📘 Statistics Which Test Differences — Simplified From the Book
1️⃣ t-tests & Other Parametric Tests
(From: t tests and other parametric tests, p.28–30)
What they are
- Parametric tests are tests used when your data follow a normal distribution (bell-shaped).
- The t-test compares mean values between two groups.
When to use
Use a t-test when:
- The outcome is numerical (e.g., blood pressure, peak flow, Hb).
- The data are normally distributed.
- You want to test if two groups differ in their mean.
What it means
- It tests the null hypothesis:
- The test produces:
- a t statistic (ignore this)
- a P value → tells you if the difference is real or just chance.
“There is no difference in the mean of the two groups.”
How to interpret (as the book says)
➡️ Do NOT worry about the t value — go straight to the P value.
- P < 0.05 → statistically significant → groups differ.
- P ≥ 0.05 → difference could be due to chance.
Example from the book
Bronchodilator vs placebo:
- Mean improvement: 96 L/min vs 70 L/min.
- t = 11.14
- P = 0.001 → highly significant
Meaning the drug works.
Key takeaway
Use t-tests only when data are normal. If data are skewed → use non-parametric.
2️⃣ Mann–Whitney & Other Non-Parametric Tests
(From: Mann–Whitney and other non-parametric tests, p.31–33)
What they are
These tests are used when:
- Data are NOT normally distributed (skewed)
- Data contain outliers
- Data measured as ranks (not true means)
How they work
Instead of comparing raw numbers, they:
- Convert all observations into ranks
- Compare which group tends to have higher ranks
When to use Mann–Whitney
- Two independent groups
- Skewed numerical data
- e.g., age, waiting times, length of stay
What it means
- Gives a U statistic (Ignore this)
- Look at the P value
- Low P = groups differ
- High P = no evidence of difference
Example from the book
Triage nurse vs GP age comparison:
- Median ages: 50 vs 46
- Skewed age distribution → non-parametric test needed
- Mann–Whitney U = 133,200
- P < 0.001 → very highly significant
Other non-parametric tests (book lists)
- Wilcoxon signed-rank → paired data
- Kruskal–Wallis → >2 groups
- Friedman test → repeated measures
Again — ignore the statistic, read the P value.
3️⃣ Chi-Squared (χ²) Test
(From: Chi-squared, p.34–36)
What it is used for
Used when your data are categories, not numbers.
Typical uses:
- Improved / Not improved
- Disease / No disease
- Male / Female
- Side effect / No side effect
Purpose
To test whether the observed counts differ from expected counts
(according to the null hypothesis of “no difference”).
When to use
- Two categorical variables
- A 2×2 table or bigger
- Large enough sample (if small → use Fisher’s exact test)
How to interpret
- χ² value itself is not important (don’t interpret it!)
- Look only at the P value.
If P < 0.05
→ significant
→ Real difference between groups
If P ≥ 0.05
→ difference could be due to chance
Example from the book
Amoxicillin vs erythromycin:
- Improvement: 60% vs 67%
- χ² = 2.3
- P = 0.13 → NOT statistically significant
Meaning → antibiotics have similar effectiveness.
Important notes from the book
- Small sample? → use Fisher’s exact test instead.
- Sometimes χ² test includes continuity corrections (Yates).
📌 Summary Table (Book-Based)
Test | Type of Data | Distribution Needed? | Compares | Example |
t-test | Numerical | Normal | Means | BP, peak flow |
Mann–Whitney | Numerical | Skewed | Ranks/medians | Age, hospital stay |
Chi-square (χ²) | Categorical | Any | Frequencies | Improved? Yes/No. |
📘 1️⃣ ANOVA (Analysis of Variance)
(Parametric test — normally distributed data)
What it is
A test used to compare means of 3 or more groups.
When to use
Use ANOVA when:
- Your outcome is numerical
- Data are normally distributed
- You have 3+ groups
(e.g., mean BP in Group A, B, C)
Why it’s needed
If you used multiple t-tests instead, you increase the chance of a Type I error.
ANOVA solves this by testing all groups in one test.
What it tells you
- P value only matters
- P < 0.05 → at least one group differs
Important
ANOVA tells you that groups differ but not which ones.
For that → “post-hoc tests” (not required in this book).
