Bessel's correction

Abstract: 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 …

Abstract: When output (x ) of a mechanistic model is compared with measurement (y ), it is common practice to calculate the correlation coefficient between x and y, and to regress y on x. There are, however, problems in this approach. The assumption of the regression, that y is linearly related to x, is not guaranteed and is unnecessary for the x-y comparison. The correlation and regression coefficients are not explicitly related to other commonly used statistics [e.g., root mean squared deviation (RMSD)]. We present an approach based on the mean squared deviation (MSD = RMSD 2 ) and show that it is better suited to the x-y comparison than regression. Mean squared deviation is the sum of three components: squared bias (SB), squared difference between standard deviations (SDSD), and lack of correlation weighted by the standard deviations (LCS). To show examples, the MSD-based analysis was applied to simulation vs. measurement comparisons in literature, and the results were compared with those from regression analysis. The analysis of MSD clearly identified the simulation vs. measurement contrasts with larger deviation than others; the correlation-regression approach tended to focus on the contrasts with lower correlation and regression line far from the equality line. It was also shown that results of the MSD-based analysis were easier to interpret than those of regression analysis. This is because the three MSD components are simply additive and all constituents of the MSD components are explicit. This approach will be useful to quantify the deviation of calculated values obtained with this model from measurements.

Abstract: We often need to know the error with which measurements are made—for example, so that we can decide whether the change in a clinical observation represents a real change in a patient's condition. We have discussed previously the within-subject standard deviation as a practical index of measurement error.1 We said that this approach should be used when the measurement error was not related to the magnitude of the measurement and recommended that we plot the subject standard deviation against the subject mean to check this. Table 1 shows some duplicate salivary cotinine measurements taken from a larger study. Figure 1 shows absolute subject difference against subject mean, which is equivalent to a standard deviation versus mean plot when we have only two measurements per subject.1 If we are to use the within-subject standard deviation as an index of measurement error we need the subject standard deviation to be …

Abstract: Large cohort studies of rare outcomes require extensive data collection, often for many relatively uninformative subjects. Sampling schemes have been proposed that oversample certain groups. For example, the case-cohort design of Prentice (1986, Biometrika 73, 1-11) provides an efficient method of analysis of failure time data. However, the variance estimate must explicitly correct for correlated score contributions. A simple robust variance estimator is proposed that allows for more complicated sampling mechanisms. The variance estimate uses a jackknife estimate of the variance of the individual influence function and is shown to be equivalent to a robust variance estimator proposed by Lin and Wei (1989, Journal of the American Statistical Association 84, 1074-1078) for the standard Cox model. Simulation results indicate excellent agreement with corrected asymptotic estimates and appropriate test size. The technique is illustrated with data evaluating the efficacy of mammography screening in reducing breast cancer mortality.