ci.prat.ak: Confidence intervals for ratios of proportions when the denominator is known

Description

It is increasingly possible that resource availabilities on a landscape will be known. For instance, in remotely sensed imagery with sub-meter resolution, the areal coverage of resources can be quantified to a high degree of precision, at even large spatial scales. Included in this function are six methods for computation of confidence intervals for a true ratio of proportions when the denominator proportion is known. The first (adjusted-Wald) results from the variance of the estimator $\hat{\sigma}_{\hat{\pi}}$ after multiplication by a constant. Similarly, the second method(Agresti-Coull-adjusted) adjusts the variance of the estimator $\hat{\sigma}_{\hat{\pi}_{AC}}$, where $\hat{\pi}_{AC}=(y+2)/(n+4)$. The third method (fixed-log) is based on delta derivations of the logged ratio. The fourth method is Bayesian and based on the beta posterior distribution derived from a binomial likelhood function and a beta prior distribution. The fifth procedure is an older method based on Noether (1959). Sixth, bootstrapping methods can also be implemented.

Usage

ci.prat.ak(y1, n1, pi2 = NULL, method = "ac", conf = 0.95, bonf = FALSE, 
bootCI.method = "perc", R = 1000, sigma.t = NULL, r = length(y1), gamma.hyper = 1, 
beta.hyper = 1)

Value

Returns a list of class = "ci". Default output is a matrix with the point and interval estimate.

Arguments

y1: The ratio numerator number of successes. A scalar or vector.
n1: The ratio numerator number of trials. A scalar or vector of length(y1)
pi2: The denominator proportion. A scalar or vector of length(y1)
method: One of "ac", "wald", "noether-fixed", "boot", "fixed-log" or "bayes" for the Agresti-Coull-adjusted, adjusted Wald, noether-fixed, bootstrapping, fixed-log and Bayes-beta, methods, respectively. Partial distinct names can be used.
conf: The level of confidence, i.e. 1 - P(type I error).
bonf: Logical, indicating whether or not Bonferroni corrections should be applied for simultaneous inference if y1, y2, n1 and n2 are vectors.
bootCI.method: If method = "boot" the type of bootstrap confidence interval to calculate. One of "norm", "basic", "perc", "BCa", or "student". See ci.boot for more information.
R: If method = "boot" the number of bootstrap samples to take. See ci.boot for more information.
sigma.t: If method = "boot" and bootCI.methd = "student" a vector of standard errors in association with studentized intervals. See ci.boot for more information.
r: The number of ratios to which family-wise inferences are being made. Assumed to be length(y1).
gamma.hyper: If method = "bayes". A scalar or vector. Value(s) for the first hyperparameter for the beta prior distribution.
beta.hyper: If method = "bayes". A scalar or vector. Value(s) for the second hyperparameter for the beta prior distribution.

Author

Ken Aho

Details

Koopman et al. (1984) suggested methods for handling extreme cases of $y_1$, $n_1$, $y_2$, and $n_2$ (see below). These are applied through exception handling here (see Aho and Bowyer 2015).

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. 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. If $\pi_{2i}$s are known, the estimator for $\theta_i$ is:

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

The function ci.prat.ak assumes that selection ratios are being specified (although other applications are certainly possible). Therefore it assume that $y_{1i}$ must be greater than 0 if $\pi_{2i} = 1$, and assumes that $y_{1i}$ must = 0 if $\pi_{2i} = 0$. Violation of these conditions will produce a warning message.

Method	Algorithm

Agresti Coull-Adjusted	${{\hat{\theta}}_{ACi}}\pm {{z}_{1-(\alpha /2)}}\sqrt{{{{\hat{\pi }}}_{AC1i}}(1-{{{\hat{\pi }}}_{AC1i}})/({{n}_{1}}+4){{{\hat{\pi }}}_{AC1i}}\pi _{2i}^{2}}$,
	where ${{\hat{\pi}}_{AC1i}}=\frac{{{y}_{1}}+z^2/2}{{{n}_{1}}+z^2}$, and ${{\hat{\theta }}_{ACi}}=\frac{{{\hat{\pi}}_{AC1i}}}{{{\pi }_{2i}}}$,
	where $z$ is the standard normal inverse cdf at probability $1 - \alpha/2$ ($\approx 2$ when $\alpha= 0.05$).

