asbio (version 1.0-5)

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 three 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 $\sigma_{\hat{\pi}}$ after multiplication by a constant. Similarly, the second method(adjusted-Agresti-Coull) adjusts the variance of the estimator $\sigma_{\hat{\pi}_{AC}}$, where $\hat{\pi}_{AC}=(y+2)/(n+4)$. The third method (fixed-log) is based on delta derivations of the logged ratio.

Usage

ci.prat.ak(y1, n1, pi2 = NULL, method = "fixed", conf = 0.95, bonf = TRUE)

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" or "fixed" for the adjusted Agresti-Coull, adjusted Wald, and log 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.

Value

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

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 in review). 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. If $\pi_{2i}$s are known, the estimator for $\theta_i$ is: $$\hat{\theta}_{i}=\frac{\hat{\pi}_{1i}}{\pi}_{2i}.$$ ll{ 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 ${{\hat{\pi}}_{AC1i}}=\frac{{{y}_{1}}+2}{{{n}_{1}}+4}$, and ${{\hat{\theta }}_{ACi}}=\frac{{{\hat{\pi}}_{AC1i}}}{{{\pi }_{2i}}}$. Fixed-log ${{\hat{\theta }}_{i}}\times \exp \left( \pm {{z}_{1-\alpha /2}}{{{\hat{\sigma }}}_{F}} \right)$, where $\hat{\sigma }_{^{F}}^{2}=(1-{{\hat{\pi}}_{1i}})/{{\hat{\pi}}_{1i}}{{n}_{1}}.$ 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. (In review) Confidence intervals for ratios of multinomial random variables (selection ratios). Environental and Ecological Statistics.

See Also

ci.prat, ci.p

Examples

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