Meta_Regression.qmd

---
title: "Meta 回归"
---

Meta-regression 通过假设混合效应模型来实现。由于抽样误差和研究之间的异质性，观察到的研究偏离了真实的总体效应。

普通最小二乘法的加权版本

meta 回归 使用一个或多个变量x来预测真实效应大小的差异。

$$
\hat\theta_k = \theta + \beta x_{k} + \epsilon_k+\zeta_k
$$

抽样误差$\epsilon_k$和研究间异质性 $\zeta_k$

固定效应（$\beta$）和 随机效应（$\zeta_k$）

## Meta回归模型

### **使用分类预测因子的meta回归**

$$
\begin{equation}   
D_g=\begin{cases}     
0: & \text{Subgroup A}\\     
1: & \text{Subgroup B.}   
\end{cases}   
\end{equation}
$$

$$
\begin{equation} \hat\theta_k = \theta + \beta D_g +\epsilon_k+\zeta_k.  \end{equation}
$$

则

$$
\begin{equation}   D_g=\begin{cases}     0: & \text{$\hat\theta_k = \theta_A + \epsilon_k+\zeta_k$}\\     1: & \text{$\hat\theta_k = \theta_B + \theta_{\Delta} +\epsilon_k+\zeta_k$}   \end{cases}   \end{equation}
$$

![](images/metaregression_subgroups.png){fig-align="center" width="419"}

### **使用连续预测因子的meta回归**

![](images/metareg_form_continuous.png){fig-align="center" width="293"}

```{r}
library(meta)
m.gen <- metagen(TE = TE,
                 seTE = seTE,
                 studlab = Author,
                 data = dmetar::ThirdWave,
                 sm = "SMD",
                 fixed = FALSE,
                 random = TRUE,
                 method.tau = "REML",
                 method.random.ci = "HK",
                 prediction = T,
                 title = "Third Wave Psychotherapies")
summary(m.gen)
```

```{r}
year <- c(2014, 1998, 2010, 1999, 2005, 2014, 
          2019, 2010, 1982, 2020, 1978, 2001,
          2018, 2002, 2009, 2011, 2011, 2013)

m.gen.reg <- metareg(m.gen, ~year)
m.gen.reg
```

$\hat\tau^2_{\text{unexplained}}=$ 0.0188

$R^2_*$ =77.08% 意味着真实效应大小的 77% 的差异可以用出版年份来解释，这是一个相当大的值。

```{r}
bubble(m.gen.reg, studlab = TRUE)
```

```{r}
# 通过meta回归进行亚组分析
metareg(m.gen, RiskOfBias)
```

RiskOfBiaslow 效应大小 g = 0.7691-0.2992 ≈0.4699

## **多元回归**

$$
\begin{equation} \hat \theta_k = \theta + \beta_1x_{1k} + ... + \beta_nx_{nk} + \epsilon_k + \zeta_k \tag{multiple-metareg} \end{equation}
$$

```{r packages-and-data}
library(metafor)
library(tidyverse)
library(dmetar)
data(MVRegressionData)
glimpse(MVRegressionData)
```

### 多重共线性

```{r}
MVRegressionData[,c("reputation", "quality", "pubyear")] %>% 
    cor()

```

```{r}
m.qual <- rma(yi = yi,
              sei = sei,
              data = MVRegressionData,
              method = "ML",
              mods = ~ quality,
              test = "knha")

m.qual
```

```{r}
m.qual.rep <- rma(yi = yi, 
                  sei = sei, 
                  data = MVRegressionData, 
                  method = "ML", 
                  mods = ~ quality + reputation, 
                  test = "knha")

m.qual.rep
```

```{r}
anova(m.qual, m.qual.rep)
```

`anova` `LRT` X2= 19.4992 ,p\<0.001

### 交互效应

```{r}
# Add factor labels to 'continent'
# 0 = Europe
# 1 = North America
levels(MVRegressionData$continent) = c("Europe", "North America")

# Fit the meta-regression model
m.qual.rep.int <- rma(yi = yi, 
                      sei = sei, 
                      data = MVRegressionData, 
                      method = "REML", 
                      mods = ~ pubyear * continent, 
                      test = "knha")

m.qual.rep.int
```

### 过拟合

$R^2_*$ = 100% ，在实践中，几乎不会解释数据中的所有异质性——事实上，如果在现实生活中的数据中找到这样的结果，这可能是模型过度拟合了。

### 排列测试

```{r}
permutest(m.qual.rep)
```

### 多模型推理

```{r}
multimodel.inference(TE = "yi", 
                     seTE = "sei",
                     data = MVRegressionData,
                     predictors = c("pubyear", "quality", 
                                    "reputation", "continent"),
                     interaction = FALSE)
```