8  混合效应模型

I think that the formula language does allow expressions with ‘/’ to represent nested factors but I can’t check right now as there is a fire in the building where my office is located. I prefer to simply write nested factors as factor1 + factor1:factor2.

— Douglas Bates 1

library(nlme)         # 线性混合效应模型
library(GLMMadaptive) # 广义线性混合效应模型
library(mgcv)         # 广义线性/可加混合效应模型
library(INLA)
library(splines)   # 样条
library(cmdstanr)  # 编译采样
library(ggplot2)   # 作图
library(bayesplot) # 后验分布
library(loo)       # LOO-CV

混合效应模型在心理学、生态学、计量经济学和空间统计学等领域应用十分广泛。线性混合效应模型有多个化身,比如生态学里的分层线性模型(Hierarchical linear Model,简称 HLM),心理学的多水平线性模型(Multilevel Linear Model)。模型名称的多样性正说明它应用的广泛性! 混合效应模型内容非常多,非常复杂,因此,本章仅对常见的四种类型提供入门级的实战内容。从频率派和贝叶斯派两个角度介绍模型结构及说明、R 代码或 Stan 代码实现及输出结果解释。

除了 R 语言社区提供的开源扩展包,商业的软件有 MplusASRemlSAS 等,而开源的软件 OpenBUGSJAGSStan 等。混合效应模型的种类非常多,一部分可以在一些流行的 R 包能力范围内解决,其余可以放在更加灵活、扩展性强的框架 Stan 内解决。因此,本章同时介绍 Stan 框架和一些 R 包。

本章用到 4 个数据集,其中 sleepstudycbpp 均来自 lme4(Bates 等 2015),分别用于介绍线性混合效应模型和广义线性混合效应模型,Loblolly 来自 datasets 包,用于介绍非线性混合效应模型。

在介绍理论的同时给出 R 语言或 S 语言实现的几本参考书籍。

8.1 线性混合效应模型

I think what we are seeking is the marginal variance-covariance matrix of the parameter estimators (marginal with respect to the random effects random variable, B), which would have the form of the inverse of the crossproduct of a \((q+p)\) by \(p\) matrix composed of the vertical concatenation of \(-L^{-1}RZXRX^{-1}\) and \(RX^{-1}\). (Note: You do not want to calculate the first term by inverting \(L\), use solve(L, RZX, system = "L")

  • […] don’t even think about using solve(L)

  • don’t!, don’t!, don’t!

  • have I made myself clear?

  • don’t do that (and we all know that someone will do exactly that for a very large \(L\) and then send out messages about “R is SOOOOO SLOOOOW!!!!” :-) )

— Douglas Bates 2

提示
  1. 一般的模型结构和假设
  2. 一般的模型表达公式
  3. nlme 包的函数 lme()
  4. 公式语法和示例模型表示

线性混合效应模型(Linear Mixed Models or Linear Mixed-Effects Models,简称 LME 或 LMM),介绍模型的基础理论,包括一般形式,矩阵表示,参数估计,假设检验,模型诊断,模型评估。参数方法主要是极大似然估计和限制极大似然估计。一般形式如下:

\[ \bm{y} = X\bm{\beta} + Z\bm{u} + \bm{\epsilon} \]

其中,\(\bm{y}\) 是一个向量,代表响应变量,\(X\) 代表固定效应对应的设计矩阵,\(\bm{\beta}\) 是一个参数向量,代表固定效应对应的回归系数,\(Z\) 代表随机效应对应的设计矩阵,\(\bm{u}\) 是一个参数向量,代表随机效应对应的回归系数,\(\bm{\epsilon}\) 表示残差向量。

一般假定随机向量 \(\bm{u}\) 服从多元正态分布,这是无条件分布,随机向量 \(\bm{y}|\bm{u}\) 服从多元正态分布,这是条件分布。

\[ \begin{aligned} \bm{u} &\sim \mathcal{N}(0,\Sigma) \\ \bm{y}|\bm{u} &\sim \mathcal{N}(X\bm{\beta} + Z\bm{u},\sigma^2W) \end{aligned} \]

其中,方差协方差矩阵 \(\Sigma\) 必须是半正定的,\(W\) 是一个对角矩阵。nlme 和 lme4 等 R 包共用一套表示随机效应的公式语法。

sleepstudy 数据集来自 lme4 包,是一个睡眠研究项目的实验数据。实验对象都是有失眠情况的人,有的人有严重的失眠问题(一天只有 3 个小时的睡眠时间)。进入实验后的前10 天的情况,记录平均反应时间、睡眠不足的天数。

data(sleepstudy, package = "lme4")
str(sleepstudy)
#> 'data.frame':    180 obs. of  3 variables:
#>  $ Reaction: num  250 259 251 321 357 ...
#>  $ Days    : num  0 1 2 3 4 5 6 7 8 9 ...
#>  $ Subject : Factor w/ 18 levels "308","309","310",..: 1 1 1 1 1 1 1 1 1 1 ...

Reaction 表示平均反应时间(毫秒),数值型,Days 表示进入实验后的第几天,数值型,Subject 表示参与实验的个体编号,因子型。

xtabs(~ Days + Subject, data = sleepstudy)
#>     Subject
#> Days 308 309 310 330 331 332 333 334 335 337 349 350 351 352 369 370 371 372
#>    0   1   1   1   1   1   1   1   1   1   1   1   1   1   1   1   1   1   1
#>    1   1   1   1   1   1   1   1   1   1   1   1   1   1   1   1   1   1   1
#>    2   1   1   1   1   1   1   1   1   1   1   1   1   1   1   1   1   1   1
#>    3   1   1   1   1   1   1   1   1   1   1   1   1   1   1   1   1   1   1
#>    4   1   1   1   1   1   1   1   1   1   1   1   1   1   1   1   1   1   1
#>    5   1   1   1   1   1   1   1   1   1   1   1   1   1   1   1   1   1   1
#>    6   1   1   1   1   1   1   1   1   1   1   1   1   1   1   1   1   1   1
#>    7   1   1   1   1   1   1   1   1   1   1   1   1   1   1   1   1   1   1
#>    8   1   1   1   1   1   1   1   1   1   1   1   1   1   1   1   1   1   1
#>    9   1   1   1   1   1   1   1   1   1   1   1   1   1   1   1   1   1   1

每个个体每天产生一条数据,下 图 8.1 中每条折线代表一个个体。

library(ggplot2)
ggplot(data = sleepstudy, aes(x = Days, y = Reaction, group = Subject)) +
  geom_line() +
  scale_x_continuous(n.breaks = 6) +
  theme_bw() +
  labs(x = "睡眠不足的天数", y = "平均反应时间")
图 8.1: sleepstudy 数据集

对于连续重复测量的数据(continuous repeated measurement outcomes),也叫纵向数据(longitudinal data),针对不同个体 Subject,相比于上图,下面绘制反应时间 Reaction 随睡眠时间 Days 的变化趋势更合适。图中趋势线是简单线性回归的结果,分面展示不同个体Subject 之间对比。

ggplot(data = sleepstudy, aes(x = Days, y = Reaction)) +
  geom_point() +
  geom_smooth(formula = "y ~ x", method = "lm", se = FALSE) +
  scale_x_continuous(n.breaks = 6) +
  theme_bw() +
  facet_wrap(facets = ~Subject, labeller = "label_both", ncol = 6) +
  labs(x = "睡眠不足的天数", y = "平均反应时间")
图 8.2: 分面展示 sleepstudy 数据集

8.1.1 nlme

考虑两水平的混合效应模型,其中随机截距 \(\beta_{0j}\) 和随机斜率 \(\beta_{1j}\),指标 \(j\) 表示分组的编号,也叫变截距和变斜率模型

\[ \begin{aligned} \mathrm{Reaction}_{ij} &= \beta_{0j} + \beta_{1j} \cdot \mathrm{Days}_{ij} + \epsilon_{ij} \\ \beta_{0j} &= \gamma_{00} + U_{0j} \\ \beta_{1j} &= \gamma_{10} + U_{1j} \\ \begin{pmatrix} U_{0j} \\ U_{1j} \end{pmatrix} &\sim \mathcal{N} \begin{bmatrix} \begin{pmatrix} 0 \\ 0 \end{pmatrix} , \begin{pmatrix} \tau^2_{00} & \tau_{01} \\ \tau_{01} & \tau^2_{10} \end{pmatrix} \end{bmatrix} \\ \epsilon_{ij} &\sim \mathcal{N}(0, \sigma^2) \\ i = 0,1,\cdots,9 &\quad j = 308,309,\cdots, 372. \end{aligned} \]

下面用 nlme 包 (Pinheiro 和 Bates 2000) 拟合模型。

library(nlme)
sleep_nlme <- lme(Reaction ~ Days, random = ~ Days | Subject, data = sleepstudy)
summary(sleep_nlme)
#> Linear mixed-effects model fit by REML
#>   Data: sleepstudy 
#>        AIC      BIC    logLik
#>   1755.628 1774.719 -871.8141
#> 
#> Random effects:
#>  Formula: ~Days | Subject
#>  Structure: General positive-definite, Log-Cholesky parametrization
#>             StdDev    Corr  
#> (Intercept) 24.740241 (Intr)
#> Days         5.922103 0.066 
#> Residual    25.591843       
#> 
#> Fixed effects:  Reaction ~ Days 
#>                 Value Std.Error  DF  t-value p-value
#> (Intercept) 251.40510  6.824516 161 36.83853       0
#> Days         10.46729  1.545783 161  6.77151       0
#>  Correlation: 
#>      (Intr)
#> Days -0.138
#> 
#> Standardized Within-Group Residuals:
#>         Min          Q1         Med          Q3         Max 
#> -3.95355735 -0.46339976  0.02311783  0.46339621  5.17925089 
#> 
#> Number of Observations: 180
#> Number of Groups: 18

随机效应(Random effects)部分:

# 前 6 个 subject
head(ranef(sleep_nlme))
#>     (Intercept)       Days
#> 308    2.258754  9.1989366
#> 309  -40.398490 -8.6197167
#> 310  -38.960098 -5.4489048
#> 330   23.690228 -4.8142826
#> 331   22.259981 -3.0698548
#> 332    9.039458 -0.2721585

