Running a One-Way ANOVA
Aniyah Thompkins
2024-04-22
1 Loading Libraries
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 select columns from data: columns(mtcars, mpg, vs:carb)##
## 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 and plot results## 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(emmeans) # for posthoc tests2 Importing Data
# import the dataset you cleaned previously
# this will be the dataset you'll use throughout the rest of the semester
# use ARC data
d <- read.csv(file="Data/arcdata_final.csv", header=T)
# for the HW, you may or may not need to use the code below this comment
# check to see if you have a variable called 'X' or 'ResponseId' in your data
names(d)## [1] "X" "trans" "ethnicity" "rse" "gad" "support"
## [7] "mfq_26"3 State Your Hypothesis
Note: You can chose to run either a one-way ANOVA (a single IV with more than 3 levels) or a two-way/factorial ANOVA (at least two IVs) for the homework. You will need to specify your hypothesis and customize your code based on the choice you make. I will run both versions of the test here for illustrative purposes.
I predict that there will be a significant effect of ethnicity on self-esteem, as measured by the Rosenberg’s Self-Esteem Scale (RSE).
4 Check Your Variables
# you only need to check the variables you're using in the current analysis
# although 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': 1210 obs. of 7 variables:
## $ X : int 1 20 30 31 33 49 57 68 81 86 ...
## $ trans : chr "no" "no" "no" "no" ...
## $ ethnicity: chr "White - British, Irish, other" "White - British, Irish, other" "White - British, Irish, other" "White - British, Irish, other" ...
## $ rse : num 2.3 1.6 3.9 1.7 3.9 2.4 1.8 1.3 3.5 2.6 ...
## $ gad : num 1.86 3.86 1.14 2 1.43 ...
## $ support : num 2.5 2.17 5 2.5 3.67 ...
## $ mfq_26 : num 4.2 3.35 4.65 4.65 4.5 4.3 5.25 5 4.7 4.05 ...# make our categorical variables factors
d$X <- as.factor(d$X) #we'll actually use our ID variable for this analysis, so make sure it's coded as a factor
d$ethnicity <- as.factor(d$ethnicity)
# check your categorical variables
table(d$ethnicity, useNA = "always")##
## Asian/Asian British - Indian, Pakistani, Bangladeshi, other
## 123
## Black/Black British - Caribbean, African, other
## 24
## Chinese/Chinese British
## 10
## Middle Eastern/Middle Eastern British - Arab, Turkish, other
## 12
## Mixed race - other
## 37
## Mixed race - White and Black/Black British
## 21
## Other ethnic group
## 10
## Prefer not to say
## 24
## White - British, Irish, other
## 949
## <NA>
## 0# also use histograms to examine your continuous variable
hist(d$rse)
5 Check Your Assumptions
5.1 ANOVA Assumptions
Assumptions checked below:
- DV should be normally distributed across levels of the IV
- All levels of the IVs should have equal number of cases and there should be no empty cells. Cells with low numbers decrease the power of the test (increase change of Type II error)
- Homogeneity of variance should be assured
- Outliers should be identified and removed
If you have confirmed everything else…
- The sampling distribution should be normal. (For a demonstration of what the sampling distribution is, go here.)
5.1.1 Check levels of IVs
# check your categorical variables and make sure they have decent cell sizes
# they should have at least 5 participants in each cell
# but larger numbers are always better
table(d$ethnicity)##
## Asian/Asian British - Indian, Pakistani, Bangladeshi, other
## 123
## Black/Black British - Caribbean, African, other
## 24
## Chinese/Chinese British
## 10
## Middle Eastern/Middle Eastern British - Arab, Turkish, other
## 12
## Mixed race - other
## 37
## Mixed race - White and Black/Black British
## 21
## Other ethnic group
## 10
## Prefer not to say
## 24
## White - British, Irish, other
## 949cross_cases(d, ethnicity)|  #Total | |
|---|---|
|  ethnicity | |
|    Asian/Asian British - Indian, Pakistani, Bangladeshi, other | 123 |
|    Black/Black British - Caribbean, African, other | 24 |
|    Chinese/Chinese British | 10 |
|    Middle Eastern/Middle Eastern British - Arab, Turkish, other | 12 |
|    Mixed race - other | 37 |
|    Mixed race - White and Black/Black British | 21 |
|    Other ethnic group | 10 |
|    Prefer not to say | 24 |
|    White - British, Irish, other | 949 |
|    #Total cases | 1210 |
d$other[d$ethnicity == "Other ethnic group"] <- "other"
d$other[d$ethnicity == "Prefer not to say"] <- NA
d$other[d$ethnicity == "Mixed race - other"] <- "other"
d$other[d$ethnicity == "Mixed race - White and Black/Black British"] <- "other"
d$other[d$ethnicity == "White - British, Irish, other"] <- "White"
d$other[d$ethnicity == "Asain/Asain British - Indian, Pakistani, Bangladeshi, other"] <- "Asain"
d$other[d$ethnicity == "Middle Eastern/Middle Eastern British - Arab, Turkish, other"] <- "Middle Eastern"
d$other[d$ethnicity == "Black/Black British - Caribbean, African, other"] <- "Black"
