# Difference between a 2 factor ANOVA and mixed effects model

The lme4 package in R includes the cake dataset.

library(lme4)
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

I've analysed the cake dataset using two different models below. The first model is a 2 factor ANOVA:

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

...and the second is a mixed effects model, with temperature as a random effect:

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

Is someone able to provide a summary of what the mixed effect model has done differently to the ANOVA?