Basic Inferential Stats in R: Correlation, T-Tests, and ANOVA

Bella Ratmelia

Today’s Outline

Refreshers on data distribution and research variables
Statistical tests: chi-square, t-test, correlations.
ANOVA

Refresher: Data Distribution

The choice of appropriate statistical tests and methods often depends on the distribution of the data. Understanding the distribution helps in selecting the right and validity of the tests.

Refresher: Research Variables

Dependent Variable (DV)

The variables that will be affected as a result of manipulation/changes in the IVs

Other names for it: Outcome, Response, Output, etc.
Often denoted as \(y\)

Independent Variable (IV)

The variables that researchers will manipulate.

Other names for it: Predictor, Covariate, Treatment, Regressor, Input, etc.
Often denoted as \(x\)

Open your project - the (possibly) easier way

Go to the folder where you put your project for this workshop
Find a file with .Rproj extension - this is the R project file that holds all the information about your project.

Double click on the file. Rstudio should launch with your project loaded! This should be easier to ensure that you are loading the correct project when opening Rstudio.

Load our data for today!

Let’s create a new R script called session-3.R, and then copy the code below to load our data for today. This code uses read_csv from readr package (part of tidyverse) to load our cleaned CSV (from the first checkpoint)

# import tidyverse library
library(tidyverse)

# read the CSV with WVS data
wvs_cleaned <- read_csv("data-output/wvs_cleaned_v1.csv")

# Convert categorical variables to factors
columns_to_convert <- c("country", "religiousity", "sex", "marital_status", "employment")

wvs_cleaned <- wvs_cleaned |> 
    mutate(across(all_of(columns_to_convert), as_factor))

# peek at the data, pay attention to the data types!
glimpse(wvs_cleaned)

Both Categorical variables - The \(X^2\) test

Chi-square test of independence

The \(X^2\) test of independence evaluates whether there is a statistically significant relationship between two categorical variables.

This is done by analyzing the frequency table (i.e., contingency table) formed by two categorical variables.

Example: Is there a relationship between religiousity and country in our WVS data?

Typically, we can start with the contigency table first, and then the visualization

table(wvs_cleaned$religiousity, wvs_cleaned$country)

                        
                          CAN  NZL  SGP
  Not a religious person 1794  310  563
  A religious person     1406  209  990
  An atheist              818   97  172
  Don't know                0   44    0

Chi-square test of independence - visualizing data

We can use barchart to visualize this (remember from last week!)

wvs_cleaned |> 
    ggplot(aes(x = country, fill = religiousity)) +
    geom_bar(position = "fill") +
    labs(title = "Proportion of religiousity for each country") +
    theme_minimal()

Chi-Square: Sample problem and results

Is there a relationship between religiosity and country?

chisq.test(table(wvs_cleaned$religiousity, wvs_cleaned$country))


    Pearson's Chi-squared test

data:  table(wvs_cleaned$religiousity, wvs_cleaned$country)
X-squared = 672.16, df = 6, p-value < 2.2e-16

X-squared = the coefficient
df = degree of freedom
p-value = the probability of getting more extreme results than what was observed. Generally, if this value is less than the pre-determined significance level (also called alpha), the result would be considered “statistically significant”

What if there is a hypothesis? How would you write this in the report?

Both Continuous variables - Correlation

Correlation

A correlation test evaluates the strength and direction of a linear relationship between two variables. The coefficient is expressed in value between -1 to 1, with 0 being no correlation at all.

Pearson’s \(r\) (r)

Measure the association between two continuous numerical variables
Sensitive to outliers
Assumes normality and/or linearity
(most likely the one that you learned in class)

Kendall’s \(\tau\) (tau)

Measure the association between two variables (ordinal-ordinal or ordinal-continuous)
less sensitive/more robust to outliers
non-parametric, does not assume normality and/or linearity

Spearman’s \(\rho\) (rho)

Measure the association between two variables (ordinal-ordinal or ordinal-continuous)
less sensitive/more robust to outliers
non-parametric, does not assume normality and/or linearity

For more info, you can refer to this reading: Measures of Association - How to Choose? (Harry Khamis, PhD)

Correlation: Sample problem and result

RQ: Is there a significant correlation between life satisfaction and financial satisfaction?

As both variables are numerical and continuous, we can use pearson correlation.

