Running a One-Way or Two-Way/Factorial ANOVA
Joshua Gritt
2024-06-10
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##
## Use magrittr pipe '%>%' to chain several operations:
## mtcars %>%
## let(mpg_hp = mpg/hp) %>%
## take(mean(mpg_hp), by = am)
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## Attaching package: 'maditr'## The following object is masked from 'package:base':
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## sort_by##
## 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 tests## Welcome to emmeans.
## Caution: You lose important information if you filter this package's results.
## See '? untidy'2 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/arc_clean.csv", header=T)
# new code! this adds a column with a number for each row. it makes it easier when we drop outliers later
d$row_id <- 1:nrow(d)3 State Your Hypothesis
Note: You can chose to run either a one-way ANOVA (a single IV with more than 2 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.
One-Way: We predict that there will be a significant effect of gender on stress, as measured by the perceived stress scale (PSS-4).
Two-Way: We predict that there will significant effects of gender and race/ethnicity on stress, as measured by the perceived stress scale (PSS-4). We also predict that race/ethnicity and gender will interact and that women of color will report significantly higher stress than men of color or white men and women.
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': 1250 obs. of 7 variables:
## $ X : int 1 20 30 31 33 57 68 81 86 104 ...
## $ gender_rc : chr "f" "m" "f" "f" ...
## $ ethnicity_rc: chr "white" "white" "white" "white" ...
## $ pss : num 3.25 3.75 1 3.25 2 4 3.75 1.25 2.5 2.5 ...
## $ phq : num 1.33 3.33 1 2.33 1.11 ...
## $ rse : num 2.3 1.6 3.9 1.7 3.9 1.8 1.3 3.5 2.6 3 ...
## $ row_id : int 1 2 3 4 5 6 7 8 9 10 ...# 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$ethnicity_rc <- as.factor(d$ethnicity_rc)
d$gender_rc <- as.factor(d$gender_rc)
d$row_id <- as.factor(d$row_id)
# we're going to recode our race/ethnicity variable into two groups: poc and white
table(d$ethnicity_rc)##
## asian black mideast multiracial other prefer_not
## 139 26 12 65 11 18
## white
## 979d$poc[d$ethnicity_rc == "asian"] <- "poc"
d$poc[d$ethnicity_rc == "black"] <- "poc"
d$poc[d$ethnicity_rc == "mideast"] <- "poc"
d$poc[d$ethnicity_rc == "multiracial"] <- "poc"
d$poc[d$ethnicity_rc == "other"] <- "poc"
d$poc[d$ethnicity_rc == "prefer_not"] <- NA
d$poc[d$ethnicity_rc == "white"] <- "white"
table(d$poc)##
## poc white
## 253 979d$poc <- as.factor(d$poc)
# you can use the describe() command on an entire dataframe (d) or just on a single variable
describe(d$pss)## vars n mean sd median trimmed mad min max range skew kurtosis se
## X1 1 1250 2.93 0.95 3 2.93 1.11 1 5 4 0.09 -0.74 0.03# we'll use the describeBy() command to view skew and kurtosis across our IVs
describeBy(d$pss, group = d$gender_rc)##
## Descriptive statistics by group
## group: f
## vars n mean sd median trimmed mad min max range skew kurtosis se
## X1 1 1020 2.97 0.95 3 2.97 1.11 1 5 4 0.04 -0.73 0.03
## ------------------------------------------------------------
## group: m
## vars n mean sd median trimmed mad min max range skew kurtosis se
## X1 1 197 2.6 0.89 2.5 2.55 0.74 1 5 4 0.5 -0.24 0.06
## ------------------------------------------------------------
## group: nb
## vars n mean sd median trimmed mad min max range skew kurtosis se
## X1 1 33 3.73 0.66 3.75 3.77 0.74 2 5 3 -0.48 -0.18 0.11describeBy(d$pss, group = d$poc)##
## Descriptive statistics by group
## group: poc
## vars n mean sd median trimmed mad min max range skew kurtosis se
## X1 1 253 3.06 0.96 3 3.06 1.11 1 5 4 0.03 -0.77 0.06
## ------------------------------------------------------------
## group: white
## vars n mean sd median trimmed mad min max range skew kurtosis se
## X1 1 979 2.9 0.95 3 2.89 1.11 1 5 4 0.1 -0.75 0.03# also use histograms to examine your continuous variable
hist(d$pss)
# and cross_cases() to examine your categorical variables
cross_cases(d, gender_rc, poc)|  poc | ||
|---|---|---|
|  poc |  white | |
|  gender_rc | ||
|    f | 205 | 799 |
|    m | 42 | 153 |
|    nb | 6 | 27 |
|    #Total cases | 253 | 979 |
5 Check Your Assumptions
5.1 ANOVA Assumptions
- 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 about, the sampling distribution should be normal. (For a demonstration of what the sampling distribution is, go here.)