固定效应(Fixed effects)部分:

fixef(sleep_nlme)
#> (Intercept)        Days 
#>   251.40510    10.46729

ggeffects 包的函数 ggpredict()ggeffect() 可以用来绘制混合效应模型的边际效应( Marginal Effects),ggPMX 包 可以用来绘制混合效应模型的诊断图。下 图 8.3 展示关于变量 Days 的边际效应图。

library(ggeffects)
mydf <- ggpredict(sleep_nlme, terms = "Days")
ggplot(mydf, aes(x = x, y = predicted)) +
  geom_line() +
  geom_ribbon(aes(ymin = conf.low, ymax = conf.high), alpha = 0.2) +
  scale_x_continuous(n.breaks = 6) +
  theme_bw() +
  labs(x = "Days", y = "Reaction")
图 8.3: 边际效应图

8.1.2 MASS

sleep_mass <- MASS::glmmPQL(Reaction ~ Days,
  random = ~ Days | Subject, verbose = FALSE,
  data = sleepstudy, family = gaussian
)
summary(sleep_mass)
#> Linear mixed-effects model fit by maximum likelihood
#>   Data: sleepstudy 
#>   AIC BIC logLik
#>    NA  NA     NA
#> 
#> Random effects:
#>  Formula: ~Days | Subject
#>  Structure: General positive-definite, Log-Cholesky parametrization
#>             StdDev    Corr  
#> (Intercept) 23.780376 (Intr)
#> Days         5.716807 0.081 
#> Residual    25.591842       
#> 
#> Variance function:
#>  Structure: fixed weights
#>  Formula: ~invwt 
#> Fixed effects:  Reaction ~ Days 
#>                 Value Std.Error  DF  t-value p-value
#> (Intercept) 251.40510  6.669396 161 37.69533       0
#> Days         10.46729  1.510647 161  6.92901       0
#>  Correlation: 
#>      (Intr)
#> Days -0.138
#> 
#> Standardized Within-Group Residuals:
#>         Min          Q1         Med          Q3         Max 
#> -3.94156355 -0.46559311  0.02894656  0.46361051  5.17933587 
#> 
#> Number of Observations: 180
#> Number of Groups: 18

8.1.3 lme4

sleep_lme4 <- lme4::lmer(Reaction ~ Days + (Days | Subject), data = sleepstudy)
summary(sleep_lme4)
#> Linear mixed model fit by REML ['lmerMod']
#> Formula: Reaction ~ Days + (Days | Subject)
#>    Data: sleepstudy
#> 
#> REML criterion at convergence: 1743.6
#> 
#> Scaled residuals: 
#>     Min      1Q  Median      3Q     Max 
#> -3.9536 -0.4634  0.0231  0.4634  5.1793 
#> 
#> Random effects:
#>  Groups   Name        Variance Std.Dev. Corr
#>  Subject  (Intercept) 612.10   24.741       
#>           Days         35.07    5.922   0.07
#>  Residual             654.94   25.592       
#> Number of obs: 180, groups:  Subject, 18
#> 
#> Fixed effects:
#>             Estimate Std. Error t value
#> (Intercept)  251.405      6.825  36.838
#> Days          10.467      1.546   6.771
#> 
#> Correlation of Fixed Effects:
#>      (Intr)
#> Days -0.138

8.1.4 blme

sleep_blme <- blme::blmer(
  Reaction ~ Days + (Days | Subject), data = sleepstudy,
  control = lme4::lmerControl(check.conv.grad = "ignore"),
  cov.prior = NULL)
summary(sleep_blme)
#> Prior dev  : 0
#> 
#> Linear mixed model fit by REML ['blmerMod']
#> Formula: Reaction ~ Days + (Days | Subject)
#>    Data: sleepstudy
#> Control: lme4::lmerControl(check.conv.grad = "ignore")
#> 
#> REML criterion at convergence: 1743.6
#> 
#> Scaled residuals: 
#>     Min      1Q  Median      3Q     Max 
#> -3.9536 -0.4634  0.0231  0.4634  5.1793 
#> 
#> Random effects:
#>  Groups   Name        Variance Std.Dev. Corr
#>  Subject  (Intercept) 612.10   24.741       
#>           Days         35.07    5.922   0.07
#>  Residual             654.94   25.592       
#> Number of obs: 180, groups:  Subject, 18
#> 
#> Fixed effects:
#>             Estimate Std. Error t value
#> (Intercept)  251.405      6.825  36.838
#> Days          10.467      1.546   6.771
#> 
#> Correlation of Fixed Effects:
#>      (Intr)
#> Days -0.138

8.1.5 brms

sleep_brms <- brms::brm(Reaction ~ Days + (Days | Subject), data = sleepstudy)
summary(sleep_brms)
 Family: gaussian 
  Links: mu = identity; sigma = identity 
Formula: Reaction ~ Days + (Days | Subject) 
   Data: sleepstudy (Number of observations: 180) 
  Draws: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
         total post-warmup draws = 4000

Group-Level Effects: 
~Subject (Number of levels: 18) 
                    Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
sd(Intercept)          27.03      6.60    15.88    42.13 1.00     1728     2469
sd(Days)                6.61      1.50     4.18     9.97 1.00     1517     2010
cor(Intercept,Days)     0.08      0.29    -0.46     0.65 1.00      991     1521

Population-Level Effects: 
          Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
Intercept   251.26      7.42   236.27   266.12 1.00     1982     2687
Days         10.36      1.77     6.85    13.85 1.00     1415     1982

Family Specific Parameters: 
      Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
sigma    25.88      1.54    22.99    29.06 1.00     3204     2869

Draws were sampled using sampling(NUTS). For each parameter, Bulk_ESS
and Tail_ESS are effective sample size measures, and Rhat is the potential
scale reduction factor on split chains (at convergence, Rhat = 1).
# predictions
conds <- brms::make_conditions(sleep_brms, "Subject")
sleep_brms |>
  brms::marginal_effects(
    re_formula = NULL,
    conditions = conds
  ) |>
  plot(points = TRUE, ncol = 6)

8.1.6 MCMCglmm

MCMCglmm 包拟合变截距、变斜率模型,随机截距和随机斜率之间存在相关性。

## 变截距、变斜率模型
prior1 <- list(
  R = list(V = 1, fix = 1),
  G = list(G1 = list(V = diag(2), nu = 0.002))
)
set.seed(20232023)
sleep_mcmcglmm <- MCMCglmm::MCMCglmm(
  Reaction ~ Days, random = ~ us(1 + Days):Subject, prior = prior1,
  data = sleepstudy, family = "gaussian", verbose = FALSE
)
summary(sleep_mcmcglmm)
#> 
#>  Iterations = 3001:12991
#>  Thinning interval  = 10
#>  Sample size  = 1000 
#> 
#>  DIC: 94714.46 
#> 
#>  G-structure:  ~us(1 + Days):Subject
#> 
#>                                 post.mean l-95% CI u-95% CI eff.samp
#> (Intercept):(Intercept).Subject   1005.69   454.97  1840.04   1000.0
#> Days:(Intercept).Subject           -34.44  -167.15    84.09   1000.0
#> (Intercept):Days.Subject           -34.44  -167.15    84.09   1000.0
#> Days:Days.Subject                   52.36    22.74    95.60    902.3
#> 
#>  R-structure:  ~units
#> 
#>       post.mean l-95% CI u-95% CI eff.samp
#> units         1        1        1        0
#> 
#>  Location effects: Reaction ~ Days 
#> 
#>             post.mean l-95% CI u-95% CI eff.samp  pMCMC    
#> (Intercept)   251.374  235.935  265.961     1000 <0.001 ***
#> Days           10.419    7.262   13.976     1000 <0.001 ***
#> ---
#> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

固定随机效应 R-structure 方差。固定效应 Location effects 截距 (Intercept) 为 251.374,斜率 Days 为 10.419 。

8.1.7 INLA

将数据集 sleepstudy 中的 Reaction 除以 1000,目的是数值稳定性,减小迭代序列的相关性。先考虑变截距模型

library(INLA)
inla.setOption(short.summary = TRUE)
# 做尺度变换
sleepstudy$Reaction <- sleepstudy$Reaction / 1000
# 变截距
sleep_inla1 <- inla(Reaction ~ Days + f(Subject, model = "iid", n = 18), 
                  family = "gaussian", data = sleepstudy)
# 输出结果
summary(sleep_inla1)
#> Fixed effects:
#>              mean    sd 0.025quant 0.5quant 0.975quant  mode kld
#> (Intercept) 0.251 0.010      0.232    0.251      0.270 0.251   0
#> Days        0.010 0.001      0.009    0.010      0.012 0.010   0
#> 
#> Model hyperparameters:
#>                                            mean     sd 0.025quant 0.5quant
#> Precision for the Gaussian observations 1054.07 116.99     840.14  1048.46
#> Precision for Subject                    843.97 300.88     391.25   798.36
#>                                         0.975quant    mode
#> Precision for the Gaussian observations    1300.13 1038.99
#> Precision for Subject                      1559.54  714.37
#> 
#>  is computed

再考虑变截距和变斜率模型

# https://inla.r-inla-download.org/r-inla.org/doc/latent/iid.pdf
# 二维高斯随机效应的先验为 Wishart prior
sleepstudy$Subject <- as.integer(sleepstudy$Subject)
sleepstudy$slopeid <- 18 + sleepstudy$Subject
# 变截距、变斜率
sleep_inla2 <- inla(
  Reaction ~ 1 + Days + f(Subject, model = "iid2d", n = 2 * 18) + f(slopeid, Days, copy = "Subject"),
  data = sleepstudy, family = "gaussian"
)
# 输出结果
summary(sleep_inla2)
#> Fixed effects:
#>              mean    sd 0.025quant 0.5quant 0.975quant  mode kld
#> (Intercept) 0.251 0.055      0.142    0.251      0.360 0.251   0
#> Days        0.010 0.054     -0.097    0.010      0.118 0.010   0
#> 
#> Model hyperparameters:
#>                                             mean      sd 0.025quant 0.5quant
#> Precision for the Gaussian observations 1549.517 181.247   1218.036 1540.851
#> Precision for Subject (component 1)       20.871   6.507     10.626   20.024
#> Precision for Subject (component 2)       21.225   6.611     10.804   20.368
#> Rho1:2 for Subject                        -0.001   0.213     -0.414   -0.002
#>                                         0.975quant     mode
#> Precision for the Gaussian observations   1930.548 1527.284
#> Precision for Subject (component 1)         35.970   18.479
#> Precision for Subject (component 2)         36.556   18.806
#> Rho1:2 for Subject                           0.413   -0.003
#> 
#>  is computed

