Chapter 6: A Tale of Two Groups

Chapter 6: A Tale of Two Groups#

Independent and Paired Samples T-Tests#

Datasets:

  1. Framingham Heart Study teaching subset (framingham_teaching.csv, n = 500) — Observational

  2. Anorexia Clinical Trial (anorexia via MASS package, n = 72) — Experimental

Learning Objectives

By the end of this chapter, you will be able to:

  • Determine whether a two-group comparison requires an independent or paired t-test

  • State and check the assumptions of each test

  • Apply Levene’s test and Welch’s correction

  • Run and interpret both tests in PSPP and R

Before You Begin: The Research Design Question#

Chapter 5 compared one group to a known value. Most health research compares two groups. Before running any test, one design question determines everything: are the two groups independent, or related (paired)?

Section 1: Independent vs Paired Designs#

1.1 The Core Distinction#

Independent design: Two completely different groups of unrelated individuals.

Framingham example: Do smokers and non-smokers have different mean systolic blood pressure? Each participant appears in exactly one group. This is classic observational epidemiology.

Paired design: The same individuals measured twice, or matched pairs.

Anorexia Clinical Trial example: 72 patients have their body weight measured before (Prewt) and after (Postwt) a psychological intervention. The exact same 72 patients contribute both measurements. This is classic experimental epidemiology.

The decision rule: Same individual (or matched pair) contributes to both groups → Paired. Completely unrelated individuals → Independent.

⚡ Common mistake: Pre/post measurements on the same patients look like two groups but are paired. Running an independent t-test on pre/post data discards the pairing, loses power, and can produce the wrong answer.

../_images/ch06_paired_vs_independent.png

Fig. 14 Figure 6.1 Independent vs paired design. In a paired design, the unit of analysis is the difference score for each individual — this eliminates between-person variability and increases statistical power.#

1.2 Why Pairing Increases Power#

When each participant serves as their own control, all the natural between-person variation (due to genetics, height, baseline metabolism) is removed from the analysis. We are asking: did the change in weight differ from zero? Because between-person variability is usually much larger than the treatment effect, pairing can dramatically increase statistical power.

Section 2: The Independent Samples T-Test#

2.1 Research Question#

Framingham: Do smokers and non-smokers have different mean systolic blood pressure at baseline?

  • \(H_0: \mu_{smoker} = \mu_{non-smoker}\)

  • \(H_1: \mu_{smoker} \neq \mu_{non-smoker}\)

2.2 The Formula#

\[t = \frac{\bar{x}_1 - \bar{x}_2}{SE_{diff}} \qquad SE_{diff} = \sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}\]

2.3 Levene’s Test and Welch’s Correction#

Before the independent t-test, check whether the two groups have similar variances.

  • Levene’s p > 0.05: Assume equal variances → standard pooled-variance t-test.

  • Levene’s p ≤ 0.05: Variances differ → use Welch’s t-test (adjusts degrees of freedom).

R’s t.test() applies Welch’s correction by default — the safer option in all cases.

2.4 Assumptions#

Assumption

How to check

Independence of groups

Design-level — different, unrelated individuals

Continuous ratio/interval outcome

Chapter 1 variable classification

Approximately Normal within each group

Histogram + Shapiro-Wilk per group; robust when n ≥ 30 per group

Approximately equal variances

Levene’s test; Welch’s correction if violated

Section 3: The Paired Samples T-Test#

💡 Plain English first: A paired t-test is essentially a one-sample t-test on the differences. For each person, calculate (after − before). Then test whether those difference scores average to zero.

3.1 Research Question#

To demonstrate the paired t-test, we turn to our experimental anorexia dataset. Patients in a clinical trial had their weight measured at baseline (Prewt) and at the end of the study (Postwt). Did their body weight significantly change?

  • \(H_0: \mu_{diff} = 0\) (mean weight change = 0 — no effect of time/intervention)

  • \(H_1: \mu_{diff} \neq 0\)

3.2 The Formula#

For each participant, compute \(d_i = \text{after}_i - \text{before}_i\).

\[t = \frac{\bar{d}}{s_d / \sqrt{n}}\]

Where \(\bar{d}\) = mean of difference scores, \(s_d\) = SD of difference scores, \(n\) = number of pairs.

3.3 Side-by-Side Comparison#

Feature

Independent

Paired

Groups

Different individuals

Same individuals twice

Unit of analysis

Individual observations

Difference scores

Formula

\(t = \frac{\bar{x}_1 - \bar{x}_2}{SE_{diff}}\)

\(t = \frac{\bar{d}}{s_d / \sqrt{n}}\)

df

\(\approx n_1+n_2-2\)

\(n-1\) (pairs)

Power

Lower

Higher (removes between-person variability)

🔬 Lab Manual — Chapter 6#

Objective#

Part 1: Independent t-test — does mean SYSBP differ between smokers and non-smokers in the Framingham cohort? Part 2: Paired t-test — did patient weight significantly change during the Anorexia clinical trial?

