| BACKGROUDPaired binary data is very common in clinical research. According to the data collected, it can form a paired2*2table. Different proper methodology is selected for different research purposes in terms of the data in the paired2*2table. Hypothesis testing and confidence interval estimation methods based on the difference and ratio of two proportions are usually used to compare the effect between the experimental group and the control group for superiority or non-inferiority or equivalence inference. The research about non-inferiority or equivalence test of paired binary data is extensive recently,and a variety of hypothesis testing and confidence interval estimation methods are developed. Because confidence interval estimation method can be used for clinical judgement more intuitionally and accurately, it is highly recommended.OBJECTIVEThe confidence interval methods of the difference and ratio of two correlated proportions are constructed in this study by considering the confidence interval of the correlation coefficient of hybridization of the sets of proportion difference and proportion ratio of the two single rate.This study is aim to simulate and verify the statistical properties of the constructed confidence interval methods of the difference and ratio of two correlated proportions by Monte Carlo methods, and explore their applicability and application requirements for supporting the analysis and evaluation of paired binary data with reasonable statistical methodology.METHODS1. Confidence interval construction of the difference between two correlated proportions. Based on the proposed so-called method of variance of estimates recovery(MOVER) thought, we selected confidence interval methods of single sample rate that have good statistical properties,such as Wilson method, AC method, Jeffreys method, to hybridize and construct confidence intervals of a difference between two correlated proportions by considering the correlation coefficient of two rates. When in hybridizing,8coefficients indicating the correlation of two correlated proportions and3different confidence interval methods of single sample rate are selected to combine to generate24interval methods of a difference between two correlated proportions.2. Confidence interval construction of the ratio of two correlated proportions. Based on the proposed so-called method of variance of estimates recovery (MOVER) thought and Fieller theory, we selected proper confidence interval methods of single sample rate,such as Wilson method, AC method, Jeffreys method, to hybridize and construct confidence intervals of the rate of two correlated proportions by considering the correlation coefficient of two rates. When in hybridizing,8coefficients indicating the correlation of two related proportions and3different confidence interval methods of single sample rate are selected to combine to develop24interval methods of the ratio of two correlated proportions.3. The simulation method and the parameters setting.In order to investigate the statistical properties of different methods,the simulation experiments are conducted with SAS9.2software by Monte Carlo method under different parameter settings.The parameters setting of Confidence intervals of the difference between two correlated proportions:(1) the marginal proportion π÷1of the positive rate in the control group at three different levels, namely,[0.05,0.1],[0.4,0.6],[0.8,0.95];(2)the difference⊿between two correlated proportions at three different levels, namely,0,[0.01,0.05],[0.1,0.2];(3)the two sets of correlation coefficient p from bivariate distribution at four different levels, namely [-0.1,0],0,[0,0.2],[0.4,0.6]. We selected the two cases when the sample size n is small (10≤n≤20) and medium size level (30≤n≤50). In terms of the different combination of the above parameters, we randomly generate1000vectors (n, π+l,⊿, p). Exact coverage probabilities(ECP), expected coverage widths(ECW), mesial non-coverage probabilities MNCP) and distal non-coverage probabilities(DNCP) of various confidence intervals were calculated. Under various conditions of the combination of different kinds of parameter values, we investigated whether the acquired coverage probabilities are closer to the pre-defined standards, whether the interval widths are narrower, and whether the left and right non-coverage area are symmetrical.The parameters setting of the confidence intervals of the ratio of two correlated proportions:(1) the marginal proportion π+l of the positive rate in the control group at three different levels, namely,[0.05,0.1],[0.4,0.6],[0.8,0.95];(2)the ratio0of two correlated proportions at three different levels, namely,0.7,1,1.1;(3)the two sets of correlation coefficient p from bivariate distribution at four different levels, namely0,0.2,0.5,0.9. We selected the two cases when the sample size n is small (10≤≤20) and medium size level (30≤n≤50). In terms of the different combination of the above parameters, we randomly generate1000vectors(n,π+l,θ,ρ). Exact coverage probabilities(ECP), expected coverage widths(ECW), mesial non-coverage probabilities(MNCP) and distal non-coverage probabilities(DNCP) of various confidence intervals were calculated. Under various conditions of the combination of different kinds of parameter values,we investigated whether the coverage probabilities are closer to the pre-assigned confidence level, whether the interval widths are narrower, and whether the left and right non-coverage area are symmetrical.RESULTSConfidence interval estimation of the difference between two correlated proportions:In all conditions of simulation, the coverage of constructed confidence intervals methods based on Wilson method combining Φ coefficient and Tau-c coefficient and Somer’D average coefficient are close to the pre-assigned confidence level;Followed by the coverage of constructed confidence intervals combining Gamma coefficient, Kappa coefficient, Contingency coefficient and improved included angle coefficient, which are generally lower than the pre-assigned confidence level.the coverage of constructed confidence intervals combining included angle coefficient is much lower than the pre-assigned confidence level in most times. In terms of the confidence interval methods based on3confidence interval methods of single sample rate, constructed confidence intervals methods based on Wilson method and AC method perform well in coverage, while confidence intervals methods based on Jeffrey method perform the worst, whose coverage is lower than the pre-assigned confidence level at most time. In terms of the widths of constructed confidence intervals, when π+l is close to0.5,the widths of constructed confidence intervals based on Wilson method is the narrowest, the next are the the widths of confidence intervals based on AC method and Jeffreys method. When π+l is not close to0.5, the widths of confidence intervals based on Jeffreys method is the narrowest, and the next are the widths of confidence intervals based on Wilson method and AC method. No obvious parrern between the different methods from the symmetry of the two-tailed non-coverage is found. Confidence interval estimation of the ratio of two correlated proportions:In all, In terms of the combined coefficients, constructed confidence intervals combining Gamma coefficient, Contingency coefficient, included angle coefficient and improved included angle coefficient are deflated in coverage, whose coverage probability is much lower than the pre-assigned confidence level at some times. In terms of the confidence interval methods based on3confidence intervals of single sample rate, constructed confidence intervals methods based on Jeffreys method perform poor in coverage, whose coverage probability is much lower than the pre-assigned confidence level at most time. The next is the confidence intervals base on AC method, whose coverage probability is much lower than the pre-assigned confidence level too. Relatively speaking, The ECPs of confidence intervals base on Wilson method are much closer to the pre-assigned confidence level. In terms of the widths of constructed confidence intervals, the ECWs of confidence intervals combining included angle coefficient and Gamma coefficient are relatively short, but the ECPs of constructed confidence intervals based on these coefficients cannot guarantee to achieve the nominal level, so this is meaningless. The ECWs of constructed confidence intervals are comparable. Among3confidence intervals of single sample rate, the ECWs of constructed confidence intervals based on Jeffreys method are much shorter than the other, and the next are those of constructed confidence intervals based on Wilson method and AC method. No obvious parrern between the different methods from the symmetry of the two-tailed non-coverage is found.CONCLUTIONSIn conclusion, the Constructed CIs for the difference between two correlated proportions of paired binary data based on Wilson method combining Φ coefficient and Tau-c coefficient and Somer’D average coefficient perform well,whose ECPs can satisfactorily achieve the nominal level, and they have advantage in interval width as well, so they are worth to be recommended. In terms of estimating the CIs for proportion rate of paired binary data, the ECPs of the Constructed CIs based on Wilson method combining Φ coefficient and Tau-c coefficient and Somer’D average coefficient and Kappa coefficient are much closer to the pre-assigned confidence level than those of the others, so we recommend these methods. |
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