Chapter 9: The Full Picture

Course Review and Examination Preparation

---

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:

• Select the correct statistical test for any scenario using the four-step decision guide

• State the key formula, assumptions, and effect size for each test in the course

• Write an APA-style statistical reporting statement for each test type

• Use the PSPP and R cheat sheets for rapid reference

• Interpret Kaplan-Meier curves and log-rank tests in a clinical report

How to Use This Chapter

This chapter consolidates and connects all methods from Chapters 1–8 using our observational and experimental datasets as the unifying threads. Read Part I for the big picture. Use Part II when selecting a test in an exam or assignment. Use Part III in the lab with PSPP or R open.

Part I — Course Summary

Chapter	Topic	Key concept	Dataset Examples
1 — The Vitals	Levels of measurement	Nominal → Ordinal → Interval → Ratio	ANYCHD/Treat = nominal; EDUC = ordinal; AGE/Prewt = ratio
2 — The Middle	Descriptive statistics	Mean, median, SD; skewness	SYSBP ≈ Normal → mean; CIGPDAY skewed → median
3 — The Margin	SE and CI	SE = s/√n; CI = x̄ ± 1.96×SE	95% CI for mean SYSBP ≈ [129.7, 133.5] mmHg
4 — Laws of Chance	Distributions and Normality	Normal, Binomial, Poisson; CLT; Q-Q	SYSBP ≈ Normal; CIGPDAY severely right-skewed
5 — Hypothesis	Hypothesis testing	H₀, H₁, p-value, Type I/II, power	Mean TOTCHOL vs 200 mg/dL reference: t-test
6 — Two Groups	Two-sample tests	Independent vs paired; Levene; Welch	SYSBP by smoking (independent); Pre/Post weight (paired)
7 — Scaling Up	ANOVA and categorical	F-statistic; Tukey; Chi-Square; RR; RD	TOTCHOL by EDUC (ANOVA); Therapy type by Weight Gain (ANOVA)
8 — Future	Correlation, regression, survival	r; ŷ = β₀+β₁x; KM curves; log-rank	AGE → SYSBP (regression); smoking → survival (KM + log-rank)

Key Formulas

Formula	What it calculates
\(\bar{x} = \sum x_i / n\)	Sample mean
\(s = \sqrt{\sum(x_i-\bar{x})^2/(n-1)}\)	Sample SD (Bessel’s correction)
\(SE = s/\sqrt{n}\)	Standard error of the mean
\(95\% \; CI = \bar{x} \pm 1.96 \times SE\)	95% confidence interval
\(z = (x-\mu)/\sigma\)	Z-score
\(t = (\bar{x}-\mu_0)/(s/\sqrt{n})\)	One-sample t-statistic
\(t = (\bar{x}_1-\bar{x}_2)/SE_{diff}\)	Independent samples t-statistic
\(t = \bar{d}/(s_d/\sqrt{n})\)	Paired samples t-statistic
\(F = MS_{Between}/MS_{Within}\)	ANOVA F-statistic
\(\eta^2 = SS_B/SS_T\)	ANOVA effect size
\(RD = p_1 - p_2\)	Risk Difference
\(RR = p_1/p_2\)	Relative Risk
\(\chi^2 = \sum(O-E)^2/E\)	Chi-Square statistic
\(V = \sqrt{\chi^2/(n(\min(r,c)-1))}\)	Cramér’s V
\(\hat{y} = \beta_0 + \beta_1 x\)	Regression prediction
\(R^2 = r^2\)	Proportion variance explained
\(S(t) = S(t{-}1) \times (1-d_t/r_t)\)	Kaplan-Meier update

Effect Size Reference

Test	Measure	Small	Medium	Large
T-tests	Cohen’s d	0.20	0.50	0.80
ANOVA	η²	0.01	0.06	0.14
Chi-Square	Cramér’s V	0.10	0.30	0.50
Correlation	|r|	0.10	0.30	0.50
Regression	R²	0.01	0.09	0.25

Part II — Test Selection Decision Guide

💡 Plain English first: Every test in this course can be selected using four questions: (1) Continuous or categorical outcome? (2) How many groups? (3) Independent or paired? (4) Are assumptions met? Work through the flowchart for any scenario.

⚡ Common mistake: Never run a test without checking assumptions first — Normality (histogram + Shapiro-Wilk), equal variances (Levene’s test), expected cell counts ≥ 5 (Chi-Square).

Fig. 20 Figure 9.1 Statistical test selection flowchart. Follow the branches: outcome type → groups → independence → assumptions. All tests in this course appear as leaf nodes.

