Running a One-Way & Two-Way ANOVA Lab
Brook Dager
2025-12-03
1 Loading Libraries
#install.packages("afex")
#install.packages("emmeans")
#install.packages("ggbeeswarm")
#install.packages("expss")
#install.packages("psych")
library(psych) # for the describe() command## Warning: package 'psych' was built under R version 4.5.2library(ggplot2) # to visualize our results## Warning: package 'ggplot2' was built under R version 4.5.2##
## Attaching package: 'ggplot2'## The following objects are masked from 'package:psych':
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## %+%, alphalibrary(expss) # for the cross_cases() command## Warning: package 'expss' was built under R version 4.5.2## Loading required package: maditr## Warning: package 'maditr' was built under R version 4.5.2##
## To modify variables or add new variables:
## let(mtcars, new_var = 42, new_var2 = new_var*hp) %>% head()##
## 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## Warning: package 'car' was built under R version 4.5.2## Loading required package: carData## Warning: package 'carData' was built under R version 4.5.2##
## 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 ## Warning: package 'afex' was built under R version 4.5.2## Loading required package: lme4## Warning: package 'lme4' was built under R version 4.5.2## Loading required package: Matrix##
## 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## Warning: package 'ggbeeswarm' was built under R version 4.5.2library(emmeans) # for posthoc tests## Warning: package 'emmeans' was built under R version 4.5.2## 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: I predict that there will be a significant difference in anxiety by peoples levels of income, between lower class, middle class and high class.
4 Check Your Variables
# you only need to check the variables you're using in the current analysis
str(d)## 'data.frame': 366 obs. of 8 variables:
## $ X : int 1 401 469 1390 2689 2752 2835 3935 4050 4058 ...
## $ gender : chr "female" "female" "female" "male" ...
## $ income : chr "3 high" "3 high" "2 middle" "3 high" ...
## $ edeq12 : num 1.58 3.08 1.83 1.5 1.5 ...
## $ gad : num 1.86 2.14 1.71 1 4 ...
## $ covid_neg: int 0 0 0 0 0 0 0 0 0 0 ...
## $ covid_pos: int 0 0 0 0 0 0 0 0 0 0 ...
## $ 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)
# we're going to recode our race variable into two groups: poc and white
# in doing so, we are creating a new variable "poc" that has 2 levels
table(d$income)##
## 1 low 2 middle 3 high prefer not to say
## 43 174 97 52# check that all our categorical variables of interest are now factors
str(d)## 'data.frame': 366 obs. of 8 variables:
