BinomRatioCI: Confidence Intervals for the Ratio of Binomial Proportions

Description

A number of methods have been develeloped for obtaining confidence intervals for the ratio of two binomial proportions. These include the Wald/Katz-log method (Katz et al. 1978), adjusted-log (Walter 1975, Pettigrew et al. 1986), Koopman asymptotic score (Koopman 1984), Inverse hyperbolic sine transformation (Newman 2001), the Bailey method (Bailey (1987), and the Noether (1957) procedure. Koopman results are found iteratively for most intervals using root finding.

Usage

BinomRatioCI(x1, n1, x2, n2, conf.level = 0.95, 
             sides = c("two.sided", "left", "right"), 
             method = c("katz.log", "adj.log", "bailey", "koopman", "noether", 
                        "sinh-1", "boot"),
             tol = .Machine$double.eps^0.25, R = 1000)

Value

A matrix with 3 columns containing the estimate, the lower and the upper confidence intervall.

Arguments

x1: number of successes for the ratio numerator.
n1: number of trials for the ratio numerator.
x2: number of successes for the ratio denominator.
n2: number of successes for the ratio denominator.
conf.level: confidence level, defaults to 0.95.
sides: a character string specifying the side of the confidence interval, must be one of "two.sided" (default), "left" or "right". You can specify just the initial letter. "left" would be analogue to a hypothesis of "greater" in a t.test.
method: confidence interval method, one of "katz.log" (default), "adj.log", "bailey", "boot", "koopman", "noether" or "sinh-1". Can be abbreviated.
tol: The desired accuracy (convergence tolerance) for the iterative root finding procedure when finding Koopman intevals. The default is taken to be the smallest positive floating-point number of the workstation implementing the function, raised to the 0.25 power, and will normally be approximately 0.0001.
R: If method "boot" is chosen, the number of bootstrap iterations.

Author

Ken Aho <kenaho1@gmail.com>, some tweaks Andri Signorell <andri@signorell.net>

Details

All arguments are being recycled.

Let $Y_1$ and $Y_2$ be multinomial random variables with parameters $n_1, \pi_{1i}$, and $n_2, \pi_{2i}$, respectively; where $i = \{1, 2, 3, \dots, r\}$. This encompasses the binomial case in which $r = 1$. We define the true selection ratio for the ith resource of r total resources to be: $$\theta_{i}=\frac{\pi _{1i}}{\pi _{2i}}$$

where $\pi_{1i}$ and $\pi_{2i}$ represent the proportional use and availability of the ith resource, respectively. Note that if $r = 1$ the selection ratio becomes relative risk. The maximum likelihood estimators for $\pi_{1i}$ and $\pi_{2i}$ are the sample proportions:

$${{\hat{\pi }}_{1i}}=\frac{{{y}_{1i}}}{{{n}_{1}}},$$ and $${{\hat{\pi }}_{2i}}=\frac{{{y}_{2i}}}{{{n}_{2}}}$$

where $y_{1i}$ and $y_{2i}$ are the observed counts for use and availability for the ith resource. The estimator for $\theta_i$ is:

$$\hat{\theta}_{i}=\frac{\hat{\pi}_{1i}}{\hat{\pi }_{2i}}.$$

Method	Algorithm

Katz-log	$\hat\theta_i\times$ exp$(\pm z_1-\alpha/2\hat{\sigma}_W)$,
	where $\hat\sigma_W^2=\frac{(1-\hat{\pi} _{1i})}{\hat{\pi}_{1i}n_1}+\frac{(1-\hat{\pi}_{2i})}{\hat{\pi}_{2i}n_2}$.

Adjusted-log	$\hat{\theta}_{Ai}\times$ exp$(\pm z_1-\alpha /2\hat{\sigma}_A)$,
	where $\hat{\theta}_{Ai}=\frac{y_{1i}+0.5/n_1+0.5}{y_{2i}+0.5/n_2+0.5}$,
	$\hat{\sigma}_A^2=\frac{1}{y_1+0.5}-\frac{1}{n_1+0.5}+\frac{1}{y_2+0.5}-\frac{1}{n_2+0.5}$.