The Four-Step Decision Guide

Step 1 — Outcome type?

• Continuous (ratio/interval) → Step 2

• Categorical (nominal/ordinal) → Chi-Square or Fisher’s Exact (Step 4)

• Time-to-event (censored) → Kaplan-Meier + log-rank test

Step 2 — How many groups?

• One (vs known value) → One-Sample T-Test

• Two groups → Step 3

• Three or more → One-Way ANOVA + Tukey’s HSD

Step 3 — Independent or paired?

• Independent (different individuals, e.g., smokers vs non-smokers) → Independent Samples T-Test

• Paired (same individuals measured twice, e.g., baseline vs follow-up weight) → Paired Samples T-Test

Step 4 — Assumptions satisfied?

• Check Normality (n ≥ 30 → CLT applies; n < 30 → Shapiro-Wilk + histogram)

• Check equal variances (Levene’s test)

• Violated → Non-parametric alternative (Mann-Whitney U, Wilcoxon, Kruskal-Wallis)

For relationships:

• Association only → Pearson’s r (or Spearman for skewed/ordinal data)

• Prediction → Simple linear regression (non-censored continuous outcome only)

• Time-to-event → Kaplan-Meier + log-rank

APA Reporting Templates

One-sample t-test:

Mean TOTCHOL (M = 235, SD = 42) was significantly higher than the reference of 200 mg/dL, t(499) = 18.6, p < .001, 95% CI [231, 239], d = 0.83.

Independent t-test:

Smokers (M = 133.2, SD = 22.4) had significantly higher SYSBP than non-smokers (M = 130.1, SD = 21.5), t(498) = 1.97, p = .049, 95% CI [0.02, 6.2], d = 0.14.

Paired t-test:

Patients’ body weight significantly increased from baseline (M = 82.4, SD = 5.1) to follow-up (M = 85.1, SD = 5.8) after the intervention, t(71) = 4.12, p < .001, d = 0.49.

One-way ANOVA:

A one-way ANOVA revealed a significant effect of education level on TOTCHOL, F(3, 496) = X.XX, p = .XXX, η² = .XX. Tukey’s HSD indicated [specific pairs].

Chi-Square:

A Chi-Square test of independence indicated a significant association between smoking and CHD incidence, χ²(1, N = 500) = X.XX, p = .XXX, V = .XX.

Pearson correlation:

There was a significant positive correlation between age and systolic blood pressure, r(498) = .XX, p = .XXX, 95% CI [.XX, .XX].

Log-rank test:

Kaplan-Meier analysis indicated that smokers had significantly shorter survival times than non-smokers, log-rank χ²(1) = X.XX, p = .XXX.

Part III — Software Cheat Sheets

PSPP Cheat Sheet

Analysis	Path
Descriptives	Analyze → Descriptive Statistics → Descriptives
Median / CI	Analyze → Descriptive Statistics → Explore
Histogram	Graphs → Chart Builder → Histogram
Normality	Analyze → Explore → Plots → tick Normality
One-sample t	Analyze → Compare Means → One-Sample T Test
Independent t	Analyze → Compare Means → Independent-Samples T Test
Paired t	Analyze → Compare Means → Paired-Samples T Test
One-Way ANOVA	Analyze → Compare Means → One-Way ANOVA
Chi-Square	Analyze → Crosstabs → Statistics → Chi-Square
Correlation	Analyze → Correlate → Bivariate
Regression	Analyze → Regression → Linear

R Cheat Sheet

# ── Load data ─────────────────────────────────────────
fram_data <- read.csv("data/framingham_teaching.csv")
library(MASS); data(anorexia)  # Experimental dataset

# ── Factor conversions ────────────────────────────────
fram_data$SEX      <- factor(fram_data$SEX, levels=c(1,2), labels=c("Male","Female"))
fram_data$EDUC     <- factor(fram_data$EDUC, levels=1:4, ordered=TRUE,
  labels=c("0-11yrs","HS_GED","Some_college","College_grad"))
fram_data$CURSMOKE <- factor(fram_data$CURSMOKE, levels=c(0,1), labels=c("Non-smoker","Smoker"))
fram_data$DIABETES <- factor(fram_data$DIABETES, levels=c(0,1), labels=c("No","Yes"))
fram_data$ANYCHD   <- factor(fram_data$ANYCHD,   levels=c(0,1), labels=c("No_CHD","CHD_event"))
fram_data$DEATH    <- factor(fram_data$DEATH,    levels=c(0,1), labels=c("Survived","Died"))