Let’s start with visualizing the data, which can be used to support the explanation.

wvs_cleaned |> 
    ggplot(aes(x = financial_satisfaction, y = life_satisfaction)) +
    geom_jitter(color="maroon", alpha=0.5) +
    geom_smooth(method = "lm", se = TRUE) # se shows the confidence interval

Correlation: Sample problem and result

Conduct the correlation test

cor.test(wvs_cleaned$life_satisfaction, 
         wvs_cleaned$financial_satisfaction, 
         method = "pearson")


    Pearson's product-moment correlation

data:  wvs_cleaned$life_satisfaction and wvs_cleaned$financial_satisfaction
t = 66.999, df = 6401, p-value < 2.2e-16
alternative hypothesis: true correlation is not equal to 0
95 percent confidence interval:
 0.6274032 0.6562061
sample estimates:
      cor 
0.6420311

cor is the correlation coefficient - this is the number that you want to report.
t is the t-test statistic
df is the degrees of freedom
p-value is the significance level of the t-test
conf.int is the confidence interval of the coefficient at 95%
sample estimates is the correlation coefficient

Learning Check #1

Is there a relationship between religiousity and age group? Calculate the chi-square test!

optional: Visualize the proportion of age group for each religiousity category.

Show answer

chisq.test(table(wvs_cleaned$religiousity, wvs_cleaned$age_group))


    Pearson's Chi-squared test

data:  table(wvs_cleaned$religiousity, wvs_cleaned$age_group)
X-squared = 97.449, df = 9, p-value < 2.2e-16

Visualization:

Show answer

wvs_cleaned |> 
    ggplot(aes(x = religiousity, fill = age_group)) +
    geom_bar(position = "fill") +
    labs(title = "Proportion of age groups for each religiousity") +
    theme_minimal()

Learning Check #1

Comparing Means Between Groups - T-Tests and ANOVA

T-Tests

A t-test is a statistical test used to compare the means of two groups/samples of continuous data type and determine if the differences are statistically significant.

The Student’s t-test is widely used when the sample size is reasonably small (less than approximately 30) or when the population standard deviation is unknown.

3 types of t-test

One-sample T-test

Test if a specific sample mean (X̄) is statistically different from a known or hypothesized population mean (μ or mu)

Two-samples / Independent Samples T-test

Used to compare the means of two independent groups (such as between-subjects research) to determine if they are significantly different.

Examples: Men vs Women group, Placebo vs Actual drugs.

Paired Samples T-Test

Used to compare the means of two related groups, such as repeated measurements on the same subjects (within-subjects research).

Examples: Before workshop vs After workshop.

T-test: One-sample T-Test

RQ: Is the average life satisfaction in our sample significantly different from the global average of 6.5?

Let’s start with visualizing the data

wvs_cleaned |> 
    ggplot(aes(y = life_satisfaction)) +
    geom_boxplot(width = 0.2) + 
    geom_hline(yintercept = 6.5, color="red")

Conduct the One-sample T-Test

global_mean_satisfaction = 6.5  

t.test(wvs_cleaned$life_satisfaction,          
       alternative = "two.sided", 
       mu = global_mean_satisfaction)


    One Sample t-test

data:  wvs_cleaned$life_satisfaction
t = 26.776, df = 6402, p-value < 2.2e-16
alternative hypothesis: true mean is not equal to 6.5
95 percent confidence interval:
 7.060083 7.148570
sample estimates:
mean of x 
 7.104326

What if there is a hypothesis? How would you write this in the report?

T-Test: Independent Samples T-Test

RQ: Is there a significant difference in life satisfaction between males and females?

Let’s first take only the necessary columns and get some summary statistics, particularly on the number of samples for each group, as well as the mean, standard deviation, and variance.

wvs_cleaned |> 
    group_by(sex) |> 
    summarise(total = n(), 
              mean = mean(life_satisfaction),
              variance = var(life_satisfaction),
              stdeviation = sd(life_satisfaction))

# A tibble: 2 × 5
  sex    total  mean variance stdeviation
  <fct>  <int> <dbl>    <dbl>       <dbl>
1 Male    3171  7.09     3.35        1.83
2 Female  3232  7.11     3.18        1.78

Visualize the differences between two samples

The variance will be easier to see when we visualize it as well. As we can see, the variance for both groups are about the same. This suggests that the variance might be homoegeneous.

wvs_cleaned |> 
    ggplot(aes(x = sex, y = life_satisfaction)) +
    geom_boxplot() +
    theme_minimal()

Conduct the independent samples T-test

Remember, the hypotheses are:

\(H_0\): There is no significant difference in the mean life satisfaction between male and female participant.

