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':
## 
##     %+%, alpha

library(expss) # for the cross_cases() command

## Loading required package: maditr

## 
## To aggregate all non-grouping columns: take_all(mtcars, mean, by = am)

## 
## Attaching package: 'maditr'

## The following object is masked from 'package:base':
## 
##     sort_by

## 
## Attaching package: 'expss'

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

library(car) # for the leveneTest() command

## Loading required package: carData

## 
## Attaching package: 'car'

## The following object is masked from 'package:expss':
## 
##     recode

## The following object is masked from 'package:psych':
## 
##     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':
## 
##     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")
## ************

## 
## Attaching package: 'afex'

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

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

# 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/mydata.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

Two-Way ANOVA: We predict that there will significant effects of gender and income on worry, as measured by the Penn State Worry Questionaire (PSWQ). We also predict that income and gender will interact and that women of low income will report significantly higher stress than men of low income, medium and high income men and women, and those who prefer not to say.

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':    329 obs. of  7 variables:
##  $ gender   : chr  "female" "female" "female" "female" ...
##  $ income   : chr  "3 high" "2 middle" "2 middle" "2 middle" ...
##  $ covid_neg: int  0 0 0 0 0 0 0 0 0 0 ...
##  $ pswq     : num  4.94 3.94 2.62 2.94 2.81 ...
##  $ iou      : num  3.19 3.37 1.7 1.89 1.11 ...
##  $ edeq12   : num  1.58 1.67 1.42 1.33 3.17 ...
##  $ row_id   : int  1 2 3 4 5 6 7 8 9 10 ...

# make our categorical variables factors
#we'll actually use our ID variable for this analysis, so make sure it's coded as a factor
d$gender <- as.factor(d$gender)
d$income <- as.factor(d$income)
d$row_id <- as.factor(d$row_id)

# you can use the describe() command on an entire dataframe (d) or just on a single variable
describe(d$pswq)

##    vars   n mean   sd median trimmed  mad  min  max range skew kurtosis   se
## X1    1 329 2.88 0.88   2.88    2.86 1.02 1.12 4.94  3.81 0.18    -0.82 0.05

# we'll use the describeBy() command to view skew and kurtosis across our IVs
describeBy(d$pswq, group = d$gender)

## 
##  Descriptive statistics by group 
## group: female
##    vars   n mean   sd median trimmed  mad  min  max range skew kurtosis   se
## X1    1 285 2.94 0.87   2.94    2.92 0.93 1.19 4.94  3.75 0.14     -0.8 0.05
## ------------------------------------------------------------ 
## group: I use another term
##    vars n mean sd median trimmed mad  min  max range skew kurtosis se
## X1    1 1 4.44 NA   4.44    4.44   0 4.44 4.44     0   NA       NA NA
## ------------------------------------------------------------ 
## group: male
##    vars  n mean   sd median trimmed  mad  min  max range skew kurtosis   se
## X1    1 39  2.4 0.86   2.25    2.35 0.65 1.12 4.19  3.06 0.66    -0.44 0.14
## ------------------------------------------------------------ 
## group: Prefer not to say
##    vars n mean   sd median trimmed mad  min  max range skew kurtosis   se
## X1    1 4 2.92 0.73   2.75    2.92 0.6 2.31 3.88  1.56 0.31     -2.1 0.37

describeBy(d$pswq, group = d$income)

## 
##  Descriptive statistics by group 
## group: 1 low
##    vars  n mean   sd median trimmed  mad  min  max range  skew kurtosis   se
## X1    1 38 3.31 0.88   3.38    3.31 0.97 1.56 4.94  3.38 -0.16    -0.95 0.14
## ------------------------------------------------------------ 
## group: 2 middle
##    vars   n mean   sd median trimmed  mad  min  max range skew kurtosis   se
## X1    1 157 2.82 0.89   2.81     2.8 1.02 1.12 4.75  3.62 0.24    -0.82 0.07
## ------------------------------------------------------------ 
## group: 3 high
##    vars  n mean   sd median trimmed  mad  min  max range skew kurtosis   se
## X1    1 86  2.8 0.86   2.69    2.77 0.93 1.19 4.94  3.75 0.26    -0.64 0.09
## ------------------------------------------------------------ 
## group: prefer not to say
##    vars  n mean   sd median trimmed  mad  min  max range skew kurtosis   se
## X1    1 48 2.87 0.85   2.91    2.86 0.97 1.25 4.62  3.38 0.07    -1.04 0.12

