Running a One-Way ANOVA
Gabrielle Seminaro
2025-06-22
1 Loading Libraries
#install.packages("afex")
#install.packages("emmeans")
#install.packages("ggbeeswarm")
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 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## 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##
## 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 at least 3 levels) OR a two-way ANOVA (two IVs, each with 2 levels). You will need to specify your hypothesis and customize your code based on the choice you make (i.e., delete code that is not relevant). We will run BOTH versions in the lab for illustrative purposes.
One-Way Hypothesis: We predict that there will be a significant difference in people’s level of felt depression based on their mental health disorders type (none or NA, anxiety disorder, other).
IV = mental health disorders
DV = depression
4 Check Your Variables
# you only need to check the variables you're using in the current analysis
# even if you checked them previously, it's always a good idea to look them over again and be sure that everything is correct
str(d)## 'data.frame': 962 obs. of 8 variables:
## $ X : int 321 401 520 1390 1422 1849 2247 2526 2609 2689 ...
## $ age : chr "1 under 18" "4 between 36 and 45" "1 under 18" "5 over 45" ...
## $ mhealth : chr "none or NA" "obsessive compulsive disorder" "none or NA" "none or NA" ...
## $ pas_covid: num 2.33 4 3 2.89 2.67 ...
## $ pss : num 2.25 2.25 2.75 2.75 4.75 3.25 2.25 3.5 1.25 4 ...
## $ support : num 2.5 3.83 2.83 2.83 2.33 ...
## $ phq : num 1.89 2.44 1.56 1.22 4 ...
## $ 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$mhealth <- as.factor(d$mhealth)
d$row_id <- as.factor(d$row_id)
# We're going to recode our race variable into two groups for the Two-Way ANOVA: poc and white
# in doing so, we are creating a new variable "poc" that has 2 levels
d <- subset(d, mhealth != "obsessive compulsive disorder")
table(d$mhealth, useNA = "always")##
## anxiety disorder bipolar
## 99 4
## depression eating disorders
## 24 18
## none or NA obsessive compulsive disorder
## 761 0
## other ptsd
## 23 16
## <NA>
## 0d$mhealth<- droplevels(d$mhealth)
table(d$mhealth, useNA = "always")##
## anxiety disorder bipolar depression eating disorders
## 99 4 24 18
## none or NA other ptsd <NA>
## 761 23 16 0d <- subset(d, mhealth != "bipolar")
table(d$mhealth, useNA = "always")##
## anxiety disorder bipolar depression eating disorders
## 99 0 24 18
## none or NA other ptsd <NA>
## 761 23 16 0d$mhealth<- droplevels(d$mhealth)
table(d$mhealth, useNA = "always")##
## anxiety disorder depression eating disorders none or NA
## 99 24 18 761
## other ptsd <NA>
## 23 16 0d <- subset(d, mhealth != "depression")
table(d$mhealth, useNA = "always")##
## anxiety disorder depression eating disorders none or NA
## 99 0 18 761
## other ptsd <NA>
## 23 16 0d$mhealth<- droplevels(d$mhealth)
table(d$mhealth, useNA = "always")##
## anxiety disorder eating disorders none or NA other
## 99 18 761 23
## ptsd <NA>
## 16 0d <- subset(d, mhealth != "eating disorders")
table(d$mhealth, useNA = "always")##
## anxiety disorder eating disorders none or NA other
## 99 0 761 23
## ptsd <NA>
## 16 0d$mhealth<- droplevels(d$mhealth)
table(d$mhealth, useNA = "always")##
## anxiety disorder none or NA other ptsd
## 99 761 23 16
## <NA>
## 0d <- subset(d, mhealth != "ptsd")
table(d$mhealth, useNA = "always")##
## anxiety disorder none or NA other ptsd
## 99 761 23 0
## <NA>
## 0d$mhealth<- droplevels(d$mhealth)
table(d$mhealth, useNA = "always")##
## anxiety disorder none or NA other <NA>
## 99 761 23 0# For your HW, you can choose to combine levels like we did here, OR you can simply choose which existing levels you want to compare/test -- to do this option, you'll need to copy/paste the "drop levels" code from the t-test lab/HW and delete the recoding code line above.