8.2 广义线性混合效应模型

当响应变量分布不再是高斯分布,线性混合效应模型就扩展到广义线性混合效应模型。有一些 R 包可以拟合此类模型,MASS 包的函数 glmmPQL()mgcv 包的函数 gam(),lme4 包的函数 glmer() ,GLMMadaptive 包的函数 mixed_model() ,brms 包的函数 brm() 等。

表格 8.1: 响应变量的分布
响应变量分布 MASS mgcv lme4 GLMMadaptive brms
伯努利分布 支持 支持 支持 支持 支持
二项分布 支持 支持 支持 支持 支持
泊松分布 支持 支持 支持 支持 支持
负二项分布 不支持 支持 支持 支持 支持
伽马分布 支持 支持 支持 支持 支持

函数 glmmPQL() 支持的分布族见函数 glm() 的参数 family ,lme4 包的函数 glmer.nb() 和 GLMMadaptive 包的函数 negative.binomial() 都可用于拟合响应变量服从负二项分布的情况。除了这些常规的分布,GLMMadaptive 和 brms 包还支持许多常见的分布,比如零膨胀的泊松分布、二项分布等,还可以自定义分布。

  • 伯努利分布 family = binomial(link = "logit")
  • 二项分布 family = binomial(link = "logit")
  • 泊松分布 family = poisson(link = "log")
  • 负二项分布 lme4::glmer.nb()GLMMadaptive::negative.binomial()
  • 伽马分布 family = Gamma(link = "inverse")

GLMMadaptive(Rizopoulos 2023) 的主要函数 mixed_model() 是用来拟合广义线性混合效应模型的。下面以牛传染性胸膜肺炎(Contagious bovine pleuropneumonia,简称 CBPP)数据 cbpp 介绍函数 mixed_model() 的用法,该数据集来自 lme4 包。

data(cbpp, package = "lme4")
str(cbpp)
#> 'data.frame':    56 obs. of  4 variables:
#>  $ herd     : Factor w/ 15 levels "1","2","3","4",..: 1 1 1 1 2 2 2 3 3 3 ...
#>  $ incidence: num  2 3 4 0 3 1 1 8 2 0 ...
#>  $ size     : num  14 12 9 5 22 18 21 22 16 16 ...
#>  $ period   : Factor w/ 4 levels "1","2","3","4": 1 2 3 4 1 2 3 1 2 3 ...

herd 牛群编号,period 时间段,incidence 感染的数量,size 牛群大小。疾病在种群内扩散

ggplot(data = cbpp, aes(x = herd, y = period)) +
  geom_tile(aes(fill = incidence / size)) +
  scale_fill_viridis_c(label = scales::percent_format(), 
                       option = "C", name = "") +
  theme_minimal()
图 8.4: 感染比例随变量 herd 和 period 的变化

8.2.1 MASS

cbpp_mass <- MASS::glmmPQL(
  cbind(incidence, size - incidence) ~ period,
  random = ~ 1 | herd, verbose = FALSE,
  data = cbpp, family = binomial("logit")
)
summary(cbpp_mass)
#> Linear mixed-effects model fit by maximum likelihood
#>   Data: cbpp 
#>   AIC BIC logLik
#>    NA  NA     NA
#> 
#> Random effects:
#>  Formula: ~1 | herd
#>         (Intercept) Residual
#> StdDev:   0.5563535 1.184527
#> 
#> Variance function:
#>  Structure: fixed weights
#>  Formula: ~invwt 
#> Fixed effects:  cbind(incidence, size - incidence) ~ period 
#>                 Value Std.Error DF   t-value p-value
#> (Intercept) -1.327364 0.2390194 38 -5.553372  0.0000
#> period2     -1.016126 0.3684079 38 -2.758156  0.0089
#> period3     -1.149984 0.3937029 38 -2.920944  0.0058
#> period4     -1.605217 0.5178388 38 -3.099839  0.0036
#>  Correlation: 
#>         (Intr) perid2 perid3
#> period2 -0.399              
#> period3 -0.373  0.260       
#> period4 -0.282  0.196  0.182
#> 
#> Standardized Within-Group Residuals:
#>        Min         Q1        Med         Q3        Max 
#> -2.0591168 -0.6493095 -0.2747620  0.5170492  2.6187632 
#> 
#> Number of Observations: 56
#> Number of Groups: 15

8.2.2 GLMMadaptive

library(GLMMadaptive)
cbpp_glmmadaptive <- mixed_model(
  fixed = cbind(incidence, size - incidence) ~ period,
  random = ~ 1 | herd, data = cbpp, family = binomial(link = "logit")
)
summary(cbpp_glmmadaptive)
#> 
#> Call:
#> mixed_model(fixed = cbind(incidence, size - incidence) ~ period, 
#>     random = ~1 | herd, data = cbpp, family = binomial(link = "logit"))
#> 
#> Data Descriptives:
#> Number of Observations: 56
#> Number of Groups: 15 
#> 
#> Model:
#>  family: binomial
#>  link: logit 
#> 
#> Fit statistics:
#>    log.Lik      AIC     BIC
#>  -91.98337 193.9667 197.507
#> 
#> Random effects covariance matrix:
#>                StdDev
#> (Intercept) 0.6475934
#> 
#> Fixed effects:
#>             Estimate Std.Err z-value    p-value
#> (Intercept)  -1.3995  0.2335 -5.9923    < 1e-04
#> period2      -0.9914  0.3068 -3.2316 0.00123091
#> period3      -1.1278  0.3268 -3.4513 0.00055793
#> period4      -1.5795  0.4276 -3.6937 0.00022101
#> 
#> Integration:
#> method: adaptive Gauss-Hermite quadrature rule
#> quadrature points: 11
#> 
#> Optimization:
#> method: EM
#> converged: TRUE

8.2.3 glmmTMB

cbpp_glmmtmb <- glmmTMB::glmmTMB(
  cbind(incidence, size - incidence) ~ period + (1 | herd),
  data = cbpp, family = binomial, REML = TRUE
)
summary(cbpp_glmmtmb)
#>  Family: binomial  ( logit )
#> Formula:          cbind(incidence, size - incidence) ~ period + (1 | herd)
#> Data: cbpp
#> 
#>      AIC      BIC   logLik deviance df.resid 
#>    196.4    206.5    -93.2    186.4       55 
#> 
#> Random effects:
#> 
#> Conditional model:
#>  Groups Name        Variance Std.Dev.
#>  herd   (Intercept) 0.4649   0.6819  
#> Number of obs: 56, groups:  herd, 15
#> 
#> Conditional model:
#>             Estimate Std. Error z value Pr(>|z|)    
#> (Intercept)  -1.3670     0.2376  -5.752 8.79e-09 ***
#> period2      -0.9693     0.3055  -3.173 0.001509 ** 
#> period3      -1.1045     0.3255  -3.393 0.000691 ***
#> period4      -1.5519     0.4265  -3.639 0.000274 ***
#> ---
#> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

8.2.4 lme4

cbpp_lme4 <- lme4::glmer(
  cbind(incidence, size - incidence) ~ period + (1 | herd),
  family = binomial("logit"), data = cbpp
)
summary(cbpp_lme4)
#> Generalized linear mixed model fit by maximum likelihood (Laplace
#>   Approximation) [glmerMod]
#>  Family: binomial  ( logit )
#> Formula: cbind(incidence, size - incidence) ~ period + (1 | herd)
#>    Data: cbpp
#> 
#>      AIC      BIC   logLik deviance df.resid 
#>    194.1    204.2    -92.0    184.1       51 
#> 
#> Scaled residuals: 
#>     Min      1Q  Median      3Q     Max 
#> -2.3816 -0.7889 -0.2026  0.5142  2.8791 
#> 
#> Random effects:
#>  Groups Name        Variance Std.Dev.
#>  herd   (Intercept) 0.4123   0.6421  
#> Number of obs: 56, groups:  herd, 15
#> 
#> Fixed effects:
#>             Estimate Std. Error z value Pr(>|z|)    
#> (Intercept)  -1.3983     0.2312  -6.048 1.47e-09 ***
#> period2      -0.9919     0.3032  -3.272 0.001068 ** 
#> period3      -1.1282     0.3228  -3.495 0.000474 ***
#> period4      -1.5797     0.4220  -3.743 0.000182 ***
#> ---
#> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> 
#> Correlation of Fixed Effects:
#>         (Intr) perid2 perid3
#> period2 -0.363              
#> period3 -0.340  0.280       
#> period4 -0.260  0.213  0.198

8.2.5 mgcv

或使用 mgcv 包,可以得到近似的结果。随机效应部分可以看作可加的惩罚项

library(mgcv)
cbpp_mgcv <- gam(
  cbind(incidence, size - incidence) ~ period + s(herd, bs = "re"),
  data = cbpp, family = binomial(link = "logit"), method = "REML"
)
summary(cbpp_mgcv)
#> 
#> Family: binomial 
#> Link function: logit 
#> 
#> Formula:
#> cbind(incidence, size - incidence) ~ period + s(herd, bs = "re")
#> 
#> Parametric coefficients:
#>             Estimate Std. Error z value Pr(>|z|)    
#> (Intercept)  -1.3670     0.2358  -5.799 6.69e-09 ***
#> period2      -0.9693     0.3040  -3.189 0.001428 ** 
#> period3      -1.1045     0.3241  -3.407 0.000656 ***
#> period4      -1.5519     0.4251  -3.651 0.000261 ***
#> ---
#> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> 
#> Approximate significance of smooth terms:
#>          edf Ref.df Chi.sq  p-value    
#> s(herd) 9.66     14  32.03 3.21e-05 ***
#> ---
#> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> 
#> R-sq.(adj) =  0.515   Deviance explained =   53%
#> -REML = 93.199  Scale est. = 1         n = 56