\(H_1\): There is a significant difference in the mean life satisfaction between male and female participant.

t.test(life_satisfaction ~ sex, 
       data = wvs_cleaned, 
       var.equal = FALSE)


    Welch Two Sample t-test

data:  life_satisfaction by sex
t = -0.41256, df = 6387.7, p-value = 0.6799
alternative hypothesis: true difference in means between group Male and group Female is not equal to 0
95 percent confidence interval:
 -0.10714841  0.06988992
sample estimates:
  mean in group Male mean in group Female 
            7.094923             7.113552

Notice that we are using Welch’s t-test instead of Students’ t-test

Welch’s t-test (also known as unequal variances t-test, is a more robust alternative to Student’s t-test. It is often used when two samples have unequal variances and possibly unequal sample sizes.

T-Test: Paired Sample T-Test

Unfortunately, our data is not suitable for paired T-Test. For demo purposes, we are going to use a built-in sample datasets called sleep from the base R dataset.

The dataset is already loaded, so you can use it right away!

type View(sleep) in your R console (bottom left), and then press enter. RStudio will open up the preview of the dataset.
type ?sleep in your R console to view the help page (a.k.a vignette) about this dataset.
type data() in your console to see what are the available datasets that you can use for practice!

sleep <- as_tibble(sleep)
print(sleep)

T-Test: Paired Sample T-Test

# A tibble: 20 × 3
   extra group ID   
   <dbl> <fct> <fct>
 1   0.7 1     1    
 2  -1.6 1     2    
 3  -0.2 1     3    
 4  -1.2 1     4    
 5  -0.1 1     5    
 6   3.4 1     6    
 7   3.7 1     7    
 8   0.8 1     8    
 9   0   1     9    
10   2   1     10   
11   1.9 2     1    
12   0.8 2     2    
13   1.1 2     3    
14   0.1 2     4    
15  -0.1 2     5    
16   4.4 2     6    
17   5.5 2     7    
18   1.6 2     8    
19   4.6 2     9    
20   3.4 2     10

Visualise the before (group 1) and after (group 2)

sleep |> 
    group_by(group) |> 
    summarise(n = n(), mean = mean(extra), sd = sd(extra), variance = var(extra))

# A tibble: 2 × 5
  group     n  mean    sd variance
  <fct> <int> <dbl> <dbl>    <dbl>
1 1        10  0.75  1.79     3.20
2 2        10  2.33  2.00     4.01

Visualization:

sleep |> 
    ggplot(aes(x = group, y = extra)) +
    geom_boxplot() +
    theme_minimal()

Visualise the before (group 1) and after (group 2)

Transform the data shape

The data is in long format. let’s transform it into wide format so that we can conduct the analysis more easily.

sleep_wide <- sleep |> 
    pivot_wider(names_from = group, values_from = extra, 
                names_prefix = "group_")
print(sleep_wide)

# A tibble: 10 × 3
   ID    group_1 group_2
   <fct>   <dbl>   <dbl>
 1 1         0.7     1.9
 2 2        -1.6     0.8
 3 3        -0.2     1.1
 4 4        -1.2     0.1
 5 5        -0.1    -0.1
 6 6         3.4     4.4
 7 7         3.7     5.5
 8 8         0.8     1.6
 9 9         0       4.6
10 10        2       3.4

Conduct the paired-sample T-test

Remember, the hypotheses are:

\(H_0\): There is no significant difference in the increase in hours of sleep.

\(H_1\): There is a significant difference in the increase in hours of sleep.

t.test(Pair(sleep_wide$group_1, sleep_wide$group_2) ~ 1,
       data = sleep)


    Paired t-test

data:  Pair(sleep_wide$group_1, sleep_wide$group_2)
t = -4.0621, df = 9, p-value = 0.002833
alternative hypothesis: true mean difference is not equal to 0
95 percent confidence interval:
 -2.4598858 -0.7001142
sample estimates:
mean difference 
          -1.58

Learning Check #2

Using the pre-loaded dataset called CO2, conduct the T-test betwen the two different Treatment to compare if there’s any difference in the carbon dioxide uptake between the two treatments. Type ?CO2 in your Console to retrieve the vignette of this data.

