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Randomized and Nonrandomized Studies 4.2 Properties of Randomization 33 30 EXPRESSING THE TREATMENT EFFECT of local recurrence of malignancy after the surgery. Patients were divided into two groups, depending upon the stage of the disease prior to surgery. Since both the risk and outcome variables are categorical, three measures of treatment effect- difference in recurrence rates, relative risk, and odds ratio-may be computed CHAPTER 4 for each stage (see the calculations in Table 3.2). It turns out that the relative risk is nearly the same for stage 1 and stage 2 patients (3.62 vs. 3.8 1). whereas the odds ratio and dif- ference in rates depend on the stage (4.02 vs. 5.92 and 0.10 vs. 0.32). In other words, there is an interaction if the treatment effect is expressed in terms of the latter two measures, but no interaction if it is measured by the relative risk. Randomized and Since the logarithm of the relative risk is equal to the difference of the log rates Nonrandomized Studies (log 19 = log rl - log r2),this is an example where an analysis in the original units (recurrence rates) show an interaction, whereas an analysis in a different scale (log - recurrence rates) does not. Often, however, interactions cannot be re- moved by changing the scale. If in the previous example, stage 1 patients had fewer recurrences with tylectomy than with mastectomy but the opposite had 32 been true for stage 2 patients, there would be no way of avoiding interaction. 4.1 Definition of Randomization 4.2 Propertits of Randomization 32 Figure 3.8 gives another example of nonremovable interaction. 34 Although it is desirable to avoid interaction since a single measure can then 4.3 Further Points on Randomization Reasons for the Use of Nonrandomized Studies 3 5 completely describe the treatment effect, sometimes, as we discussed in Section 4.4 37 3.1, because one measure of treatment effect is more useful than others, tp 4.5 Types of Comparative Studies 3 8 measure should be used even if it does result in interaction. 4.5.1 Cohort Studies 4.5.2 Case-Control Studies 39 4.5.3 Cross-Sectional Studies 42 Our Attitude toward Nonrandomized Studies 43 REFERENCES 4.6 Appendix 4A The Odds Ratio and the Relative Risk in Case-Control Studies 43 Atkins, H., Mayward, J. L., Klugman, D. J.. and Wayte, A. B. (1972), Treatment of Early Breast References 44 Cancer: A Report after 10 Years of a Clinical Trial, British Medical Journal, 423-429. Berkson, J. (1958), Smoking and Lung Cancer: Some Observations on Two Recent Reports, Journal of the American Statistical Association, 53, 28-38. Colton, T. (1974). Siatistlcs in Medicine. Boston: Little, Brown. T Daniel, D., and Wood, F. S. (1971), Fitting Equatrons to Data, New York: Wiley. Estimating a treatment effect requires the construction of a standard of com- Fleiss, J S. (1973). Statistical Methods for Rates and Proportions. New York: Wiley. parison. As we have seen in Chapter 1, this involves a comparison group which Manushek, E. A,, & Jackson. J. E. (1977), Statistical Methods forSocial Scientists, New York: Academic Press. does not receive the treatment of interest. In this chapter we will explore several Hill, A. B. (1965). The Environment and Disease: Associat~onor Causation? Proceedings of the ways of establishing such a comparison group, emphasizing the difference be- Royal Society of Medicine. 58,295-300. tween randomization and other methods. It will be seen that a randomized al- MacMahon, B., and Pugh. T F. (1970). Epidemiology. Principles and Methods, Boston: Little, location of subjects to a treatment and control group generally ensures that the Brown. latter is an adequate standard of comparison for the former. Mosteller, F., and Tukey, J. W. ( 1977). Data Analys~sand Regression. A Second Course in Sta- We will start by defining randomization and discussing the properties that tistics, Reading, MA: Addison-Wesley. make this method particularly attractive. We will then give reasons for doing Sheps, M. C. (1959),an Examination of Some Methods of Comparing Rates or Proportions. Bio- nonrandomized studies, and distinguish the different types of studies involving mevics, 15, 87-97. a comparison group. For simplicity of presentation, this chapter will be confined Tufte, E. R. (1974), Data Analysis for Politics and Policy, Englewood Cliffs, N.J.: Prentice- mainly to studies with a dichotomous risk factor. Hall. 31 32 RANDOMIZED AND NONRANDOMIZED STUDIES 4.2 PROPERTIES OF RANDOMIZATION 33 4.1 DEFINITION OF RANDOMIZATION teristics in each group and thereby facilitates causal inference. If the number of subjects in a randomized study is large, it is unlikely that the two groups differ Randomization is a method whereby subjects are allocated to one of the two with respect to any characteristic that can affect the outcome under study, risk factor groups by a random mechanism which assures that each individual whether or not these characteristics