Premed · Premed · Statistics Biostatistics
Lecture 28: Course Review
Statistics / Biostatistics
Learning Objectives
By the end of this lecture, students will be able to:
- Integrate the major topics of the course into a unified statistical reasoning framework
- Select the appropriate statistical test for a given research scenario
- Synthesize concepts from descriptive statistics, probability, inference, and regression
- Apply biostatistical concepts (sensitivity, specificity, relative risk, odds ratios) to clinical problems
- Critically evaluate statistical claims in medical literature
Lecture Content
I. Foundations: Data, Design, and Description
Data in clinical research are classified as either categorical (nominal for unordered categories, ordinal for ordered categories) or numerical (discrete for countable values, continuous for values measurable on a continuum). The type of data determines which summary statistics and tests are appropriate.
Study design falls into two broad categories. Observational designs include cross-sectional, case-control, and cohort studies (both prospective and retrospective). Experimental designs center on the randomized controlled trial (RCT), which is the gold standard for causal inference. Key design elements include randomization, blinding (single, double, or triple), control groups, and placebo use. The choice between intention-to-treat (ITT) and per-protocol analysis has important implications for interpreting results.
For descriptive statistics of central tendency, the mean is sensitive to outliers and appropriate for symmetric distributions, the median is resistant to outliers and preferred for skewed distributions, and the mode identifies the most frequent value and is useful for categorical data. For variability, the range, interquartile range (IQR), variance, and standard deviation each serve different purposes. The standard deviation represents the average distance from the mean and is the most commonly reported measure of spread. The coefficient of variation (CV = SD/mean) allows comparison of variability across different scales.
Distribution shape is described as symmetric, left-skewed, or right-skewed, and skew can be identified from the relationship between the mean and median. Data visualization -- through histograms, boxplots, scatterplots, and bar charts -- is essential for choosing the right analysis, with the graph type matched to the data type.
II. Probability
The basic probability rules form the foundation of statistical inference. P(A) lies between 0 and 1, and P(sample space) = 1. The complement rule states P(not A) = 1 - P(A). The addition rule gives P(A or B) = P(A) + P(B) - P(A and B). The multiplication rule gives P(A and B) = P(A) P(B|A), which simplifies to P(A) P(B) for independent events.
Conditional probability is P(A|B) = P(A and B) / P(B). Bayes' theorem provides the crucial link: P(A|B) = [P(B|A) * P(A)] / P(B). Its most important clinical application is updating the probability of disease given a positive test result, where the prior probability (prevalence) is critical to interpretation.
Among discrete distributions, the binomial models a fixed number of independent trials with two outcomes and constant probability, with PMF P(X = k) = C(n,k) p^k (1-p)^(n-k). The Poisson models rare events in a fixed interval, with the distinctive property that the mean equals the variance (both equal lambda).
The normal distribution is symmetric and bell-shaped, fully defined by its mean (mu) and standard deviation (sigma). The empirical rule states that approximately 68% of values fall within 1 SD, 95% within 2 SD, and 99.7% within 3 SD of the mean. The standard normal (Z) has mu = 0 and sigma = 1, with Z = (X - mu) / sigma providing the transformation for probability calculations using z-tables.
<image>A four-panel review figure summarizing probability and distributions. Panel A: Venn diagram illustrating the addition and multiplication rules of probability with two overlapping events A and B, with the formulas written below. Panel B: A decision tree illustrating Bayes' theorem applied to diagnostic testing -- branching from Disease+/Disease- to Test+/Test-, with probabilities at each branch and the calculation of positive predictive value highlighted. Panel C: A binomial distribution bar chart (n = 10, p = 0.3) with the mean marked. Panel D: The standard normal curve with shaded areas under the curve at 1, 2, and 3 standard deviations, and the corresponding percentages (68%, 95%, 99.7%) labeled. A caption reads: "Core probability concepts and distributions form the foundation for all statistical inference."</image>
III. Sampling Distributions and the Central Limit Theorem
The sampling distribution is the distribution of a sample statistic (such as the sample mean) across all possible samples of the same size from a population. The Central Limit Theorem (CLT) states that regardless of the population distribution, the sampling distribution of the sample mean approaches a normal distribution as sample size increases, with mean equal to the population mean (mu) and standard error SE = sigma / sqrt(n). Larger n produces smaller SE and more precise estimates. The CLT justifies the use of z- and t-based inference methods.
