cancer.Rmd

# 吸烟导致癌症？ {#cancer}

```{r cancer_setup,message=FALSE,cache=FALSE}
library(tidyverse)
library(rstan)
```


从肺癌患者和无癌个体中随机抽取两组样本[^cancer]，目的是考察两组人吸烟习惯的差异。
数据如下

```{r cancer01}
cancer <- tribble(
  ~group, ~n, ~smokers,
  "Cancer patients", 86, 82,
  "Control group", 86, 72
)
cancer
```



## 二项模型


为完成这个二项抽样模型，将每组数据里吸烟人数的比例$\pi_i$作为参数，并使用uniform先验概率分布
$$
\begin{aligned}[t]
r_i &\sim \mathsf{Binomial}(n_i, \pi_i)
\end{aligned}
$$
吸烟比例的差值为
$$
\delta = \pi_1 - \pi_2 ,
$$


并让$\pi$使用uniform先验概率分布
$$
\begin{aligned}
\pi_i &\sim \mathsf{Beta}(1, 1)
\end{aligned}
$$



在`generated quantities` block中计算了比例log-odds ratio之差
$$
\lambda = \log\left(\frac{\pi_1}{1 - \pi_1}\right) - \log \left( \frac{\pi_2}{1 - \pi_2} \right) ,
$$



具体 Stan 模型如下
```{r}
cancer_mod1 <- "
data {
  int<lower = 0> r[2];
  int<lower = 1> n[2];
}
parameters {
  vector<lower = 0, upper = 1>[2] p;
}
model {
  p ~ beta(1, 1);
  r ~ binomial(n, p);
}
generated quantities {
  real delta;
  int delta_up;
  real lambda;
  int lambda_up;

  delta = p[1] - p[2];
  delta_up = delta > 0;
  lambda = logit(p[1]) - logit(p[2]);
  lambda_up = lambda > 0;

}
"


cancer_data <- list(
  r = cancer$smokers,
  n = cancer$n
)


cancer_fit1 <- stan(model_code = cancer_mod1, data = cancer_data)
```



```{r}
cancer_fit1
```

## 二项Logit模型


另外一种方法，就是直接模拟比例之差，这里需要用到参数$\alpha$ 和 $\beta$，每组数据的比例都是参数的log-odds

$$
\begin{aligned}[t]
r_i &\sim \mathsf{Binomial}(n_i, \pi_i) \\
\pi_1 &= \frac{1}{1 + \exp(-(\alpha + \beta)} \\
\pi_2 &= \frac{1}{1 + \exp(-\alpha))}
\end{aligned}
$$
这里设定$\alpha$ 和$\beta$ 弱先验信息

$$
\begin{aligned}
\alpha &\sim N(0, 10)\\
\beta &\sim N(0, 2.5)
\end{aligned}
$$

```{r cancer_mod2}
stan_program <- "
data {
  int<lower = 0> r[2];
  int<lower = 1> n[2];
}
parameters {
  real a;
  real b;
}
transformed parameters {
  vector<lower = 0., upper = 1.>[2] p;
  p[1] = inv_logit(a + b);
  p[2] = inv_logit(a);
}
model {
  a ~ normal(0, 10);
  b ~ normal(0, 2.5);
  r ~ binomial(n, p);
}
generated quantities {
  real delta;
  int delta_up;
  real lambda;
  int lambda_up;

  delta = p[1] - p[2];
  delta_up = delta > 0;
  lambda = logit(p[1]) - logit(p[2]);
  lambda_up = lambda > 0;

}
"

cancer_data <- list(
  r = cancer$smokers,
  n = cancer$n
)


cancer_fit2 <- stan(model_code = stan_program, data = cancer_data)
```


```{r}
cancer_fit2
```



## 参考

[^cancer]: This example is derived from Simon Jackman,
    "[Cancer: difference in two binomial proportions](https://web-beta.archive.org/web/20070601000000*/http://jackman.stanford.edu:80/mcmc/cancer.odc)",
    *BUGS Examples,* 2007-07-24, This examples comes from @JohnsonAlbert1999a, using data from @Dorn1954a.