Paired Samples T-Test

Lecture 14

Dave Brocker

Farmingdale State College

Introduction to Paired Samples t-Test

What is a Paired Samples t-Test?

A statistical method used to compare the means of two related groups.
Commonly used when measuring the same group before and after an intervention or over two different conditions.

Purpose of Paired Samples t-Test

Why Use Paired Samples t-Test?

To determine if there is a statistically significant difference between two related group means.
Useful in psychology for assessing changes due to an intervention, therapy, or experiment.

Formula for Paired Samples t-Test

Mathematical Representation

Formula: \(\large{t = \frac{\bar{D}}{\frac{s_D}{\sqrt{n}}}}\)
- \(\bar{D}\): Mean of the differences between paired observations.
- \(s_D\): Standard deviation of the differences.
- \(n\): Number of pairs.

Theoretical Formula Explanation

Understanding Each Component

Difference Mean \(\bar{D}\): Average of the differences between paired scores.
Standard Deviation of Differences \(s_D\): Indicates how much individual differences deviate from the mean difference.
Sample Size \(n\): Number of pairs, affecting the significance of results.

Application of Paired Samples t-Test

When to Apply This Test

Pretest-Posttest Design: Measure changes in the same group over time.
Matched Pairs: Compare two related groups, such as twins or matched participants.
Repeated Measures: Test the same group under different conditions or at different times.

Paired Samples t-Test Example

Paired Samples t-Test

Application in Psychology Research

A psychologist measures stress levels of a group before and after a mindfulness program.

Steps:

Calculate the difference between pre- and post-intervention scores.
Find the mean \(\bar{D}\) and standard deviation \(s_D\) of the differences.
Apply the formula to calculate the t-value and check for significance.

Paired Samples t-Test

Application in Psychology Research

Code

# Load Packages
library(dplyr)
library(broom)
library(gt)

# Random Seed
set.seed(20)

# Create data
data <- 
  tibble(
  # Make a variable, `group` with 30 T1 and 30 T2
  grp = rep(c("T1","T2"), each = 30),
  # Make a variable, `score` with 60 scores
  score = c(rnorm(30,8,2),rnorm(30,4,1)) |> round(0)
)

Paired Samples t-Test

Data Preview

Code

set.seed(20)
pt_data <- 
  tibble(
  grp = rep(c("T1","T2"), each = 30),
  score = c(rnorm(30,8,2),rnorm(30,4,1)) |> round(0)
) 

pt_data|> 
  slice_sample(n = 5) |> 
  gt()

grp	score
T1	8
T2	5
T2	4
T2	5
T1	6

Paired Samples t-Test

Compare Scores

Code

data_new <- 
  pt_data |> 
  group_by(grp) |> 
  mutate(
    id = row_number()
  ) |> 
  tidyr::pivot_wider(
                     names_from = grp,
                     values_from = score
                     ) |> 
  ungroup() |> 
  mutate(
    D = T2-T1,
    Dfi = ifelse(stringr::str_detect(D,"^-"),paste0("(",D,")"),D)
  )

data_new |> 
  select(-D) |> 
  gt_preview()

	id	T1	T2	Dfi
1	1	10	4	(-6)
2	2	7	6	(-1)
3	3	12	6	(-6)
4	4	5	5	0
5	5	7	3	(-4)
6..29
30	30	7	4	(-3)

Paired Samples t-Test

Construct the Formula (\(\bar{D}\) | \(\sigma_{\bar{D}}\))

\[\large{t = \frac{D}{\frac{\sigma}{\sqrt{n}}} = \frac{-3.26}{\frac{2.083}{\sqrt{30}}} = (-8.57)}\]

Paired Samples t-Test

Construct the Formula (\(\bar{D}\) | \(\sigma_{\bar{D}}\))

Code

t.test(pt_data$score~pt_data$grp) |> 
  tidy() |> 
  rename(
    D = estimate,
    t = statistic, 
    p = p.value,
    df = parameter) |> 
  mutate(
    `CI[ll,uu]` = paste0("[",conf.low |> round(2),", ",conf.high |> round(2),"]"),
    p = p |> round(10)
  ) |> 
  select(-conf.low:-alternative) |> 
  select(D,t,df,p,`CI[ll,uu]`) |> 
  gt()

D	t	df	p	CI[ll,uu]
3.266667	7.58112	41.65179	2.3e-09	[2.4, 4.14]

D = mean difference of T2-T1
t = t-test statistic
df = n-1 (30-1)
p = probability results came from the Null Distribution
CI[ll,uu] = 95% CI of Mean Difference (D)

Paired T-Tests

Visualizing Results: Option 1

Code

library(ggplot2)
library(viridis)

data |> 
  ggplot(aes(grp, score, fill = grp)) + 
  stat_summary(
    fun = "mean",
    geom = "bar"
  ) + 
  stat_summary(
    fun.data = "mean_se",
    geom = "errorbar",
    width=.2
  ) +
  theme_minimal() +
  labs(
    x = "\nTreatment Group",
    y = "Stress Score\n",
    fill = "Time",
    title = "Mindfulness Intervention decreases stress levels\n"
  ) + 
  theme(
    plot.title.position =  "plot",
    plot.title = element_text(face = "bold")
  ) +
  scale_fill_viridis(discrete = TRUE, option = "E")

Paired T-Tests

Visualizing Results: Option 2

Code

library(ggplot2)
library(viridis)

data |> 
  ggplot(aes(grp, score, color = grp)) + 
  geom_jitter(aes(alpha = .2)) + 
  stat_summary(
    fun.data = "mean_se",
    geom = "line",
    group = 1,
    color = "black"
  ) + 
  stat_summary(
    fun.data = "mean_se",
    geom = "pointrange"
  ) +
  theme_minimal() +
  labs(
    x = "\nTreatment Group",
    y = "Stress Score\n",
    fill = "Time",
    title = "Mindfulness Intervention decreases stress levels\n"
  ) + 
  theme(
    plot.title.position =  "plot",
    plot.title = element_text(face = "bold"),
    legend.position = "none"
  ) +
  scale_color_manual(values = c("darkgreen","darkred"))

Summary of Paired Samples t-Test

Key Takeaways

Paired samples t-tests are ideal for comparing two related groups.
They help determine if interventions or conditions produce significant changes.
Remember to check assumptions: normality of differences and appropriate pairing.