## The null hypothesis and p-values

The null hypothesis (H₀) states that there is no difference or association (e.g. drug A lowers IOP no more than drug B). The alternative hypothesis (H₁) states that there is one
The p-value is the probability of getting a result at least as extreme as the one observed if the null hypothesis were true

It is not the probability that the null hypothesis is true, nor the probability that the result is due to chance alone
If p is below the chosen significance level (α), usually 0.05, the null hypothesis is rejected

A two-tailed test looks for a difference in either direction and is the default. A one-tailed test needs a strong prior reason to ignore one direction
Statistical significance is not clinical significance: a very large trial can find a statistically significant IOP difference of 0.3 mmHg that does not matter to patients

## Errors and power

	H₀ true (no real difference)	H₀ false (real difference)
H₀ rejected	Type I error (false positive). Probability = α	Correct. Probability = power (1 − β)
H₀ not rejected	Correct	Type II error (false negative). Probability = β

Power = 1 − β: the probability of detecting a real effect of a given size. Studies usually aim for 80% or 90%
An underpowered study that finds "no significant difference" has not shown that there is no difference ("absence of evidence is not evidence of absence")
Multiple comparisons: testing many outcomes raises the chance of a false positive. With 20 independent tests at α = 0.05, about one is expected to be "significant" by chance. The Bonferroni correction divides α by the number of tests: 5 tests need p below 0.05 ÷ 5 = 0.01

## Sample size

a smaller difference to detect (the minimum clinically important difference)
greater variability (a larger SD)
a lower α (a stricter significance level)
higher power
expected dropouts, which are added on afterwards

For comparing two means: n per group ≈ 2 × (z_α/2 + z_β)² × SD² ÷ difference². For α = 0.05 (two-tailed) and 80% power, (1.96 + 0.84)² ≈ 7.85

Worked example: to detect a 2 mmHg difference in IOP with an SD of 4 mmHg: n = 2 × 7.85 × 4² ÷ 2² = 2 × 7.85 × 16 ÷ 4 = 62.8, so 63 per group, before allowing for dropouts
Halving the difference to 1 mmHg quadruples the sample size, to 252 per group

## Confidence intervals and p-values

A 95% CI that excludes the value of no effect (0 for a difference, 1 for a ratio) corresponds to p < 0.05
The CI is more informative than the p-value: it shows the likely size of the effect and how precisely it is known
Worked example: a mean IOP difference of 1.5 mmHg (95% CI 0.4 to 2.6) is significant, because the CI excludes 0. An RR of 0.8 (95% CI 0.6 to 1.1) is not, because it includes 1

## Choosing a test

Parametric tests assume a distribution, usually normal, and compare means. Non-parametric tests use ranks, make fewer assumptions, and suit skewed or ordinal data and small samples, but have less power when the data are normal
Paired data are two measurements on the same subject: before and after treatment, or fellow eyes

Comparison	Parametric	Non-parametric
Two independent groups (continuous)	Unpaired (Student's) t-test	Mann–Whitney U test
Two paired measurements (continuous)	Paired t-test	Wilcoxon signed-rank test
Three or more independent groups	One-way ANOVA	Kruskal–Wallis test
Association between two continuous variables	Pearson correlation (r)	Spearman rank correlation (ρ)
Two or more independent groups (categorical)	Chi-squared (χ²) test; Fisher's exact test when expected counts are small (under 5)
Paired categorical data	McNemar's test (e.g. the same patients tested before and after)
Time to an event	Kaplan–Meier curves compared with the log-rank test; Cox regression gives the hazard ratio

## Correlation, agreement and regression

The correlation coefficient (r) runs from −1 (perfect negative) through 0 (no linear relationship) to +1 (perfect positive)

r² is the proportion of the variation in one variable explained by the other: r = 0.7 explains 49%
Correlation does not show causation, and only measures a linear relationship

Correlation is not agreement. Two tonometers can be perfectly correlated while one always reads 5 mmHg higher
Bland–Altman plot: the difference between two methods plotted against their mean

Bias = the mean difference. 95% limits of agreement = mean difference ± 1.96 × SD of the differences
Worked example: a rebound tonometer reads on average 1.2 mmHg higher than Goldmann, with an SD of the differences of 2.0 mmHg. Limits of agreement = 1.2 ± 3.92 = −2.7 to +5.1 mmHg. Whether that is acceptable is a clinical judgement

Kappa (κ) measures agreement between observers on categorical data beyond chance (e.g. two graders of retinopathy). About 0.61–0.80 is substantial and above 0.80 almost perfect
Regression predicts an outcome from one or more variables, and adjusts for confounders

## Survival analysis

Used for time-to-event outcomes, such as time to glaucoma progression or graft failure
Censoring: patients who leave the study or reach its end without the event still contribute data for the time they were followed
The Kaplan–Meier curve steps down at each event. The hazard ratio is the ratio of event rates over time between groups (e.g. HR 0.5 means half the rate at any time)

Type I error: α, a false positive. Type II error: β, a false negative. Power = 1 − β
Comparing a new instrument with a reference standard: Bland–Altman, not correlation
Two eyes of one person are correlated: treating them as independent breaks the assumptions of these tests (see Study Design)