In A-Farina/pl462: Data package and Templates for PL462

knitr::opts_chunk$set(echo = TRUE, 
                      include = TRUE, 
                      message = FALSE, 
                      warning = FALSE)
# Loading required packages
if (!require("pacman")) install.packages("pacman")
pacman::p_load(tidyverse, ggpubr, rstatix, palmerpenguins)
pacman::p_load_gh("A-Farina/pl462")

Today we will discuss the Analysis of Variance (ANOVA) from a factorial perspective.

Review:

What is a One-Way ANOVA?

What are the statistical assumptions of an ANOVA?

What is a One-Way Repeated Measures ANOVA?

New Material:

What is a Two-Way ANOVA?

What is a Three-Way ANOVA?

Analysis Framework

1. Load the Data

2. Understand the data (Distribution, Variability, Outliers)
  * Summary Statistics
  * Data Visualization
  * Outliers

3. Check Assumptions
  * Normality of Residuals or DV by group
  * Homogeneity (Equality) of Variance or Sphericity
  * Independent Samples

4. Conduct the Statistical Test
  * Run ANOVA

5. Conduct Post-Hoc Analyses as needed
  * Pairwise *t*-tests
  * One-Way ANOVAs

One-Way ANOVA (Between-Subjects)

1. Load the Data

# Loading the Data
penguins <- palmerpenguins::penguins # Loads the dataset called 'penguins' from the palmerpengins package and assigns it the name "penguins"
?penguins # Loads the help menu giving some descriptive information for the dataset
glimpse(penguins) # Gives a quick view of the structure of the dataset.

Research Question: We are interested in determining if different species of penguins have different flipper lengths and whether that varies according to sex.

What is the DV?

What is/are the IV(s)? How many levels do they have?

2. Understand the data (Distribution, Variability, Outliers)

• Summary Statistics

penguins %>% 
  group_by(species, sex) %>% 
  get_summary_stats(flipper_length_mm, type = "mean_sd")

penguins_complete <- penguins %>% filter(!is.na(sex))

What are your observations given the descriptive statistics?

• Data Visualization

# Boxplot
penguins_complete %>% 
  ggplot() + 
  aes(x = species, y = flipper_length_mm, color = sex) + 
  geom_boxplot() +
  geom_jitter(alpha = 0.2, width = .1)

# Continuous Density Plot
penguins_complete %>% ggplot() + 
  aes(x = flipper_length_mm, fill = sex) + 
  geom_density() + 
  facet_grid(species ~ .)

What are your observations (distribution, variability) given these data visualizations?

• Outliers

penguins_complete %>% 
  group_by(species, sex) %>% 
  rstatix::identify_outliers(flipper_length_mm) # Produces a table of outliers and extreme cases.

Are there any outliers in these data?

3. Check Assumptions

• Normality of Residuals or DV by group

a. Analyzing the ANOVA model residuals. This approach is generally easier to do and helpful if there are many groups. This is the approach we will take.

# Assumption testing- Normality of Residuals- QQ Plot
model <- lm(flipper_length_mm ~ species * sex, data = penguins_complete)
ggqqplot(residuals(model)) # Graphically depicts the correlation of these data and a normal distribution.

# Shapiro-Wilkes Test
shapiro_test(residuals(model)) # Statistically determines normality of these data.

b. Checking Normality for each group separately. This might be helpful if you have only a few groups.

# Assumption testing- Normality DV by groups
ggqqplot(penguins_complete, "flipper_length_mm") +
  facet_grid(sex ~ species)# Graphically depicts the correlation of these data and a normal distribution.

# Shapiro-Wilkes Test
penguins_complete %>%
  group_by(species, sex) %>%
  shapiro_test(flipper_length_mm) # Statistically determines normality of these data. 

Note: may be overly sensitive if data is more than 50. In that case, the qq plot is preferred.

Are these data normally distributed?

If the data are not normally distributed, a Kruskal-Wallis test is a non-parametric alternative to the One-Way ANOVA.

• Homogeneity (Equality) of Variance

# Residuals vs Fitted Plot
plot(model, 1) # Graphically compares the residuals to fitted values (mean of each group)

# Assumption testing- Homogeneity of Variance
penguins_complete %>% levene_test(flipper_length_mm ~ species*sex) # Statistically shows homogeneity of variance.

If the data do not have homogeneity of variance, a welch one way ANOVA test can be performed (welch_anova_test())

Do these data meet the assumption of homogeneity of variance?

• Independent Samples

Are these independent samples?

4. Conduct the Statistical Test

• Run ANOVA

# One-Way ANOVA- Tests of Between-Subject with equal variance
penguins_complete %>% 
  anova_test(flipper_length_mm ~ species * sex)

Interpret the interaction effect between species and sex, is this effect significant and how would you interpret it?