5.1.1 Check levels of IVs
#one way ANOVA:
table(d$gender_rc)##
## f m nb
## 1020 197 33#two way
cross_cases(d, gender_rc, poc)|  poc | ||
|---|---|---|
|  poc |  white | |
|  gender_rc | ||
|    f | 205 | 799 |
|    m | 42 | 153 |
|    nb | 6 | 27 |
|    #Total cases | 253 | 979 |
# our number of small nb participants is going to hurt us for the two-way anova, but it should be okay for the one-way anova
# so we'll create a new dataframe for the two-way analysis and call it d2
d2 <- subset(d, gender_rc != "nb")
d2$gender_rc <- droplevels(d2$gender_rc)
# to double-check any changes we made
cross_cases(d2, gender_rc, poc)|  poc | ||
|---|---|---|
|  poc |  white | |
|  gender_rc | ||
|    f | 205 | 799 |
|    m | 42 | 153 |
|    #Total cases | 247 | 952 |
5.1.2 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(pss~gender_rc, data = d)## Levene's Test for Homogeneity of Variance (center = median)
## Df F value Pr(>F)
## group 2 5.1218 0.006091 **
## 1247
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1leveneTest(pss~gender_rc*poc, data = d2)## Levene's Test for Homogeneity of Variance (center = median)
## Df F value Pr(>F)
## group 3 1.2775 0.2807
## 11955.1.3 Check for outliers using Cook’s distance and Residuals vs Leverage plot
5.1.3.1 Run a Regression
# use this commented out section only if you need to remove outliers
# to drop a single outlier, remove the # at the beginning of the line and use this code:
d <- subset(d, row_id!=c(1108))
# to drop multiple outliers, remove the # at the beginning of the line and 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, and c is our covariate
reg_model <- lm(pss ~ gender_rc, data = d) #for one-way
reg_model2 <- lm(pss ~ gender_rc*poc, data = d2) #for two-way5.1.3.2 Check for outliers (One-Way)
# Cook's distance
plot(reg_model, 4)
# Residuals vs Leverage
plot(reg_model, 5)
5.1.3.3 Check for outliers (Two-Way)
# Cook's distance
plot(reg_model2, 4)
# Residuals vs Leverage
plot(reg_model2, 5)
5.2 Issues with My Data
Our cell sizes are very unbalanced. 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 is significant for our three-level gender variable. We are ignoring this and continuing with the analysis anyway, but in the real world this is something we would have to correct for.
We identified and removed a single outlier.