# ── Descriptives ──────────────────────────────────────
summary(fram_data)
tapply(fram_data$SYSBP, fram_data$CURSMOKE, mean)

# ── SE and CI ─────────────────────────────────────────
t.test(fram_data$SYSBP)

# ── Normality ─────────────────────────────────────────
hist(fram_data$SYSBP)
qqnorm(fram_data$SYSBP); qqline(fram_data$SYSBP)
shapiro.test(fram_data$SYSBP)

# ── One-sample t ──────────────────────────────────────
t.test(fram_data$TOTCHOL, mu = 200)

# ── Independent t ─────────────────────────────────────
library(car)
leveneTest(SYSBP ~ CURSMOKE, data = fram_data)
t.test(SYSBP ~ CURSMOKE, data = fram_data)

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

# ── One-Way ANOVA ─────────────────────────────────────
anova_res <- aov(TOTCHOL ~ EDUC, data = fram_data)
summary(anova_res); TukeyHSD(anova_res)

# ANOVA for Anorexia Weight Gain
anorexia$Weight_Change <- anorexia$Postwt - anorexia$Prewt
anova_anorexia <- aov(Weight_Change ~ Treat, data = anorexia)

# ── Chi-Square and proportions ────────────────────────
tbl <- table(fram_data$CURSMOKE, fram_data$ANYCHD)
chisq.test(tbl)
prop.test(tbl)   # Two-proportion z-test + RD/RR from prop.table

# ── Correlation ───────────────────────────────────────
cor.test(fram_data$AGE, fram_data$SYSBP, method="pearson")
cor.test(fram_data$CIGPDAY, fram_data$TOTCHOL, method="spearman")

# ── Regression ────────────────────────────────────────
model <- lm(SYSBP ~ AGE, data = fram_data)
summary(model); confint(model)
par(mfrow=c(2,2)); plot(model); par(mfrow=c(1,1))

# Regression for Anorexia weights
model_wt <- lm(Postwt ~ Prewt, data = anorexia)

# ── Survival analysis ─────────────────────────────────
library(survival)
surv_obj <- Surv(fram_data$TIMEDTH, fram_data$DEATH)
km_fit   <- survfit(surv_obj ~ CURSMOKE, data=fram_data)
plot(km_fit, col=c("blue","red"), lwd=2,
     xlab="Days", ylab="Survival probability")
survdiff(surv_obj ~ CURSMOKE, data=fram_data)   # Log-rank test

Part IV — Epidemiological Measures Reference

Health science courses require familiarity with measures beyond standard test statistics. The following table summarises the key epidemiological measures and where they appear in this course.

Measures of Disease Frequency

Measure	Formula	Framingham example
Incidence rate	Events / person-time	155 CHD events / 4000 person-years = 38.8/1000 PY
Cumulative incidence	Events / population at risk	155/500 = 31% over 10 years
Prevalence	Cases at one time / population	PREVCHD: 25/500 = 5% at baseline

Measures of Association

Measure	Formula	Use	Study design
Risk Difference (RD)	p₁ − p₂	Absolute impact	Cohort / RCT
Relative Risk (RR)	p₁ / p₂	Relative impact	Cohort / RCT
Odds Ratio (OR)	ad/bc	When RR not available	Case-control; logistic regression
Attributable Risk (AR)	RD × 100	Extra events per 100 exposed	Cohort
NNH / NNT	1/RD	Clinical impact	Cohort / RCT

OR vs RR: OR ≈ RR when the outcome is rare (< 10%). When common (like CHD at 31%), OR > RR and they should not be used interchangeably.

Screening and Diagnostic Test Measures

When a test (e.g., a high-glucose screening test for diabetes) is applied to a population, four measures describe its performance:

Measure	Formula	Plain language
Sensitivity (Se)	True positives / All actual positives	How good is the test at catching real cases?
Specificity (Sp)	True negatives / All actual negatives	How good is the test at ruling out non-cases?
Positive Predictive Value (PPV)	True positives / All test positives	If the test is positive, how likely is the patient truly positive?
Negative Predictive Value (NPV)	True negatives / All test negatives	If the test is negative, how likely is the patient truly negative?

Example — glucose screening for diabetes (Framingham context): Suppose we use a glucose threshold of ≥ 110 mg/dL to screen for diabetes. The 2×2 table of true diabetes status vs test result would allow calculation of all four measures.

⚡ Common mistake: Sensitivity and specificity are properties of the test. PPV and NPV are properties of the test + the population prevalence. A test with high sensitivity and specificity can have low PPV if disease prevalence is very low (most positives will be false positives).