Show answer

t.test(uptake ~ Treatment, data = CO2)


    Welch Two Sample t-test

data:  uptake by Treatment
t = 3.0485, df = 80.945, p-value = 0.003107
alternative hypothesis: true difference in means between group nonchilled and group chilled is not equal to 0
95 percent confidence interval:
  2.382366 11.336682
sample estimates:
mean in group nonchilled    mean in group chilled 
                30.64286                 23.78333

Visualize the data:

Show answer

CO2 |> ggplot(aes(x = Treatment, y = uptake)) +
    geom_boxplot(width = 0.2) +
    theme_minimal()

Learning Check #2

ANOVA (Analysis of Variance)

ANOVA (Analysis of Variance) is a statistical test used to compare the means of three or more groups or samples and determine if the differences are statistically significant.

There are two ‘mainstream’ ANOVA that will be covered in this workshop:

One-Way ANOVA: comparing means across two or more independent groups (levels) of a single independent variable.
Two-Way ANOVA: comparing means across two or more independent groups (levels) of two independent variable.
Other types of ANOVA that you may encounter: Repeated measures ANOVA, Multivariate ANOVA (MANOVA), ANCOVA, etc.

One-Way ANOVA: Sample problem and result

RQ: Is there a significant difference in life satisfaction between different country?

Let’s visualize the data first!

wvs_cleaned |> 
    ggplot(aes(x = country, y = life_satisfaction)) +
    geom_boxplot() +
    theme_minimal() +
    theme(axis.text.x = element_text(angle = 45, hjust = 1))

One-Way ANOVA: Sample problem and result

Conduct the one-way Anova test

satisfaction_country_anova <- aov(life_satisfaction ~ country, data = wvs_cleaned)
summary(satisfaction_country_anova)

              Df Sum Sq Mean Sq F value   Pr(>F)    
country        2    179   89.55   27.68 1.07e-12 ***
Residuals   6400  20701    3.23                     
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

F-value: the coefficient value
Pr(>F): the p-value
Sum Sq: Sum of Squares
Mean Sq : Mean Squares
Df: Degrees of Freedom

Post-hoc test (only when result is significant)

If your ANOVA test indicates significant result, the next step is to figure out which category pairings are yielding the significant result.

Tukey’s Honest Significant Difference (HSD) can help us figure that out! Other alternative is the pairwise.t.test, but let’s try Tukey’s for now.

TukeyHSD(satisfaction_country_anova)

  Tukey multiple comparisons of means
    95% family-wise confidence level

Fit: aov(formula = life_satisfaction ~ country, data = wvs_cleaned)

$country
               diff        lwr        upr     p adj
NZL-CAN  0.55540673  0.3783288  0.7324847 0.0000000
SGP-CAN  0.02046602 -0.1008956  0.1418276 0.9174716
SGP-NZL -0.53494071 -0.7279104 -0.3419711 0.0000000

ANOVA Assumptions

The Dependent variable should be a continuous variable
The Independent variable should be a categorical variable
The observations for Independent variable should be independent of each other
The Dependent Variable distribution should be approximately normal – even more crucial if sample size is small.
- You can verify this by visualizing your data in histogram, or use Shapiro-Wilk Test, among other things
The variance for each combination of groups should be approximately equal – also referred to as “homogeneity of variances” or homoskedasticity.
- One way to verify this is using Levene’s Test
No significant outliers

Verifying the assumption: Test for Homogeneity of variance

Levene’s Test to test for homogeneity of variance i.e. homoskedasticity

library(car)
leveneTest(life_satisfaction ~ country, 
           data = wvs_cleaned)

Levene's Test for Homogeneity of Variance (center = median)
        Df F value Pr(>F)
group    2  1.9408 0.1437
      6400

The results indicate that the p-value is more than the significance level of 0.05, suggesting that there is NO significant difference in variance across the groups. Consequently, the assumption of homogeneity of variances is NOT violated.

