The following data were generated from a 2 2 factorial experiment with 3 replica

Question

The following data were generated from a 2 2 factorial experiment with 3 replicates,
where factor A levels represent two different injection procedures of an anesthetic to
the occipital nerve (located in the back of the neck), and factor B levels represent two
different drugs, which physicians recommend to increase the effectiveness of the
injections. Three headache patients were randomly selected for each combination of
injection and drug.

(a) Create the ANOVA table.
(b) Test at the 5% significance level to determine if factors A and B interact.
(c) Test at the 5% significance level to determine if differences exist among the levels of factor A.
(d) Test at the 5% significance level to determine if differences exist among the levels of factor B.

Explanation / Answer

here is an R code of the above problem:

data=scan()
7 10 8 10 11 6 13 11 12 16 15 11
factorA<-c(rep("1",3),rep("2",3),rep("1",3),rep("2",3))
factorB<-c(rep("1",6),rep("2",6))
df<-data.frame(data,factorA,factorB)
annova<-aov(data~factorA*factorB,data=df)
annova
summary(annova)

OUTPUT:

annova
Call:
aov(formula = data ~ factorA * factorB, data = df)

Terms:
factorA factorB factorA:factorB Residuals
Sum of Squares 5.33333 56.33333 1.33333 34.66667
Deg. of Freedom 1 1 1 8

Residual standard error: 2.081666
Estimated effects may be unbalanced
> summary(annova)
Df Sum Sq Mean Sq F value Pr(>F)   
factorA 1 5.33 5.33 1.231 0.29949   
factorB 1 56.33 56.33 13.000 0.00693 **
factorA:factorB 1 1.33 1.33 0.308 0.59426   
Residuals 8 34.67 4.33   
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Inferences:

1, We do not find any interaction between Factor A and Factor B at 5% level of significance.

2,There is no significant difference among the level of factor A.

3,There is significant difference among levels of factor B.

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