# check that all our categorical variables of interest are now factors
str(d)## 'data.frame': 883 obs. of 8 variables:
## $ X : int 321 520 1390 1422 1849 2247 2526 2609 2689 2752 ...
## $ age : chr "1 under 18" "1 under 18" "5 over 45" "1 under 18" ...
## $ mhealth : Factor w/ 3 levels "anxiety disorder",..: 2 2 2 2 1 2 2 2 3 2 ...
## $ pas_covid: num 2.33 3 2.89 2.67 2.56 ...
## $ pss : num 2.25 2.75 2.75 4.75 3.25 2.25 3.5 1.25 4 3.25 ...
## $ support : num 2.5 2.83 2.83 2.33 1.5 ...
## $ phq : num 1.89 1.56 1.22 4 3.67 ...
## $ row_id : Factor w/ 962 levels "1","2","3","4",..: 1 3 4 5 6 7 8 9 10 11 ...# check our DV skew and kurtosis
describe(d$phq)## vars n mean sd median trimmed mad min max range skew kurtosis se
## X1 1 883 2 0.85 1.78 1.91 0.82 1 4 3 0.75 -0.44 0.03# we'll use the describeBy() command to view our DV's skew and kurtosis across our IVs' levels
describeBy(d$phq, group = d$mhealth)##
## Descriptive statistics by group
## group: anxiety disorder
## vars n mean sd median trimmed mad min max range skew kurtosis se
## X1 1 99 2.62 0.87 2.56 2.62 1.15 1 4 3 0.01 -1.14 0.09
## ------------------------------------------------------------
## group: none or NA
## vars n mean sd median trimmed mad min max range skew kurtosis se
## X1 1 761 1.9 0.8 1.67 1.8 0.82 1 4 3 0.9 -0.06 0.03
## ------------------------------------------------------------
## group: other
## vars n mean sd median trimmed mad min max range skew kurtosis se
## X1 1 23 2.76 0.86 2.78 2.78 1.15 1 4 3 -0.15 -1.07 0.18# also use histograms to examine your continuous variable
hist(d$phq)
# 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 assured (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$mhealth)##
## anxiety disorder none or NA other
## 99 761 235.1.2 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(phq~mhealth, data = d)## Levene's Test for Homogeneity of Variance (center = median)
## Df F value Pr(>F)
## group 2 2.1652 0.1153
## 8805.1.3 Check for outliers using Cook’s distance and Residuals VS Leverage plot
5.1.3.1 Run a Regression to get these 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(195))
# 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.
reg_model <- lm(phq~mhealth, data = d) #for One-Way5.1.3.2 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 mental health disorders 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.
We identified and removed a single outlier for the One-Way ANOVA.
# Run an ANOVA
``` r
# One-Way
aov_model <- aov_ez(data = d,
id = "X",
between = c("mhealth"),
dv = "phq",
anova_table = list(es = "pes"))## Contrasts set to contr.sum for the following variables: mhealth6 View Output
nice(aov_model)## Anova Table (Type 3 tests)
##
## Response: phq
## Effect df MSE F pes p.value
## 1 mhealth 2, 879 0.65 46.92 *** .096 <.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 = "mhealth")
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="mhealth", adjust="sidak")## mhealth emmean SE df lower.CL upper.CL
## anxiety disorder 2.62 0.0808 879 2.43 2.81
## none or NA 1.90 0.0292 879 1.83 1.97
## other 2.84 0.1710 879 2.43 3.25
##
## Confidence level used: 0.95
## Conf-level adjustment: sidak method for 3 estimatespairs(emmeans(aov_model, specs="mhealth", adjust="sidak"))## contrast estimate SE df t.ratio p.value
## anxiety disorder - none or NA 0.717 0.0859 879 8.347 <.0001
## anxiety disorder - other -0.219 0.1900 879 -1.154 0.4809
## none or NA - other -0.936 0.1740 879 -5.382 <.0001
##
## 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 felt depression based on their mental health disorders type (none or NA, anxiety disorder, other), we used a one-way ANOVA. Our data was unbalanced, with many more people who have none or NA mental health disorders participating in our survey (n = 761) than who have anxiety disorder (n = 99) or have other (n = 23). 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 non-significant Levene’s test (p = .115) indicates that the assumption of homogeneity of variance was met.
We found a significant effect of mental health disorder type, F(2, 879) = 46.92, p < .001, ηp2 = .096 (medium effect size; Cohen, 1988). Posthoc tests using Sidak’s adjustment revealed that participants who have anxiety disorder (M = 2.62, SE = 0.08) reported significantly more depression than those who have none or NA mental health disorders (M = 1.90, SE = 0.03) but significantly less depression than those who have other mental health disorders (M = 2.84, SE = 0.17); participants who have other mental health disorders reported the highest amount of depression overall (see Figure 1 for a comparison).
References
Cohen J. (1988). Statistical Power Analysis for the Behavioral Sciences. New York, NY: Routledge Academic.