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 several columns with one summary: take(mtcars, mpg, hp, fun = 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':
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##     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 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")
## ************

## 
## Attaching package: 'afex'

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

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 parental marriage status on satisfaction of life.

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':    2115 obs. of  7 variables:
##  $ age       : chr  "1 between 18 and 25" "1 between 18 and 25" "1 between 18 and 25" "1 between 18 and 25" ...
##  $ marriage5 : chr  "are currently divorced from one another" "are currently married to one another" "are currently married to one another" "are currently married to one another" ...
##  $ moa_safety: num  2.75 3.25 3 1.25 2.25 2.5 4 3.25 2.75 3.5 ...
##  $ swb       : num  4.33 4.17 1.83 5.17 3.67 ...
##  $ support   : num  6 6.75 5.17 5.58 6 ...
##  $ socmeduse : int  47 23 34 35 37 13 37 43 37 29 ...
##  $ 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$marriage5 <- as.factor(d$marriage5)
# d$swb <- as.factor(d$swb)
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$marriage5)

## 
##             are currently divorced from one another 
##                                                 509 
##                are currently married to one another 
##                                                1401 
##       never married each other and are not together 
##                                                 166 
## never married each other but are currently together 
##                                                  39

# d$lom[d$marriage5 == "never married each other and are not together"] <- "divorced"
# d$lom[d$marriage5 == "are currently divorced from one another"] <- "divorced"
# d$lom[d$marriage5 == "are currently married to one another"] <- "together"
# d$lom[d$marriage5 == "never married each other and currently together"] <- "together"
# table(d$lom)
# d$lom <- as.factor(d$lom)

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

##    vars    n mean   sd median trimmed  mad min max range  skew kurtosis   se
## X1    1 2115 4.44 1.33    4.5     4.5 1.48   1   7     6 -0.36    -0.48 0.03

# we'll use the describeBy() command to view skew and kurtosis across our IVs
describeBy(d$swb, group = d$marriage5)

## 
##  Descriptive statistics by group 
## group: are currently divorced from one another
##    vars   n mean   sd median trimmed  mad min max range  skew kurtosis   se
## X1    1 509 4.14 1.34   4.33    4.18 1.48   1   7     6 -0.22    -0.53 0.06
## ------------------------------------------------------------ 
## group: are currently married to one another
##    vars    n mean   sd median trimmed  mad min max range  skew kurtosis   se
## X1    1 1401 4.59 1.29   4.83    4.66 1.24   1   7     6 -0.44    -0.43 0.03
## ------------------------------------------------------------ 
## group: never married each other and are not together
##    vars   n mean   sd median trimmed  mad min max range  skew kurtosis  se
## X1    1 166 4.04 1.35      4    4.08 1.48   1   7     6 -0.19    -0.54 0.1
## ------------------------------------------------------------ 
## group: never married each other but are currently together
##    vars  n mean   sd median trimmed  mad  min max range  skew kurtosis   se
## X1    1 39 4.51 1.42   4.67    4.55 1.24 1.33   7  5.67 -0.21    -0.51 0.23

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

# and cross_cases() to examine your categorical variables
cross_cases(d, marriage5, swb)

	swb
	1	1.16666666666667	1.33333333333333	1.5	1.66666666666667	1.83333333333333	2	2.16666666666667	2.33333333333333	2.5	2.66666666666667	2.83333333333333	3	3.16666666666667	3.33333333333333	3.5	3.66666666666667	3.83333333333333	4	4.16666666666667	4.33333333333333	4.5	4.66666666666667	4.83333333333333	5	5.16666666666667	5.33333333333333	5.5	5.66666666666667	5.83333333333333	6	6.16666666666667	6.33333333333333	6.5	6.66666666666667	6.83333333333333	7
marriage5
are currently divorced from one another	5	5	6	1	8	6	13	7	9	10	12	16	13	18	26	13	22	20	26	18	27	23	26	20	26	22	21	15	16	15	12	7	11	2	4	3	5
are currently married to one another	5	2	5	9	11	8	13	25	16	22	20	37	34	35	36	41	39	41	61	57	60	55	65	67	67	78	78	68	67	65	63	42	32	29	14	11	23
never married each other and are not together	3	1	3	1	1	5	1	1	5	2	8	6	3	4	5	8	9	8	10	9	9	7	3	7	8	7	1	6	5	4	9	1	2	1	2		1
never married each other but are currently together			1			1	1		1	1			1		1		4	3	1	1	2	1	2	3	1	2	3	1		2	1		1		1	1	2
#Total cases	13	8	15	11	20	20	28	33	31	35	40	59	51	57	68	62	74	72	98	85	98	86	96	97	102	109	103	90	88	86	85	50	46	32	21	15	31

