary Report. Effect of encainide and flecainide on mortality in a randomized trial of arrhythmia suppression after myocardial infarction. N Engl J Med 1989;321:406-12. 7 Dickert N, Grady C. What’s the price of a research subject? Approaches to payment for research participation. N Engl J Med 1999;341:198-203. 8 Grady C. Money for research participation: does it jeopardize informed consent? Am J Bioethics 2001;1:40-4. 9 Macklin R. “Due” and “undue” inducements: On paying money to research subjects. IRB: a review of human subjects research 1981;3:1-6. 10 McGee G. Subject to payment? JAMA 1997;278:199-200. 11 McNeil P. Paying people to participate in research. Bioethics 1997;11:390-6. 12 Wilkenson M, Moore A. Inducement in research. Bioethics 1997;11: 373-89. 13 Halpern SD, Karlawish JHT, Casarett D, Berlin JA, Asch DA. Empirical assessment of whether moderate payments are undue or unjust inducements for participation in clinical trials. Arch Intern Med 2004;164:801-3. 14 Bentley JP, Thacker PG. The influence of risk and monetary payment on the research participation decision making process. J Med Ethics 2004;30:293-8. 15 Viens AM. Socio-economic status and inducement to participate. Am J Bioethics 2001;1. Statistics Notes Standard deviations and standard errors Douglas G Altman, J Martin Bland The terms “standard error” and “standard deviation” are often confused. 1 The contrast between these two terms reflects the important distinction between data description and inference, one that all researchers should appreciate. The standard deviation (often SD) is a measure of variability. When we calculate the standard deviation of a sample, we are using it as an estimate of the variability of the population from which the sample was drawn. For data with a normal distribution, 2 about 95% of individuals will have values within 2 standard deviations of the mean, the other 5% being equally scattered above and below these limits. Contrary to popular misconception, the standard deviation is a valid measure of variability regardless of the distribution. About 95% of observations of any distribution usually fall within the 2 standard deviation limits, though those outside may all be at one end. We may choose a different summary statistic, however, when data have a skewed distribution. 3 When we calculate the sample mean we are usually interested not in the mean of this particular sample, but in the mean for individuals of this type—in statistical terms, of the population from which the sample comes. We usually collect data in order to generalise from them and so use the sample mean as an estimate of the mean for the whole population. Now the sample mean will vary from sample to sample; the way this variation occurs is described by the “sampling distribution” of the mean. We can estimate how much sample means will vary from the standard deviation of this sampling distribution, which we call the standard error (SE) of the estimate of the mean. As the standard error is a type of standard deviation, confusion is understandable. Another way of considering the standard error is as a measure of the precision of the sample mean. The standard error of the sample mean depends on both the standard deviation and the sample size, by the simple relation SE = SD/√(sample size). The standard error falls as the sample size increases, as the extent of chance variation is reduced—this idea underlies the sample size calculation for a controlled trial, for BMJ VOLUME 331 15 OCTOBER 2005 bmj.com Downloaded from bmj.com on 28 February 2008 16 Beauchamp TL, Childress JF. Respect for autonomy. Principles of biomedical ethics. 4th ed. New York: Oxford University Press, 1994:120-88. 17 Roberts LW. Evidence-based ethics and informed consent in mental illness research. Arch Gen Psychiatry 2000;57:540-2. 18 Bayer R, Oppenheimer GM. Toward a more democratic medicine: sharing the burden of ignorance. AIDS Doctors: voices from the epidemic. New York: Oxford University Press, 2000:156-69. 19 Coulter A, Rozansky D. Full engagement in health. BMJ 2004;329:1197-8. 20 Joffe S, Manocchia M, Weeks JC, Cleary PD. What do patients value in their hospital care? An empirical perspective on autonomy centred bioethics. J Med Ethics 2003;29:103-8. 21 Heesen C, Kasper J, Segal J, Kopke S, Muhlhauser I. Decisional role preferences, risk knowledge and information interests in patients with multiple sclerosis. Mult Scler 2004;10:643-50. 22 Azoulay E, Pochard F, Chevret S, et al. Half the family members of intensive care unit patients do not want to share in the decision-making process: a study in 78 French intensive care units. Crit Care Med 2004;32:1832-8. 23 Dunn LB, Gordon NE. Improving informed consent and enhancing recruitment for research by understanding economic behavior. JAMA 2005;293:609-12. example. By contrast the standard deviation will not tend to change as we increase the size of our sample. So, if we want to say how widely scattered some measurements are, we use the standard deviation. If we want to indicate the uncertainty around the estimate of the mean measurement, we quote the standard error of the mean. The standard error is most useful as a means of calculating a confidence interval. For a large sample, a 95% confidence interval is obtained as the values 1.96×SE either side of the mean. We will discuss confidence intervals in more detail in a subsequent Statistics Note. The standard error is also used to calculate P values in many circumstances. The principle of a sampling distribution applies to other quantities that we may estimate from a sample, such as a proportion or regression coefficient, and to contrasts between two samples, such as a risk ratio or the difference between two means or proportions. All such quantities have uncertainty due to sampling variation, and for all such estimates a standard error can be calculated to indicate the degree of uncertainty. In many publications a ± sign is used to join the standard deviation (SD) or standard error (SE) to an observed mean—for example, 69.4±9.3 kg. That notation gives no indication whether the second figure is the standard deviation or the standard error (or indeed something else). A review of 88 articles published in 2002 found that 12 (14%) failed to identify which measure of dispersion was reported (and three failed to report any measure of variability). 4 The policy of the BMJ and many other journals is to remove ± signs and request authors to indicate clearly whether the standard deviation or standard error is being quoted. All journals should follow this practice. Competing interests: None declared. 1 Nagele P. Misuse of standard error of the mean (SEM) when reporting variability of a sample. A critical evaluation of four anaesthesia journals. Br J Anaesthesiol 2003;90:514-6. 2 Altman DG, Bland JM. The normal distribution. BMJ 1995;310:298. 3 Altman DG, Bland JM. Quartiles, quintiles, centiles, and other quantiles. BMJ 1994;309:996. 4 Olsen CH. Review of the use of statistics in Infection and Immunity. Infect Immun 2003;71:6689-92. Education and debate Cancer Research UK/NHS Centre for Statistics in Medicine, Wolfson College, Oxford OX2 6UD Douglas G Altman professor of statistics in medicine Department of Health Sciences, University of York, York YO10 5DD J Martin Bland professor of health statistics Correspondence to: Prof Altman doug.altman@ cancer.org.uk BMJ 2005;331:903 903
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