It is essential to distinguish between the standard deviation and the standard error. The SD measures variability of individual observations, while the SE measures variability of the sample statistic -- that is, the precision of the estimate.
IV. Confidence Intervals
A confidence interval provides a range of plausible values for a population parameter. For a mean, the formula is x-bar +/- t (s / sqrt(n)), where t is the critical value from the t-distribution with df = n - 1. As n increases, the t-distribution approaches the standard normal. For a proportion, the formula is p-hat +/- z* sqrt[p-hat(1 - p-hat) / n], requiring np >= 10 and n(1-p) >= 10 for the normal approximation.
A 95% CI means that if the sampling process were repeated many times, 95% of the calculated intervals would contain the true parameter. It does NOT mean there is a 95% probability the true value is in this specific interval. The width of a confidence interval is affected by the confidence level (higher means wider), sample size (larger means narrower), and variability (greater means wider).
V. Hypothesis Testing Framework
The null hypothesis (H0) is the default claim of no difference or no effect. The alternative hypothesis (Ha) is the claim being tested (one-sided or two-sided). The test statistic standardizes how far the sample result is from the null value. The p-value is the probability of obtaining a result as extreme as or more extreme than observed, assuming H0 is true. A small p-value provides evidence against H0, with alpha = 0.05 as the common threshold. The decision is to reject H0 if p < alpha and fail to reject otherwise.
Type I error (alpha) is rejecting H0 when true (false positive). Type II error (beta) is failing to reject H0 when false (false negative). Power = 1 - beta is the probability of correctly rejecting a false H0, and it increases with larger sample size, larger effect size, higher alpha, and lower variability. Statistical significance and clinical significance are not the same -- a statistically significant result may be clinically trivial. Always consider effect size and confidence intervals alongside p-values.
VI. Common Statistical Tests: Choosing the Right Test
The one-sample t-test compares a sample mean to a known or hypothesized value. The two-sample t-test (independent) compares means of two independent groups, assuming independent observations, approximately normal distributions (robust with large n), and equal variances (or use Welch's t-test if unequal). The paired t-test compares means from two related measurements such as before/after or matched pairs.
The one-proportion z-test compares a sample proportion to a hypothesized value. The two-proportion z-test compares proportions between two independent groups. The chi-square test of independence tests the association between two categorical variables in a contingency table, with chi-square = sum of [(observed - expected)^2 / expected] and df = (rows - 1)(columns - 1).
ANOVA compares means across three or more groups using the F-statistic (between-group variance / within-group variance). A significant F-test indicates at least one group differs, and post-hoc tests (Tukey, Bonferroni) identify which groups are different. ANOVA assumes independence, normality, and equal variances.
Nonparametric alternatives are used when assumptions are violated: the Mann-Whitney U replaces the independent t-test, the Wilcoxon signed-rank replaces the paired t-test, and the Kruskal-Wallis replaces one-way ANOVA. These methods use ranks instead of raw data, require fewer assumptions, but have less power when parametric assumptions hold.
<image>A decision-tree flowchart for selecting the appropriate statistical test. Start: "What type of outcome variable?" Branch 1 -- "Continuous": "How many groups?" -> "One group" -> one-sample t-test; -> "Two groups" -> "Independent or paired?" -> independent two-sample t-test or paired t-test; -> "Three or more groups" -> ANOVA. Branch 2 -- "Categorical": "How many variables?" -> "One variable (one proportion)" -> one-proportion z-test; -> "Two variables (association)" -> chi-square test of independence. A note at the bottom: "If assumptions are violated (non-normal, small n, ordinal data), use nonparametric alternatives." Arrows from each parametric test point to its nonparametric counterpart. A caption reads: "Selecting the correct statistical test depends on the type of data, the number of groups, and whether observations are independent or paired."</image>
VII. Correlation and Regression
Pearson's r measures the strength and direction of the linear relationship between two continuous variables, ranging from -1 to +1, with r = 0 indicating no linear relationship. The coefficient of determination r^2 gives the proportion of variance in Y explained by X. Correlation does not imply causation.