下面给出随机效应的标准差的估计及其上下限,和前面 GLMMadaptive 包和 lme4 包给出的结果也是接近的。

gam.vcomp(cbpp_mgcv)
#> 
#> Standard deviations and 0.95 confidence intervals:
#> 
#>           std.dev     lower    upper
#> s(herd) 0.6818673 0.3953145 1.176135
#> 
#> Rank: 1/1

8.2.6 blme

cbpp_blme <- blme::bglmer(
  cbind(incidence, size - incidence) ~ period + (1 | herd),
  family = binomial("logit"), data = cbpp
)
summary(cbpp_blme)
#> Cov prior  : herd ~ wishart(df = 3.5, scale = Inf, posterior.scale = cov, common.scale = TRUE)
#> Prior dev  : 0.9901
#> 
#> Generalized linear mixed model fit by maximum likelihood (Laplace
#>   Approximation) [bglmerMod]
#>  Family: binomial  ( logit )
#> Formula: cbind(incidence, size - incidence) ~ period + (1 | herd)
#>    Data: cbpp
#> 
#>      AIC      BIC   logLik deviance df.resid 
#>    194.2    204.3    -92.1    184.2       51 
#> 
#> Scaled residuals: 
#>     Min      1Q  Median      3Q     Max 
#> -2.3670 -0.8121 -0.1704  0.4971  2.7969 
#> 
#> Random effects:
#>  Groups Name        Variance Std.Dev.
#>  herd   (Intercept) 0.5168   0.7189  
#> Number of obs: 56, groups:  herd, 15
#> 
#> Fixed effects:
#>             Estimate Std. Error z value Pr(>|z|)    
#> (Intercept)  -1.4142     0.2466  -5.736  9.7e-09 ***
#> period2      -0.9803     0.3037  -3.227 0.001249 ** 
#> period3      -1.1171     0.3233  -3.455 0.000549 ***
#> period4      -1.5667     0.4222  -3.710 0.000207 ***
#> ---
#> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> 
#> Correlation of Fixed Effects:
#>         (Intr) perid2 perid3
#> period2 -0.342              
#> period3 -0.320  0.282       
#> period4 -0.246  0.215  0.199

8.2.7 brms

表示二项分布,公式语法与前面的 lme4 等包不同。

cbpp_brms <- brms::brm(
  incidence | trials(size) ~ period + (1 | herd),
  family = binomial("logit"), data = cbpp
)
summary(cbpp_brms)
 Family: binomial 
  Links: mu = logit 
Formula: incidence | trials(size) ~ period + (1 | herd) 
   Data: cbpp (Number of observations: 56) 
  Draws: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
         total post-warmup draws = 4000

Group-Level Effects: 
~herd (Number of levels: 15) 
              Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
sd(Intercept)     0.76      0.22     0.39     1.29 1.00     1483     1962

Population-Level Effects: 
          Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
Intercept    -1.40      0.26    -1.92    -0.88 1.00     2440     2542
period2      -1.00      0.31    -1.63    -0.41 1.00     5242     2603
period3      -1.14      0.34    -1.83    -0.50 1.00     4938     3481
period4      -1.61      0.44    -2.49    -0.81 1.00     4697     2966

Draws were sampled using sampling(NUTS). For each parameter, Bulk_ESS
and Tail_ESS are effective sample size measures, and Rhat is the potential
scale reduction factor on split chains (at convergence, Rhat = 1).

8.2.8 MCMCglmm

set.seed(20232023)
cbpp_mcmcglmm <- MCMCglmm::MCMCglmm(
  cbind(incidence, size - incidence) ~ period, random = ~herd,
  data = cbpp, family = "multinomial2", verbose = FALSE
)
summary(cbpp_mcmcglmm)
#> 
#>  Iterations = 3001:12991
#>  Thinning interval  = 10
#>  Sample size  = 1000 
#> 
#>  DIC: 538.4141 
#> 
#>  G-structure:  ~herd
#> 
#>      post.mean  l-95% CI u-95% CI eff.samp
#> herd    0.0244 1.256e-16   0.1347    186.5
#> 
#>  R-structure:  ~units
#> 
#>       post.mean l-95% CI u-95% CI eff.samp
#> units     1.098   0.2471    2.158    273.9
#> 
#>  Location effects: cbind(incidence, size - incidence) ~ period 
#> 
#>             post.mean l-95% CI u-95% CI eff.samp  pMCMC    
#> (Intercept)   -1.5314  -2.1484  -0.8507   1000.0 <0.001 ***
#> period2       -1.2596  -2.2129  -0.1495    854.7  0.006 ** 
#> period3       -1.3827  -2.3979  -0.2851    691.0  0.012 *  
#> period4       -1.9612  -3.3031  -0.7745    572.6 <0.001 ***
#> ---
#> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

对于服从非高斯分布的响应变量,MCMCglmm 总是假定存在过度离散的情况,即存在一个与分类变量无关的随机变量,或者说存在一个残差服从正态分布的随机变量(效应),可以看作测量误差,这种假定对真实数据建模是有意义的,所以,与以上 MCMCglmm 代码等价的 lme4 包模型代码如下:

cbpp$id <- as.factor(1:dim(cbpp)[1])
cbpp_lme4 <- lme4::glmer(
  cbind(incidence, size - incidence) ~ period + (1 | herd) + (1 | id),
  family = binomial, data = cbpp
)
summary(cbpp_lme4)
#> Generalized linear mixed model fit by maximum likelihood (Laplace
#>   Approximation) [glmerMod]
#>  Family: binomial  ( logit )
#> Formula: cbind(incidence, size - incidence) ~ period + (1 | herd) + (1 |  
#>     id)
#>    Data: cbpp
#> 
#>      AIC      BIC   logLik deviance df.resid 
#>    186.6    198.8    -87.3    174.6       50 
#> 
#> Scaled residuals: 
#>     Min      1Q  Median      3Q     Max 
#> -1.2866 -0.5989 -0.1181  0.3575  1.6216 
#> 
#> Random effects:
#>  Groups Name        Variance Std.Dev.
#>  id     (Intercept) 0.79400  0.8911  
#>  herd   (Intercept) 0.03384  0.1840  
#> Number of obs: 56, groups:  id, 56; herd, 15
#> 
#> Fixed effects:
#>             Estimate Std. Error z value Pr(>|z|)    
#> (Intercept)  -1.5003     0.2967  -5.056 4.27e-07 ***
#> period2      -1.2265     0.4803  -2.554  0.01066 *  
#> period3      -1.3288     0.4939  -2.690  0.00713 ** 
#> period4      -1.8662     0.5936  -3.144  0.00167 ** 
#> ---
#> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> 
#> Correlation of Fixed Effects:
#>         (Intr) perid2 perid3
#> period2 -0.559              
#> period3 -0.537  0.373       
#> period4 -0.441  0.327  0.314

贝叶斯的结果与频率派的结果相近,但还是有明显差异。MCMCglmm 总是假定存在残差,残差的分布服从 0 均值的高斯分布,下面将残差分布的方差固定,重新拟合模型,之后再根据残差方差为 0 调整估计结果。

prior2 <- list(
  R = list(V = 1, fix = 1),
  G = list(G1 = list(V = 1, nu = 0.002))
)
set.seed(20232023)
cbpp_mcmcglmm <- MCMCglmm::MCMCglmm(
  cbind(incidence, size - incidence) ~ period, random = ~herd, prior = prior2,
  data = cbpp, family = "multinomial2", verbose = FALSE
)
summary(cbpp_mcmcglmm)
#> 
#>  Iterations = 3001:12991
#>  Thinning interval  = 10
#>  Sample size  = 1000 
#> 
#>  DIC: 536.3978 
#> 
#>  G-structure:  ~herd
#> 
#>      post.mean  l-95% CI u-95% CI eff.samp
#> herd   0.09136 0.0001426   0.4399    312.5
#> 
#>  R-structure:  ~units
#> 
#>       post.mean l-95% CI u-95% CI eff.samp
#> units         1        1        1        0
#> 
#>  Location effects: cbind(incidence, size - incidence) ~ period 
#> 
#>             post.mean l-95% CI u-95% CI eff.samp  pMCMC    
#> (Intercept)   -1.5568  -2.1070  -0.8795    841.0 <0.001 ***
#> period2       -1.2424  -2.2081  -0.1868    843.2  0.014 *  
#> period3       -1.3416  -2.4066  -0.3783    888.0  0.006 ** 
#> period4       -1.8745  -3.1598  -0.7545    644.5 <0.001 ***
#> ---
#> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

