1 Loading Libraries

library(psych) # for the describe() command

## Warning: package 'psych' was built under R version 4.1.3

library(ggplot2) # to visualize our results

## 
## Attaching package: 'ggplot2'

## The following objects are masked from 'package:psych':
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##     %+%, alpha

library(expss) # for the cross_cases() command

## Warning: package 'expss' was built under R version 4.1.3

## Loading required package: maditr

## Warning: package 'maditr' was built under R version 4.1.3

## 
## To modify variables or add new variables:
##              let(mtcars, new_var = 42, new_var2 = new_var*hp) %>% head()

## 
## Attaching package: 'expss'

## The following object is masked from 'package:ggplot2':
## 
##     vars

library(car) # for the leveneTest() command

## Warning: package 'car' was built under R version 4.1.3

## 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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##     logit

library(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
## - NEWS: emmeans() for ANOVA models now uses model = 'multivariate' as default.
## - Get and set global package options with: afex_options()
## - Set orthogonal sum-to-zero contrasts globally: set_sum_contrasts()
## - For example analyses see: browseVignettes("afex")
## ************

## 
## Attaching package: 'afex'

## The following object is masked from 'package:lme4':
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##     lmer

library(emmeans) # for posthoc tests

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)

# create the ID variable
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 3 or more 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$row_id <- as.factor(d$row_id) #we'll actually use our ID variable for this analysis, so make sure it's coded as a factor
d$gender_rc <- as.factor(d$gender_rc)
d$ethnicity_rc <- as.factor(d$ethnicity_rc)

# check your categorical variables
table(d$gender_rc)

## 
##    f    m   nb 
## 1020  197   33

table(d$ethnicity_rc, useNA = "always")

## 
##       asian       black     mideast multiracial       other  prefer_not 
##         139          26          12          65          11          18 
##       white        <NA> 
##         979           0

# also use histograms to examine your continuous variable
hist(d$pss)

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$gender_rc)

## 
##    f    m   nb 
## 1020  197   33

table(d$ethnicity_rc)

## 
##       asian       black     mideast multiracial       other  prefer_not 
##         139          26          12          65          11          18 
##       white 
##         979

cross_cases(d, gender_rc, ethnicity_rc)

	ethnicity_rc
	asian	black	mideast	multiracial	other	prefer_not	white
gender_rc
f	110	24	9	54	8	16	799
m	27	2	2	8	3	2	153
nb	2		1	3			27
#Total cases	139	26	12	65	11	18	979

# we're going to recode our race/ethnicity variable into two groups: poc and white
# you may or may not need to combine and/or drop groups for the HW
table(d$ethnicity_rc)

## 
##       asian       black     mideast multiracial       other  prefer_not 
##         139          26          12          65          11          18 
##       white 
##         979

d$poc[d$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, useNA = "always")

## 
##   poc white  <NA> 
##   253   979    18

d$poc <- as.factor(d$poc)

# 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 == "m" | gender_rc == "f")
d2$gender_rc <- droplevels(d2$gender_rc)

# to double-check any changes we made
table(d2$gender_rc)

## 
##    f    m 
## 1020  197

cross_cases(d2, gender_rc, poc)

	poc
	poc	white
gender_rc
f	205	799
m	42	153
#Total cases	247	952

# 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 0.11

describeBy(d2$pss, group = d2$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

describeBy(d2$pss, group = d2$poc)

## 
##  Descriptive statistics by group 
## group: poc
##    vars   n mean   sd median trimmed  mad min max range skew kurtosis   se
## X1    1 247 3.05 0.96      3    3.05 1.11   1   5     4 0.04    -0.75 0.06
## ------------------------------------------------------------ 
## group: white
##    vars   n mean   sd median trimmed  mad min max range skew kurtosis   se
## X1    1 952 2.88 0.94   2.75    2.86 1.11   1   5     4 0.13    -0.72 0.03

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 ' ' 1

leveneTest(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
##       1195

5.1.3 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(pss~gender_rc, data = d) #for one-way
reg_model2 <- lm(pss~gender_rc*poc, data = d2) #for two-way

5.1.3.2 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.

6 Run an ANOVA

aov_model <- aov_ez(data = d,
                    id = "row_id",
                    between = c("gender_rc"),
                    dv = "pss",
                    anova_table = list(es = "pes"))

## Contrasts set to contr.sum for the following variables: gender_rc

aov_model2 <- aov_ez(data = d2,
                    id = "row_id",
                    between = c("gender_rc","poc"),
                    dv = "pss",
                    anova_table = list(es = "pes"))

## Warning: Missing values for following ID(s):
## 67, 68, 153, 159, 184, 319, 376, 470, 677, 828, 837, 879, 932, 943, 986, 1042, 1048, 1250
## Removing those cases from the analysis.

## Contrasts set to contr.sum for the following variables: gender_rc, poc

7 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, 1247 0.87 26.20 *** .040   <.001
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '+' 0.1 ' ' 1

nice(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 ' ' 1

8 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 0.0292 1247     2.90     3.04
##  m           2.60 0.0663 1247     2.44     2.75
##  nb          3.73 0.1621 1247     3.35     4.12
## 
## Confidence level used: 0.95 
## Conf-level adjustment: sidak method for 3 estimates

pairs(emmeans(aov_model, specs="gender_rc", adjust="tukey"))

##  contrast estimate     SE   df t.ratio p.value
##  f - m       0.379 0.0725 1247   5.231  <.0001
##  f - nb     -0.761 0.1647 1247  -4.618  <.0001
##  m - nb     -1.140 0.1751 1247  -6.507  <.0001
## 
## P value adjustment: tukey method for comparing a family of 3 estimates

10 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_model2, specs="gender_rc", 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

##  gender_rc emmean     SE   df lower.CL upper.CL
##  f           3.04 0.0365 1195     2.96     3.12
##  m           2.57 0.0813 1195     2.38     2.75
## 
## Results are averaged over the levels of: poc 
## Confidence level used: 0.95 
## Conf-level adjustment: sidak method for 2 estimates

pairs(emmeans(aov_model2, specs="gender_rc", adjust="tukey"))

## NOTE: Results may be misleading due to involvement in interactions

##  contrast estimate     SE   df t.ratio p.value
##  f - m       0.474 0.0892 1195   5.320  <.0001
## 
## Results are averaged over the levels of: poc

emmeans(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 estimates

pairs(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_rc

emmeans(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 estimates

pairs(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  0.0002

emmeans(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 estimates

pairs(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  0.5463

11 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 did not identify any outliers 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(2,1247) = 26.20, p < .001, η_p² = .040 (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, η_p² .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, η_p² = .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.

Running a One-Way or Two-Way/Factorial ANOVA

Heather Perkins

2024-04-08