Introduction to Biostatistics

What is Biostatistics?

• It applies statistical methods to biological and health sciences.
• Deals with data collection, summarization, and analysis.
• Used for drawing inferences and making decisions.

Why is Biostatistics Needed?

• It helps understand and quantify variation, as biological processes are subject to random variation.
• Statistical methods are used to make valid inferences from data and make informed decisions.

Applications of Biostatistics

• Used in Epidemiology, Clinical trials, Genomics, Public Health, Forensics and Ecology

Examples of Biostatistics Use

• Determining if cancer risk is higher near nuclear power plants.
• Comparing the effectiveness of a new drug versus an existing one.
• Identifying genes associated with specific diseases.
• Finding correlations between air pollution and respiratory illnesses.
• Designing surveys to estimate the prevalence of diseases like diabetes.

Types of Data

Quantitative Data

• Measured numerically
  • Continuous: Takes any value within a range (height, weight, temperature).
  • Discrete: Takes on specific values (number of patients, pregnancies).

Qualitative Data

• Data is categorized
  • Nominal: Categories without order (gender, race, marital status).
  • Ordinal: Categories with a specific order (pain scale, satisfaction level).

Descriptive Statistics

• Summarize and describe data set characteristics.

Measures of Central Tendency

• Mean: Average value, sensitive to outliers. Formula: $\bar{x} = \frac{\sum_{i=1}^{n} x_i}{n}$
• Median: Middle value, not sensitive to outliers.
• Mode: Most frequent value.

Measures of Dispersion

• Range: Difference between largest and smallest values.
• Variance: How spread out data are from the mean. Formula: $s^2 = \frac{\sum_{i=1}^{n} (x_i - \bar{x})^2}{n-1}$
• Standard deviation: Square root of the variance. Formula: $s = \sqrt{s^2}$
• Interquartile range: Difference between the 75th and 25th percentiles.

Inferential Statistics

• Used to make inferences about a population from a sample of data.

Hypothesis Testing

• Tests a statement about a population parameter.
• Determines whether there is sufficient evidence to reject the null hypothesis.
  • Null hypothesis ($H_0$): Statement of no effect or difference.
  • Alternative hypothesis ($H_1$): Contradicts the null hypothesis.

P-value

• It is the probability of observing a test statistic as extreme as, or more extreme than, the observed one, assuming the null hypothesis is true.
  • A small p-value (typically < 0.05) indicates strong evidence against the null hypothesis.

Confidence Intervals

• It is a range of values likely to contain the true population parameter with a certain confidence level.
  • For instance, a 95% confidence interval means if sampling is repeated, 95% of the resulting intervals would contain the true population parameter.

Bayes' Theorem

• In probability theory and statistics, Bayes' Theorem describes the probability of an event based on prior knowledge of conditions related to the event, formally expressed as:
  • $P(A|B) = \frac{P(B|A)P(A)}{P(B)}$
• $P(A|B)$: Conditional probability of A given B is true.
• $P(B|A)$: Conditional probability of B given A is true.
• $P(A)$ and $P(B)$: Probabilities of A and B being true independently.

Theorem Deduction

• Bayes' Theorem is derived from the basic definitions of conditional probability:
  • $P(A|B) = \frac{P(A \cap B)}{P(B)}$
  • $P(B|A) = \frac{P(B \cap A)}{P(A)}$

Probability Calculations

• Since $P(A \cap B) = P(B \cap A)$ we can rewrite the equations:
  • $P(A \cap B) = P(A|B)P(B)$
  • $P(B \cap A) = P(B|A)P(A)$
• By equating the two expressions, yields
  • $P(A|B)P(B) = P(B|A)P(A)$
• Bayes' Theorem is obtained by dividing both sides by $P(B)$:
  • $P(A|B) = \frac{P(B|A)P(A)}{P(B)}$

Example

• Consider a test with 99% accuracy. If the result is positive, the probability of actually having the disease can be calculated using prior information.
  • Test accuracy: $P(\text{positive}|\text{diseases}) = 0.99$ and $P(\text{negative}|\text{healthy}) = 0.99$
  • Prevalence of the disease: $P(\text{diseased}) = 0.001$

Calculations

• To calculate $P(\text{diseased}|\text{positive})$ using Bayes' Theorem:
  • $P(\text{diseased}|\text{positive}) = \frac{P(\text{positive}|\text{diseases})P(\text{diseased})}{P(\text{positive})}$
• $P(\text{positive})$ is calculated as:
  • $P(\text{positive}) = P(\text{positive}|\text{diseased})P(\text{diseased}) + P(\text{positive}|\text{healthy})P(\text{healthy})$
• Using the values $P(\text{positive}|\text{healthy}) = 1 - P(\text{negative}|\text{healthy}) = 0.01$:
  • $P(\text{positive}) = (0.99 \times 0.001) + (0.01 \times 0.999) = 0.01098$
• Thus
  • $P(\text{diseases}|\text{positive}) = \frac{0.99 \times 0.001}{0.01098} \approx 0.089$
• Therefore, the probability of an individual actually having the disease given a positive test result is approximately 8.9%., illustrating how Bayes’ Theorem accounts for disease prevalence.