ci.prat.ak {asbio}  R Documentation 
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 submeter 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 (adjustedWald)
results from the variance of the estimator \hat{\sigma}_{\hat{\pi}}
after multiplication by a constant.
Similarly, the second method(AgrestiCoulladjusted) adjusts the variance of the estimator \hat{\sigma}_{\hat{\pi}_{AC}}
,
where \hat{\pi}_{AC}=(y+2)/(n+4)
. The third method (fixedlog) 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)
Arguments
y1 
The ratio numerator number of successes. A scalar or vector. 
n1 
The ratio numerator number of trials. A scalar or vector of 
pi2 
The denominator proportion. A scalar or vector of 
method 
One of 
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 
bootCI.method 
If 
R 
If 
sigma.t 
If 
r 
The number of ratios to which familywise inferences are being made. Assumed to be 
gamma.hyper 
If 
beta.hyper 
If 
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 CoullAdjusted  {{\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 ). 

Bayesbeta  (\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}) . 

Fixedlog  {{\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}}. 

Noetherfixed  \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}}} . 

Waldadjusted  {{\hat{\theta }}_{i}}\pm {{z}_{1(\alpha /2)}}\sqrt{{{{\hat{\pi }}}_{1i}}(1{{{\hat{\pi }}}_{1i}})/{{n}_{1}}{{{\hat{\pi }}}_{1i}}\pi _{2i}^{2}}. 
Value
Returns a list of class = "ci"
. Default output is a matrix with the point and interval estimate.
Author(s)
Ken Aho
References
Aho, K., and Bowyer, T. 2015. Confidence intervals for ratios of proportions: implications for selection ratios. Methods in Ecology and Evolution 6: 121132.
See Also
Examples
ci.prat.ak(3,4,.5)