d$other[d$ethnicity == "Chinese/Chinese British"] <- "Chinese"
table(d$other)##
## Black Chinese Middle Eastern other White
## 24 10 12 68 949d$other <- as.factor(d$other)
# you can use the describe() command on an entire dataframe (d) or just on a single variable
describe(d$rse)## vars n mean sd median trimmed mad min max range skew kurtosis se
## X1 1 1210 2.63 0.72 2.7 2.64 0.74 1 4 3 -0.22 -0.72 0.02# we'll use the describeBy() command to view skew and kurtosis across our IVs
describeBy(d$rse, group = d$other)##
## Descriptive statistics by group
## group: Black
## vars n mean sd median trimmed mad min max range skew kurtosis se
## X1 1 24 2.56 0.75 2.6 2.56 0.96 1.1 4 2.9 0.04 -1.03 0.15
## ------------------------------------------------------------
## group: Chinese
## vars n mean sd median trimmed mad min max range skew kurtosis se
## X1 1 10 2.66 0.55 3 2.69 0.22 1.9 3.2 1.3 -0.33 -1.98 0.18
## ------------------------------------------------------------
## group: Middle Eastern
## vars n mean sd median trimmed mad min max range skew kurtosis se
## X1 1 12 2.79 0.79 2.9 2.8 0.67 1.5 4 2.5 -0.06 -1.28 0.23
## ------------------------------------------------------------
## group: other
## vars n mean sd median trimmed mad min max range skew kurtosis se
## X1 1 68 2.57 0.65 2.6 2.57 0.59 1.2 4 2.8 -0.05 -0.61 0.08
## ------------------------------------------------------------
## group: White
## vars n mean sd median trimmed mad min max range skew kurtosis se
## X1 1 949 2.64 0.73 2.7 2.66 0.74 1 4 3 -0.24 -0.74 0.025.1.2 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
leveneTest(rse~other, data = d)## Levene's Test for Homogeneity of Variance (center = median)
## Df F value Pr(>F)
## group 4 0.8838 0.473
## 10585.1.3 Check for outliers using Cook’s distance and Residuals vs Leverage plot
5.1.3.1 Run a Regression
# use the lm() command to run the regression
# formula is y~x1*x2, where y is our DV, x1 is our first IV and x2 is our second IV
reg_model <- lm(rse~other, 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)
5.2 Issues with My Data
My cell sizes are very unbalanced. A small sample size for three of the levels of my variable limits my power and increases my Type II error rate.
Levene’s test is not significant for my five-level other variable, showing homogeneity of variance.
After running Cook’s Distance, there seems to be an observation of one outlier above the 0.5 cutoff. I am ignoring this and continuing with the analysis anyway, but in the real world this is something I would have to correct for.
6 Run an ANOVA
aov_model <- aov_ez(data = d,
id = "X",
between = c("other"),
dv = "rse",
anova_table = list(es = "pes"))## Warning: Missing values for 147 ID(s), which were removed before analysis:
## 119, 140, 490, 520, 580, 824, 851, 1065, 1180, 1224, ... [showing first 10 only]
## Below the first few rows (in wide format) of the removed cases with missing data.
## X other .
## # 14 119 <NA> 3.4
## # 20 140 <NA> 3.6
## # 66 490 <NA> 2.9
## # 67 520 <NA> 2.6
## # 75 580 <NA> 2.7
## # 100 824 <NA> 3.2## Contrasts set to contr.sum for the following variables: other7 View Output
Effect size cutoffs from Cohen (1988):
- η2 = 0.01 indicates a small effect
- η2 = 0.06 indicates a medium effect
- η2 = 0.14 indicates a large effect
nice(aov_model)## Anova Table (Type 3 tests)
##
## Response: rse
## Effect df MSE F pes p.value
## 1 other 4, 1058 0.52 0.35 .001 .846
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '+' 0.1 ' ' 18 Visualize Results
afex_plot(aov_model, x = "other")
9 Run Posthoc Tests (One-Way)
Only run posthocs if the test is significant! E.g., only run the posthoc tests on gender if there is a main effect for gender.
10 Write Up Results
10.1 One-Way ANOVA
To test my hypothesis that there would be a significant effect of ethnicity on self-esteem, I used a one-way ANOVA. My data was unbalanced, with many more White participants participating in my survey (n = 949), than Black participants (n = 24), Chinese participants (n = 10), Middle Eastern participants (n = 12), and Others (n = 68). This significantly reduces the power of my test and increases the chances of a Type II error. I did not identify any outliers following visual analysis of a Residuals vs Leverage plot, however I did observe an outlier that was above the threshold on 0.5 utilizng Cook’s Distance. An insignificant Levene’s test (p = .473) indicates that my data follows the assumption of homogeneity of variance. I continued with my analysis for the purpose of this class.
I found a very small effect of ethnicity on self-esteem, F(4,1058) = 0.35, p .846, ηp2 = .001 (small effect size; Cohen, 1988). Since there was no main effect found for ethnicity on self-esteem, Posthoc tests using Tukey’s HSD was not ran.
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References
Cohen J. (1988). Statistical Power Analysis for the Behavioral Sciences. New York, NY: Routledge Academic.