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

table(d$marriage5)

## 
##             are currently divorced from one another 
##                                                 509 
##                are currently married to one another 
##                                                1401 
##       never married each other and are not together 
##                                                 166 
## never married each other but are currently together 
##                                                  39

# 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)

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(swb~marriage5, data = d)

## Levene's Test for Homogeneity of Variance (center = median)
##         Df F value Pr(>F)
## group    3  0.4368 0.7267
##       2111

# leveneTest(pss~gender_rc*poc, data = d2)

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(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(swb ~ marriage5, 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)

5.2 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 4 level marriage 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 = "row_id",
                    between = c("marriage5"),
                    dv = "swb",
                    anova_table = list(es = "pes"))

## Contrasts set to contr.sum for the following variables: marriage5

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: swb
##      Effect      df  MSE         F  pes p.value
## 1 marriage5 3, 2111 1.72 19.81 *** .027   <.001
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '+' 0.1 ' ' 1

8 Visualize Results

afex_plot(aov_model, x = "marriage5")

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="marriage5", adjust="tukey")

## Note: adjust = "tukey" was changed to "sidak"
## because "tukey" is only appropriate for one set of pairwise comparisons

##  marriage5                                           emmean     SE   df
##  are currently divorced from one another               4.14 0.0581 2111
##  are currently married to one another                  4.59 0.0350 2111
##  never married each other and are not together         4.04 0.1017 2111
##  never married each other but are currently together   4.51 0.2099 2111
##  lower.CL upper.CL
##      4.00     4.29
##      4.50     4.68
##      3.79     4.30
##      3.99     5.04
## 
## Confidence level used: 0.95 
## Conf-level adjustment: sidak method for 4 estimates

pairs(emmeans(aov_model, specs="marriage5", adjust="tukey"))

##  contrast                                                                                           
##  are currently divorced from one another - are currently married to one another                     
##  are currently divorced from one another - never married each other and are not together            
##  are currently divorced from one another - never married each other but are currently together      
##  are currently married to one another - never married each other and are not together               
##  are currently married to one another - never married each other but are currently together         
##  never married each other and are not together - never married each other but are currently together
##  estimate     SE   df t.ratio p.value
##   -0.4460 0.0678 2111  -6.574  <.0001
##    0.0983 0.1172 2111   0.839  0.8360
##   -0.3704 0.2178 2111  -1.701  0.3236
##    0.5442 0.1076 2111   5.058  <.0001
##    0.0756 0.2128 2111   0.355  0.9847
##   -0.4686 0.2333 2111  -2.009  0.1849
## 
## P value adjustment: tukey method for comparing a family of 4 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.

11 Write Up Results

11.1 One-Way ANOVA

To test our hypothesis that there would be a significant effect of parental marital status on life satisfaction, we used a one-way ANOVA. Our unevel levels of participants with the different parental marital status significantly reduces the power of our test and increases the chances of a Type II error. A significant Levene’s test (p = .73) 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 marital status, F(3,2111) = 19.81, p < .001, η_p² = .072 (large effect size; Cohen, 1988). Posthoc tests using Sidak’s test revealed that those with seperated parents reported lower life satisfaction than those with parents that were still together, while those with parents who were married and still together, showed the highest life satisfaction. (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

Claudia Siegel

2024-07-25