Question

4. Lessons in Data Stealing: RBD Edition. Many textbook authors collect data to use in their books. Often, they are kind enough to package these together into an R library that anyone can freely download. Install the package BHH2, which is based on a book by Box, Hunter, and Hunter, 2nd edition. In this package is a data set called penicillin.data. Get this loaded into R. This data set records how much penicillin can be created (yield) via 4 different processes (treat) on 5 different blocks (blend, you could think about this as 5 different days). Run an ANOVA to determine if there is a significant difference among the treatments or the blocks (using a0.05). Include your code, a copy of the printout, and your hypothesis test results. After this, explain what the run column is all about (which we didn't use here).

Answer 1

Here our model is

$y_{ij}=\mu+\tau_i+\beta_j+e_{ij}$,

where

General mean effect 11

Treatment effect Ti -

$\beta_j= j^{th} \text{ Block effect}$

Random error component ty =

Hol : T1 = T2 = T3 = T4 011
vs.
H11 : Not Ho1

$H_{02} : \beta_1=\beta_2=\beta_3=\beta_4=\beta_5$ vs.

H12: Not Ho2

R-Code

install.packages("BHH2")

library(BHH2)
data(penicillin.data)
attach(penicillin.data)
aa=penicillin.data
block=as.factor(aa$blend)
treatment=as.factor(aa$treat)
y=aa$yield
summary(aov(y~block+treatment))

Output:

> summary(aov(y~block+treatment))
Df Sum Sq Mean Sq F value Pr(>F)
block 4 264 66.00 3.504 0.0407 *
treatment 3 70 23.33 1.239 0.3387
Residuals 12 226 18.83   
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

CONCLUSION:

Here, we can see that the p-value for block is < 0.05, i.e. we reject the null and conclude on the basis of the given data that block effects are significant. On the other hand in case of treatment the p-value is > 0.05. So, we accept the null hypothesis and conclude that there is no significant differences among the treatments.

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General mean effect 11

Treatment effect Ti -

We were unable to transcribe this image

Random error component ty =

Hol : T1 = T2 = T3 = T4 011

H11 : Not Ho1

We were unable to transcribe this image

H12: Not Ho2