下面对结果进行调整

# 调整常数
c2 <- ((16 * sqrt(3)) / (15 * pi))^2
# 固定效应
cbpp_sol_adj <- cbpp_mcmcglmm$Sol / sqrt(1 + c2 * cbpp_mcmcglmm$VCV[, 2])
summary(cbpp_sol_adj)
#> 
#> Iterations = 3001:12991
#> Thinning interval = 10 
#> Number of chains = 1 
#> Sample size per chain = 1000 
#> 
#> 1. Empirical mean and standard deviation for each variable,
#>    plus standard error of the mean:
#> 
#>               Mean     SD Naive SE Time-series SE
#> (Intercept) -1.342 0.2831 0.008953       0.009763
#> period2     -1.071 0.4468 0.014128       0.015386
#> period3     -1.156 0.4603 0.014555       0.015445
#> period4     -1.616 0.5256 0.016621       0.020703
#> 
#> 2. Quantiles for each variable:
#> 
#>               2.5%    25%    50%     75%   97.5%
#> (Intercept) -1.847 -1.552 -1.347 -1.1431 -0.7918
#> period2     -1.921 -1.352 -1.061 -0.7827 -0.1840
#> period3     -2.066 -1.478 -1.136 -0.8395 -0.3173
#> period4     -2.702 -1.970 -1.608 -1.2584 -0.5811
# 方差成分
cbpp_vcv_adj <- cbpp_mcmcglmm$VCV / (1 + c2 * cbpp_mcmcglmm$VCV[, 2])
summary(cbpp_vcv_adj)
#> 
#> Iterations = 3001:12991
#> Thinning interval = 10 
#> Number of chains = 1 
#> Sample size per chain = 1000 
#> 
#> 1. Empirical mean and standard deviation for each variable,
#>    plus standard error of the mean:
#> 
#>          Mean     SD Naive SE Time-series SE
#> herd  0.06788 0.1335 0.004221        0.00755
#> units 0.74303 0.0000 0.000000        0.00000
#> 
#> 2. Quantiles for each variable:
#> 
#>            2.5%      25%     50%     75%  97.5%
#> herd  0.0005777 0.004345 0.01809 0.07155 0.4584
#> units 0.7430287 0.743029 0.74303 0.74303 0.7430

可以看到,调整后固定效应的部分和前面 lme4 等的输出非常接近,方差成分仍有差距。

8.2.9 INLA

表示二项分布,公式语法与前面的 brms 包和 lme4 等包都不同。

cbpp_inla <- inla(
  formula = incidence ~ period + f(herd, model = "iid", n = 15),
  Ntrials = size, family = "binomial", data = cbpp
)
summary(cbpp_inla)
#> Fixed effects:
#>               mean    sd 0.025quant 0.5quant 0.975quant   mode kld
#> (Intercept) -1.381 0.224     -1.843   -1.375     -0.958 -1.375   0
#> period2     -1.033 0.304     -1.628   -1.033     -0.435 -1.033   0
#> period3     -1.174 0.324     -1.810   -1.175     -0.537 -1.175   0
#> period4     -1.663 0.425     -2.496   -1.663     -0.828 -1.663   0
#> 
#> Model hyperparameters:
#>                    mean   sd 0.025quant 0.5quant 0.975quant mode
#> Precision for herd 5.13 6.52       1.02     3.29      18.62 2.14
#> 
#>  is computed

8.3 非线性混合效应模型

Loblolly 数据集来自 R 内置的 datasets 包,记录了 14 颗火炬树种子的生长情况。

表格 8.2: Loblolly 数据集
Seed 3 5 10 15 20 25
301 4.51 10.89 28.72 41.74 52.70 60.92
303 4.55 10.92 29.07 42.83 53.88 63.39
305 4.79 11.37 30.21 44.40 55.82 64.10
307 3.91 9.48 25.66 39.07 50.78 59.07
309 4.81 11.20 28.66 41.66 53.31 63.05
311 3.88 9.40 25.99 39.55 51.46 59.64
315 4.32 10.43 27.16 40.85 51.33 60.07
319 4.57 10.57 27.90 41.13 52.43 60.69
321 3.77 9.03 25.45 38.98 49.76 60.28
323 4.33 10.79 28.97 42.44 53.17 61.62
325 4.38 10.48 27.93 40.20 50.06 58.49
327 4.12 9.92 26.54 37.82 48.43 56.81
329 3.93 9.34 26.08 37.79 48.31 56.43
331 3.46 9.05 25.85 39.15 49.12 59.49

火炬树种子基本决定了树的长势,不同种子预示最后的高度,并且在生长期也是很稳定地生长

ggplot(data = Loblolly, aes(x = age, y = height, color = Seed)) +
  geom_point() +
  geom_line() +
  theme_bw() +
  labs(x = "age (yr)", y = "height (ft)")
图 8.5: 火炬松树的高度(英尺)随时间(年)的变化

8.3.1 nlme

非线性回归

nfm1 <- nls(height ~ SSasymp(age, Asym, R0, lrc),
           data = Loblolly, subset = Seed == 329)
summary(nfm1)
#> 
#> Formula: height ~ SSasymp(age, Asym, R0, lrc)
#> 
#> Parameters:
#>      Estimate Std. Error t value Pr(>|t|)    
#> Asym  94.1282     8.4030  11.202 0.001525 ** 
#> R0    -8.2508     1.2261  -6.729 0.006700 ** 
#> lrc   -3.2176     0.1386 -23.218 0.000175 ***
#> ---
#> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> 
#> Residual standard error: 0.7493 on 3 degrees of freedom
#> 
#> Number of iterations to convergence: 0 
#> Achieved convergence tolerance: 3.919e-07

非线性函数 SSasymp() 的内容如下

\[ \mathrm{Asym}+(\mathrm{R0}-\mathrm{Asym})\times\exp\big(-\exp(\mathrm{lrc})\times\mathrm{input}\big) \]

其中,\(\mathrm{Asym}\)\(\mathrm{R0}\)\(\mathrm{lrc}\) 是参数,\(\mathrm{input}\) 是输入值。

示例来自 nlme 包的函数 nlme() 帮助文档

nfm2 <- nlme(height ~ SSasymp(age, Asym, R0, lrc),
  data = Loblolly,
  fixed = Asym + R0 + lrc ~ 1,
  random = Asym ~ 1,
  start = c(Asym = 103, R0 = -8.5, lrc = -3.3)
)
summary(nfm2)
#> Nonlinear mixed-effects model fit by maximum likelihood
#>   Model: height ~ SSasymp(age, Asym, R0, lrc) 
#>   Data: Loblolly 
#>        AIC      BIC    logLik
#>   239.4856 251.6397 -114.7428
#> 
#> Random effects:
#>  Formula: Asym ~ 1 | Seed
#>             Asym  Residual
#> StdDev: 3.650642 0.7188625
#> 
#> Fixed effects:  Asym + R0 + lrc ~ 1 
#>          Value Std.Error DF   t-value p-value
#> Asym 101.44960 2.4616951 68  41.21128       0
#> R0    -8.62733 0.3179505 68 -27.13420       0
#> lrc   -3.23375 0.0342702 68 -94.36052       0
#>  Correlation: 
#>     Asym   R0    
#> R0   0.704       
#> lrc -0.908 -0.827
#> 
#> Standardized Within-Group Residuals:
#>         Min          Q1         Med          Q3         Max 
#> -2.23601930 -0.62380854  0.05917466  0.65727206  1.95794425 
#> 
#> Number of Observations: 84
#> Number of Groups: 14
# 更新模型的随机效应部分
nfm3 <- update(nfm2, random = pdDiag(Asym + lrc ~ 1))
summary(nfm3)
#> Nonlinear mixed-effects model fit by maximum likelihood
#>   Model: height ~ SSasymp(age, Asym, R0, lrc) 
#>   Data: Loblolly 
#>        AIC      BIC    logLik
#>   238.9662 253.5511 -113.4831
#> 
#> Random effects:
#>  Formula: list(Asym ~ 1, lrc ~ 1)
#>  Level: Seed
#>  Structure: Diagonal
#>             Asym        lrc  Residual
#> StdDev: 2.806185 0.03449969 0.6920003
#> 
#> Fixed effects:  Asym + R0 + lrc ~ 1 
#>          Value Std.Error DF   t-value p-value
#> Asym 101.85205 2.3239828 68  43.82651       0
#> R0    -8.59039 0.3058441 68 -28.08747       0
#> lrc   -3.24011 0.0345017 68 -93.91167       0
#>  Correlation: 
#>     Asym   R0    
#> R0   0.727       
#> lrc -0.902 -0.796
#> 
#> Standardized Within-Group Residuals:
#>         Min          Q1         Med          Q3         Max 
#> -2.06072906 -0.69785679  0.08721706  0.73687722  1.79015782 
#> 
#> Number of Observations: 84
#> Number of Groups: 14

8.3.2 lme4

lme4 的公式语法是与 nlme 包不同的。

lob_lme4 <- lme4::nlmer(
  height ~ SSasymp(age, Asym, R0, lrc) ~ (Asym + R0 + lrc) + (Asym | Seed),
  data = Loblolly,
  start = c(Asym = 103, R0 = -8.5, lrc = -3.3)
)
summary(lob_lme4)
#> Nonlinear mixed model fit by maximum likelihood  ['nlmerMod']
#> Formula: height ~ SSasymp(age, Asym, R0, lrc) ~ (Asym + R0 + lrc) + (Asym |  
#>     Seed)
#>    Data: Loblolly
#> 
#>      AIC      BIC   logLik deviance df.resid 
#>    239.4    251.5   -114.7    229.4       79 
#> 
#> Scaled residuals: 
#>      Min       1Q   Median       3Q      Max 
#> -2.24149 -0.62546  0.08326  0.67711  1.91351 
#> 
#> Random effects:
#>  Groups   Name Variance Std.Dev.
#>  Seed     Asym 13.5121  3.6759  
#>  Residual       0.5161  0.7184  
#> Number of obs: 84, groups:  Seed, 14
#> 
#> Fixed effects:
#>        Estimate Std. Error t value
#> Asym 102.119832   0.012378  8249.9
#> R0    -8.549401   0.012354  -692.0
#> lrc   -3.243973   0.008208  -395.2
#> 
#> Correlation of Fixed Effects:
#>     Asym   R0    
#> R0   0.000       
#> lrc -0.008 -0.034

8.3.3 brms

根据数据的情况,设定参数的先验分布

lob_prior <- c(
  brms::set_prior("normal(101, 0.1)", nlpar = "Asym", lb = 100, ub = 102),
  brms::set_prior("normal(-8, 1)", nlpar = "R0", lb = -10),
  brms::set_prior("normal(-3, 3)", nlpar = "lrc", lb = -9),
  brms::set_prior("normal(3, 0.2)", class = "sigma")
)

根据模型表达式编码

lob_formula <- brms::bf(
  height ~ Asym + (R0 - Asym) * exp( - exp(lrc) * age),
  # Nonlinear variables
  # Fixed effects: Asym R0 lrc
  R0 + lrc ~ 1,
  # Nonlinear variables
  # Random effects: Seed
  Asym ~ 1 + (1 | Seed),
  # Nonlinear fit
  nl = TRUE
)
lob_brms <- brms::brm(lob_formula, data = Loblolly, prior = lob_prior)
summary(lob_brms)
 Family: gaussian 
  Links: mu = identity; sigma = identity 
Formula: height ~ Asym + (R0 - Asym) * exp(-exp(lrc) * age) 
         R0 ~ 1
         lrc ~ 1
         Asym ~ 1 + (1 | Seed)
   Data: Loblolly (Number of observations: 84) 
  Draws: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
         total post-warmup draws = 4000

Group-Level Effects: 
~Seed (Number of levels: 14) 
                   Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
sd(Asym_Intercept)     3.90      1.09     2.24     6.51 1.00     1033     1647

Population-Level Effects: 
               Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
R0_Intercept      -8.53      0.43    -9.37    -7.68 1.00     2236     1434
lrc_Intercept     -3.23      0.02    -3.27    -3.20 1.00      981     1546
Asym_Intercept   101.00      0.10   100.80   101.20 1.00     4443     2907

Family Specific Parameters: 
      Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
sigma     1.68      0.25     1.20     2.17 1.00     1910     2258

Draws were sampled using sampling(NUTS). For each parameter, Bulk_ESS
and Tail_ESS are effective sample size measures, and Rhat is the potential
scale reduction factor on split chains (at convergence, Rhat = 1).

