Confidence Limits Made Easy: Interval Estimation Using a Substitution Method

Department of Public Health Medicine and Epidemiology, University College Dublin, Ireland.
American Journal of Epidemiology (Impact Factor: 5.23). 04/1998; 147(8):783-90. DOI: 10.1093/oxfordjournals.aje.a009523
Source: PubMed


The use of confidence intervals has become standard in the presentation of statistical results in medical journals. Calculation of confidence limits can be straightforward using the normal approximation with an estimate of the standard error, and in particular cases exact solutions can be obtained from published tables. However, for a number of commonly used measures in epidemiology and clinical research, formulae either are not available or are so complex that calculation is tedious. The author describes how an approach to confidence interval estimation which has been used in certain specific instances can be generalized to obtain a simple and easily understood method that has wide applicability. The technique is applicable as long as the measure for which a confidence interval is required can be expressed as a monotonic function of a single parameter for which the confidence limits are available. These known confidence limits are substituted into the expression for the measure--giving the required interval. This approach makes fewer distributional assumptions than the use of the normal approximation and can be more accurate. The author illustrates his technique by calculating confidence intervals for Levin's attributable risk, some measures in population genetics, and the "number needed to be treated" in a clinical trial. Hitherto the calculation of confidence intervals for these measures was quite problematic. The substitution method can provide a practical alternative to the use of complex formulae when performing interval estimation, and even in simpler situations it has major advantages.

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    • "In the following sections first the method of substitution will be explained, subsequently a new method of deriving the confidence and prediction interval limits for risk differences from a meta-analyses of odds ratios will be discussed. Converting odds ratios into risk ratios A method which has become known as the method of substitution can be used to convert odds ratios into risk ratios (Daly, 1998; Zhang & Yu, 1998). This method can be used to convert the combined effect size and the limits of both the confidence interval and the prediction interval, using the following formula: í µí±…í µí±–í µí± í µí±˜ í µí±Ÿí µí±Ží µí±¡í µí±–í µí±œ = í µí±‚í µí±‘í µí±‘í µí± í µí±Ÿí µí±Ží µí±¡í µí±–í µí±œ (1 − í µí°´í µí± í µí± í µí±¢í µí±ší µí±’í µí±‘ í µí±í µí±œí µí±›í µí±¡í µí±Ÿí µí±œí µí±™ í µí±Ÿí µí±–í µí± í µí±˜) + (í µí°´í µí± í µí± í µí±¢í µí±ší µí±’í µí±‘ í µí±í µí±œí µí±›í µí±¡í µí±Ÿí µí±œí µí±™ í µí±Ÿí µí±–í µí± í µí±˜ × í µí±‚í µí±‘í µí±‘í µí± í µí±Ÿí µí±Ží µí±¡í µí±–í µí±œ) "
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    ABSTRACT: This paper describes a method to convert meta-analytic results in (log) Odds Ratio to either Risk Ratio or Risk Difference. It has been argued that odds ratios are mathematically superior for meta-analysis, but risk ratios and risk differences are shown to be easier to interpret. Therefore, the proposed method enables the calculation of meta-analytic results in (log) odds ratio and to transform them afterwards in risk ratio and risk difference. This transformation is based on the assumption of equal significance of the results. It is implemented Meta-Essentials: Workbooks for meta-analyses.
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    • "The PAF estimates the proportional amount that risk of death would be reduced if a specific MMSE stage were eliminated of population (Rockhill et al. 1998). To estimate the PAF of death due to specific MMSE stage, the following calculation was performed: [ px(HR − 1)/(1 + px(HR − 1))] × 100 ('p' represents the proportion of subjects who were exposed to the specific MMSE stage and 'HR' represents the hazard ratio of the specific MMSE stage in the multivariate model) (Rockhill et al. 1998; Daly, 1998). All p values were two-tailed and we used bootstrap resampling to compute all CI at the 95% level (95% CI). "
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    Full-text · Article · Jun 2014 · Epidemiology and Psychiatric Sciences
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    • "SRM was interpreted by calculating the probability of change statistic P^, which represents the cumulative normal distribution function of the derived SRM. The P^ statistic denotes the probability the instrument detects a change and ranges from 0.5 (no ability to detect change) to 1 (perfect ability to detect change) [27], 95% CI was estimated using the substitution method [30]. "
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    ABSTRACT: Background Mental well-being now features prominently in UK and international health policy. However, progress has been hampered by lack of valid measures that are responsive to change. The objective of this study was to evaluate the responsiveness of the Warwick Edinburgh Mental Well-being Scale (WEMWBS) at both the individual and group level. Methods Secondary analysis of twelve different interventional studies undertaken in different populations using WEMWBS as an outcome measure. Standardised response mean (SRM), probability of change statistic (P̂) and standard error of measurement (SEM) were used to evaluate whether WEMWBS detected statistically important changes at the group and individual level, respectively. Results Mean change in WEMWBS score ranged from −0.6 to 10.6. SRM ranged from −0.10 (95% CI: -0.35, 0.15) to 1.35 (95% CI: 1.06, 1.64). In 9/12 studies the lower limit of the 95% CI for P̂ was greater than 0.5, denoting responsiveness. SEM ranged from 2.4 to 3.1 units, and at the threshold 2.77 SEM, WEMWBS detected important improvement in at least 12.8% to 45.7% of participants (lower limit of 95% CI>5.0%). Conclusions WEMWBS is responsive to changes occurring in a wide range of mental health interventions undertaken in different populations. It offers a secure base for research and development in this rapidly evolving field. Further research using external criteria of change is warranted.
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