Option A — PSPP#

Independent t-test (Framingham):

  1. Analyze → Compare Means → Independent-Samples T Test.

  2. Move SYSBP to Test Variables; CURSMOKE to Grouping Variable.

  3. Define Groups: Group 1 = 0, Group 2 = 1. Continue → OK.

Option B — R / RStudio#

# -------------------------------------------------------
# Chapter 6 Lab: Independent and Paired T-Tests
# -------------------------------------------------------

# ══ Part 1: Independent t-test (Framingham Data) ═════
fram_data <- read.csv("data/framingham_teaching.csv")
fram_data$CURSMOKE <- factor(fram_data$CURSMOKE, levels=c(0,1),
  labels=c("Non-smoker","Smoker"))

# Descriptives by smoking group
tapply(fram_data$SYSBP, fram_data$CURSMOKE, mean)
tapply(fram_data$SYSBP, fram_data$CURSMOKE, sd)
tapply(fram_data$SYSBP, fram_data$CURSMOKE, length)

# Normality check per group
by(fram_data$SYSBP, fram_data$CURSMOKE, shapiro.test)

# Levene's test
library(car)
leveneTest(SYSBP ~ CURSMOKE, data = fram_data)

# Independent t-test (Welch by default)
t.test(SYSBP ~ CURSMOKE, data = fram_data)

# Effect size: Cohen's d
m1 <- mean(fram_data$SYSBP[fram_data$CURSMOKE=="Smoker"],   na.rm=TRUE)
m2 <- mean(fram_data$SYSBP[fram_data$CURSMOKE=="Non-smoker"],na.rm=TRUE)
s_pool <- sd(fram_data$SYSBP, na.rm=TRUE)
d <- (m1 - m2) / s_pool
cat("Cohen's d =", round(d, 3), "\n")

# Visualise
boxplot(SYSBP ~ CURSMOKE, data = fram_data,
        main = "Systolic BP by Smoking Status",
        xlab = "Smoking Status", ylab = "SYSBP (mmHg)",
        col = c("#BDD7EE","#FEE0D2"))

# ══ Part 2: Paired t-test (Anorexia Data) ════════════
library(MASS)
data(anorexia)

# Calculate difference scores: Weight Gain (After - Before)
difference_scores <- anorexia$Postwt - anorexia$Prewt

# Check Normality of DIFFERENCES (not raw values)
shapiro.test(difference_scores)
hist(difference_scores,
     main = "Weight Change in Trial (Postwt − Prewt)",
     xlab = "Change in Weight (lbs)",
     col = "#C7E9C0", border = "white")

# Paired t-test
t.test(anorexia$Postwt, anorexia$Prewt, paired = TRUE)

What to examine:

  • Independent t-test: Is mean SYSBP higher in smokers than non-smokers? Is the difference statistically significant? Is Cohen’s d > 0.2 (small), > 0.5 (medium)?

  • Levene’s test: If p < 0.05, Welch’s correction (default in R) is appropriate.

  • Paired t-test: Look at the p-value. Did the patients’ weight significantly change from baseline to follow-up? The paired design eliminates natural variations in height and body type, focusing entirely on whether individuals changed.

🧪 Test Your Knowledge#

Compare mean TOTCHOL between diabetic (DIABETES=1) and non-diabetic (DIABETES=0) participants in the Framingham dataset. (a) Is this independent or paired? (b) State H₀ and H₁. (c) Run the test and interpret the output including Cohen’s d.

Show Solution
# (a) Independent — diabetic and non-diabetic are two different groups
#     of different individuals.

# (b) H_0: \mu_{TOTCHOL(diabetes)} = \mu_{TOTCHOL(no diabetes)}
#     H_1: \mu_{TOTCHOL(diabetes)} \neq \mu_{TOTCHOL(no diabetes)}

# (c) R code:
fram_data$DIABETES <- factor(fram_data$DIABETES, levels=c(0,1),
  labels=c("No diabetes","Diabetes"))

tapply(fram_data$TOTCHOL, fram_data$DIABETES, mean)
t.test(TOTCHOL ~ DIABETES, data = fram_data)

# Effect size:
m_diab  <- mean(fram_data$TOTCHOL[fram_data$DIABETES=="Diabetes"], na.rm=TRUE)
m_nodiab<- mean(fram_data$TOTCHOL[fram_data$DIABETES=="No diabetes"], na.rm=TRUE)
d_chol  <- abs(m_diab - m_nodiab) / sd(fram_data$TOTCHOL, na.rm=TRUE)
cat("Cohen's d =", round(d_chol, 3))

Key Terms#

Term

Definition

Independent t-test

Compares means of two unrelated groups.

Paired t-test

Compares two measurements from same individuals or matched pairs. Unit = difference scores.

Levene’s test

Tests equality of variances. p > 0.05 → assume equal variances.

Welch’s correction

Adjusts df when group variances are unequal. Default in R.

Difference score (d)

Per-person change: after − before. Paired t-test analyses these directly.

Review Questions#

  1. A Framingham researcher compares mean BMI between participants with prevalent hypertension and those without. Is this independent or paired? State H₀ and H₁, run the test in R, and interpret.

  2. Explain why Welch’s t-test is preferred over the standard equal-variance t-test as a default in modern practice.

  3. In the paired t-test example using the Anorexia dataset, why is the Shapiro-Wilk test applied to the difference scores rather than to the Prewt and Postwt measurements separately?

  4. Run t.test(SYSBP ~ CURSMOKE, data=fram_data) in R. Report t, df, p, and the 95% CI for the difference. Is the difference clinically meaningful given the SD of SYSBP (~22 mmHg)?

  5. Explain, using the concept of between-person variability, why a paired t-test for a pre/post weight study would have higher power than an independent t-test comparing the baseline group to the follow-up group.

Key Takeaways

  • Independent: two unrelated groups; t = (x̄₁ − x̄₂)/SE_diff.

  • Paired: same individuals twice; t = d̄/(s_d/√n). Higher power.

  • Levene’s test → check equal variances. Welch’s correction → safer default.

  • Never run independent t-test on pre/post data from the same individuals.

  • Always report Cohen’s d — statistical significance alone is not enough.

Next: Chapter 7 — Scaling Up extends to three or more groups and categorical associations.

Part II — Testing What We Think We Know

Contents