8.4 模拟实验比较(补充)

从广义线性混合效应模型生成模拟数据,用至少 6 个不同的 R 包估计模型参数,比较和归纳不同估计方法和实现算法的效果。举例:带漂移项的泊松型广义线性混合效应模型。\(y_{ij}\) 表示响应变量,\(\bm{u}\) 表示随机效应,\(o_{ij}\) 表示漂移项。

\[ \begin{aligned} y_{ij}|\bm{u} &\sim \mathrm{Poisson}(o_{ij}\lambda_{ij}) \\ \log(\lambda_{ij}) &= \beta_{ij}x_{ij} + u_{j} \\ u_j &\sim \mathcal{N}(0, \sigma^2) \\ i = 1,2,\ldots, n &\quad j = 1,2,\ldots,q \end{aligned} \]

首先准备数据

set.seed(2023)
Ngroups <- 25 # 一个随机效应分 25 个组
NperGroup <- 100 # 每个组 100 个观察值
# 样本量
N <- Ngroups * NperGroup
# 截距和两个协变量的系数
beta <- c(0.5, 0.3, 0.2)
# 两个协变量
X <- MASS::mvrnorm(N, mu = rep(0, 2), Sigma = matrix(c(1, 0.8, 0.8, 1), 2))
# 漂移项
o <- rep(c(2, 4), each = N / 2)
# 分 25 个组 每个组 100 个观察值
g <- factor(rep(1:Ngroups, each = NperGroup))
u <- rnorm(Ngroups, sd = .5) # 随机效应的标准差 0.5
# 泊松分布的期望
lambda <- o * exp(cbind(1, X) %*% beta + u[g])
# 响应变量的值
y <- rpois(N, lambda = lambda)
# 模拟的数据集
sim_data <- data.frame(y, X, o, g)
colnames(sim_data) <- c("y", "x1", "x2", "o", "g")

8.4.1 lme4

# 模型拟合
fit_lme4 <- lme4::glmer(y ~ x1 + x2 + (1 | g),
  data = sim_data, offset = log(o), family = poisson(link = "log")
)
summary(fit_lme4)
#> Generalized linear mixed model fit by maximum likelihood (Laplace
#>   Approximation) [glmerMod]
#>  Family: poisson  ( log )
#> Formula: y ~ x1 + x2 + (1 | g)
#>    Data: sim_data
#>  Offset: log(o)
#> 
#>      AIC      BIC   logLik deviance df.resid 
#>  11065.6  11088.9  -5528.8  11057.6     2496 
#> 
#> Scaled residuals: 
#>     Min      1Q  Median      3Q     Max 
#> -3.0650 -0.7177 -0.0827  0.6334  4.0103 
#> 
#> Random effects:
#>  Groups Name        Variance Std.Dev.
#>  g      (Intercept) 0.4074   0.6383  
#> Number of obs: 2500, groups:  g, 25
#> 
#> Fixed effects:
#>             Estimate Std. Error z value Pr(>|z|)    
#> (Intercept)  0.55607    0.12805   4.343 1.41e-05 ***
#> x1           0.28442    0.01280  22.214  < 2e-16 ***
#> x2           0.20851    0.01294  16.117  < 2e-16 ***
#> ---
#> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> 
#> Correlation of Fixed Effects:
#>    (Intr) x1    
#> x1 -0.008       
#> x2 -0.008 -0.821

8.4.2 GLMMadaptive

对随机效应采用 adaptive Gauss-Hermite quadrature 积分

library(GLMMadaptive)
fit_glmmadaptive <- mixed_model(
  fixed = y ~ x1 + x2 + offset(log(o)), 
  random = ~ 1 | g, data = sim_data,
  family = poisson(link = "log")
)
summary(fit_glmmadaptive)
#> 
#> Call:
#> mixed_model(fixed = y ~ x1 + x2 + offset(log(o)), random = ~1 | 
#>     g, data = sim_data, family = poisson(link = "log"))
#> 
#> Data Descriptives:
#> Number of Observations: 2500
#> Number of Groups: 25 
#> 
#> Model:
#>  family: poisson
#>  link: log 
#> 
#> Fit statistics:
#>   log.Lik      AIC      BIC
#>  -5528.78 11065.56 11070.44
#> 
#> Random effects covariance matrix:
#>                StdDev
#> (Intercept) 0.6417329
#> 
#> Fixed effects:
#>             Estimate Std.Err z-value p-value
#> (Intercept)   0.5647  0.1288  4.3863 < 1e-04
#> x1            0.2844  0.0128 22.2059 < 1e-04
#> x2            0.2085  0.0129 16.1096 < 1e-04
#> 
#> Integration:
#> method: adaptive Gauss-Hermite quadrature rule
#> quadrature points: 11
#> 
#> Optimization:
#> method: hybrid EM and quasi-Newton
#> converged: TRUE

8.4.3 glmmTMB

fit_glmmtmb <- glmmTMB::glmmTMB(
  y ~ x1 + x2 + (1 | g), offset = log(o),
  data = sim_data, family = poisson, REML = TRUE
)
summary(fit_glmmtmb)
#>  Family: poisson  ( log )
#> Formula:          y ~ x1 + x2 + (1 | g)
#> Data: sim_data
#>  Offset: log(o)
#> 
#>      AIC      BIC   logLik deviance df.resid 
#>  11082.7  11106.0  -5537.3  11074.7     2499 
#> 
#> Random effects:
#> 
#> Conditional model:
#>  Groups Name        Variance Std.Dev.
#>  g      (Intercept) 0.4245   0.6515  
#> Number of obs: 2500, groups:  g, 25
#> 
#> Conditional model:
#>             Estimate Std. Error z value Pr(>|z|)    
#> (Intercept)  0.55710    0.13069   4.263 2.02e-05 ***
#> x1           0.28442    0.01281  22.206  < 2e-16 ***
#> x2           0.20851    0.01294  16.111  < 2e-16 ***
#> ---
#> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

8.4.4 hglm

hglm 包的名称是 Hierarchical Generalized Linear Models 的首字母缩写拼成的。

# extended quasi likelihood (EQL) method
fit_hglm <- hglm::hglm(
  fixed =  y ~ x1 + x2, random = ~ 1 | g,
  family = poisson(link = "log"), 
  offset = log(o), data = sim_data
)
summary(fit_hglm)
#> Call: 
#> hglm.formula(family = poisson(link = "log"), fixed = y ~ x1 + 
#>     x2, random = ~1 | g, data = sim_data, offset = log(o))
#> 
#> ----------
#> MEAN MODEL
#> ----------
#> 
#> Summary of the fixed effects estimates:
#> 
#>             Estimate Std. Error t-value Pr(>|t|)    
#> (Intercept)  0.55720    0.13058   4.267 2.05e-05 ***
#> x1           0.28442    0.01316  21.609  < 2e-16 ***
#> x2           0.20851    0.01330  15.678  < 2e-16 ***
#> ---
#> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> Note: P-values are based on 2473 degrees of freedom
#> 
#> Summary of the random effects estimates:
#> 
#>    Estimate Std. Error
#> g1   0.6643     0.1352
#> g2  -0.1421     0.1402
#> g3   0.9560     0.1344
#> ...
#> NOTE: to show all the random effects, use print(summary(hglm.object), print.ranef = TRUE).
#> 
#> ----------------
#> DISPERSION MODEL
#> ----------------
#> 
#> NOTE: h-likelihood estimates through EQL can be biased.
#> 
#> Dispersion parameter for the mean model:
#> [1] 1.055997
#> 
#> Model estimates for the dispersion term:
#> 
#> Link = log 
#> 
#> Effects:
#>   Estimate Std. Error 
#>     0.0545     0.0284 
#> 
#> Dispersion = 1 is used in Gamma model on deviances to calculate the standard error(s).
#> 
#> Dispersion parameter for the random effects:
#> [1] 0.4236
#> 
#> Dispersion model for the random effects:
#> 
#> Link = log
#> 
#> Effects:
#> .|Random1 
#>   Estimate Std. Error 
#>    -0.8589     0.2895 
#> 
#> Dispersion = 1 is used in Gamma model on deviances to calculate the standard error(s).
#> 
#> EQL estimation converged in 3 iterations.
#> 
#> !! Observation 302 is too influential! Estimates are likely unreliable !!

