cake.RdData on the breakage angle of chocolate cakes made with three different recipes and baked at six different temperatures. This is a split-plot design with the recipes being whole-units and the different temperatures being applied to sub-units (within replicates). The experimental notes suggest that the replicate numbering represents temporal ordering.
A data frame with 270 observations on the following 5 variables.
replicatea factor with levels 1 to 15
recipea factor with levels A, B and C
temperaturean ordered factor with levels 175
< 185 < 195 < 205 < 215 < 225
anglea numeric vector giving the angle at which the cake broke.
tempnumeric value of the baking temperature (degrees F).
Original data were presented in Cook (1938), and reported in Cochran and Cox (1957, p. 300). Also cited in Lee, Nelder and Pawitan (2006).
The replicate factor is nested within the
recipe factor, and temperature is nested
within replicate.
Cook, F. E. (1938) Chocolate cake, I. Optimum baking temperature. Master's Thesis, Iowa State College.
Cochran, W. G., and Cox, G. M. (1957) Experimental designs, 2nd Ed. New York, John Wiley & Sons.
Lee, Y., Nelder, J. A., and Pawitan, Y. (2006) Generalized linear models with random effects. Unified analysis via H-likelihood. Boca Raton, Chapman and Hall/CRC.
str(cake)
#> 'data.frame': 270 obs. of 5 variables:
#> $ replicate : Factor w/ 15 levels "1","2","3","4",..: 1 1 1 1 1 1 1 1 1 1 ...
#> $ recipe : Factor w/ 3 levels "A","B","C": 1 1 1 1 1 1 2 2 2 2 ...
#> $ temperature: Ord.factor w/ 6 levels "175"<"185"<"195"<..: 1 2 3 4 5 6 1 2 3 4 ...
#> $ angle : int 42 46 47 39 53 42 39 46 51 49 ...
#> $ temp : num 175 185 195 205 215 225 175 185 195 205 ...
## 'temp' is continuous, 'temperature' an ordered factor with 6 levels
(fm1 <- lmer(angle ~ recipe * temperature + (1|recipe:replicate), cake, REML= FALSE))
#> Linear mixed model fit by maximum likelihood ['lmerMod']
#> Formula: angle ~ recipe * temperature + (1 | recipe:replicate)
#> Data: cake
#> AIC BIC logLik -2*log(L) df.resid
#> 1719.0519 1791.0203 -839.5259 1679.0519 250
#> Random effects:
#> Groups Name Std.Dev.
#> recipe:replicate (Intercept) 6.249
#> Residual 4.371
#> Number of obs: 270, groups: recipe:replicate, 45
#> Fixed Effects:
#> (Intercept) recipeB recipeC
#> 33.12222 -1.47778 -1.52222
#> temperature.L temperature.Q temperature.C
#> 6.43033 -0.71285 -2.32551
#> temperature^4 temperature^5 recipeB:temperature.L
#> -3.35128 -0.15119 0.45419
#> recipeC:temperature.L recipeB:temperature.Q recipeC:temperature.Q
#> 0.08765 -0.23277 1.21475
#> recipeB:temperature.C recipeC:temperature.C recipeB:temperature^4
#> 2.69322 2.63856 3.02372
#> recipeC:temperature^4 recipeB:temperature^5 recipeC:temperature^5
#> 3.13711 -0.66354 -1.62525
(fm2 <- lmer(angle ~ recipe + temperature + (1|recipe:replicate), cake, REML= FALSE))
#> Linear mixed model fit by maximum likelihood ['lmerMod']
#> Formula: angle ~ recipe + temperature + (1 | recipe:replicate)
#> Data: cake
#> AIC BIC logLik -2*log(L) df.resid
#> 1709.5822 1745.5665 -844.7911 1689.5822 260
#> Random effects:
#> Groups Name Std.Dev.
#> recipe:replicate (Intercept) 6.237
#> Residual 4.475
#> Number of obs: 270, groups: recipe:replicate, 45
#> Fixed Effects:
#> (Intercept) recipeB recipeC temperature.L temperature.Q
#> 33.1222 -1.4778 -1.5222 6.6109 -0.3855
#> temperature.C temperature^4 temperature^5
#> -0.5483 -1.2977 -0.9141
(fm3 <- lmer(angle ~ recipe + temp + (1|recipe:replicate), cake, REML= FALSE))
#> Linear mixed model fit by maximum likelihood ['lmerMod']
#> Formula: angle ~ recipe + temp + (1 | recipe:replicate)
#> Data: cake
#> AIC BIC logLik -2*log(L) df.resid
#> 1708.1578 1729.7483 -848.0789 1696.1578 264
#> Random effects:
#> Groups Name Std.Dev.
#> recipe:replicate (Intercept) 6.229
#> Residual 4.540
#> Number of obs: 270, groups: recipe:replicate, 45
#> Fixed Effects:
#> (Intercept) recipeB recipeC temp
#> 1.516 -1.478 -1.522 0.158
## and now "choose" :
anova(fm3, fm2, fm1)
#> Data: cake
#> Models:
#> fm3: angle ~ recipe + temp + (1 | recipe:replicate)
#> fm2: angle ~ recipe + temperature + (1 | recipe:replicate)
#> fm1: angle ~ recipe * temperature + (1 | recipe:replicate)
#> npar AIC BIC logLik -2*log(L) Chisq Df Pr(>Chisq)
#> fm3 6 1708.2 1729.8 -848.08 1696.2
#> fm2 10 1709.6 1745.6 -844.79 1689.6 6.5755 4 0.1601
#> fm1 20 1719.0 1791.0 -839.53 1679.0 10.5304 10 0.3953