R 中的lme4包包括cake数据集。

library(lme4)
head(cake[,2:4], 20)
   recipe temperature angle
1       A         175    42
2       A         185    46
3       A         195    47
4       A         205    39
5       A         215    53
6       A         225    42
7       B         175    39
8       B         185    46
9       B         195    51
10      B         205    49
11      B         215    55
12      B         225    42
13      C         175    46
14      C         185    44
15      C         195    45
16      C         205    46
17      C         215    48
18      C         225    63
19      A         175    47
20      A         185    29

我cake使用以下两种不同的模型分析了数据集。第一个模型是 2 因子方差分析：

summary(aov(angle ~ temperature + recipe, cake))
             Df Sum Sq Mean Sq F value   Pr(>F)    
temperature   5   2100   420.1   6.918 4.37e-06 ***
recipe        2    135    67.5   1.112     0.33    
Residuals   262  15908    60.7                     
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

...第二个是混合效应模型，具有temperature随机效应：

lmer(angle ~ recipe + (1| temperature), data=cake, REML=F)
Linear mixed model fit by maximum likelihood 
Formula: angle ~ recipe + (1 | temperature) 
   Data: cake 
  AIC  BIC logLik deviance REMLdev
 1893 1911 -941.7     1883    1877
Random effects:
 Groups      Name        Variance Std.Dev.
 temperature (Intercept)  6.4399  2.5377  
 Residual                60.2560  7.7625  
Number of obs: 270, groups: temperature, 6

Fixed effects:
            Estimate Std. Error t value
(Intercept)   33.122      1.320  25.093
recipeB       -1.478      1.157  -1.277
recipeC       -1.522      1.157  -1.315

Correlation of Fixed Effects:
        (Intr) recipB
recipeB -0.438       
recipeC -0.438  0.500

有人能够提供混合效应模型与 ANOVA 不同之处的总结吗？