are known to the investigators. To illustrate has an equal chance of being assigned to either group. Tossing a fair coin and this property, we consider a hypothetical study to determine whether.drug X allocating an individual to one group or the other based on-the appearance of is effective, as compared to drug Y, in reducing blood pressure for patients with "heads" or the use of a table of random numbers are examples of such processes. hypertension. The investigators identify a number of hypertensive patients. They For instance, in studying the efficacy of a new medication relative to a standard randomize the patients into a drug X group and a drug Y group. What has been one, the names of the patients could be entered sequentially, line by line, in a gained by randomization here? book, and a number from a table of random numbers could be assigned to each By employing randomization, the investigators assure themselves that the line. The patients assigned even numbers would be allocated to the new medi- groups are likely to have similar distributions of variables which can affect blood cation, those with odd numbers to a standard one. A more sophisticated ran- pressure. More precisely, the probability is small that potentially confounding domized design should be used if we require equal numbers of patients in the variables differ in the two grodps by a large amount. If the drug X group sub- two medication groups. After the randomization process has determined that sequently exhibits a substantially lower average blood pressure than does the a particular subject should be assigned to a particular group, the investigator drug Y group, randomization makes it unlikely that the difference is caused by must have enough control to implement that assignment. There is clearly no way a factor other than the drug. For example, it is unlikely that the drug X group to conduct a randomized study if the investigator must accept the assignment has lower blood pressure because it consists of younger people. of people to treatment or comparison groups as determined by nature or by some 2. Randomization eliminates selection effects. If individuals found eligible institutional process (some examples will be given in Sections 4.4 and 4.5). for a study are randomized into groups, there is no possibility that the investi- The primary virtue of randomization is that with high probability the two gators' initial biases or preferences about which subjects should receive what groups will be similar. Indeed, the only initial systematic difference betweed program could influence the results. For example, in the blood pressure illus- the two groups will be that one received the treatment and the other did not. tration, the investigator, had he or she not randomized, may have tended to give Therefore, if the treatment has no effect, the distribution of the outcome variable the drug X to the more severe hypertensives. Thus a crude comparison of sub- in the two groups would be quite similar. In the next section we provide a more sequent blood pressures between the two groups would not give a fair comparison extensive discussion of the properties of randomization. of the two drugs. Or consider the Lanarkshire experiment carried out in schools Although randomization offers important advantages, the investigator may to study the effect of milk on growth of children. It was criticized by Student sometimes want to consider nonrandom allocations. For example, it is possible (1931) because a loose design allowed the investigators to allocate, perhaps to use systematic processes such as allocating every second subject or all the unconsciously, more milk to the poorer and ill-nourished children than to the subjects with odd birth years to one of the two groups. Such processes may be well-fed children. Although the number of children participating in the study much easier to administer than is randomization, and generally these systematic was large, this failure in the design prevented a clear inference abbut the effect processes will be essentially equivalent to randomization. However, there is al- of milk. ways a risk that the characteristic on which the allocation is done (order of ar- If the individuals are considered sequentially for admission to the study, the rival, birth year) is related to the outcome under study, so that its effect cannot randomization scheme should be kept secret from the investigator. Otherwise, be disentangled from that of the treatment. Haphazard processes, where no a medical researcher who knows that the next patient arriving at the hospital well-defined method is used to form the groups, are even more dangerous since will be assigned by the randomization scheme to drug Y may declare that patient the investigator may allocate, often unconsciously, a particular type of subject ineligible for the study if he or she would have favored drug X for this patient.
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