# Compute a test of trend in prevalences based on a likelihood-ratio statistic

### Description

This function takes a series of point estimates and their associated standard errors and
computes the p-value for the test of a monotone decrease in the
population prevalences (in sequence order).
The p-value for a monotone increase is
also reported.
More formally, let the *K* population prevalences in sequence order be *p_1, …, p_K*.
We test the null hypothesis:

*H_0 : p_1 = … = p_K*

vs

*H_1 : p_1 ≥ p_2 … ≥ p_K*

with at least one equality strict. A likelihood ratio statistic for this test has
been derived (Bartholomew 1959).
The null distribution of the likelihood ratio statistic is very complex
but can be determined by a simple Monte Carlo process.

We also test the null hypothesis:

*H_0 : p_1 ≥ p_2 … ≥ p_K*

vs

*H_1 : \overline{H_0}*

The null distribution of the likelihood ratio statistic is very complex but can be determined by a simple Monte Carlo process. The function requires the isotone library.

### Usage

1 | ```
LRT.value.trend(x, sigma)
``` |

### Arguments

`x` |
A vector of prevalence estimates in the order (e.g., time). |

`sigma` |
A vector of standard error estimates corresponding to |

### Value

A list with components

`pvalue.increasing`

: The p-value for the test of a monotone increase in population prevalence.`pvalue.decreasing`

: The p-value for the test of a monotone decrease in population prevalence.`L`

: The value of the likelihood-ratio statistic.`x`

: The passed vector of prevalence estimates in the order (e.g., time).`sigma`

The passed vector of standard error estimates corresponding to`x`

.

### Author(s)

Mark S. Handcock

### References

Bartholomew, D. J. (1959). A test of homogeneity for ordered alternatives. Biometrika 46 36-48.

### Examples

1 2 3 4 5 6 | ```
## Not run:
x <- c(0.16,0.15,0.3)
sigma <- c(0.04,0.04,0.1)
LRT.value.trend(x,sigma)
## End(Not run)
``` |