One-way ANOVA

Question 1

Differences in mean weight between 2 cattle breeds?

Answer 1

cattle
## # A tibble: 30 x 2
##    breed  weight
##       
##  1 Breed1   188.
##  2 Breed2   148.
##  3 Breed1   180.
##  4 Breed2   146.
##  5 Breed1   199.
##  6 Breed2   153.
##  7 Breed1   191.
##  8 Breed2   135.
##  9 Breed1   196.
## 10 Breed2   151.
## # ... with 20 more rows

Question 2

Two sample t-test

Answer 2

fit

Question 3

Code for manual calculation of mean and varience?

Answer 3

cattlesummary %  # using the cattle data,
  group_by(breed) %>%  # group by the breed variable, then
  # calculate the mean and sd per group:
  summarise(mean_wt = mean(x = weight, na.rm = TRUE),
            sd_wt = sd(weight, na.rm = TRUE))
cattlesummary
## # A tibble: 2 x 3
##   breed  mean_wt sd_wt
##        
## 1 Breed1    196.  10.6
## 2 Breed2    154.  12.3

Question 4

Model assumptions (2-sample t-test)

Answer 4

• equal variances
• normality
• independance of observations

Question 5

Model assumptions (Normality) code

Answer 5

1. ggplot(cattle, aes(breed, weight)) +
   geom_boxplot()
2. hist(cattle$weight)
3. shapiro.test(cattle$weight)

##     Shapiro-Wilk normality test
## 
## data:  cattle$weight
## W = 0.93704, p-value = 0.103

If p > 0.05, the distribution of the cattle data is not significantly different from a normal distribution, i.e. we can assume normality.

Question 6

What happens if assumptions are not met?

Answer 6

• equal variences –> perform test with unequal variences (Welch t-test)
• Normality –> if N>30 assume normality anyway (Central Limit Theorem)
• Independence of observations –> if not independant, use paired t-test

Question 7

What do we need to consider before running a t-test

Answer 7

1. differences between the treatment effects (e.g. the difference between 4 diets of chickens)
2. differences within the treatment effects (e.g. differences within each chicken diet)

Question 8

When do you use a 1-way anova?

Answer 8

When there is only 1 factor

e.g. chicken diet (4 diet options) but only “factor” is the diet

Question 9

Code for normality (model assumption)

Answer 9

chicks %
  pivot_longer(cols = starts_with("Diet"), 
    names_to = "diet", 
    values_to = "weight") %>%
  mutate(diet = as.factor(diet))

ggplot(chicks, aes(diet, weight)) +
geom_boxplot()

hist(chicks$weight)

shapiro.test(chicks$weight)

Question 10

Code for equal variences (model assumptions)

Answer 10

bartlett.test(weight ~ diet, data = chicks)

Question 11

How are the concepts of ANOVA divided into components?

Answer 11

Total Sums-of-Square (SS) = Treatment SS + Residual SS

Question 12

ANOVA code in R

Answer 12

model

Question 13

If null hypothesis is true what does that mean for the F statistic?

Answer 13

It should have a value around 1

Question 14

What does a large F value suggest?

Answer 14

Null hypothesis is false

Question 15

emmeans plots (ANOVA)

Answer 15

library(emmeans)

emm

Question 16

Summary of ANOVA

Answer 16

2-sample t-tests are limited to situation when we have experiments with only 2 levels

The ANOVA allows us to analyse experiments with 2 or more treatment levels

It can be generalised to analyse any experiment, e.g. more than 1 treatment factors

The ANOVA table helps us determine whether there is a significant different between at least one pair of treatment means