Bayes-beta	$(\frac{X_{\alpha/2}}{\pi_{2i}}$ , $\frac{X_{1-(\alpha/2)}}{\pi_{2i}})$,
	where $X \sim BETA(y_{1i} + \gamma_{i}, n_1 + \beta - y_{1i})$.

Fixed-log	${{\hat{\theta }}_{i}}\times \exp \left( \pm {{z}_{1-\alpha /2}}{{{\hat{\sigma }}}_{F}} \right)$,
	where $\hat{\sigma}_{^{F}}^{2}=(1-{{\hat{\pi}}_{1i}})/{{\hat{\pi}}_{1i}}{{n}_{1}}.$

Noether-fixed	$\frac{{{{\hat{\pi }}}_{1i}}/{{\pi }_{2}}}{1+z_{1-(\alpha /2)}^{2}}1+\frac{z_{1-(\alpha /2)}^{2}}{2{{y}_{1i}}}\pm z_{1-(\alpha /2)}^{2}\sqrt{\hat{\sigma}_{NF}^{2}+\frac{z_{1-(\alpha /2)}^{2}}{4y_{1i}^{2}}}$,
	where $\hat{\sigma }_{NF}^{2}=\frac{1-{{{\hat{\pi }}}_{1i}}}{{{n}_{1}}{{{\hat{\pi }}}_{1i}}}$.

Wald-adjusted	${{\hat{\theta }}_{i}}\pm {{z}_{1-(\alpha /2)}}\sqrt{{{{\hat{\pi }}}_{1i}}(1-{{{\hat{\pi }}}_{1i}})/{{n}_{1}}{{{\hat{\pi }}}_{1i}}\pi _{2i}^{2}}.$

References

Aho, K., and Bowyer, T. 2015. Confidence intervals for ratios of proportions: implications for selection ratios. Methods in Ecology and Evolution 6: 121-132.

Examples

Run this code

ci.prat.ak(3,4,.5)

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Method	Algorithm

Agresti Coull-Adjusted	\({{\hat{\theta}}_{ACi}}\pm {{z}_{1-(\alpha /2)}}\sqrt{{{{\hat{\pi }}}_{AC1i}}(1-{{{\hat{\pi }}}_{AC1i}})/({{n}_{1}}+4){{{\hat{\pi }}}_{AC1i}}\pi _{2i}^{2}}\),
	where \({{\hat{\pi}}_{AC1i}}=\frac{{{y}_{1}}+z^2/2}{{{n}_{1}}+z^2}\), and \({{\hat{\theta }}_{ACi}}=\frac{{{\hat{\pi}}_{AC1i}}}{{{\pi }_{2i}}}\),
	where \(z\) is the standard normal inverse cdf at probability \(1 - \alpha/2\) (\(\approx 2\) when \(\alpha= 0.05\)).

Bayes-beta	\((\frac{X_{\alpha/2}}{\pi_{2i}}\) , \(\frac{X_{1-(\alpha/2)}}{\pi_{2i}})\),
	where \(X \sim BETA(y_{1i} + \gamma_{i}, n_1 + \beta - y_{1i})\).

Fixed-log	\({{\hat{\theta }}_{i}}\times \exp \left( \pm {{z}_{1-\alpha /2}}{{{\hat{\sigma }}}_{F}} \right)\),
	where \(\hat{\sigma}_{^{F}}^{2}=(1-{{\hat{\pi}}_{1i}})/{{\hat{\pi}}_{1i}}{{n}_{1}}.\)

Noether-fixed	\(\frac{{{{\hat{\pi }}}_{1i}}/{{\pi }_{2}}}{1+z_{1-(\alpha /2)}^{2}}1+\frac{z_{1-(\alpha /2)}^{2}}{2{{y}_{1i}}}\pm z_{1-(\alpha /2)}^{2}\sqrt{\hat{\sigma}_{NF}^{2}+\frac{z_{1-(\alpha /2)}^{2}}{4y_{1i}^{2}}}\),
	where \(\hat{\sigma }_{NF}^{2}=\frac{1-{{{\hat{\pi }}}_{1i}}}{{{n}_{1}}{{{\hat{\pi }}}_{1i}}}\).

Wald-adjusted	\({{\hat{\theta }}_{i}}\pm {{z}_{1-(\alpha /2)}}\sqrt{{{{\hat{\pi }}}_{1i}}(1-{{{\hat{\pi }}}_{1i}})/{{n}_{1}}{{{\hat{\pi }}}_{1i}}\pi _{2i}^{2}}.\)