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

# and cross_cases() to examine your categorical variables
cross_cases(d, gender, income)

	income
	1 low	2 middle	3 high	prefer not to say
gender
female	34	139	71	41
I use another term		1
male	4	15	14	6
Prefer not to say		2	1	1
#Total cases	38	157	86	48

5 Check Your Assumptions

5.1 ANOVA Assumptions

DV should be normally distributed across levels of the IV (skew and kurtosis)
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) (lowest cell determines power of whole test)
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

table(d$gender)

## 
##             female I use another term               male  Prefer not to say 
##                285                  1                 39                  4

cross_cases(d, gender, income) #2 way anova

	income
	1 low	2 middle	3 high	prefer not to say
gender
female	34	139	71	41
I use another term		1
male	4	15	14	6
Prefer not to say		2	1	1
#Total cases	38	157	86	48

# 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 != "I use another term")
d2$gender <- droplevels(d2$gender)
d3 <- subset(d2, gender != "Prefer not to say")
d3$gender <- droplevels(d3$gender)

# to double-check any changes we made
cross_cases(d3, gender, income)

	income
	1 low	2 middle	3 high	prefer not to say
gender
female	34	139	71	41
male	4	15	14	6
#Total cases	38	154	85	47

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
# significant means no homogeneity of variance, acknowledge issues for this class but don't change anything

leveneTest(pswq~gender*income, data = d3) # 2 way anova

## Levene's Test for Homogeneity of Variance (center = median)
##        Df F value Pr(>F)
## group   7  0.4187 0.8905
##       316

5.1.3 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(296))

# 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(pswq ~ gender*income, data = d3) #for two-way

5.1.3.2 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 (male low income) limits our power and increases our Type II error rate.

Levene’s test is was not significant, so we have met the assumptions for homogeneity of variance.

We checked for outliers and decided to not remove any, as there are none reaching Cook’s distance of 0.5. Although, there are two at or above 0.1.

6 Run an ANOVA

aov_model2 <- aov_ez(data = d3,
                    id = "row_id",
                    between = c("gender","income"),
                    dv = "pswq",
                    anova_table = list(es = "pes"))

## Contrasts set to contr.sum for the following variables: gender, income

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_model2)

## Anova Table (Type 3 tests)
## 
## Response: pswq
##          Effect     df  MSE       F  pes p.value
## 1        gender 1, 316 0.74 8.95 ** .028    .003
## 2        income 3, 316 0.74    2.08 .019    .102
## 3 gender:income 3, 316 0.74    0.85 .008    .467
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '+' 0.1 ' ' 1

8 Visualize Results

afex_plot(aov_model2, x = "gender", trace = "income")

afex_plot(aov_model2, x = "income", trace = "gender")

9 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", adjust="tukey") #removes effect of race

## 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 emmean     SE  df lower.CL upper.CL
##  female   3.01 0.0587 316     2.87     3.14
##  male     2.50 0.1597 316     2.14     2.86
## 
## Results are averaged over the levels of: income 
## Confidence level used: 0.95 
## Conf-level adjustment: sidak method for 2 estimates

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

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

##  contrast      estimate   SE  df t.ratio p.value
##  female - male    0.509 0.17 316   2.991  0.0030
## 
## Results are averaged over the levels of: income

# emmeans(aov_model2, specs="income", adjust="tukey")
# pairs(emmeans(aov_model2, specs="income", adjust="tukey"))
 
# emmeans(aov_model2, specs="gender", by="income", adjust="sidak")
# pairs(emmeans(aov_model2, specs="gender", by="income", adjust="sidak"))

# emmeans(aov_model2, specs="income", by="gender", adjust="sidak")
# pairs(emmeans(aov_model2, specs="income", by="gender", adjust="sidak"))

10 Write Up Results

10.1 Two-Way ANOVA

To test our hypothesis that gender and income 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 = 285) than men (n = 39). We removed those who used another term or preferred not to specify their gender, due to the small number of participants. We did not remove any outliers from the data set.

As predicted, we found a significant main effect for gender, F(1,315) = 11.64, p < .001, η_p² .036 (small to medium effect size; Cohen, 1988). As predicted, women reported significantly more stress (mean = 3.01) than men (mean = 2.39). Contrary to our expectations, we did not find a significant main effect for income (p = .376).

Lastly, we did not find a significant interaction between gender and income (p = .370) (see Figure 2).

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

Ellie Kemper

2024-07-22