Factorial Completely Randomized Design
Definition
Factorial Completely Randomized Design is an experimental design in
which the treatment is formed by a combination of the levels of several
factors. Factorial treatment structures are useful for examining the
effects of two or more factors on a response y,
whether or not interaction exists. The treatment design of
the multi-factor experiment was differentiated based on the level of
importance and restrictions on randomization of the levels of each
factor making up the treatment.
Model
with,
General Data Table of Factorial Completely Randomized Design :
Hypothesis Testing
Hypothesis:
Factor A Effect:
\(H_{0}:\alpha_{1}=\alpha_{2}=...=\alpha_{a}=0\) (Factor A has no effect on response)
\(H_{1}:\) there is at least one i where \(\alpha_{i} \neq 0\)
Factor B Effect:
\(H_{0}:\beta_{1}=\beta_{2}=...=\beta_{b}\) (Factor B has no effect on response)
\(H_{1}:\) there is at least one j where \(\beta_{j} \neq 0\)
Interaction of Factor A and B Effect:
\(H_{0}:\alpha\beta_{1}=\alpha\beta_{2}=...=\alpha\beta_{ab}\) (Interaction of Factor A and Factor B has no effect on response)
\(H_{1}\): there are at least a pair (i,j) where \(\alpha\beta_{ij} \neq 0\)
Analysis of variance table for a Factorial Completely Randomized Design:
with,
or you can use this formula, it will give same result:
with
Rejection Region:
Hypothesis null will be rejected if \(F > F (df1;df2;alpha)\).
Example
An experiment was conducted to determine the effects of four different pesticides on the yield of fruit from three different varieties (B1, B2, B3) of a citrus tree. Eight trees from each variety were randomly selected from an orchard. The four pesticides were then randomly assigned to two trees of each variety and applications were made according to recommended levels. Yields of fruit (in bushels per tree) were obtained after the test period. The data appear in Table below.
Set up an analysis of variance table and conduct the appropriate F-tests of main effects and interactions using alpha 0.05.
Answer:
I.Hypotheses:
Interaction of Factor A and B Effect:
\(H_{0}:\alpha\beta_{1}=\alpha\beta_{2}=...=\alpha\beta_{ab}\) (Interaction of Factor A and Factor B has no effect on response)
\(H_{1}\): there are at least a pair (i,j) where \(\alpha\beta_{ij} \neq 0\)
Factor A Effect:
\(H_{0}:\alpha_{1}=\alpha_{2}=...=\alpha_{a}=0\) (Factor A has no effect on response)
\(H_{1}:\) there is at least one i where \(\alpha_{i} \neq 0\)
Factor B Effect:
\(H_{0}:\beta_{1}=\beta_{2}=...=\beta_{b}\) (Factor B has no effect on response)
\(H_{1}:\) there is at least one j where \(\beta_{j} \neq 0\)
II.Significance Level: 5%
III.Test Statistics
Because of rounding errors, the values for TSS, SSA, SSB, SSAB, and SSE are somewhat different from the values obtained from a computer program.
The main effect sums of squares are
The interaction sum of squares is
The sum of squares error is obtained as
Complete ANOVA Table:
Complete ANOVA Table using R:
library(readxl)## Warning: package 'readxl' was built under R version 4.1.3example1<-read_excel("D:/MATERI KULIAH S2 IPB/ASPRAK 2/factorial.xlsx")
example1## # A tibble: 24 x 3
## pesticide_A variety_B response
## <chr> <chr> <dbl>
## 1 a1 b1 49
## 2 a2 b1 50
## 3 a3 b1 43
## 4 a4 b1 53
## 5 a1 b1 39
## 6 a2 b1 55
## 7 a3 b1 38
## 8 a4 b1 48
## 9 a1 b2 55
## 10 a2 b2 67
## # ... with 14 more rowsmodel_aov <- aov(response ~ pesticide_A*variety_B, data = example1)
summary(model_aov)## Df Sum Sq Mean Sq F value Pr(>F)
## pesticide_A 3 2227 742.5 17.556 0.00011 ***
## variety_B 2 3996 1998.0 47.244 2.05e-06 ***
## pesticide_A:variety_B 6 457 76.2 1.801 0.18168
## Residuals 12 508 42.3
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1IV.Rejection Region
Reject H0 if:
Factor A : \(F_{test}>F_{3;12;0.05}=3.49\)
Factor A : \(F_{test}>F_{2;12;0.05}=3.89\)
Interaction AB : \(F_{test}>F_{6;12;0.05}=3.00\)
V.Conclusion
The first test of significance must be to test for an interaction between factors A and B, because if the interaction is significant then the main effects may have no interpretation
Interaction AB:
The computed value of F=1.801 doesn’t exceed the tabulated value of 3.00 for alpha=0.05, df1=6, and df2=12 in the F tables. Hence, we have insufficient evidence to indicate an interaction between pesticide levels and variety of trees levels.
Because the interaction is not significant, we can next test the main effects of the two factors.
Factor A:
The computed value of F=17.556 does exceed the tabulated value of 3.49 for alpha=0.05, df1=3, and df2=12 in the F tables. Hence, we have sufficient evidence to indicate a significant difference in the mean yields among the four pesticide levels (Factor A).
Factor B:
The computed value of F=47.244 does exceed the tabulated value of 3.89 for alpha=0.05, df1=2, and df2=12 in the F tables. Hence, we have sufficient evidence to indicate a significant difference in the mean yields among the three varieties of citrus trees. (Factor B).