Simple linear regression models Y = b0 + b1*X + error, where b0 is the y-intercept and b1 is the slope (change in Y per one-unit change in X). Coefficients are estimated by least squares, minimizing the sum of squared residuals. Assumptions include linearity, independence, normality of residuals, and constant variance (homoscedasticity). The hypothesis test for the slope tests H0: beta1 = 0 (no linear relationship).
Multiple regression extends the model to Y = b0 + b1X1 + b2X2 + ... + bp*Xp + error, controlling for confounders by including multiple predictors. Each coefficient represents the effect of that predictor while holding all others constant. Adjusted R^2 accounts for the number of predictors. Multicollinearity, where predictors are highly correlated, makes coefficients unstable.
VIII. Biostatistics: Clinical Measures and Epidemiology
In diagnostic testing, sensitivity = P(test+ | disease+) identifies how well a test detects disease, while specificity = P(test- | disease-) identifies how well it rules out disease. Positive predictive value (PPV) = P(disease+ | test+) and negative predictive value (NPV) = P(disease- | test-) both depend on prevalence. The ROC curve plots sensitivity against (1 - specificity) at various thresholds, with AUC measuring overall test performance.
For measures of association, the relative risk (RR) = risk in exposed / risk in unexposed is used in cohort studies and RCTs. The odds ratio (OR) = odds in exposed / odds in unexposed is used in case-control studies and approximates RR when disease is rare. The absolute risk reduction (ARR) = risk in control - risk in treatment, and NNT = 1 / ARR. The hazard ratio (HR), from survival analysis, represents the ratio of instantaneous event rates.
In survival analysis, Kaplan-Meier curves estimate the survival function while properly handling censored data. The log-rank test compares survival curves between groups. Cox proportional hazards regression estimates hazard ratios while adjusting for covariates.
<image>A three-panel figure summarizing clinical biostatistics measures. Panel A: A 2x2 contingency table (Disease+ / Disease- vs. Test+ / Test-) with cells labeled a (true positive), b (false positive), c (false negative), d (true negative). Formulas for sensitivity (a / a+c), specificity (d / b+d), PPV (a / a+b), and NPV (d / c+d) are written alongside. Panel B: A 2x2 table for a cohort study (Exposed / Unexposed vs. Disease+ / Disease-) with formulas for RR and OR annotated. Below, the formulas for ARR and NNT are shown. Panel C: A Kaplan-Meier survival curve showing two groups (Treatment vs. Control) diverging over time, with tick marks indicating censored observations and the log-rank test p-value displayed. A caption reads: "These clinical biostatistics measures are essential tools for evaluating diagnostic tests, treatment effects, and patient outcomes."</image>
IX. Interpreting Medical Literature and Avoiding Pitfalls
The critical appraisal framework evaluates validity (internal), results (magnitude, precision), and applicability (generalizability). Systematic reviews and meta-analyses are summarized with forest plots, heterogeneity assessed with I^2, and publication bias evaluated with funnel plots. The Bradford Hill criteria guide the evaluation of whether an association is causal, emphasizing strength, consistency, temporality, biological gradient, plausibility, and other factors.
Common statistical errors to recognize include multiple comparisons without correction (inflating Type I error), p-hacking and selective reporting, confusing statistical significance with clinical significance, reporting only relative risk (masking small absolute differences), underpowered studies producing false negatives, and the ecological fallacy and Simpson's paradox.
The key integrative skills from this course are to always report effect sizes alongside p-values, to recognize that confidence intervals are more informative than p-values alone, to consider the clinical context (who was studied, what was measured, and how it applies to the patient), and to understand that evidence-based medicine combines the best available evidence with clinical expertise and patient values.
X. Key Concepts for Final Examination
Students should be prepared to calculate and interpret descriptive statistics, probability, confidence intervals, and test statistics. They should be able to select the correct statistical test for a given scenario and interpret output from t-tests, chi-square tests, ANOVA, and regression. Computation and interpretation of sensitivity, specificity, PPV, NPV, RR, OR, ARR, and NNT are essential skills, as is the ability to read and interpret forest plots, ROC curves, and Kaplan-Meier curves. Students should be able to identify statistical errors and biases in published research, apply Bayes' theorem to diagnostic testing scenarios, distinguish between correlation and causation, explain the Central Limit Theorem and its role in inference, and evaluate the quality of evidence using structured frameworks such as GRADE and the Bradford Hill criteria.