Plotting Residuals: Residual vs Fitted

When we plot the residuals¹, we can see some outliers as well:

plot(satisfaction_country_anova, 1)

Verifying the assumptions: Test for Normality

Shapiro-Wilk Test to test for normality.

library(car)
set.seed(123) # set seed for reproducibility, make sure it samples the same way everytime.
shapiro.test(sample(residuals(satisfaction_country_anova), 5000))


    Shapiro-Wilk normality test

data:  sample(residuals(satisfaction_country_anova), 5000)
W = 0.92998, p-value < 2.2e-16

The p-value from the Shapiro-Wilk test is less than the significance level of 0.05, indicating that the data significantly deviates from normality. Therefore, the assumption of normality is NOT satisfied.

Plotting Residuals: Q-Q Plot

We can see this better from when we plot the residuals:

plot(satisfaction_country_anova, 2)

When the assumptions are not met…

We can use Kruskal-Wallis rank sum test as an non-parametric alternative to One-Way ANOVA!

kruskal.test(life_satisfaction ~ country, data = wvs_cleaned)


    Kruskal-Wallis rank sum test

data:  life_satisfaction by country
Kruskal-Wallis chi-squared = 80.746, df = 2, p-value < 2.2e-16

Other alternative: Welch’s ANOVA for when the homoskedasticity assumption is not met.

oneway.test(life_satisfaction ~ country, data = wvs_cleaned, var.equal = FALSE)


    One-way analysis of means (not assuming equal variances)

data:  life_satisfaction and country
F = 27.783, num df = 2.0, denom df = 1710.5, p-value = 1.336e-12

Two-Way ANOVA: Sample problem and result

RQ: Is there a significant difference in life satisfaction across religiousity and countries?

Let’s visualize the data!

wvs_cleaned |> 
    ggplot(aes(x = religiousity, y = life_satisfaction)) +
    geom_boxplot() +
    facet_wrap(~ country) +
    theme_minimal() +
    theme(axis.text.x = element_text(angle = 45, hjust = 1))

Two-Way ANOVA: Sample problem and result

Conduct the Two-way ANOVA test (Additive model)

satisfaction_relig_country_anova <- aov(life_satisfaction ~ religiousity + country, 
                                    data = wvs_cleaned)
summary(satisfaction_relig_country_anova)

               Df Sum Sq Mean Sq F value   Pr(>F)    
religiousity    3    128   42.82   13.32 1.15e-08 ***
country         2    192   95.83   29.82 1.29e-13 ***
Residuals    6397  20560    3.21                     
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Post-hoc test for Two-way ANOVA

TukeyHSD(satisfaction_relig_country_anova)

  Tukey multiple comparisons of means
    95% family-wise confidence level

Fit: aov(formula = life_satisfaction ~ religiousity + country, data = wvs_cleaned)

$religiousity
                                                diff        lwr         upr
A religious person-Not a religious person  0.2272818  0.1003760  0.35418751
An atheist-Not a religious person         -0.1400866 -0.3058657  0.02569256
Don't know-Not a religious person          0.3302655 -0.3699567  1.03048776
An atheist-A religious person             -0.3673683 -0.5337177 -0.20101894
Don't know-A religious person              0.1029838 -0.5973737  0.80334122
Don't know-An atheist                      0.4703521 -0.2380815  1.17878572
                                              p adj
A religious person-Not a religious person 0.0000253
An atheist-Not a religious person         0.1313305
Don't know-Not a religious person         0.6191993
An atheist-A religious person             0.0000001
Don't know-A religious person             0.9816251
Don't know-An atheist                     0.3204859

$country
               diff        lwr         upr     p adj
NZL-CAN  0.53301723  0.3565021  0.70953237 0.0000000
SGP-CAN -0.04499362 -0.1659695  0.07598224 0.6580925
SGP-NZL -0.57801085 -0.7703672 -0.38565452 0.0000000

Conduct the Two-way ANOVA test (with Interaction)

“With interaction” means we are testing whether the effect of one variable (religiosity) on the outcome (life satisfaction) depends on the level of the other variable (country), or vice versa. For the R code, we use religiousity * country instead of religiousity + country

satisfaction_relig_country_anova <- aov(life_satisfaction ~ religiousity * country, 
                                    data = wvs_cleaned)
summary(satisfaction_relig_country_anova)

