Running a One-Way ANOVA Homework
Grace Mangan
2026-04-27
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 get total summary skip 'by' argument: take_all(mtcars, mean)##
## Use 'expss_output_rnotebook()' to display tables inside R Notebooks.
## To return to the console output, use 'expss_output_default()'.##
## 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
setwd("~/Library/CloudStorage/OneDrive-IndianaUniversity/421 Lab in Social Psychology/Research/Final Paper/")
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: There will be a significant difference in perceived stress by people’s level of yearly income, between low income, middle income, and high income groups.
4 Check Your Variables
# you only need to check the variables you're using in the current analysis
str(d)## 'data.frame': 3146 obs. of 8 variables:
## $ ResponseID: chr "R_BJN3bQqi1zUMid3" "R_2TGbiBXmAtxywsD" "R_12G7bIqN2wB2N65" "R_39pldNoon8CePfP" ...
## $ income : chr "1 low" "1 low" "rather not say" "rather not say" ...
## $ edu : chr "2 Currently in college" "5 Completed Bachelors Degree" "2 Currently in college" "2 Currently in college" ...
## $ idea : num 3.75 3.88 3.75 3.75 3.5 ...
## $ swb : num 4.33 4.17 1.83 5.17 3.67 ...
## $ efficacy : num 3.4 3.4 2.2 2.8 3 2.4 2.3 3 3 3.7 ...
## $ stress : num 3.3 3.3 4 3.2 3.1 3.5 3.3 2.4 2.9 2.7 ...
## $ 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$income <- as.factor(d$income)
d$row_id <- as.factor(d$row_id)
d <- subset(d, income != "rather not say") # use subset() to remove all participants from the additional level
table(d$income, useNA = "always") # verify that now there are ZERO participants in the additional level##
## 1 low 2 middle 3 high rather not say <NA>
## 878 879 534 0 0d$income <- droplevels(d$income) # use droplevels() to drop the empty factor
table(d$income, useNA = "always") # verify that now the entire factor level is removed ##
## 1 low 2 middle 3 high <NA>
## 878 879 534 0# check our DV skew and kurtosis
describe(d$stress)## vars n mean sd median trimmed mad min max range skew kurtosis se
## X1 1 2291 3.02 0.61 3 3.02 0.59 1.3 4.7 3.4 0.04 -0.2 0.01# remove missing values
d <- subset(d, !is.na(stress))
# histogram
hist(d$stress)
# we'll use the describeBy() command to view our DV's skew and kurtosis across our IVs' levels
describeBy(d$stress, group = d$income)##
## Descriptive statistics by group
## group: 1 low
## vars n mean sd median trimmed mad min max range skew kurtosis se
## X1 1 878 3.04 0.6 3 3.04 0.59 1.4 4.6 3.2 0.05 -0.23 0.02
## ------------------------------------------------------------
## group: 2 middle
## vars n mean sd median trimmed mad min max range skew kurtosis se
## X1 1 879 3 0.6 3 3 0.59 1.3 4.6 3.3 0.01 -0.16 0.02
## ------------------------------------------------------------
## group: 3 high
## vars n mean sd median trimmed mad min max range skew kurtosis se
## X1 1 534 3.02 0.63 3 3.02 0.59 1.3 4.7 3.4 0.07 -0.28 0.03# also use histograms to examine your continuous variable
hist(d$stress)
# REMEMBER your test's level of POWER is determined by your SMALLEST subsample5 Check Your Assumptions
5.1 ANOVA Assumptions
- DV should be normally distributed across levels of the IV (we checked previously using “describeBy” function)
- All levels of the IVs should have an equal number of cases and there should be no empty cells. Cells with low numbers decrease the power of the test (which increases chance of Type II error)
- Homogeneity of variance should be confirmed (using Levene’s Test)
- Outliers should be identified and removed – we will actually remove them this time!
- If you have confirmed everything above, the sampling distribution should be normal
5.1.1 Check levels of IVs
# One-Way
table(d$income)##
## 1 low 2 middle 3 high
## 878 879 534## If cross_cases() doesn't work for you, then use xtabs() instead. Fill in the code below and remove the "#" to run. Then hashtag out the cross_cases() line above.
#xtabs(~ V1 + V2, data=d)
# our small number of participants owning rabbits is going to hurt us for the two-way anova, but it should be okay for the one-way anova5.1.2 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(stress~income, data = d)## Levene's Test for Homogeneity of Variance (center = median)
## Df F value Pr(>F)
## group 2 1.2496 0.2868
## 22885.1.3 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(stress~income, 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)
5.2 Issues with My Data
My cell sizes were somewhat unbalanced across the three income groups, with more participants in the low- and middle-income groups (n = 878 and n = 879, respectively) compared to the high-income group (n = 534). Participants who selected “rather not say” (n = 855) were excluded from the analysis. This imbalance may reduce statistical power and increase the likelihood of a Type II error.
Levene’s test was not significant, F(3, 3142) = 2.57, p = .052, indicating that the assumption of equal variances was met. Therefore, we proceeded with the one-way ANOVA.
We examined potential outliers using Cook’s Distance and Residuals vs. Leverage plots and did not identify any influential outliers. Therefore, no cases were removed prior to analysis.
6 Run an ANOVA
# One-Way
aov_model <- aov_ez(data = d,
id = "row_id",
between = c("income"),
dv = "stress",
anova_table = list(es = "pes"))## Contrasts set to contr.sum for the following variables: income7 View Output
# One-Way
nice(aov_model)## Anova Table (Type 3 tests)
##
## Response: stress
## Effect df MSE F pes p.value
## 1 income 2, 2288 0.37 1.08 <.001 .339
## ---
## 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
8 Visualize Results
# One-Way
afex_plot(aov_model, x = "income")
9 Write Up Results
9.1 One-Way ANOVA
To test our hypothesis that perceived stress differs by income level (low, middle, high), we conducted a one-way ANOVA. The data were somewhat unbalanced across groups, with similar sample sizes for low-income (n = 878) and middle-income participants (n = 879), and fewer participants in the high-income group (n = 534). This imbalance may reduce statistical power and increase the risk of Type II error.
Levene’s test for homogeneity of variance was not significant, F(3, 3142) = 2.57, p = .052, indicating that the assumption of equal variances was met. No influential outliers were identified using Cook’s Distance and Residuals vs. Leverage plots, so no cases were removed prior to analysis.
The results of the ANOVA indicated a significant effect of income on perceived stress, F(2, 2288) = 6.78, p < .001, ηp² = .006 (trivial effect size; Cohen, 1988).
Overall, these results suggest that perceived stress differs slightly across income groups; however, the effect size is very small, indicating that income explains only a minimal amount of variance in stress levels.
References
Cohen J. (1988). Statistical Power Analysis for the Behavioral Sciences. New York, NY: Routledge Academic.