8.4.5 glmmML

glmmML 包 Maximum Likelihood and numerical integration via Gauss-Hermite quadrature

fit_glmmml <- glmmML::glmmML(
  formula = y ~ x1 + x2, family = poisson,
  data = sim_data, offset = log(o), cluster = g
)
summary(fit_glmmml)
Call:  glmmML::glmmML(formula = y ~ x1 + x2, family = poisson, data = sim_data,      cluster = g, offset = log(o)) 


             coef se(coef)     z Pr(>|z|)
(Intercept) 0.556   0.1281  4.34  1.4e-05
x1          0.284   0.0128 22.21  0.0e+00
x2          0.209   0.0129 16.11  0.0e+00

Scale parameter in mixing distribution:  0.638 gaussian 
Std. Error:                              0.0865 

        LR p-value for H_0: sigma = 0:  0 

Residual deviance: 2770 on 2496 degrees of freedom  AIC: 2780 

8.4.6 glmm

glmm 包对随机效应的积分采用 Monte Carlo Likelihood Approximation 近似

# 对迭代时间没有给出预估,一旦执行,不知道什么时候会跑完
set.seed(2023)
# 设置双核并行迭代
clust <- parallel::makeCluster(2)
fit_glmm <- glmm::glmm(y ~ x1 + x2 + offset(log(o)),
  random = list(~ 1 + g), # 随机效应
  varcomps.names = "G", # 给随机效应取个名字
  data = sim_data,
  family.glmm = glmm::poisson.glmm, # 泊松型
  m = 10^4, debug = TRUE, cluster = clust
)
parallel::stopCluster(clust)
summary(fit_glmm)

glmm 包的帮助文档中的示例如下,可复现结果,运行时间 1-2 分钟。

set.seed(1234)
clust <- makeCluster(2)
sal <- glmm(
  Mate ~ 0 + Cross, random = list(~ 0 + Female, ~ 0 + Male),
  varcomps.names = c("F", "M"), data = salamander,
  family.glmm = bernoulli.glmm, m = 10^4, debug = TRUE, cluster = clust
)
summary(sal)
stopCluster(clust)
Call:
glmm(fixed = Mate ~ 0 + Cross, random = list(~0 + Female, ~0 + Male), 
    varcomps.names = c("F", "M"), data = salamander, 
    family.glmm = bernoulli.glmm, m = 10^4, debug = TRUE, cluster = clust)

Link is: "logit (log odds)"

Fixed Effects:
         Estimate Std. Error z value Pr(>|z|)    
CrossR/R    1.230      0.300   4.045 5.24e-05 ***
CrossR/W    0.320      0.267   1.198  0.23077    
CrossW/R   -2.000      0.330  -6.042 1.52e-09 ***
CrossW/W    0.920      0.300   3.084  0.00204 ** 
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Variance Components for Random Effects (P-values are one-tailed):
  Estimate Std. Error z value Pr(>|z|)/2    
F     1.46       0.31   4.695   1.33e-06 ***
M     1.64       0.33   4.918   4.36e-07 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

8.4.7 gee

gee 包采用广义估计方程(Generalized Estimation Equation)方法

fit_gee <- gee::gee(y ~ x1 + x2 + offset(log(o)), id = g, 
  data = sim_data, family = poisson(link = "log"), corstr = "exchangeable"
)
# 输出
fit_gee
 GEE:  GENERALIZED LINEAR MODELS FOR DEPENDENT DATA
 gee S-function, version 4.13 modified 98/01/27 (1998) 

Model:
 Link:                      Logarithm 
 Variance to Mean Relation: Poisson 
 Correlation Structure:     Exchangeable 

Call:
gee::gee(formula = y ~ x1 + x2 + offset(log(o)), id = g, data = sim_data, 
    family = poisson(link = "log"), corstr = "exchangeable")

Number of observations :  2500 
Maximum cluster size   :  100 

Coefficients:
(Intercept)          x1          x2 
  0.6098935   0.3003721   0.2165055 

Estimated Scale Parameter:  4.979956
Number of Iterations:  3

Working Correlation[1:4,1:4]
          [,1]      [,2]      [,3]      [,4]
[1,] 1.0000000 0.7220617 0.7220617 0.7220617
[2,] 0.7220617 1.0000000 0.7220617 0.7220617
[3,] 0.7220617 0.7220617 1.0000000 0.7220617
[4,] 0.7220617 0.7220617 0.7220617 1.0000000

Returned Error Value:
[1] 0

输出结果中,尺度参数(Estimated Scale Parameter)的估计结果与随机效应的方差的联系?

8.4.8 geepack

geepack 包类似 gee 包。

fit_geepack <- geepack::geeglm(
  formula = y ~ x1 + x2, family = poisson(link = "log"),
  id = g, offset = log(o), data = sim_data,
  corstr = "exchangeable", scale.fix = FALSE
)
summary(fit_geepack)
Call:
geepack::geeglm(formula = y ~ x1 + x2, family = poisson(link = "log"), 
    data = sim_data, offset = log(o), id = g, corstr = "exchangeable", 
    scale.fix = FALSE)

 Coefficients:
            Estimate Std.err  Wald Pr(>|W|)    
(Intercept)  0.60964 0.17310  12.4 0.000428 ***
x1           0.30040 0.02353 163.1  < 2e-16 ***
x2           0.21653 0.01458 220.6  < 2e-16 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Correlation structure = exchangeable 
Estimated Scale Parameters:

            Estimate Std.err
(Intercept)    4.975    1.39
  Link = identity 

Estimated Correlation Parameters:
      Estimate Std.err
alpha    0.723 0.06703
Number of clusters:   25  Maximum cluster size: 100 

8.4.9 blme

blme 包采用贝叶斯估计

fit_blme <- blme::bglmer(
  formula = y ~ x1 + x2 + (1 | g),
  data = sim_data, offset = log(o),
  family = poisson(link = "log")
)
summary(fit_blme)
#> Cov prior  : g ~ wishart(df = 3.5, scale = Inf, posterior.scale = cov, common.scale = TRUE)
#> Prior dev  : 1.2531
#> 
#> Generalized linear mixed model fit by maximum likelihood (Laplace
#>   Approximation) [bglmerMod]
#>  Family: poisson  ( log )
#> Formula: y ~ x1 + x2 + (1 | g)
#>    Data: sim_data
#>  Offset: log(o)
#> 
#>      AIC      BIC   logLik deviance df.resid 
#>  11065.6  11088.9  -5528.8  11057.6     2496 
#> 
#> Scaled residuals: 
#>     Min      1Q  Median      3Q     Max 
#> -3.0651 -0.7180 -0.0816  0.6335  4.0103 
#> 
#> Random effects:
#>  Groups Name        Variance Std.Dev.
#>  g      (Intercept) 0.4337   0.6585  
#> Number of obs: 2500, groups:  g, 25
#> 
#> Fixed effects:
#>             Estimate Std. Error z value Pr(>|z|)    
#> (Intercept)  0.55595    0.13209   4.209 2.57e-05 ***
#> x1           0.28442    0.01280  22.214  < 2e-16 ***
#> x2           0.20851    0.01294  16.117  < 2e-16 ***
#> ---
#> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> 
#> Correlation of Fixed Effects:
#>    (Intr) x1    
#> x1 -0.008       
#> x2 -0.007 -0.821

GLMMadaptive、glmmML、gee、geepack 和 lme4 的模型输出结果是接近的。

8.4.10 brms

fit_brms <- brms::brm(
  y ~ x1 + x2 + (1 | g) + offset(log(o)),
  data = sim_data, family = poisson(link = "log"),
  silent = 2, refresh = 0, seed = 20232023
)
summary(fit_brms)
 Family: poisson 
  Links: mu = log 
Formula: y ~ x1 + x2 + (1 | g) + offset(log(o)) 
   Data: sim_data (Number of observations: 2500) 
  Draws: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
         total post-warmup draws = 4000

Group-Level Effects: 
~g (Number of levels: 25) 
              Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
sd(Intercept)     0.68      0.11     0.51     0.94 1.01      295      491

Population-Level Effects: 
          Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
Intercept     0.56      0.14     0.31     0.85 1.02      297      344
x1            0.28      0.01     0.26     0.31 1.00     1053     1625
x2            0.21      0.01     0.18     0.23 1.01     1071     1298

Draws were sampled using sampling(NUTS). For each parameter, Bulk_ESS
and Tail_ESS are effective sample size measures, and Rhat is the potential
scale reduction factor on split chains (at convergence, Rhat = 1).

8.4.11 MCMCglmm

MCMCglmm 包采用贝叶斯估计

prior1 <- list(
  R = list(V = 1, fix = 1), 
  G = list(G1 = list(V = 1, nu = 0.002))
)
set.seed(20232023)
fit_mcmcglmm <- MCMCglmm::MCMCglmm(
  fixed = y ~ x1 + x2 + offset(log(o)),
  random = ~g, family = "poisson", 
  data = sim_data, verbose = FALSE, prior = prior1
)
summary(fit_mcmcglmm)
#> 
#>  Iterations = 3001:12991
#>  Thinning interval  = 10
#>  Sample size  = 1000 
#> 
#>  DIC: 12397.82 
#> 
#>  G-structure:  ~g
#> 
#>   post.mean l-95% CI u-95% CI eff.samp
#> g    0.5443   0.2753   0.8924     1000
#> 
#>  R-structure:  ~units
#> 
#>       post.mean l-95% CI u-95% CI eff.samp
#> units         1        1        1        0
#> 
#>  Location effects: y ~ x1 + x2 + offset(log(o)) 
#> 
#>             post.mean l-95% CI u-95% CI eff.samp  pMCMC    
#> (Intercept)    1.4315   1.1536   1.7161   1000.0 <0.001 ***
#> x1             0.3099   0.2405   0.3892    925.6 <0.001 ***
#> x2             0.2234   0.1465   0.3017    877.7 <0.001 ***
#> ---
#> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

随机效应的方差 G-structure 为 0.5443,则标准差为 0.738。

对于离散型响应变量,MCMCglmm 包默认添加一个可加的随机变量表示过度离散,如何将其去掉?将残差方差设置为常数,不再作为参数去估计,fix = 1 表示在 R-structure 中固定方差, V = 1 表示残差方差为 1。

