Confidence intervals

Probability basics

• Probability runs from 0 (impossible) to 1 (certain); it may be written as a fraction, decimal or percentage.
• The probabilities of all possible outcomes add up to 1, so P(not A) = 1 − P(A).
• Odds = p ÷ (1 − p); probability = odds ÷ (1 + odds). A probability of 0.2 equals odds of 0.25 (1 to 4).

The addition rule: "or"

• For mutually exclusive events (cannot happen together): P(A or B) = P(A) + P(B).
• Blood groups are mutually exclusive: if 40% of people are group O and 45% group A, the chance that a random person is O or A = 0.40 + 0.45 = 0.85.
• For events that can happen together, subtract the overlap: P(A or B) = P(A) + P(B) − P(A and B). If 30% smoke and 20% are diabetic, and 6% are both, P(smoker or diabetic) = 0.30 + 0.20 − 0.06 = 0.44.

Exam trap Before adding, check whether the events can occur together; if they can, subtract the overlap.

The multiplication rule: "and"

• For independent events (one does not affect the other): P(A and B) = P(A) × P(B).
• Two children both boys = 0.5 × 0.5 = 0.25; three girls in a row = 0.5³ = 0.125.
• Conditional probability: P(A given B) = P(A and B) ÷ P(B). Bayes' theorem turns pre-test probability into post-test probability and underlies predictive values.

Remember "Or" means add (if exclusive); "and" means multiply (if independent).

Probability in genetics

• Each pregnancy is independent: having one affected child does not change the risk for the next. Two carrier parents have a 1/4 × 1/4 = 1/16 chance that two children are both affected.
• Carrier frequency from disease frequency (Hardy–Weinberg): if an autosomal recessive disease affects 1 in 10,000, q² = 1/10,000, so q = 1/100, and carrier frequency 2pq is about 2q = 1 in 50. For CF at 1 in 2,500, q = 1/50 and carriers are about 1 in 25.

Estimation

• A point estimate is the single best value from the sample (mean, proportion, RR). An interval estimate (the confidence interval) gives a range around it showing how precise it is.
• Sample statistics estimate population parameters; they vary from sample to sample by chance.

Standard error

• Standard error of the mean (SE) = SD ÷ √n. It is the SD of the means you would get if you repeated the sampling many times, so it measures the precision of the sample mean.
• SD describes the scatter of individuals; SE describes the uncertainty in the estimate. A study of 10,000 people has the same SD as one of 100 but a much smaller SE.
• SE of a proportion = √[p(1 − p) ÷ n].
• SE falls with the square root of n: quadrupling the sample halves the SE.
• The main use of the SE is to calculate confidence intervals (and test statistics).
• Central limit theorem: means of large samples (over about 30) are approximately normally distributed even when the data are not, which is why ±1.96 SE works.

Confidence intervals

• 95% CI = estimate ± 1.96 × SE; 99% CI = estimate ± 2.58 × SE; 90% CI = estimate ± 1.645 × SE.
• Meaning: if the study were repeated many times, 95% of the intervals calculated would contain the true population value. Informally, we are 95% confident the true value lies inside.
• A CI is narrower with a larger sample, less variability and a lower confidence level (a 90% CI is narrower than a 99% CI).
• The CI shows both statistical significance and clinical importance: its position shows the size of the effect and its width shows precision, which a p value alone cannot.

• Non-overlapping CIs for two groups imply a significant difference; overlapping CIs do not by themselves prove there is none, and a formal test is needed.

Worked examples

• Mean BP: sample mean 130 mmHg, SD 20, n = 100. SE = 20 ÷ 10 = 2; 95% CI = 130 ± 1.96 × 2 = 130 ± 3.9, so about 126 to 134 mmHg. With n = 400, SE = 1 and the CI narrows to about 128 to 132.
• Prevalence: 20% in a sample of 400. SE = √(0.2 × 0.8 ÷ 400) = 0.02; 95% CI = 20% ± 3.9%, about 16% to 24%.
• Ratios: RR 1.5 (95% CI 0.9 to 2.4) is not significant because it includes 1; RR 1.5 (1.1 to 2.0) is significant.
• Differences: mean fall in BP 5 mmHg (95% CI −1 to 11) is not significant because it includes 0.

Exam trap A confidence interval is about the population value, not about where 95% of individual patients lie; that is mean ± 1.96 SD.

Degrees of freedom

• The number of values free to vary once a statistic is fixed: n − 1 for a single sample SD or one-sample t-test; (rows − 1) × (columns − 1) for chi-squared, so a 2 × 2 table has 1 degree of freedom.

How FCPS asks about this

• Probability of group O or group A when 40% are O and 45% are A: 0.85 (mutually exclusive, so add).
• Two independent events together: multiply the probabilities.
• Both parents CF carriers: each child has a 1 in 4 chance of being affected.
• Autosomal recessive disease at 1 in 10,000: carrier frequency about 1 in 50.
• Primary use of the standard error of the mean: calculating confidence intervals; SE = SD ÷ √n.
• 95% CI from mean 130, SD 20, n 100: about 126 to 134.
• RR with 95% CI including 1: not statistically significant.