6 Run an ANOVA
aov_model <- aov_ez(data = d,
id = "X",
between = c("gender_rc"),
dv = "pss",
anova_table = list(es = "pes"))## Contrasts set to contr.sum for the following variables: gender_rcaov_model2 <- aov_ez(data = d2,
id = "X",
between = c("gender_rc","poc"),
dv = "pss",
anova_table = list(es = "pes"))## Warning: Missing values for 18 ID(s), which were removed before analysis:
## 490, 520, 1224, 1276, 1472, 2550, 3030, 3648, 5262, 6431, ... [showing first 10 only]
## Below the first few rows (in wide format) of the removed cases with missing data.
## X gender_rc poc .
## # 67 490 f <NA> 2.50
## # 68 520 f <NA> 2.75
## # 151 1224 f <NA> 1.25
## # 157 1276 f <NA> 1.50
## # 182 1472 f <NA> 2.00
## # 315 2550 f <NA> 3.00## Contrasts set to contr.sum for the following variables: gender_rc, poc7 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: pss
## Effect df MSE F pes p.value
## 1 gender_rc 2, 1246 0.87 27.54 *** .042 <.001
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '+' 0.1 ' ' 1nice(aov_model2)## Anova Table (Type 3 tests)
##
## Response: pss
## Effect df MSE F pes p.value
## 1 gender_rc 1, 1195 0.87 28.31 *** .023 <.001
## 2 poc 1, 1195 0.87 0.56 <.001 .453
## 3 gender_rc:poc 1, 1195 0.87 3.43 + .003 .064
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '+' 0.1 ' ' 18 Visualize Results
afex_plot(aov_model, x = "gender_rc")
afex_plot(aov_model2, x = "gender_rc", trace = "poc")
afex_plot(aov_model2, x = "poc", trace = "gender_rc")
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.
emmeans(aov_model, specs="gender_rc", adjust="tukey")## Note: adjust = "tukey" was changed to "sidak"
## because "tukey" is only appropriate for one set of pairwise comparisons## gender_rc emmean SE df lower.CL upper.CL
## f 2.97 0.0291 1246 2.90 3.04
## m 2.60 0.0663 1246 2.44 2.75
## nb 3.79 0.1644 1246 3.40 4.18
##
## Confidence level used: 0.95
## Conf-level adjustment: sidak method for 3 estimatespairs(emmeans(aov_model, specs="gender_rc", adjust="tukey"))## contrast estimate SE df t.ratio p.value
## f - m 0.379 0.0724 1246 5.237 <.0001
## f - nb -0.815 0.1670 1246 -4.879 <.0001
## m - nb -1.194 0.1773 1246 -6.734 <.0001
##
## P value adjustment: tukey method for comparing a family of 3 estimates10 Run Posthoc Tests (Two-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.
emmeans(aov_model, specs="gender_rc", adjust="tukey")## Note: adjust = "tukey" was changed to "sidak"
## because "tukey" is only appropriate for one set of pairwise comparisons## gender_rc emmean SE df lower.CL upper.CL
## f 2.97 0.0291 1246 2.90 3.04
## m 2.60 0.0663 1246 2.44 2.75
## nb 3.79 0.1644 1246 3.40 4.18
##
## Confidence level used: 0.95
## Conf-level adjustment: sidak method for 3 estimatespairs(emmeans(aov_model, specs="gender_rc", adjust="tukey"))## contrast estimate SE df t.ratio p.value
## f - m 0.379 0.0724 1246 5.237 <.0001
## f - nb -0.815 0.1670 1246 -4.879 <.0001
## m - nb -1.194 0.1773 1246 -6.734 <.0001
##
## P value adjustment: tukey method for comparing a family of 3 estimatesemmeans(aov_model2, specs="poc", adjust="tukey")## NOTE: Results may be misleading due to involvement in interactions
## Note: adjust = "tukey" was changed to "sidak"
## because "tukey" is only appropriate for one set of pairwise comparisons## poc emmean SE df lower.CL upper.CL
## poc 2.84 0.0791 1195 2.66 3.01
## white 2.77 0.0412 1195 2.68 2.86
##
## Results are averaged over the levels of: gender_rc
## Confidence level used: 0.95
## Conf-level adjustment: sidak method for 2 estimatespairs(emmeans(aov_model2, specs="poc", adjust="tukey"))## NOTE: Results may be misleading due to involvement in interactions## contrast estimate SE df t.ratio p.value
## poc - white 