Interpreting the interaction effect

If a significant two-way interaction exists, then the effect that one IV has on the DV is dependent on the level of a second IV. If you have a significant two-way interaction, you would then find the simple main effects (run one-way ANOVAs to determine if there is a simple main effect, if there is, a pairwise comparison would be used to determine which groups are different).

For a non-significant two-way interaction you can determine if there is a significant main effect from the ANOVA table and follow it up with a pairwise comparison between groups to determine where the differences exist.

ANOVA tests determine whether there is a difference among groups. Multiple Comparisons Tables (pairwise_t_test) are used to understand where the differences exist. We will interpret the adjusted p-value (p.adj). This adjusted value uses a Holm correction to account for multiple tests (adjusts the $\alpha).$

We have a significant interaction effect. Therefore, we will run multiple one-way ANOVAs to determine the simple main effects and visualize the means to determine where the difference exists.

5. Conduct Post-Hoc Analyses as needed

• Pairwise t-tests

penguins_complete %>% 
  group_by(sex) %>% 
  anova_test(flipper_length_mm ~ species)

penguins_complete %>% 
  group_by(sex) %>% 
  pairwise_t_test(flipper_length_mm ~ species)

penguins_complete %>% 
  group_by(species) %>% 
  anova_test(flipper_length_mm ~ sex)

# Data Visualizaiton
penguins_complete %>% 
  group_by(species, sex) %>% 
  get_summary_stats(flipper_length_mm, type = "mean_sd") %>% 
  ggplot() + 
  aes(x = species, y = mean, color = sex) + 
  geom_path(aes(group = sex))

Interpret the results

---

Two-Way Mixed-Measures ANOVA

1. Load the Data

# Loading the Data
grit <- pl462::grit # Loads the dataset called 'grit' from the pl462 package and assigns it the name "penguins"
?grit # Loads the help menu giving some descriptive information for the dataset
glimpse(grit) # Gives a quick view of the structure of the dataset.

Research Question: We are interested in determining whether grit changes based on when the measure is taken (matriculation, freshmen year, sophomore year) and the biological sex of the participant.

What is the DV?

What is/are the IV(s)? How many levels do they have?

2. Understand the data (Distribution, Variability, Outliers)

• Summary Statistics

grit %>% 
  group_by(class, sex) %>% 
  get_summary_stats(grit, type = "mean_sd")

What are your observations given the descriptive statistics?

• Data Visualization

# Boxplot
grit %>% 
  ggplot() + 
  aes(x = class, y = grit, fill = sex) + 
  geom_boxplot() 

# Continuous Density Plot
grit %>% ggplot() + 
  aes(x = grit, fill = sex, group = sex) + 
  geom_density(alpha = .5) + 
  facet_grid(class ~ .)

What are your observations (distribution, variability) given these data visualizations?

• Outliers

grit %>% 
  group_by(class, sex) %>% 
  rstatix::identify_outliers(grit) # Produces a table of outliers and extreme cases.

Are there any outliers in these data?

3. Check Assumptions

• Normality of Residuals or DV by group

a. Analyzing the ANOVA model residuals. This approach is generally easier to do and helpful if there are many groups. This is the approach we will take.

# Assumption testing- Normality of Residuals- QQ Plot
model <- lm(grit ~ class*sex, data = grit)
ggqqplot(residuals(model)) # Graphically depicts the correlation of these data and a normal distribution.

# Shapiro-Wilkes Test
shapiro_test(residuals(model)) # Statistically determines normality of these data.

b. Checking Normality for each group separately. This might be helpful if you have only a few groups.

# Assumption testing- Normality DV by groups

ggqqplot(grit, "grit") + facet_grid(sex ~ class) # Graphically depicts the correlation of these data and a normal distribution.

# Shapiro-Wilkes Test
grit %>%
  group_by(class, sex) %>%
  shapiro_test(grit) # Statistically determines normality of these data. 

Note: may be overly sensitive if data is more than 50. In that case, the qq plot is preferred.

Are these data normally distributed?

If the data are not normally distributed, a Kruskal-Wallis test is a non-parametric alternative to the One-Way ANOVA (kruskal_test()).

• Sphericity variance of the differences between groups should be equal. This can be checked using the Mauchly’s test of sphericity. This is automatically reported and adjusted with anova_test()

4. Conduct the Statistical Test

• Run ANOVA

# One-Way Repeated Measures ANOVA- Tests of Within-Subject.
grit %>% 
  anova_test(wid = id, dv = grit, within = class, between = sex)

Interpret the results

5. Conduct Post-Hoc Analyses as needed

• Pairwise t-tests

grit %>% 
  pairwise_t_test(grit ~ class,
                  paired = TRUE)

Interpret the results

A-Farina/pl462 documentation built on May 9, 2022, 1:52 p.m.