# 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. An optional plot of the estimates and the null distribution of the test statistics is provided.
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. The alternatie hypothesis is for a monotone decreasing trend.
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.

Alternatively, we can 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.
In both cases we also test for:

*H : p_1 ≤ p_2 … ≤ p_K*

that is, a monotonically increasing trend. The function requires the isotone library.

### Usage

1 2 3 |

### Arguments

`data` |
A two row matrix or data.frame of prevalence estimates and
their standard errors. The first row is the prevalence estimates and the
second are the standard errors. The column are the comparison groups in the
order (e.g., time) there are to be assessed. The row names of |

`variables` |
A character vector of column names it select from |

`null` |
A character string indicating the null hypothesis to use. The value |

`confidence.level` |
The confidence level for the confidence intervals. The default is 0.95 for 95%. |

`number.of.bootstrap.samples` |
The number of Monte Carlo draws to determine the null distribution of the likelihood ratio statistic. |

`plot` |
A character vector of choices, a subset of |

`seed` |
The value of the random number seed. Preset by default to allow reproducibility. |

### 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 | ```
d <- t(data.frame(estimate=c(0.16,0.15,0.3), sigma=c(0.04,0.04,0.1)))
colnames(d) <- c("time_1","time_2","time_3")
LRT.trend.test(d,number.of.bootstrap.samples=1000)
``` |