0.067 0.0892 1195 0.751 0.4527
##
## Results are averaged over the levels of: gender_rcemmeans(aov_model2, specs="gender_rc", by="poc", adjust="sidak")## poc = poc:
## gender_rc emmean SE df lower.CL upper.CL
## f 3.16 0.0652 1195 3.01 3.30
## m 2.52 0.1441 1195 2.20 2.84
##
## poc = white:
## gender_rc emmean SE df lower.CL upper.CL
## f 2.93 0.0330 1195 2.85 3.00
## m 2.62 0.0755 1195 2.45 2.79
##
## Confidence level used: 0.95
## Conf-level adjustment: sidak method for 2 estimatespairs(emmeans(aov_model2, specs="gender_rc", by="poc", adjust="sidak"))## poc = poc:
## contrast estimate SE df t.ratio p.value
## f - m 0.639 0.1581 1195 4.044 0.0001
##
## poc = white:
## contrast estimate SE df t.ratio p.value
## f - m 0.309 0.0824 1195 3.753 0.0002emmeans(aov_model2, specs="poc", by="gender_rc", adjust="sidak")## gender_rc = f:
## poc emmean SE df lower.CL upper.CL
## poc 3.16 0.0652 1195 3.01 3.30
## white 2.93 0.0330 1195 2.85 3.00
##
## gender_rc = m:
## poc emmean SE df lower.CL upper.CL
## poc 2.52 0.1441 1195 2.20 2.84
## white 2.62 0.0755 1195 2.45 2.79
##
## Confidence level used: 0.95
## Conf-level adjustment: sidak method for 2 estimatespairs(emmeans(aov_model2, specs="poc", by="gender_rc", adjust="sidak"))## gender_rc = f:
## contrast estimate SE df t.ratio p.value
## poc - white 0.2321 0.0731 1195 3.175 0.0015
##
## gender_rc = m:
## contrast estimate SE df t.ratio p.value
## poc - white -0.0982 0.1626 1195 -0.604 0.546311 Write Up Results
11.1 One-Way ANOVA
To test our hypothesis that there would be a significant effect of gender on stress, we used a one-way ANOVA. Our data was unbalanced, with many more women participating in our survey (n = 1004) than men (n = 195) or non-binary and other gender participants (n = 32). 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 a Residuals vs Leverage plot. A significant Levene’s test (p = .002) 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 gender, F(2,1246) = 27.54, p < .001, ηp2 = .042 (large effect size; Cohen, 1988). Posthoc tests using Tukey’s HSD revealed that women reported more stress than men but less stress than non-binary and other gender participants, while non-binary and other gender participants reported the highest amount of stress overall (see Figure 1 for a comparison).
11.2 Two-Way ANOVA
To test our hypothesis that gender and race would impact stress and would interact significantly, we used a two-way/factorial ANOVA. Our data met most of the assumptions of the test, although our data was unbalanced, with many more women participating in our survey (n = 1004) than men (n = 195). We identified and removed a single outlier following visual analysis of a Residuals vs Leverage plot.
As predicted, we found a significant main effect for gender, F(1,1195) = 28.31, p < .001, ηp2 .023 (small effect size; Cohen, 1988). As predicted, women reported significantly more stress than men. Contrary to our expectations, we did not find a significant main effect for race (p = .453).
Lastly, we found a significant interaction between gender and race (see Figure 2), F(1,1195) = 3.43, p = .064, ηp2 = .003 (trivial effect size; Cohen, 1988). When comparing by race, women of color (M = 3.16, SE = .07) reported significantly more stress than men of color (M = 2.52, SE = .14; p < .001), as did white women (M = 2.93, SE = .03) compared to white men (M = 2.62, SE = .08; p < .001). When comparing by gender, women of color reported significantly more stress than white women (p = .002), while men of color and white men reported similar levels of stress (p = .546).
References
Cohen J. (1988). Statistical Power Analysis for the Behavioral Sciences. New York, NY: Routledge Academic.