                       Df Sum Sq Mean Sq F value   Pr(>F)    
religiousity            3    128   42.82  13.339 1.12e-08 ***
country                 2    192   95.83  29.853 1.25e-13 ***
religiousity:country    4     38    9.42   2.936   0.0194 *  
Residuals            6393  20522    3.21                     
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Interaction plot

To better see this effect, let’s plot the interaction.

wvs_cleaned |> 
    ggplot(aes(x = religiousity, y = life_satisfaction, 
                          group = country, color = country)) + # lines will be grouped and colored by country
    stat_summary(fun = mean, geom = "point") + # Add points to show mean life_satisfaction for each religiosity by country
    stat_summary(fun = mean, geom = "line") + # Connect the points with lines 
    theme_minimal()

Interaction plot

Interpreting our interaction plot

Some observations that we can make:

New Zealand’s pattern is distinctly different from the other two countries; NZ hows highest life satisfaction for “Not a religious person” but drops sharply for “A religious person”, while for Canada and Singapore, there is a similar patterns with peaks at “A religious person”
The interaction is most visible in how religious vs non-religious people differ across countries, with the largest difference between countries appears among non-religious people.
The relationship between religiosity and life satisfaction clearly varies by country. We know this because the lines are not parallel with each other.

Learning Check #3

Is there a significant difference in political leaning between different age groups?

Visualize the data as well
Test for normality and homoskedasticity, and choose the appropriate test

Show answer

poli_age_anova <- aov(political_scale ~ age_group, data = wvs_cleaned)
summary(poli_age_anova)

              Df Sum Sq Mean Sq F value   Pr(>F)    
age_group      3    260   86.62   19.87 8.16e-13 ***
Residuals   6399  27898    4.36                     
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

TukeyHSD:

Show answer

TukeyHSD(poli_age_anova)

  Tukey multiple comparisons of means
    95% family-wise confidence level

Fit: aov(formula = political_scale ~ age_group, data = wvs_cleaned)

$age_group
                  diff          lwr       upr     p adj
29-44-18-28 0.43031509  0.218917533 0.6417127 0.0000010
45-60-18-28 0.54558808  0.333612739 0.7575634 0.0000000
61+-18-28   0.60813361  0.393806178 0.8224610 0.0000000
45-60-29-44 0.11527298 -0.061269949 0.2918159 0.3354767
61+-29-44   0.17781852 -0.001541767 0.3571788 0.0530024
61+-45-60   0.06254554 -0.117495372 0.2425864 0.8086582

Reporting result with apaTables and gtsummary

Reporting with apaTables

apaTables is a package that will generate APA-formatted report table for correlation, ANOVA, and regression. It has limited customisations and few varity of tables. The documentation online is for the “development” version which is not what we will get if we install normally with install.packages(), so we need to rely on the vignette more. View the documentation here

Example: get the correlation table for political_scale, life_satisfaction, and financial_satisfaction

library(apaTables)

wvs_cleaned |> 
    select(life_satisfaction, financial_satisfaction, political_scale) |> 
    apa.cor.table( table.number = 1, filename = "fig-output/table-cor.doc")

Reporting with gtsummary

Another popular packages is gt (stands for “great tables”) and its ‘add-on’, gtsummary. It has lots of customizations (which can get overwhelming!) but fortunately the documentation is pretty good and there are plenty of code samples. View the documentation here

Example: get the mean differences table for political_scale, life_satisfaction, and financial_satisfaction, grouped by sex

library(gtsummary)

wvs_cleaned |> 
    dplyr::select(life_satisfaction, financial_satisfaction, political_scale, sex) |> 
    tbl_summary(by = sex) |> 
    add_difference()

Reporting with gtsummary

Characteristic	Male N = 3,171¹	Female N = 3,232¹	Difference²	95% CI^2,3	p-value²
life_satisfaction	7.00 (6.00, 8.00)	7.00 (6.00, 8.00)	-0.02	-0.11, 0.07	0.7
financial_satisfaction	7.00 (5.00, 8.00)	7.00 (5.00, 8.00)	0.07	-0.04, 0.17	0.2
political_scale	5.00 (4.00, 7.00)	5.00 (4.00, 6.00)	0.44	0.34, 0.54	<0.001
¹ Median (Q1, Q3)
² Welch Two Sample t-test
³ CI = Confidence Interval

End of Session 4!

Next session: Linear and Logistic Regressions!