Types of Variables & Descriptive Statistics

Types of Variables

• Variables can be categorized as either categorical or numerical.

Categorical Variables

• Nominal variables are unordered categories, such as blood type or gender.
• Ordinal variables represent ordered categories, like pain scales or satisfaction ratings.

Numerical Variables

• Discrete variables include whole numbers such as number of children.
• Continuous variables can take any value within a range, examples include height and temperature.

Descriptive Statistics

• Descriptive statistics include measures of central tendency and measures of dispersion.

Measures of Central Tendency

• The Mean is the average value, calculated using the formula: $$\mu = \frac{\sum_{i=1}^{N} X_i}{N}$$
• Median is the middle value in a dataset.
• Mode is the most frequently occurring value.

Measures of Dispersion

• Range represents the difference between the maximum and minimum values.
• Variance is the average squared difference from the mean, calculated as: $$\sigma^2 = \frac{\sum_{i=1}^{N} (X_i - \mu)^2}{N}$$
• Standard Deviation is the square root of the variance and indicates the spread of data around the mean: $$\sigma = \sqrt{\frac{\sum_{i=1}^{N} (X_i - \mu)^2}{N}}$$
• Interquartile Range (IQR) is the range of the middle 50% of the data

Probability

• Probability measures the likelihood of an event occurring, expressed as: $$P(A) = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}}$$
• Conditional Probability is the likelihood of an event given another event has occurred: $$P(A|B) = \frac{P(A \cap B)}{P(B)}$$

Probability Distributions

• Distributions include normal, binomial, and poisson.

Normal Distribution

• Normal Distribution is symmetrical and bell-shaped, where Mean = Median = Mode.
• Approximately 68% of data falls within 1 standard deviation of the mean in a normal distribution.
• About 95% of data points are within 2 standard deviations of the mean.
• Around 99.7% of data is within 3 standard deviations of the mean.

Binomial Distribution

• Binomial Distribution indicates the probability of successes in a fixed number of trials: $$P(X=k) = \binom{n}{k} p^k (1-p)^{n-k}$$

Poisson Distribution

• Poisson Distribution provides the probability of a number of events occurring in a fixed interval: $$P(X=k) = \frac{e^{-\lambda} \lambda^k}{k!}$$

Hypothesis Testing

• Hypothesis testing involves a series of steps to determine the validity of a claim.

Steps for Hypothesis Testing

• State the hypotheses.
  • Null Hypothesis ($H_0$) indicates no effect or no difference.
  • Alternative Hypothesis ($H_1$ or $H_a$) indicates that there is an effect or a difference.
• Set the Significance Level ($\alpha$).
  • Typically, $\alpha = 0.05$, representing a 5% chance of Type I error.
• Calculate the Test Statistic.
  • Examples include t-statistic, z-statistic, and chi-square statistic.
• Determine the p-value.
  • P-value refers to the probability of observing the test statistic if the null hypothesis is true.
• Make a Decision.
  • If p-value $\leq \alpha$, the null hypothesis is rejected.

Types of Errors in Hypothesis Testing

• Type I Error (False Positive): Rejecting the null hypothesis when true.
• Type II Error (False Negative): Failing to reject the null hypothesis when false.

Common Statistical Tests

• Various statistical tests are used depending on the data type and purpose of the analysis.
• T-tests are for comparing the means of two groups using continuous data.
• ANOVA is used to compare the means of three or more groups with continuous data.
• Chi-Square Tests are for testing the association between categorical variables.
• Correlations measure the strength and direction of a linear relationship between two variables using continuous data.
• Regression Analysis models the relationship between dependent and independent variables.
• Non-parametric Tests are applicable when data do not meet parametric assumptions.
• Mann-Whitney U Tests compare two independent groups using ordinal/non-normal continuous data.
• Kruskal-Wallis Tests compare three or more independent groups using ordinal/non-normal continuous data.
• Wilcoxon Signed-Rank Tests compare two related groups using ordinal/non-normal continuous data.

Confidence Intervals

• Confidence Intervals define a range of values likely to contain the true population parameter.
• Typically expressed as a 95% confidence interval.
• Calculated Using: $$CI = \text{Sample Statistic} \pm (\text{Critical Value} \times \text{Standard Error})$$

Regression Analysis

• Models the relationship between variables.

Linear Regression

• Linear Regression models the linear relationship between a dependent variable (Y) and one or more independent variables (X): $$Y = \beta_0 + \beta_1 X + \epsilon$$
  • $\beta_0$ is the intercept.
  • $\beta_1$ is the slope.
  • $\epsilon$ is the error term.

Multiple Regression

• Multiple Regression is an extension of linear regression with multiple independent variables.

Logistic Regression

• Logistic Regression models the probability of a binary outcome: $$\text{logit}(p) = \ln\left(\frac{p}{1-p}\right) = \beta_0 + \beta_1 X$$

Survival Analysis

• Survival analysis Analyzes the expected duration of time until an event occurs.
• Kaplan-Meier Curve is a Non-parametric statistic estimating the survival function from lifetime data.
• Cox Proportional Hazards Model relates the time until an event occurs to explanatory variables.

Study Designs

• Include observational and experimental studies.

Observational Studies

• Cohort Study follows a group of people over time to see who develops a particular outcome.
• Case-Control Study compares people (cases) with a condition to those without (controls).
• Cross-Sectional Study involves Data collected at a single point in time.

Experimental Studies

• Randomized Controlled Trial (RCT): Participants are randomly assigned to different treatment groups.

Bias

• Systematic errors that can affect study results.

Types of Bias

• Selection Bias: Systematic difference between baseline characteristics of compared groups.
• Information Bias: Systematic difference in how data is collected from different study groups.
• Confounding Bias: A third variable is associated with both the exposure and the outcome.