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: 4.98). 04/1998; 147(8):783-90. DOI: 10.1093/oxfordjournals.aje.a009523
Source: PubMed

ABSTRACT 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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    • "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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    ABSTRACT: Background. To test the hypothesis that cognitive impairment in older adults is associated with all-cause mortality risk and the risk increases when the degree of cognitive impairment augments; and then, if this association is confirmed, to report the population-attributable fraction (PAF) of mortality due to cognitive impairment. Method. A representative random community sample of individuals aged over 55 was interviewed, and 4557 subjects remaining alive at the end of the first year of follow-up were included in the analysis. Instruments used in the assessment included the Mini-Mental Status Examination (MMSE), the History and Aetiology Schedule (HAS) and the Geriatric Mental State (GMS)-AGECAT. For the standardised degree of cognitive impairment Perneczky et al's MMSE criteria were applied. Mortality information was obtained from the official population registry. Multivariate Cox proportional hazard models were used to test the association between MMSE degrees of cognitive impairment and mortality risk. We also estimated the PAF of mortality due to specific MMSE stages. Results. Cognitive impairment was associated with mortality risk, the risk increasing in parallel with the degree of cognitive impairment (Hazard ratio, HR: 1.18 in the 'mild' degree of impairment; HR: 1.29 in the 'moderate' degree; and HR: 2.08 in the 'severe' degree). The PAF of mortality due to severe cognitive impairment was 3.49%. Conclusions. A gradient of increased mortality-risk associated with severity of cognitive impairment was observed. The results support the claim that routine assessment of cognitive function in older adults should be considered in clinical practice.
    Epidemiology and Psychiatric Sciences 06/2014; DOI:10.1017/S2045796014000390 · 3.36 Impact Factor
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    • "For these computations, we approximated the bone marrow dose distribution by using percentiles. Confidence limits for PAR were based on the substitution method (Daly 1998). "
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    ABSTRACT: BACKGROUND: Risks of most types of leukemia from exposure to acute high doses of ionizing radiation are well known, but risks associated with protracted exposures, as well as associations between radiation and chronic lymphocytic leukemia (CLL), are not clear. OBJECTIVES: We estimated relative risks of CLL and non-CLL from protracted exposures to low-dose ionizing radiation. METHODS: A nested case-control study was conducted in a cohort of 110,645 Ukrainian cleanup workers of the 1986 Chornobyl nuclear power plant accident. Cases of incident leukemia diagnosed in 1986-2006 were confirmed by a panel of expert hematologists/hematopathologists. Controls were matched to cases on place of residence and year of birth. We estimated individual bone marrow radiation doses by the Realistic Analytical Dose Reconstruction with Uncertainty Estimation (RADRUE) method. We then used a conditional logistic regression model to estimate excess relative risk of leukemia per gray (ERR/Gy) of radiation dose. RESULTS: We found a significant linear dose response for all leukemia [137 cases, ERR/Gy = 1.26 (95% CI: 0.03, 3.58]. There were nonsignificant positive dose responses for both CLL and non-CLL (ERR/Gy = 0.76 and 1.87, respectively). In our primary analysis excluding 20 cases with direct in-person interviews < 2 years from start of chemotherapy with an anomalous finding of ERR/Gy = -0.47 (95% CI: < -0.47, 1.02), the ERR/Gy for the remaining 117 cases was 2.38 (95% CI: 0.49, 5.87). For CLL, the ERR/Gy was 2.58 (95% CI: 0.02, 8.43), and for non-CLL, ERR/Gy was 2.21 (95% CI: 0.05, 7.61). Altogether, 16% of leukemia cases (18% of CLL, 15% of non-CLL) were attributed to radiation exposure. CONCLUSIONS: Exposure to low doses and to low dose-rates of radiation from post-Chornobyl cleanup work was associated with a significant increase in risk of leukemia, which was statistically consistent with estimates for the Japanese atomic bomb survivors. Based on the primary analysis, we conclude that CLL and non-CLL are both radiosensitive.
    Environmental Health Perspectives 01/2013; 121(1):59-65. DOI:10.1289/ehp.1204996 · 7.03 Impact Factor
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    • "For example, the radiation dose ratio studied by Farrier et al. (2009) is proportional to 1 + p 1 //1 + p 2 where p i , i = 1 2 are independently estimated proportions. The MOVER process is then applied to the confidence intervals for ln1 + p i , i = 1 2. A similar approach may be used to obtain intervals for Levin's attributable risk (Daly, 1998), the population risk difference and population impact number (Bender and Grouven, 2008), and cause-specific standardised mortality ratios adjusted for missing (at random) cause data (Rittgen and Becker, 2000). "
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    ABSTRACT: The Propagating Imprecision (PropImp) approach is a general, easily implemented method for calculating confidence intervals for a function of two or more parameters which are estimated separately. It extends the single-parameter substitution method of Daly (1998) and is very flexible and widely applicable. It does not assume linearity, but normally presupposes monotonicity over the working range of the parameters being estimated. We show several examples, and obtain some coverage, location, and width results for intervals derived by this approach. In many two-parameter applications, the MOVER approach yields a simpler solution, but PropImp is sometimes applicable when MOVER cannot be used.
    Communication in Statistics- Theory and Methods 09/2011; 40(17):3154-3180. DOI:10.1080/03610921003764225 · 0.28 Impact Factor


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