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The Effectiveness of Adjustment by Subclassification in Removing Bias in Observational Studies

W. G. Cochran
Biometrics
Vol. 24, No. 2 (Jun., 1968), pp. 295-313
DOI: 10.2307/2528036
Stable URL: http://www.jstor.org/stable/2528036
Page Count: 19
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The Effectiveness of Adjustment by Subclassification in Removing Bias in Observational Studies
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Abstract

In some investigations, comparison of the means of a variate y in two study groups may be biased because y is related to a variable x whose distribution differs in the two groups. A frequently used device for trying to remove this bias is adjustment by subclassification. The range of x is divided into c subclasses. Weighted means of the subclass means of y are compared, using the same weights for each study group. The effectiveness of this procedure in removing bias depends on several factors, but for monotonic relations between y and x, an analytical approach suggests that for c = 2, 3, 4, 5, and 6 the percentages of bias removed are roughly 64%, 79%, 86%, 90%, and 92%, respectively. These figures should also serve as a guide when x is an ordered classification (e.g. none, slight, moderate, severe) that can be regarded as a grouping of an underlying continuous variable. The extent to which adjustment reduces the sampling error of the estimated difference between the y means is also examined. An interesting side result is that for x normal, the percentage reduction in the bias of $\bar x_2$-$\bar x_1$ due to adjustment equals the percentage reduction in its variance. Under a simple mathematical model, errors of measurement in x reduce the amount of bias removed to a fraction 1/(1 + h) of its value, where h is the ratio of the variance of the errors of measurement to the variance of the correct measurements. Since ordered classifications are often used because x is difficult to measure, h may be substantial in such cases, though more information is needed on the values of h that are typical in practice.

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