---
title: "Trend and heterogeneity tests"
output: rmarkdown::html_vignette
vignette: >
  %\VignetteIndexEntry{Trend and heterogeneity tests}
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteEncoding{UTF-8}
---

```{r, include = FALSE}
knitr::opts_chunk$set(
  class.output = "outp",
  class.message = "outp",
  class.warning = "outp",
  class.error = "outp",
  comment = "#"
)
library(ckbtools)
```

# Trend test

The `trend()` function carries out a chi-squared test for trend (on 1 degree of freedom) on a vector of independent estimates. The function arguments are a vector `beta` of independent estimates and a vector `se` of their standard errors. The function returns the test statistic and p-value in a named list.

## Example

These data are taken from Figure 3 of the paper 'Fresh fruit consumption and major cardiovascular disease in the China Kadoorie Biobank' by Du, H., Li, L., Bennett, D. et al. N Engl J Med 2016; 374: 1332--1343.

```{r}
b <- c(-0.59784, -0.28768, -0.4943)
s <- c(0.127929, 0.121515, 0.094959)

trend(b, s)
```



# Heterogeneity test

The `heterogeneity()` function carries out a a chi-squared test (on $k - 1$ degrees of freedom) for heterogeneity on a vector of $k$ independent estimates. The function arguments are a vector `beta` of independent estimates and a vector `se` of their standard errors. The function returns the test statistic, degrees of freedom and p-value in a named list.

## Example

These data are taken from Figure 3 of Du et al.

```{r}
beta <- c(-0.47804, -0.46204, -0.40048)
se <- c(0.090005, 0.176823, 0.106565)

heterogeneity(beta, se)
```

# Methods

## Trend

Suppose that we want to check whether estimates change progressively from one stratum (subgroup) to the next then we could use a chi-squared test for trend (on 1 degree of freedom).

The test statistic for trend given a vector of $k$ independent estimates $\hat{\beta}_i$ and their standard errors $\sigma_i$ is

$$ \frac{\sum_{i = 1}^{k} \left(w_i \hat{\beta}_i (i - A) \right)^2}{\sum_{i = 1}^{k}{w_i (i - A)^2}}$$
where

$$A = \frac{\sum_{j = 1}^{k}{j w_j}}{\sum_{j = 1}^{k}{w_j}}$$

and $w_i = 1/\sigma^2$. This test statistic has a chi-squared distribution with 1 degree of freedom under the null hypothesis of no linear trend.


## Heterogeneity

Suppose that information on the estimates for different strata (subgroups) are to be assessed in order to see if they differ between strata (i.e. effect modification) then a chi-squared test (on $k - 1$ degrees of freedom) for heterogeneity between the estimates for the different strata can used.

The test statistic for heterogeneity given a vector of $k$ independent estimates $\hat{\beta}_i$ and their standard errors $\sigma_i$ is

$$ \left(\sum_{i=1}^{k} w_i \hat{\beta}_i^2 \right) - \frac{\left(\sum_{i=1}^{k} w_i \hat{\beta}_i \right)^2}{\sum_{i=1}^{k} w_i}, $$

where $w_i = 1/\sigma^2$, which has a chi-squared distribution with $k-1$ degrees of freedom under the null hypothesis.

The test statistic can also be written as:

$$
\sum_{i=1}^{k} \left[ w_i \left( \hat{\beta}_i - \frac{\sum_{j=1}^{k} w_j\hat{\beta}_j}{\sum_{j=1}^{k} w_j} \right)^2 \right]
$$

If there are only two strata (subgroups) then the tests for trend and heterogeneity are identical.



## Notes

Both these formulae can be expressed in terms of $(o-e)$ and $v$ using
$$\hat{\beta}_i = (o_i - e_i) / v_i$$
and
$$\mathrm{se}(\hat{\beta}_i)^2 = 1 / w_i = 1 / v_i$$

First calculate the logrank statistic $(o-e)$ and its variance $v$ in each separate stratum (subgroup), and their sums:
$$O - E = \sum_{i=1}^{k} (o_i - e_i)$$
and
$$V = \sum_{i=1}^{k} v_i$$

### Trend
Define

$$m_i = \sum_{i = 1}^{k} i v_i/V$$

and

$$T = \sum_{i = 1}^{k} (i-m_i)(o_i - e_i)$$

The variance of $T$, $\mathrm{var}(T)$, is then

$$\mathrm{var}(T) = \sum_{i=1}^{k} (i - m)^2v$$

and the chi-squared test statistic (1 degree of freedom) for trend is
$$\frac{T^2}{\mathrm{var}(T)}$$

### Heterogeneity

The chi-squared test statistic ($k - 1$ degrees of freedom) for heterogeneity is

$$\sum_{i=1}^k \left( \frac{(o_i-e_i)^2}{v_i} \right) - \frac{(O-E)^2}{V}$$

# References

[Early Breast Cancer Trialists' Collaborative Group, Section 5 in "Treatment of Early Breast Cancer. Volume 1. Worldwide Evidence 1985-1990"](https://www.ctsu.ox.ac.uk/research/the-early-breast-cancer-trialists-collaborative-group-ebctcg/original-methods-for-ebctcg-meta-analyses/section-5-statistical-methods)

[Effects of chemotherapy and hormonal therapy for early breast cancer on recurrence and 15-year survival: an overview of the randomised trials. The Lancet 365, 1687–1717 (2005).](https://doi.org/10.1016/S0140-6736(05)66544-0) (Not open access)

[Armitage, P. Tests for Linear Trends in Proportions and Frequencies. Biometrics 11, 375–386 (1955).](https://doi.org/10.2307/3001775) (Not open access)

[Du, H. et al. Fresh Fruit Consumption and Major Cardiovascular Disease in China. New England Journal of Medicine 374, 1332–1343 (2016).](https://doi.org/10.1056/NEJMoa1501451)