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':
## 
##     %+%, alpha
library(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':
## 
##     vars
library(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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##     logit
library(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")
## ************
## 
## Attaching package: 'afex'
## The following object is masked from 'package:lme4':
## 
##     lmer
library(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).

One-Way Hypothesis: There will be a significant difference in feelings of maturity by people’s level of political party, between democrat, republican, and independent.

IV = political party DV = maturity

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':    3112 obs. of  8 variables:
##  $ ResponseID  : chr  "R_BJN3bQqi1zUMid3" "R_2TGbiBXmAtxywsD" "R_12G7bIqN2wB2N65" "R_39pldNoon8CePfP" ...
##  $ gender      : chr  "f" "m" "m" "f" ...
##  $ party_rc    : chr  "democrat" "independent" "apolitical" "apolitical" ...
##  $ moa_maturity: num  3.67 3.33 3.67 3 3.67 ...
##  $ npi         : num  0.6923 0.1538 0.0769 0.0769 0.7692 ...
##  $ exploit     : num  2 3.67 4.33 1.67 4 ...
##  $ efficacy    : num  3.4 3.4 2.2 2.8 3 2.4 2.3 3 3 3.7 ...
##  $ 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$party_rc <- as.factor(d$party_rc)
d$row_id <- as.factor(d$row_id)

# We're going to recode our party variable into 3 groups for the One-Way ANOVA: democrat, independent, republican

table(d$party_rc)
## 
##  apolitical    democrat independent  republican 
##         436        1579         322         775
d <- subset(d, party_rc != "apolitical") # use subset() to remove all participants from the additional level

table(d$party_rc, useNA = "always") # verify that now there are ZERO participants in the additional level
## 
##  apolitical    democrat independent  republican        <NA> 
##           0        1579         322         775           0
d$party_rc <- droplevels(d$party_rc) # use droplevels() to drop the empty factor

table(d$party_rc, useNA = "always") # verify that now the entire factor level is removed 
## 
##    democrat independent  republican        <NA> 
##        1579         322         775           0
table(d$party_rc)
## 
##    democrat independent  republican 
##        1579         322         775
d$party_rc <- as.factor(d$party_rc)

# "drop levels" code copy/pasted from the t-test lab/HW.


# check that all our categorical variables of interest are now factors
str(d)
## 'data.frame':    2676 obs. of  8 variables:
##  $ ResponseID  : chr  "R_BJN3bQqi1zUMid3" "R_2TGbiBXmAtxywsD" "R_2Quh0h3wxTjZjKP" "R_2CfdmFw1NTliv4e" ...
##  $ gender      : chr  "f" "m" "f" "f" ...
##  $ party_rc    : Factor w/ 3 levels "democrat","independent",..: 1 2 2 1 1 1 2 2 1 1 ...
##  $ moa_maturity: num  3.67 3.33 3.67 4 4 ...
##  $ npi         : num  0.6923 0.1538 0.6154 0.0769 0.0769 ...
##  $ exploit     : num  2 3.67 5.33 1 3.33 ...
##  $ efficacy    : num  3.4 3.4 2.3 3 3 3 3.1 3.6 2.9 2.5 ...
##  $ row_id      : Factor w/ 3112 levels "1","2","3","4",..: 1 2 7 8 9 12 13 15 17 18 ...
# check our DV skew and kurtosis
describe(d$moa_maturity)
##    vars    n mean   sd median trimmed  mad  min max range  skew kurtosis   se
## X1    1 2676 3.59 0.43   3.67    3.65 0.49 1.33   4  2.67 -1.17     1.63 0.01
# we'll use the describeBy() command to view our DV's skew and kurtosis across our IVs' levels
describeBy(d$moa_maturity, group = d$party_rc)
## 
##  Descriptive statistics by group 
## group: democrat
##    vars    n mean   sd median trimmed  mad  min max range  skew kurtosis   se
## X1    1 1579 3.58 0.43   3.67    3.63 0.49 1.67   4  2.33 -1.02     0.95 0.01
## ------------------------------------------------------------ 
## group: independent
##    vars   n mean   sd median trimmed  mad  min max range skew kurtosis   se
## X1    1 322  3.6 0.46   3.67    3.67 0.49 1.33   4  2.67 -1.5     3.37 0.03
## ------------------------------------------------------------ 
## group: republican
##    vars   n mean   sd median trimmed  mad  min max range  skew kurtosis   se
## X1    1 775 3.62 0.43   3.67    3.68 0.49 1.33   4  2.67 -1.31     2.11 0.02
# also use histograms to examine your continuous variable
hist(d$moa_maturity)