📘 2️⃣ Wilcoxon Signed-Rank Test
(Non-parametric equivalent of paired t-test)
What it is
A test for comparing two paired / repeated measurements
BUT when the data are not normally distributed.
Examples
- Before vs After treatment
- Left vs Right measurement
- Pre-op vs Post-op scores
How it works
- Converts the before/after differences into ranks
- Compares the direction of change
Interpretation
- Ignore the W statistic
- Go straight to P value
- P < 0.05 → significant change
- P ≥ 0.05 → no evidence of change
Use when
- Paired sample
- Skewed data
- Non-normal differences
📘 3️⃣ Kruskal–Wallis Test
(Non-parametric version of ANOVA)
What it is
A test comparing 3 or more groups when the data are NOT normally distributed.
Examples
- Median waiting times in 3 clinics
- Skewed lengths of hospital stay
- Pain scores that are ordinal (0–10)
How it works
- Ranks all values in all groups
- Checks if one group consistently has higher/lower ranks
Interpretation
- Ignore the H statistic
- Look only at P
- P < 0.05 → at least one group differs
- P ≥ 0.05 → no detectable difference
📘 4️⃣ Fisher’s Exact Test
(For categorical data with small sample sizes)
When to use
Use Fisher’s test instead of χ² when:
- Any expected cell count < 5
- Very small sample (e.g., 2×2 table with small numbers)
Examples
- Rare diseases
- Small pilot studies
- Very small groups
Why it’s important
It gives the exact P value, making it more accurate than χ² in small samples.
Interpretation
- Ignore all calculations
- Just read the P value
Tip
If numbers are tiny → use Fisher
If numbers are large → χ² is fine
📘 5️⃣ FULL DECISION TABLE — Choosing the Right Statistical Test
Use this table in exams — it mirrors the book perfectly:
A. Outcome is NUMERICAL
Are data normally distributed?
YES → Parametric tests
Design | Groups | Test |
Unpaired | 2 groups | t-test |
Unpaired | 3+ groups | ANOVA |
Paired / repeated | 2 groups | Paired t-test |
Paired / repeated | 3+ groups | Repeated Measures ANOVA |
NO → Non-parametric tests(skewed)
Design | Groups | Test |
Unpaired | 2 groups | Mann–Whitney U |
Unpaired | 3+ groups | Kruskal–Wallis |
Paired / repeated | 2 groups | Wilcoxon Signed-Rank |
Paired / repeated | 3+ groups | Friedman Test |
B. Outcome is CATEGORICAL
Table type | Purpose | Best test |
2×2 table | Compare proportions | Chi-square (χ²) |
2×2 small numbers | Expected cell <5 | Fisher’s Exact Test |
>2 categories | Compare categories | Chi-square (χ²) |
C. Correlation (relationship)
Data type | Test |
Both variables normal | Pearson correlation |
One or both non-normal | Spearman correlation |
D. Prediction
Aim | Test |
Predict a number | Linear regression |
Predict category (yes/no) | Logistic regression |
E. Survival (time-to-event)
Aim | Test |
Compare survival curves | Log-rank test |
Adjust for multiple variables | Cox regression model |
📘 6️⃣ Additional Tests Already in the Book
Below is a quick summary of the remaining ones:
📘 Logistic Regression
Predicts a binary outcome (e.g., disease/no disease).
Output → Odds Ratio.
📘 Pearson vs Spearman
Condition | Test |
Linear + normal | Pearson r |
Non-normal or ranks | Spearman rs |
📘 Cox Regression Model
For survival analysis.
Output → Hazard Ratio (HR).
HR > 1 → higher risk
HR < 1 → protective
📘 Survival/Time-to-event
- Kaplan–Meier curve → visual survival
- Log-rank test → compares 2 curves
- Cox regression → adjusts for covariates
📘 Chi-Square Extensions
- Yates correction → makes χ² more accurate
- Mantel–Haenszel → combines several 2×2 tables
(common in meta-analyses)
📘 Friedman Test
Non-parametric repeated-measures ANOVA
(e.g., same patients measured 3 times with skewed data)
📘 Summary of What to Ignore in Exams
The book repeats this many times:
- Ignore χ² value → look at P
- Ignore U statistic → look at P
- Ignore W statistic → look at P
- Ignore t value → P again
- Ignore regression standard error → look at P
- Ignore HR coefficient → look at CI & P