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Example 5-1
Section* *

Consider a study of Battery Life, measured in hours, where 4 brands of batteries are evaluated using 4 replications in a completely randomized design (Battery Data):

Brand A |
Brand B |
Brand C |
Brand D |

110 | 118 | 108 | 117 |

113 | 116 | 107 | 112 |

108 | 112 | 112 | 115 |

115 | 117 | 108 | 119 |

The question arises: should the brand of battery be considered a fixed or, alternatively, a random effect? If the researchers were interested in comparing the performance of the specific brands they chose for the study, then we have a fixed effect. But let’s say this was not what happened. Instead, the researchers were actually interested in a study of variation in lifetimes of batteries, the results of which would be applicable to all brands of batteries. In this case, they would have recognized that there many manufacturers of batteries, and may have chosen (presumably with a random sampling process) a sample of 4 of the many brands out there. They test 4 batteries of each of these brands, and they want to characterize the variability in battery life. In this latter case, the battery brand is a *random effect.*

Although the ANOVA calculations are the same, this is a fundamentally different sort of study.

Using Minitab we can compare the results for considering battery brand as a fixed vs. random effect:

- Fixed Effect model:
##### One-Way ANOVA: lifetime versus trt

Source DF SS MS F P trt 3 141.69 47.23 6.21 0.009 Errro 12 91.25 7.60 Total 16 232.94 -
Random Effect model (now running in

`Stat`>`ANOVA`>`General Linear Model`in Minitab:#### General Linear Model: lifetime versus trt

Factor Type Levels Values trt random 4 BrandA, BrandB, BrandC, BrandD ##### Analysis of Variance for lifetime, using Adjusted SS for Tests

Source DF Seq SS Adj SS Adj MS F P trt 3 141.688 141.688 47.229 6.21 0.009 Error 12 91.250 91.250 7.604 Total 15 232.938 ##### Expected Mean Squares, using Adjusted SS

Source Expected Mean Square

for each term1 trt (2) + 4.0000 (1) 2 Error (2) Error Terms for Tests, using Adjusted SS

Source Error DF Error MS Synthesis

of Error MS1 trt 12.00 7.604 (2) Variance Components, using Adjusted SS

Source Estimated

valuetrt 9.906 Error 7.604 We can verify the estimated variance component (arrow above) for the random treatment effect as:

\(s_{\text{among trts}}^{2}=\dfrac{MS_{trt}-MS_{error}}{n}=\dfrac{47.229-7.604}{4}=9.9063\)

With this, we can calculate the ICC as

\(ICC= \dfrac{9.9063}{9.9063+7.604}=0.5657\)

The key points in comparing these two ANOVAs is 1) *the scope of inference* and 2) *the hypothesis being tested.* For a fixed effect, the scope of inference is restricted to only 4 brands chosen for comparison and the Null hypothesis is a statement of equality of means. In contrast, as a random effect, the scope of inference is the larger population of battery brands and the Null hypothesis is a statement that the variance due to battery brand is 0.