# REMEMBER your test's level of POWER is determined by your SMALLEST subsample

5 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$party_rc)
## 
##    democrat independent  republican 
##        1579         322         775

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

# One-Way
leveneTest(moa_maturity~party_rc, data = d)
## Levene's Test for Homogeneity of Variance (center = median)
##         Df F value Pr(>F)
## group    2    0.76 0.4678
##       2673
# Not Significant

5.1.3 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(1108))

# to drop multiple outliers, 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.
reg_model <- lm(moa_maturity~party_rc, data = d) #for One-Way

5.1.3.2 Check for outliers (One-Way)

# Cook's distance
plot(reg_model, 4)

# Residuals VS Leverage
plot(reg_model, 5)

5.2 Issues with My Data

Our cell sizes are unbalanced between the party type group levels. Our apolitical level of our variable slightly limits our power and increases our Type II error rate. It is not relevant to the hypothesis so we must move forward and remove it. Levene’s test was not significant (p= 0.5) for our three-level party-type variable with the One-Way ANOVA.

We identified and removed zero outliers for the One-Way ANOVA.

6 Run an ANOVA

# One-Way
aov_model <- aov_ez(data = d,
                    id = "row_id",
                    between = c("party_rc"),
                    dv = "moa_maturity",
                    anova_table = list(es = "pes"))
## Contrasts set to contr.sum for the following variables: party_rc

7 View Output

nice(aov_model)
## Anova Table (Type 3 tests)
## 
## Response: moa_maturity
##     Effect      df  MSE      F  pes p.value
## 1 party_rc 2, 2673 0.18 2.80 + .002    .061
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '+' 0.1 ' ' 1

ANOVA 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

8 Visualize Results

# One-Way
afex_plot(aov_model, x = "party_rc")

9 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="party_rc", adjust="sidak")
##  party_rc    emmean     SE   df lower.CL upper.CL
##  democrat      3.58 0.0108 2673     3.55     3.60
##  independent   3.60 0.0239 2673     3.55     3.66
##  republican    3.62 0.0154 2673     3.58     3.66
## 
## Confidence level used: 0.95 
## Conf-level adjustment: sidak method for 3 estimates
pairs(emmeans(aov_model, specs="party_rc", adjust="sidak"))
##  contrast                 estimate     SE   df t.ratio p.value
##  democrat - independent    -0.0289 0.0263 2673  -1.099  0.5149
##  democrat - republican     -0.0432 0.0188 2673  -2.294  0.0568
##  independent - republican  -0.0144 0.0285 2673  -0.504  0.8692
## 
## P value adjustment: tukey method for comparing a family of 3 estimates

10 Write Up Results

10.1 One-Way ANOVA

To test our hypothesis that there will be a significant difference in feelings of maturity by people’s level of political party, between democrat, republican, and independent, we used a one-way ANOVA. Our data was unbalanced, with many more people who identify as democrats participating in our survey (n = 1579) than who are republican (n = 775) or independent (n = 332). This reduces the power of our test and increases the chances of a Type II error. We removed no outliers following visual analysis of Cook’s Distance and Residuals VS Leverage plots. A nonsignificant Levene’s test (p = 0.5) also indicates that our data does not violate the assumption of homogeneity of variance; see Figure 1 for comparison.

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References

Cohen J. (1988). Statistical Power Analysis for the Behavioral Sciences. New York, NY: Routledge Academic.