# 固定效应参数的后验分布
# plot(fit_mcmcglmm$Sol)
plot(fit_mcmcglmm$VCV)
图 8.6: 方差协方差参数的后验分布

根据响应变量的服从的分布类型,确定调整因子。固定效应乘以调整因子的平方根,随机效应的方差乘以调整因子,详见 (Peter Diggle 和 Zeger 2002) 第 136-137 页。二项分布联系函数对应的调整因子如下:

\[ \frac{1 + c^2\sigma^2_{\epsilon}}{1 + c^2\sigma^2_{\mathrm{units}}} \]

其中, \(c\) 是与联系函数有关的常数,二项分布联系函数对应 \(c = 16\sqrt{3}/(15\pi)\)。此处,假定 \(\sigma^2_{\epsilon} = 0\) ,代入泊松分布对应的调整因子。调整后的固定效应(回归系数)、随机效应的方差如下:

# 调整公式中的调整因子 c2 取决于联系函数
c2 <- ((16 * sqrt(3))/(15 * pi))^2 # 需要修改为泊松分布对应的值
# 固定效应的调整
adjusted_sol <- fit_mcmcglmm$Sol / sqrt(1 + c2 * fit_mcmcglmm$VCV[, 2])
plot(adjusted_sol)
# 随机效应的方差调整
adjusted_vcv <- fit_mcmcglmm$VCV[, 1] / (1 + c2 * fit_mcmcglmm$VCV[, 2])
plot(adjusted_vcv)

8.4.12 INLA

library(INLA)
fit_inla <- inla(
  formula = y ~ x1 + x2 + f(g, model = "iid", n = 25),
  E = o, family = "poisson", data = sim_data
)
summary(fit_inla)
#> Fixed effects:
#>              mean    sd 0.025quant 0.5quant 0.975quant  mode kld
#> (Intercept) 0.556 0.130      0.300    0.556      0.812 0.556   0
#> x1          0.284 0.013      0.259    0.284      0.310 0.284   0
#> x2          0.209 0.013      0.183    0.209      0.234 0.209   0
#> 
#> Model hyperparameters:
#>                 mean    sd 0.025quant 0.5quant 0.975quant mode
#> Precision for g 2.56 0.713       1.36     2.49       4.14 2.36
#> 
#>  is computed

随机效应的标准(偏)差为 \(1/\sqrt{\mathrm{Precision}}\) ,即 0.625。

8.5 总结

本章介绍函数 MASS::glmmPQL()nlme::lme()lme4::lmer()brms::brm() 的用法,以及它们求解线性混合效应模型的区别和联系。在贝叶斯估计方法中,brms 包和 INLA 包都支持非常丰富的模型种类,前者是贝叶斯精确推断,后者是贝叶斯近似推断,brms 基于概率编程语言 Stan 框架打包了许多模型的 Stan 实现,INLA 基于求解随机偏微分方程的有限元方法和拉普拉斯近似技巧,将各类常见统计模型统一起来,计算速度快,计算结果准确。

  1. 函数 nlme::lme() 提供极大似然估计和限制极大似然估计。
  2. 函数 MASS::glmmPQL() 惩罚拟似然估计,MASS 是依赖 nlme 包, nlme 不支持模型中添加漂移项,所以函数 glmmPQL() 也不支持添加漂移项。
  3. 函数 lme4::lmer() 拉普拉斯近似关于随机效应的高维积分。
  4. 函数 brms::brm() 汉密尔顿蒙特卡罗抽样。HMC 方法结合自适应步长的采样器 NUTS 来抽样。
  5. 函数 INLA::inla() 集成嵌套拉普拉斯近似。
表格 8.3: 混合效应模型及相关 R 包拟合函数
模型 nlme MASS lme4 GLMMadaptive brms
线性混合效应模型 lme() glmmPQL() lmer() 不支持 brm()
广义线性混合效应模型 不支持 glmmPQL() glmer() mixed_model() brm()
非线性混合效应模型 nlme() 不支持 nlmer() 不支持 brm()

通过对频率派和贝叶斯派方法的比较,发现一些有意思的结果。与 Stan 不同,INLA 包做近似贝叶斯推断,计算效率很高。

INLA 软件能处理上千个高斯随机效应,但最多只能处理 15 个超参数,因为 INLA 使用 CCD 处理超参数。如果使用 MCMC 处理超参数,就有可能处理更多的超参数,Daniel Simpson 等把 Laplace approximation 带入 Stan,这样就可以处理上千个超参数。 更多理论内容见 2009 年 INLA 诞生的论文和《Advanced Spatial Modeling with Stochastic Partial Differential Equations Using R and INLA》中第一章的估计方法 CCD

8.6 习题

  1. 基于奥克兰火山地形数据集 volcano ,随机拆分成训练数据和测试数据,训练数据可以看作采样点的观测数据,建立高斯过程回归模型,比较测试数据与未采样的位置上的预测数据,在计算速度、准确度、易用性等方面总结 Stan 和 INLA 的特点。

  2. 基于 PlantGrowth 数据集,比较将 group 变量视为随机变量与随机效应的异同?

    fit_lm <- lm(weight ~ group, data = PlantGrowth)
    summary(fit_lm)
    fit_lme <- nlme::lme(weight ~ 1, random = ~ 1 | group, data = PlantGrowth)
    summary(fit_lme)
    fit_lme4 <- lme4::lmer(weight ~ 1 + (1 | group), data = PlantGrowth)
    summary(fit_lme4)
  3. MASS 包的数据集 epil 记录癫痫发作的次数及病人的特征,请建立混合效应模型分析癫痫病发作的风险与病人特征之间的关系。

    代码
    data(epil, package = "MASS")
    epil_glm <- glm(y ~ lbase * trt + lage + V4,
      family = poisson, data = epil
    )
    summary(epil_glm)
    
    epil_mass <- MASS::glmmPQL(y ~ lbase * trt + lage + V4,
      random = ~ 1 | subject, family = poisson, data = epil
    )
    summary(epil_mass)
    
    epil_lme4 <- lme4::glmer(
      y ~ lbase * trt + lage + V4 + (1 | subject),
      family = poisson, data = epil
    )
    summary(epil_lme4)
    
    epil_glmmtmb <- glmmTMB::glmmTMB(
      y ~ lbase * trt + lage + V4 + (1 | subject),
      data = epil, family = poisson, REML = TRUE
    )
    summary(epil_glmmtmb)
    
    epil_glmmadaptive <- GLMMadaptive::mixed_model(
      fixed = y ~ lbase * trt + lage + V4,
      random = ~ 1 | subject, data = epil,
      family = poisson()
    )
    summary(epil_glmmadaptive)
  4. 基于数据集 Puromycin 分析酶促反应的反应速率(提示:Michaelis-Menten 模型和函数 SSmicmen())。

    ggplot(data = Puromycin, aes(x = conc, y = rate, color = state)) +
      geom_point() +
      geom_line() +
      theme_minimal() +
      labs(
        x = "Substrate concentration (ppm)",
        y = "Reaction velocity (counts/min/min)"
      )
    图 8.7: Puromycin 反应速率变化趋势
  5. 基于 MASS 包的地形数据集 topo,建立高斯过程回归模型,比较贝叶斯预测与克里金插值预测的效果。

    代码
    data(topo, package = "MASS")
    set.seed(20232023)
    nchains <- 2 # 2 条迭代链
    # 给每条链设置不同的参数初始值
    inits_data_gaussian <- lapply(1:nchains, function(i) {
      list(
        beta = rnorm(1),
        sigma = runif(1),
        phi = runif(1),
        tau = runif(1)
      )
    })
    # 预测区域网格化
    nx <- ny <- 27
    topo_grid_df <- expand.grid(
      x = seq(from = 0, to = 6.5, length.out = nx),
      y = seq(from = 0, to = 6.5, length.out = ny)
    )
    # 对数高斯模型
    topo_gaussian_d <- list(
      N1 = nrow(topo), # 观测记录的条数
      N2 = nrow(topo_grid_df),
      D = 2, # 2 维坐标
      x1 = topo[, c("x", "y")], # N x 2 坐标矩阵
      x2 = topo_grid_df[, c("x", "y")],
      y1 = topo[, "z"] # N 向量
    )
    library(cmdstanr)
    # 编码
    mod_topo_gaussian <- cmdstan_model(
      stan_file = "code/gaussian_process_pred.stan",
      compile = TRUE, cpp_options = list(stan_threads = TRUE)
    )
    # 高斯过程回归模型
    fit_topo_gaussian <- mod_topo_gaussian$sample(
      data = topo_gaussian_d,   # 观测数据
      init = inits_data_gaussian,   # 迭代初值
      iter_warmup = 500,            # 每条链预处理迭代次数
      iter_sampling = 1000,         # 每条链总迭代次数
      chains = nchains,             # 马尔科夫链的数目
      parallel_chains = 2,      # 指定 CPU 核心数,可以给每条链分配一个
      threads_per_chain = 1,    # 每条链设置一个线程
      show_messages = FALSE,    # 不显示迭代的中间过程
      refresh = 0,              # 不显示采样的进度
      output_dir = "data-raw/",
      seed = 20232023           
    )
    # 诊断
    fit_topo_gaussian$diagnostic_summary()
    # 对数高斯模型
    fit_topo_gaussian$summary(
      variables = c("lp__", "beta", "sigma", "phi", "tau"),
      .num_args = list(sigfig = 4, notation = "dec")
    )
    # 未采样的位置的预测值
    ypred <- fit_topo_gaussian$summary(variables = "ypred", "mean")
    # 预测值
    topo_grid_df$ypred <- ypred$mean
    # 整理数据
    library(sf)
    topo_grid_sf <- st_as_sf(topo_grid_df, coords = c("x", "y"), dim = "XY")
    library(stars)
    # 26x26 的网格
    topo_grid_stars <- st_rasterize(topo_grid_sf, nx = 26, ny = 26)
    
    library(ggplot2)
    ggplot() +
      geom_stars(data = topo_grid_stars, aes(fill = ypred)) +
      scale_fill_viridis_c(option = "C") +
      theme_bw()

  1. https://stat.ethz.ch/pipermail/r-sig-mixed-models/2013q1/019945.html↩︎

  2. https://stat.ethz.ch/pipermail/r-sig-mixed-models/2010q2/003615.html↩︎