Bailey	$\hat{\theta} _i\left[\frac{1\pm z_1-\left( \alpha /2 \right)\left( \hat{\pi}_{1i}'/y_{1i}+\hat{\pi}_{2i}'/y_{2i}-z_1-\left(\alpha/2 \right)^2\hat{\pi} _{1i}'\hat{\pi}_{2i}'/9y_{1i}y_{2i} \right)^{1/2}/3}{1-z_{1-\left(\alpha/2 \right)^2}\hat{\pi} _{2i}'/9y_{2i}} \right]^3$,
	where $\hat{\pi_{1i}}'$ = 1 - $\hat{\pi}_{1i}$, and $\hat{\pi}_{2i}'$ = 1 - $\hat{\pi}_{2i}$.

Inv. hyperbolic sine	$\ln({{\hat{\theta }}_{i}})\pm \left[ 2sin{{h}^{-1}}\left( \frac{{{z}_{(1-\alpha /2)}}}{2}\sqrt{\frac{1}{{{y}_{1i}}}-\frac{1}{{{n}_{1}}}+\frac{1}{{{y}_{2i}}}-\frac{1}{{{n}_{2}}}} \right) \right]$,

Koopman	Find $X^2(\theta_0)$ = $\chi _1^2(1 - \alpha)$, where
	${{\tilde{\pi }}_{1i}}=\frac{{{\theta }_{0}}({{n}_{1}}+{{y}_{2i}})+{{y}_{1i}}+{{n}_{2}}-{{[{{\{{{\theta }_{0}}({{n}_{1}}+{{y}_{2i}})+{{y}_{1i}}+ {{n}_{2}}\}}^{2}}-4{{\theta }_{0}}({{n}_{1}}+{{n}_{2}})({{y}_{1i}}+{{y}_{2i}})]}^{0.5}}}{2({{n}_{1}}+{{n}_{2}})}$,
	${{\tilde{\pi }}_{2i}}=\frac{{{{\tilde{\pi }}}_{1i}}}{{{\theta }_{0}}}, and {{X}^{2}}({{\theta}_{0}})=\frac{{{\left( {{y}_{1i}}-{{n}_{1}}{{{\tilde{\pi }}}_{1i}} \right)}^{2}}} {{{n}_{1}}{{{\tilde{\pi }}}_{1i}}(1-{{{\tilde{\pi }}}_{1i}})}\left\{ 1+\frac{{{n}_{1}}({{\theta}_{0}}-{{{\tilde{\pi }}}_{1i}})}{{{n}_{2}}(1-{\tilde{\pi}_{1i}})} \right\}$.

Noether	$\hat{\theta}_i\pm z_1-\alpha/2\hat{\sigma}_N$,
	where $\hat{\sigma }_{N}^{2}=\hat{\theta }_{i}^{2}\left( \frac{1}{{{y}_{1i}}}-\frac{1}{{{n}_{1}}}+\frac{1}{{{y}_{2i}}}-\frac{1}{{{n}_{2}}} \right)$.

Exception handling strategies are generally necessary in the cases $x_1$ = 0, $n_1$ = $x_1$, $x_2$ = 0, and $n_2$ = $x_2$ (see Aho and Bowyer, in review).

The bootstrap method currently employs percentile confidence intervals.

References

Agresti, A., Min, Y. (2001) On small-sample confidence intervals for parameters in discrete distributions. Biometrics 57: 963-97.

Aho, K., and Bowyer, T. (In review) Confidence intervals for ratios of multinomial proportions: implications for selection ratios. Methods in Ecology and Evolution.

Bailey, B.J.R. (1987) Confidence limits to the risk ratio. Biometrics 43(1): 201-205.

Katz, D., Baptista, J., Azen, S. P., and Pike, M. C. (1978) Obtaining confidence intervals for the risk ratio in cohort studies. Biometrics 34: 469-474

Koopman, P. A. R. (1984) Confidence intervals for the ratio of two binomial proportions. Biometrics 40:513-517.

Manly, B. F., McDonald, L. L., Thomas, D. L., McDonald, T. L. and Erickson, W.P. (2002) Resource Selection by Animals: Statistical Design and Analysis for Field Studies. 2nd edn. Kluwer, New York, NY

Newcombe, R. G. (2001) Logit confidence intervals and the inverse sinh transformation. The American Statistician 55: 200-202.

Pettigrew H. M., Gart, J. J., Thomas, D. G. (1986) The bias and higher cumulants of the logarithm of a binomial variate. Biometrika 73(2): 425-435.

