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Two-way ANOVA test is used to evaluate simultaneously the effect of two grouping variables (for example group (A) and category(B)) on a response variable.

Two-way ANOVA test hypotheses

1. There is no difference in the means of factor A

2. There is no difference in means of factor B

3. There is no interaction between factors A and B

(a) The alternative hypothesis for cases 1 and 2 is: the means are not equal.

(b) The alternative hypothesis for case 3 is: there is an interaction between A and B.

Assumptions of two-way ANOVA test Two-way ANOVA, like all ANOVA tests, assumes that the observations within each cell are normally distributed and have equal variances. Well show you how to check these assumptions after fitting ANOVA.

  1. Store the default data “ToothGrowth” in the variable “d” for future use.
attach(ToothGrowth)
d<- (ToothGrowth)
View(d)
  1. Know the names of the variables/arrays
names(d)
## [1] "len"  "supp" "dose"
  1. How many types of supplementary “supp” and dosage “dose” are there?
str(as.factor(supp))
##  Factor w/ 2 levels "OJ","VC": 2 2 2 2 2 2 2 2 2 2 ...
str(as.factor(dose))
##  Factor w/ 3 levels "0.5","1","2": 1 1 1 1 1 1 1 1 1 1 ...
  1. Convert dose as a factor and recode the levels as “D0.5”, “D1”, “D2”.
d$dose <- factor(d$dose, 
                  levels = c(0.5, 1, 2),
                  labels = c("D0.5", "D1", "D2"))
  1. Print/view the data one more time to observe the change in “dose” according to factors.
View(d)
head(d)
##    len supp dose
## 1  4.2   VC D0.5
## 2 11.5   VC D0.5
## 3  7.3   VC D0.5
## 4  5.8   VC D0.5
## 5  6.4   VC D0.5
## 6 10.0   VC D0.5
  1. Make a frequency table with “supp” and “dose”.
table(d$supp, d$dose)
##     
##      D0.5 D1 D2
##   OJ   10 10 10
##   VC   10 10 10
  1. Install if(!require(devtools)) install.packages(“devtools”) devtools::install_github(“kassambara/ggpubr”) or, install.packages(“ggpubr”)
if(!require(devtools)) install.packages("devtools")
## Loading required package: devtools
## Warning: package 'devtools' was built under R version 3.5.3
## Warning: package 'usethis' was built under R version 3.5.3
devtools::install_github("kassambara/ggpubr")
## Skipping install of 'ggpubr' from a github remote, the SHA1 (de26170d) has not changed since last install.
##   Use `force = TRUE` to force installation

8. Box plot with multiple groups:

(a) Plot tooth length by groups (“dose”)

(b) Color box plot by a second group: “supp”

library("ggpubr")
## Loading required package: ggplot2
## Loading required package: magrittr
ggboxplot(d, x = "dose", y = "len", color = "supp",
          palette = c("#00AFBB", "#E7B800"))

library("ggpubr")
ggline(d, x = "dose", y = "len", color = "supp",
       add = c("mean_se", "dotplot"),
       palette = c("#00AFBB", "#E7B800"))
## `stat_bindot()` using `bins = 30`. Pick better value with `binwidth`.

9. Next we want to know if tooth length depends on supp and dose.

A. The R function aov() can be used to answer this question.

B. Summarize the analysis of variance model using the summary(). Write down the conclusion.

aov(len~supp + dose, data=d)
## Call:
##    aov(formula = len ~ supp + dose, data = d)
## 
## Terms:
##                     supp     dose Residuals
## Sum of Squares   205.350 2426.434   820.425
## Deg. of Freedom        1        2        56
## 
## Residual standard error: 3.82759
## Estimated effects may be unbalanced
  1. Do the two way ANOVA with interaction effect and summarize the result.
aov(len~supp*dose, data=d)
## Call:
##    aov(formula = len ~ supp * dose, data = d)
## 
## Terms:
##                     supp     dose supp:dose Residuals
## Sum of Squares   205.350 2426.434   108.319   712.106
## Deg. of Freedom        1        2         2        54
## 
## Residual standard error: 3.631411
## Estimated effects may be unbalanced

**11. How do you interpret the p values of supp, dose, and the interaction between supp and dose: supp*dose**

The p-value of supp is 0.000429, which indicates that the levels of supp are associated with significant different tooth length.

The p-value of dose is < 2e-16, which indicates that the levels of dose are associated with significant different tooth length.

**The p-value for the interaction between supp*dose is 0.02, which indicates that the relationships between dose and tooth length depends on the supp method.**