Study Design Reference

Design	Direction	What you measure	Key strength	Key limitation
RCT / Experimental	Forward	Incidence, RR, efficacy	Randomisation controls confounding	Expensive; ethical constraints
Cohort (Observational)	Forward	Incidence, RR, survival	Temporal precedence established	Time and cost; loss to follow-up
Case-control	Backward	OR (not RR)	Efficient for rare diseases	Recall bias; cannot calculate incidence
Cross-sectional	Snapshot	Prevalence, OR	Fast and cheap	Cannot establish temporality

The Framingham Heart Study is a prospective observational cohort — the strongest observational design for identifying risk factors. The Anorexia trial is an Experimental RCT — the gold standard for testing active interventions.

---

Non-Parametric Equivalents — Quick Reference

When parametric assumptions are violated (non-Normal data, small samples, or ordinal outcomes), use the non-parametric equivalent. These tests rank the data rather than assuming a specific distribution.

Parametric Test	When to use it	Non-Parametric Equivalent	When to switch
One-Sample T-Test	Continuous outcome vs known value	Wilcoxon Signed-Rank Test (one-sample)	n < 30 and non-Normal
Independent Samples T-Test	Two unrelated groups, continuous outcome	Mann-Whitney U Test (Wilcoxon Rank-Sum)	Non-Normal, small n, or ordinal outcome
Paired Samples T-Test	Same individuals measured twice	Wilcoxon Signed-Rank Test (paired)	Difference scores non-Normal and n < 30
One-Way ANOVA	Three or more groups, continuous outcome	Kruskal-Wallis Test	Non-Normal within groups or ordinal outcome
Pearson’s r	Linear association, two continuous variables	Spearman’s Rho (ρ)	Skewed data, outliers, or ordinal variables

⚡ Common mistake: Many students automatically reach for non-parametric tests whenever data “looks skewed.” Remember the Central Limit Theorem — with n ≥ 30 per group, parametric tests are robust to moderate non-Normality. Non-parametric tests have lower statistical power than their parametric equivalents, so switching unnecessarily increases the risk of a Type II error.

In R:

# Mann-Whitney U (independent groups)
wilcox.test(SYSBP ~ CURSMOKE, data = fram_data)

# Wilcoxon signed-rank (paired)
wilcox.test(after, before, paired = TRUE)

# Kruskal-Wallis (3+ groups)
kruskal.test(TOTCHOL ~ EDUC, data = fram_data)

# Spearman's rho
cor.test(fram_data$AGE, fram_data$SYSBP, method = "spearman")

---

A Final Word

In 1948, researchers in Framingham, Massachusetts enrolled 5,209 people and asked: what causes heart disease?

They did not know the answer. They had hypotheses — smoking, diet, stress, genetics — but no proof. What they had was a commitment to measuring carefully, following up rigorously, and applying rigorous statistical methods to what they found.

Over the next 75 years, the Framingham Heart Study produced over 3,000 publications. It established that high blood pressure damages arteries. It proved that smoking causes heart attacks. It showed that high cholesterol predicts myocardial infarction. It coined the term risk factor. It built the evidence base for every cardiovascular prevention guideline in use today.

But observation is only half the story. Once a public health problem is identified, we must test a solution. That is why this book paired the Framingham cohort with the Anorexia Clinical Trial. The 72 patients in that experimental dataset represent the other pillar of health science: the clinical intervention. When you ran a paired t-test or an ANOVA on their outcomes, you were evaluating the very psychological therapies that save lives and restore well-being.

The statistical methods in this course are not academic exercises. They are the tools that produced that evidence. The t-tests, ANOVA, Chi-Square, regression, and Kaplan-Meier analyses in these chapters are, in their essentials, the same analyses that Framingham investigators and clinical researchers rely on every day.

The participants in our teaching datasets represent real lives. When you calculate the Relative Risk of CHD for smokers, or measure the efficacy of Family Therapy, you are not solving a textbook problem. You are replicating — in miniature — the analyses that eventually convinced governments to ban smoking in public spaces, shaped modern psychiatric care, and saved millions of lives.

That is what statistics is for.

Key Takeaways

• Test selection: outcome type → number of groups → independence → assumptions.

• Effect size is always required alongside p-value.

• Survival data (censored) requires KM curves + log-rank, not OLS regression.

• Correlation ≠ causation — confounding, reverse causation, and coincidence are always alternatives.

• The Datasets: Between the observational Framingham cohort and the experimental Anorexia trial, you have now practiced every core statistical method required in modern health science.

---

Part IV — Putting It All Together