Walter, S. D. (1975) The distribution of Levins measure of attributable risk. Biometrika 62(2): 371-374.

Examples

Run this code

# From Koopman (1984)

BinomRatioCI(x1 = 36, n1 = 40, x2 = 16, n2 = 80, method = "katz")
BinomRatioCI(x1 = 36, n1 = 40, x2 = 16, n2 = 80, method = "koop")

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Method	Algorithm

Katz-log	\(\hat\theta_i\times\) exp\((\pm z_1-\alpha/2\hat{\sigma}_W)\),
	where \(\hat\sigma_W^2=\frac{(1-\hat{\pi} _{1i})}{\hat{\pi}_{1i}n_1}+\frac{(1-\hat{\pi}_{2i})}{\hat{\pi}_{2i}n_2}\).

Adjusted-log	\(\hat{\theta}_{Ai}\times\) exp\((\pm z_1-\alpha /2\hat{\sigma}_A)\),
	where \(\hat{\theta}_{Ai}=\frac{y_{1i}+0.5/n_1+0.5}{y_{2i}+0.5/n_2+0.5}\),
	\(\hat{\sigma}_A^2=\frac{1}{y_1+0.5}-\frac{1}{n_1+0.5}+\frac{1}{y_2+0.5}-\frac{1}{n_2+0.5}\).

Bailey	\(\hat{\theta} _i\left[\frac{1\pm z_1-\left( \alpha /2 \right)\left( \hat{\pi}_{1i}'/y_{1i}+\hat{\pi}_{2i}'/y_{2i}-z_1-\left(\alpha/2 \right)^2\hat{\pi} _{1i}'\hat{\pi}_{2i}'/9y_{1i}y_{2i} \right)^{1/2}/3}{1-z_{1-\left(\alpha/2 \right)^2}\hat{\pi} _{2i}'/9y_{2i}} \right]^3\),
	where \(\hat{\pi_{1i}}'\) = 1 - \(\hat{\pi}_{1i}\), and \(\hat{\pi}_{2i}'\) = 1 - \(\hat{\pi}_{2i}\).

Inv. hyperbolic sine	\(\ln({{\hat{\theta }}_{i}})\pm \left[ 2sin{{h}^{-1}}\left( \frac{{{z}_{(1-\alpha /2)}}}{2}\sqrt{\frac{1}{{{y}_{1i}}}-\frac{1}{{{n}_{1}}}+\frac{1}{{{y}_{2i}}}-\frac{1}{{{n}_{2}}}} \right) \right]\),

Koopman	Find \(X^2(\theta_0)\) = \(\chi _1^2(1 - \alpha)\), where
	\({{\tilde{\pi }}_{1i}}=\frac{{{\theta }_{0}}({{n}_{1}}+{{y}_{2i}})+{{y}_{1i}}+{{n}_{2}}-{{[{{\{{{\theta }_{0}}({{n}_{1}}+{{y}_{2i}})+{{y}_{1i}}+ {{n}_{2}}\}}^{2}}-4{{\theta }_{0}}({{n}_{1}}+{{n}_{2}})({{y}_{1i}}+{{y}_{2i}})]}^{0.5}}}{2({{n}_{1}}+{{n}_{2}})}\),
	\({{\tilde{\pi }}_{2i}}=\frac{{{{\tilde{\pi }}}_{1i}}}{{{\theta }_{0}}}, and {{X}^{2}}({{\theta}_{0}})=\frac{{{\left( {{y}_{1i}}-{{n}_{1}}{{{\tilde{\pi }}}_{1i}} \right)}^{2}}} {{{n}_{1}}{{{\tilde{\pi }}}_{1i}}(1-{{{\tilde{\pi }}}_{1i}})}\left\{ 1+\frac{{{n}_{1}}({{\theta}_{0}}-{{{\tilde{\pi }}}_{1i}})}{{{n}_{2}}(1-{\tilde{\pi}_{1i}})} \right\}\).

Noether	\(\hat{\theta}_i\pm z_1-\alpha/2\hat{\sigma}_N\),
	where \(\hat{\sigma }_{N}^{2}=\hat{\theta }_{i}^{2}\left( \frac{1}{{{y}_{1i}}}-\frac{1}{{{n}_{1}}}+\frac{1}{{{y}_{2i}}}-\frac{1}{{{n}_{2}}} \right)\).