Biostatistics Lectures 1-4 Summary

Biostatistics Lectures 1-4 Summary

• Biostatistics is the collection, classification, analysis, and interpretation of data from biomedical research. It helps create medical knowledge.
• Science is empirical, relying on observations and experiences. Inductive reasoning draws general conclusions from specific observations.
• Basic research and clinical research are interconnected. Clinical research uses randomized studies to avoid biased results.
• In biostatistics, samples are studied because they are subsets of populations. However, the sample itself is not the main focus.
• Samples are used to make inferences about larger populations. Larger samples have a higher likelihood of accurately reflecting the population. Small, biased samples are more prone to random error, which can lead to inaccurate population representations.
• Random error (sampling error) is the difference between the sample mean and the population mean due to sampling.
• Sample quantities are known, measured values. Population quantities are unknown and have to be estimated.
• Clinical research involves study design, data collection, data processing, and data analysis.
• Statistical software, like R, facilitates reproducible research, which is a crucial aspect of science. Reproducible research means the results can be verified and repeated by others.
• Reproducible research requires using code throughout the process.
• Rectangular data is used in most studies. This is a tabular structure where rows represent cases (observations, records) and columns represent characteristics or variables.
• Primary key is a unique identifier for each case

Types of Variables

• Categorical variables:
  • Nominal: No inherent order (e.g., blood type, sex).
  • Ordinal: Inherent order (e.g., educational level, satisfaction).
  • Dichotomous: Only two categories (e.g., yes/no, diseased/healthy).
• Numerical variables:
  • Continuous: Measured on a continuum with infinite possible values (e.g. temperature).
  • Discrete: Counted, with finite possible values (e.g., number of children).

Frequency Tables

• Frequency tables present the number or percentage of participants in each category.
• Useful for categorical variables and grouped numeric data like "age group".
• Can also be used for ordinal variables.
• Cumulative frequencies can be displayed if the number of categories is limited.

Contingency Tables

• Contingency tables (cross-tabulations) are used to explore the association between two categorical variables.
• Examining the relationship between two categorical variables (e.g., exposure and outcome).
• Similar marginals and category-specific proportions suggest that the variables are not associated.

Plotting Categorical Variables

• Pie charts and bar plots illustrate relative frequencies. Pie charts show percentages, bar charts show counts.
• Bar graphs, arranged horizontally or vertically, are useful to show categorical data.

Plotting Numeric Variables

• Histograms and box plots are used for numeric data.
• Histograms show the distribution of data across different ranges of values or classes, graphically depicting data frequency patterns.
• Box plots visualize data spread and identify outliers as the values that are significantly different from most of the data.

Measures of Location

• Mean: Average of a set of values. Sensitive to outliers.
• Median: The middle value when data is sorted. Not sensitive to outliers.
• Quantiles: Values that divide the data into segments based on proportions (e.g., 10th percentile, median = 50th percentile, quartiles).
• Mode: Most frequent value.

Measures of Spread

• Variance: The average of squared deviations from the mean.
• Standard deviation: Square root of the variance.
• Range: Difference between maximum and minimum values.
• Interquartile range (IQR): Difference between 75th and 25th percentile.

Distribution as a Concept

• Probability distributions describe the probability of different values in a variable.
• Density plots show the total area under the curve equal to 100%.

Population versus Sample

• Population involves the entire group or collection of data of interest. The mean and variance are unknown
• Samples are parts of populations. Sample means and sample variances are known and are used to estimate their population counterparts.

Describing Numerical Variables

• Median and quantiles are useful for skewed data.

Probability Distributions

• Probability distributions are used to describe variation in numeric data, giving the probability of each possible numerical outcome
• Probability distributions are typically empirically described by mean, standard deviation, median, and quantiles.

The Normal Distribution

• Defined by its mean and standard deviation (i.e., μ and σ)
• Its symmetrical which makes it useful for calculations
• A large volume of data follows this curve, making it a useful analytical tool
• Most probability falls within 1, 2, and 3 standard deviations from the mean
• Used in many biostatistical calculations and test to create a standardized framework

Normal Distribution Tests

• Contextual knowledge—knowing the variable.
• Shapiro-Wilk test: Determines if a variable's distribution deviates significantly from normal. Lower p-value (p<0.05) suggests deviation
• Q-Q plots (quantile-quantile plots): Compare the quantiles of the variable to the quantiles of a normal distribution. Straight line points to a normal distribution.

Importance of the Normal Distribution

• Many variables follow a typical normal distribution.
• Central limit theorem: The average of repeated samples approximates a normal distribution even if the underlying distribution isn't normal.
• The standard error measures how accurate sample means are in estimating the population mean; a smaller SE shows greater accuracy

The Three Distributions

• Population distribution: Entire group including mean and stdev, unknown
• Sample distribution: Portion of the population, including mean and stdev, known.
• Sampling distribution: Distribution of sample means, including mean and standard error.

Confidence Intervals

• Confidence intervals provide a range within which the true population mean is likely to fall.
• 95% CI means that 95% of repeated intervals would contain the true value if the experiment were repeated many times.
• For small samples, t-distribution is used to account for the uncertainty in the sample deviation, and standard deviation, using a t-value instead of a z-value (larger t-values show greater uncertainty).

Assumptions of t-tests

• Data comes from normally distributed populations.
• Sample data must be independent.
• Populations have equal standard deviations.

Comparing Two Independent Sample Means

• Different sample means are often because of random error but also may reflect actual population differences in means.
• Testing involves a null hypothesis that sample means are equally different (H0) or different means (H1)

P-values and Hypothesis Testing

• P-value: The probability of observing the data or more extreme results if the null hypothesis is true.
• A small p-value (typically less than 0.05) suggests the null hypothesis is unlikely, and we reject it in favor of the alternative hypothesis.

Proportions

• Proportion: Part of the whole; a fraction of a total.
• Proportion values are limited to 0-1.
• We cannot use typical test methods for analyzing proportions.

Binomial Distribution

• Used for discrete variables with binary outcomes (success/failure).
• Shaped in a skewed pattern and defined by p, probability of success and n or number of trials
• This is used to define the probability that an outcome occurs at least x times in n trials.

Confidence Intervals for Proportions

• Confidence intervals for proportions estimate the range in which the true population proportion lies, with a given probability for repeated trials.
• Exact methods are used when sample sizes are small, while larger sample sizes allow the use of the normal distribution methodology.