Running a One-Way & Two-Way ANOVA Lab
Caroline Cobb
2026-05-01
1 Loading Libraries
#install.packages("afex")
#install.packages("emmeans")
#install.packages("ggbeeswarm")
#install.packages("expss")
library(psych) # for the describe() command
library(ggplot2) # to visualize our results##
## Attaching package: 'ggplot2'## The following objects are masked from 'package:psych':
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## %+%, alphalibrary(expss) # for the cross_cases() command## Loading required package: maditr##
## To aggregate data: take(mtcars, mean_mpg = mean(mpg), by = am)##
## Attaching package: 'expss'## The following object is masked from 'package:ggplot2':
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## varslibrary(car) # for the leveneTest() command## Loading required package: carData##
## Attaching package: 'car'## The following object is masked from 'package:expss':
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## recode## The following object is masked from 'package:psych':
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## logitlibrary(afex) # to run the ANOVA ## Loading required package: lme4## Loading required package: Matrix## Registered S3 method overwritten by 'lme4':
## method from
## na.action.merMod car##
## Attaching package: 'lme4'## The following object is masked from 'package:expss':
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## dummy## ************
## Welcome to afex. For support visit: http://afex.singmann.science/## - Functions for ANOVAs: aov_car(), aov_ez(), and aov_4()
## - Methods for calculating p-values with mixed(): 'S', 'KR', 'LRT', and 'PB'
## - 'afex_aov' and 'mixed' objects can be passed to emmeans() for follow-up tests
## - Get and set global package options with: afex_options()
## - Set sum-to-zero contrasts globally: set_sum_contrasts()
## - For example analyses see: browseVignettes("afex")
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## Attaching package: 'afex'## The following object is masked from 'package:lme4':
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## lmerlibrary(ggbeeswarm) # to run plot results
library(emmeans) # for posthoc tests## Welcome to emmeans.
## Caution: You lose important information if you filter this package's results.
## See '? untidy'2 Importing Data
# For HW, import the project dataset you cleaned previously this will be the dataset you'll use throughout the rest of the semester
d <- read.csv(file="Data/projectdata.csv", header=T)
# new code! this adds a column with a number for each row. It will make it easier if we need to drop outliers later
d$row_id <- 1:nrow(d)3 State Your Hypothesis
Note: For your HW, you will choose to run EITHER a one-way ANOVA (a single IV with 3 or more levels) OR a two-way/factorial ANOVA (at least two IVs with 2 or 3 levels each). You will need to specify your hypothesis and customize your code based on the choice you make. We will run BOTH versions of the test in the lab for illustrative purposes.
One-Way: We predict that there will be a significant difference in people’s level of independence based on their gender (male, female, other).
4 Check Your Variables
# you only need to check the variables you're using in the current analysis
str(d)## 'data.frame': 3092 obs. of 8 variables:
## $ ResponseID : chr "R_BJN3bQqi1zUMid3" "R_2TGbiBXmAtxywsD" "R_12G7bIqN2wB2N65" "R_39pldNoon8CePfP" ...
## $ gender : chr "f" "m" "m" "f" ...
## $ sibling : chr "at least one sibling" "at least one sibling" "at least one sibling" "at least one sibling" ...
## $ moa_independence: num 3.67 3.67 3.5 3 3.83 ...
## $ swb : num 4.33 4.17 1.83 5.17 3.67 ...
## $ belong : num 2.8 4.2 3.6 4 3.4 4.2 3.9 3.6 2.9 2.5 ...
## $ socmeduse : int 47 23 34 35 37 13 37 43 37 29 ...
## $ row_id : int 1 2 3 4 5 6 7 8 9 10 ...# make our categorical variables of interest "factors"
# because we'll use our newly created row ID variable for this analysis, so make sure it's coded as a factor, too.
d$gender <- as.factor(d$gender)
d$row_id <- as.factor(d$row_id)
# check that all our categorical variables of interest are now factors
str(d)## 'data.frame': 3092 obs. of 8 variables:
## $ ResponseID : chr "R_BJN3bQqi1zUMid3" "R_2TGbiBXmAtxywsD" "R_12G7bIqN2wB2N65" "R_39pldNoon8CePfP" ...
## $ gender : Factor w/ 3 levels "f","m","nb": 1 2 2 1 2 1 1 1 1 1 ...
## $ sibling : chr "at least one sibling" "at least one sibling" "at least one sibling" "at least one sibling" ...
## $ moa_independence: num 3.67 3.67 3.5 3 3.83 ...
## $ swb : num 4.33 4.17 1.83 5.17 3.67 ...
## $ belong : num 2.8 4.2 3.6 4 3.4 4.2 3.9 3.6 2.9 2.5 ...
## $ socmeduse : int 47 23 34 35 37 13 37 43 37 29 ...
## $ row_id : Factor w/ 3092 levels "1","2","3","4",..: 1 2 3 4 5 6 7 8 9 10 ...# check our DV skew and kurtosis
describe(d$moa_independence)## vars n mean sd median trimmed mad min max range skew kurtosis se
## X1 1 3092 3.54 0.46 3.67 3.61 0.49 1 4 3 -1.44 2.52 0.01# we'll use the describeBy() command to view our DV's skew and kurtosis across our IVs' levels
describeBy(d$moa_independence, group = d$gender)##
## Descriptive statistics by group
## group: f
## vars n mean sd median trimmed mad min max range skew kurtosis se
## X1 1 2270 3.55 0.45 3.67 3.62 0.49 1.17 4 2.83 -1.41 2.33 0.01
## ------------------------------------------------------------
## group: m
## vars n mean sd median trimmed mad min max range skew kurtosis se
## X1 1 769 3.51 0.5 3.67 3.58 0.49 1 4 3 -1.52 2.81 0.02
## ------------------------------------------------------------
## group: nb
## vars n mean sd median trimmed mad min max range skew kurtosis se
## X1 1 53 3.31 0.43 3.33 3.35 0.49 2.17 4 1.83 -0.72 0.36 0.06# also use histograms to examine your continuous variable
hist(d$moa_independence)
# REMEMBER your test's level of POWER is determined by your SMALLEST subsample5 Check Your Assumptions
5.1 ANOVA Assumptions
- DV should be normally distributed across levels of the IV (we checked previously using “describeBy” function)
- All levels of the IVs should have an equal number of cases and there should be no empty cells. Cells with low numbers decreases the power of the test (which increases chance of Type II error)
- Homogeneity of variance should be confirmed (using Levene’s Test)
- Outliers should be identified and removed – we will actually remove them this time!
- If you have confirmed everything above, the sampling distribution should be normal
5.1.1 Check levels of IVs
# One-Way
table(d$gender)##
## f m nb
## 2270 769 535.1.2 Check homogeneity of variance
# use the leveneTest() command from the car package to test homogeneity of variance
# uses the 'formula' setup: formula is y~x1*x2, where y is our DV and x1 is our first IV and x2 is our second IV
# One-Way
leveneTest(moa_independence~gender, data = d)## Levene's Test for Homogeneity of Variance (center = median)
## Df F value Pr(>F)
## group 2 2.9117 0.05454 .
## 3089
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 15.1.3 Check for outliers using Cook’s distance and Residuals VS Leverage plot
5.1.3.1 Run a Regression to get both outlier plots
# use this commented out section below ONLY IF if you need to remove outliers
# to drop a single outlier, use this code:
#d <- subset(d, row_id!=c(1108))
# to drop multiple outliers, use this code:
# d <- subset(d, row_id!=c(1108) & row_id!=c(602))
# use the lm() command to run the regression
# formula is y~x1*x2 + c, where y is our DV, x1 is our first IV, x2 is our second IV.
# One-Way
reg_model <- lm(moa_independence~gender, data = d) 5.1.3.2 Check for outliers (One-Way)
# Cook's distance
plot(reg_model, 4)
# Residuals VS Leverage
plot(reg_model, 5)
## Issues with My Data
Our cell sizes are very unbalanced between the gender group levels. A small sample size for one of the levels of our variable limits our power and increases our Type II error rate.
Levene's test was not significant for our three-level variable with the One-Way ANOVA, so we are continuing with the analysis.
No outliers were removed for the One-Way ANOVA.
# Run an ANOVA
``` r
# One-Way
aov_model <- aov_ez(data = d,
id = "row_id",
between = c("gender"),
dv = "moa_independence",
anova_table = list(es = "pes"))## Contrasts set to contr.sum for the following variables: gender6 View Output
# One-Way
nice(aov_model)## Anova Table (Type 3 tests)
##
## Response: moa_independence
## Effect df MSE F pes p.value
## 1 gender 2, 3089 0.21 9.46 *** .006 <.001
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '+' 0.1 ' ' 1ANOVA Effect Size [partial eta-squared] cutoffs from Cohen (1988): * η^2 < 0.01 indicates a trivial effect * η^2 >= 0.01 indicates a small effect * η^2 >= 0.06 indicates a medium effect * η^2 >= 0.14 indicates a large effect
7 Visualize Results
# One-Way
afex_plot(aov_model, x = "gender")
8 Run Posthoc Tests (One-Way)
ONLY run posthoc IF the ANOVA test is SIGNIFICANT! E.g., only run the posthoc tests on pet type if there is a main effect for pet type
emmeans(aov_model, specs="gender", adjust="sidak")## gender emmean SE df lower.CL upper.CL
## f 3.55 0.00973 3089 3.53 3.58
## m 3.51 0.01670 3089 3.47 3.55
## nb 3.31 0.06370 3089 3.16 3.46
##
## Confidence level used: 0.95
## Conf-level adjustment: sidak method for 3 estimatespairs(emmeans(aov_model, specs="gender", adjust="sidak"))## contrast estimate SE df t.ratio p.value
## f - m 0.0478 0.0193 3089 2.473 0.0359
## f - nb 0.2419 0.0644 3089 3.757 0.0005
## m - nb 0.1941 0.0658 3089 2.949 0.0090
##
## P value adjustment: tukey method for comparing a family of 3 estimates9 Write Up Results
9.1 One-Way ANOVA
To test our hypothesis that there will be a significant difference in people’s level of independence based on their gender (male, female, other), we used a one-way ANOVA. Our data was unbalanced, with many more people identifying as female (n = 2270) than male (n = 769) than other (n = 53). This significantly reduces the power of our test and increases the chances of a Type II error. Levene’s test was not significant (p = .055), indicating that the assumption of homogeneity of variance was met.
We found a significant effect of gender, F(2, 3089) = 9.46, p < .001, ηp2 = .006 (trivial effect size; Cohen, 1988). Posthoc tests using Sidak’s adjustment revealed that participants who identify as female (M = 3.55, SE = 0.01) reported slightly higher levels of independence than those who identify as male (M = 3.51, SE = 0.02) and those who identfied as other (M = 3.31, SE = 0.06). Those who identified as other reported the lowest level of independence overall. (see Figure 1 for a comparison).
References
Cohen J. (1988). Statistical Power Analysis for the Behavioral Sciences. New York, NY: Routledge Academic.