## $ X : int 1 401 469 1390 2689 2752 2835 3935 4050 4058 ...
## $ gender : chr "female" "female" "female" "male" ...
## $ income : Factor w/ 4 levels "1 low","2 middle",..: 3 3 2 3 2 3 1 2 1 2 ...
## $ edeq12 : num 1.58 3.08 1.83 1.5 1.5 ...
## $ gad : num 1.86 2.14 1.71 1 4 ...
## $ covid_neg: int 0 0 0 0 0 0 0 0 0 0 ...
## $ covid_pos: int 0 0 0 0 0 0 0 0 0 0 ...
## $ row_id : Factor w/ 366 levels "1","2","3","4",..: 1 2 3 4 5 6 7 8 9 10 ...# check our DV skew and kurtosis
describe(d$gad)## vars n mean sd median trimmed mad min max range skew kurtosis se
## X1 1 366 1.55 0.61 1.43 1.44 0.42 1 4 3 1.64 2.77 0.03# we'll use the describeBy() command to view our DV's skew and kurtosis across our IVs' levels
describeBy(d$gad, group = d$income)##
## Descriptive statistics by group
## group: 1 low
## vars n mean sd median trimmed mad min max range skew kurtosis se
## X1 1 43 1.85 0.76 1.71 1.76 0.85 1 4 3 0.93 0.01 0.12
## ------------------------------------------------------------
## group: 2 middle
## vars n mean sd median trimmed mad min max range skew kurtosis se
## X1 1 174 1.55 0.63 1.43 1.43 0.42 1 4 3 1.79 3.49 0.05
## ------------------------------------------------------------
## group: 3 high
## vars n mean sd median trimmed mad min max range skew kurtosis se
## X1 1 97 1.46 0.5 1.29 1.38 0.42 1 3.14 2.14 1.32 1.17 0.05
## ------------------------------------------------------------
## group: prefer not to say
## vars n mean sd median trimmed mad min max range skew kurtosis se
## X1 1 52 1.48 0.54 1.43 1.38 0.42 1 3.43 2.43 1.67 2.79 0.08# also use histograms to examine your continuous variable
hist(d$gad)
# 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 of cases and there should be no empty cells. Cells with low numbers decrease the power of the test (which increases chance of Type 2 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$income)##
## 1 low 2 middle 3 high prefer not to say
## 43 174 97 52# our small number of participants owning rabbits is going to hurt us for the two-way anova, but it should be okay for the one-way anova
# We will create a new dataframe for the two-way analysis and call it d_twoway and remove the pet owning Ps.
d_oneway <- subset(d, income != "prefer not to say")
d_oneway$income <- droplevels(d_oneway$income)
table(d_oneway$income)##
## 1 low 2 middle 3 high
## 43 174 975.1.2 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(gad~income, data = d_oneway)## Levene's Test for Homogeneity of Variance (center = median)
## Df F value Pr(>F)
## group 2 3.371 0.03561 *
## 311
## ---
## 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(gad~income, data = d_oneway) 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
Our cell sizes are very unbalanced between the income type 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 significant for our three-level income type variable with the One-way ANOVA.
I identified and removed a single outlier for the One-Way ANOVA.
6 Run an ANOVA
# One-Way
aov_model <- aov_ez(data = d_oneway,
id = "X",
between = c("income"),
dv = "gad",
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: gad
## Effect df MSE F pes p.value
## 1 income 2, 311 0.38 6.21 ** .038 .002
## ---
## 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 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="income", adjust="sidak")## income emmean SE df lower.CL upper.CL
## 1 low 1.85 0.0935 311 1.63 2.07
## 2 middle 1.55 0.0465 311 1.44 1.66
## 3 high 1.46 0.0622 311 1.31 1.61
##
## Confidence level used: 0.95
## Conf-level adjustment: sidak method for 3 estimatespairs(emmeans(aov_model, specs="income", adjust="sidak"))## contrast estimate SE df t.ratio p.value
## 1 low - 2 middle 0.3037 0.1040 311 2.910 0.0108
## 1 low - 3 high 0.3925 0.1120 311 3.495 0.0016
## 2 middle - 3 high 0.0888 0.0777 311 1.143 0.4884
##
## P value adjustment: tukey method for comparing a family of 3 estimates10 Write Up Results
10.1 One-Way ANOVA
To test our hypothesis that there will be a significant difference in anxiety by peoples level of income between lower class, middle class and high class, we used a one-way ANOVA. Our data was unbalanced, with many more people who are middle class participating in our survey (n = 174) than who are high class (n = 97) or low class (n = 43). This significantly reduces the power of our test and increases the chances of a Type II error. We also identified and removed a single outlier following visual analysis of Cook’s Distance and Residuals VS Leverage plots. A significant Levene’s test (p = .036) also indicates that our data violates the assumption of homogeneity of variance. This suggests that there is an increased chance of Type I error. We continued with our analysis for the purpose of this class.
We found a significant effect of income type, F(2, 311) = 6.21, p <.05, ηp2 = .038 (small effect size; Cohen, 1988). Posthoc tests using Sidak’s adjustment revealed that participants who were low class (M = 1.85, SE = .09) reported more anxiety than those who were middle class (M = 1.55, SE = .05) and more anxiety than those who are high class (M = 1.46, SE = .06); middle and high class did not significantly differ. (see Figure 1 for a comparison).
References
Cohen J. (1988). Statistical Power Analysis for the Behavioral Sciences. New York, NY: Routledge Academic.