ci.impt {asbio} | R Documentation |

## Confidence interval for the product of two proportions

### Description

Provides one and two-tailed confidence intervals for the true product of two proportions.

### Usage

```
ci.impt(y1, n1, y2 = NULL, n2 = NULL, avail.known = FALSE, pi.2 = NULL,
conf = .95, x100 = TRUE, alternative = "two.sided", bonf = TRUE, wald = FALSE)
```

### Arguments

`y1` |
The number of successes associated with the first proportion. |

`n1` |
The number of trials associated with the first proportion. |

`y2` |
The number of successes associated with the second proportion. Not used if |

`n2` |
The number of trials associated with the first proportion. Not used if |

`avail.known` |
Logical. Are the proportions |

`pi.2` |
Proportions for |

`conf` |
Confidence level, i.e., 1 - |

`x100` |
Logical. If true, estimate is multiplied by 100. |

`alternative` |
One of |

`bonf` |
Logical. If |

`wald` |
Logical. If |

### Details

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,\dots, r`

.
Under delta derivation, the log of the products of `\pi_{1i}`

and `\pi_{2i}`

(or the log of a product of `\pi_{1i}`

and `\pi_{2i}`

and a constant) is asymptotically normal with mean
`log(\pi_{1i} \times \pi_{2i})`

and variance `(1 - \pi_{1i})/\pi_{1i}n_1 + (1 - \pi_{2i})/ \pi_{2i}n_2`

. Thus, an asymptotic `(1 - \alpha)100`

percent confidence interval for `\pi_{1i} \times \pi_{2i}`

is given by:

```
\hat{\pi}_{1i} \times \hat{\pi}_{2i} \times \exp(\pm z_{1-(\alpha/2)}\hat{\sigma}_i)
```

where: `\hat{\sigma}^2_i = \frac{(1 - \hat{\pi}_{1i})}{\hat{\pi}_{1i}n_1} + \frac{(1 - \hat{\pi}_{2i})}{\hat{\pi}_{2i}n_2}`

and `z_{1-(\alpha/2)}`

is the standard normal inverse CDF at probability `1 - \alpha`

.

### Value

Returns a list of `class = "ci"`

. Printed results are the parameter estimate and confidence bounds.

### Note

Method will perform poorly given unbalanced sample sizes.

### Author(s)

Ken Aho

### References

Aho, K., and Bowyer, T. 2015. Confidence intervals for a product of proportions: Implications for importance values. *Ecosphere* 6(11): 1-7.

### See Also

### Examples

```
ci.impt(30,40, 25,40)
```

*asbio*version 1.9-7 Index]