cake: Breakage Angle of Chocolate Cakes

Description Format Details Source References Examples

Description

Data 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.

Format

A data frame with 270 observations on the following 5 variables.

replicate

a factor with levels 1 to 15

recipe

a factor with levels A, B and C

temperature

an ordered factor with levels 175 < 185 < 195 < 205 < 215 < 225

angle

a numeric vector giving the angle at which the cake broke.

temp

numeric value of the baking temperature (degrees F).

Details

The replicate factor is nested within the recipe factor, and temperature is nested within replicate.

Source

Original data were presented in Cook (1938), and reported in Cochran and Cox (1957, p. 300). Also cited in Lee, Nelder and Pawitan (2006).

References

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.

Examples

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str(cake)
## 'temp' is continuous, 'temperature' an ordered factor with 6 levels

(fm1 <- lmer(angle ~ recipe * temperature + (1|recipe:replicate), cake, REML= FALSE))
(fm2 <- lmer(angle ~ recipe + temperature + (1|recipe:replicate), cake, REML= FALSE))
(fm3 <- lmer(angle ~ recipe + temp        + (1|recipe:replicate), cake, REML= FALSE))

## and now "choose" :
anova(fm3, fm2, fm1)

Example output

Loading required package: Matrix
'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 ...
Linear mixed model fit by maximum likelihood  ['lmerMod']
Formula: angle ~ recipe * temperature + (1 | recipe:replicate)
   Data: cake
      AIC       BIC    logLik  deviance  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  
Linear mixed model fit by maximum likelihood  ['lmerMod']
Formula: angle ~ recipe + temperature + (1 | recipe:replicate)
   Data: cake
      AIC       BIC    logLik  deviance  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  
Linear mixed model fit by maximum likelihood  ['lmerMod']
Formula: angle ~ recipe + temp + (1 | recipe:replicate)
   Data: cake
      AIC       BIC    logLik  deviance  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  
Data: cake
Models:
fm3: angle ~ recipe + temp + (1 | recipe:replicate)
fm2: angle ~ recipe + temperature + (1 | recipe:replicate)
fm1: angle ~ recipe * temperature + (1 | recipe:replicate)
    Df    AIC    BIC  logLik deviance   Chisq Chi 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

lme4 documentation built on June 22, 2021, 9:07 a.m.