STT152 Reviewer: Matrices and Statistical Inference

• Study notes on STT152 Reviewer

Concepts

• A matrix is an array of numbers with r rows and c columns.
• Dimension/Order refers to the size of a matrix, such as the number of rows and columns.
• A vector is an array of numbers arranged in rows or columns.
• A square matrix has an equal number of rows and columns.
• Diagonal elements are those found on the diagonal of a square matrix, while non-diagonal elements are off the diagonal.

Matrix Operations

• Transpose of a matrix.
• Trace of a matrix.
• Addition of conformable matrices.
• Scalar multiplication.
• Multiplication of conformable matrices.
• Determinant of a matrix.

Other Concepts in Matrix

• Special matrices include symmetric, idempotent, null, identity, and J matrices.
• Matrix concepts encompass invertibility, singularity, linear dependence, ranks, eigenvalues, eigenvectors, decomposition, and calculus.

Review of Statistical Inference

• Statistical inference involves methods for making generalizations about population characteristics based on sample information.

Approaches to Inference

• Estimation aims to estimate the value of a parameter of interest.
• Point estimation calculates a single number to guess the unknown parameter.
• Confidence interval estimation creates an interval expected to contain the unknown parameter with a specific confidence level.
• Hypothesis testing determines whether the sample agrees with a researcher's assertion about population characteristics.
• Parametric tests concern specific distributional characteristics of the population.
• Non-parametric tests infer about the population without assuming a specific distribution.

Example

• An "OBJECTIVE" aims to determine "WHAT IS THE IMPACT OF EXERCISE ON LOWERING BLOOD PRESSURE IN ADULTS WITH HYPOERTENSION?".
• Specific objectives support this goal.
• Point estimation estimates the population mean decrease in systolic blood pressure after a 12-week exercise regimen.
• Hypothesis testing determines if exercise leads to a statistically significant reduction in blood pressure compared to a control group (assuming no reduction in the control group).

Basic Definitions

• A random variable, denoted by Y, has a probability density function, f(y), or a probability mass function, p(y) for discrete variables.

Variance and Covariance

• The variance of a random variable Y, Var(Y).
• The covariance of Z and Y, Cov(Y, Z).
• Random variables Z and Y are independent if their joint function meets certain criteria.
• Expectation of a random matrix.

The Normal Distribution

• If Y follows a normal distribution with mean mu and variance sigma^2, then E(Y) = mu, Var(Y) = sigma^2, and m(t) is the moment generating function.
• The normal distribution reasonably describes the relative frequency distribution of several random variables.
• Procedures in inferential statistics commonly assume the population is normally distributed.
• One assumption is errors are normally distributed and expected to average to 0.

The Model Building Process

• A model is a set of assumptions or simplified representations that summarize the structure, relationships, or behaviors within a system.
• Models describe, explain, and sometimes predict real-world systems by capturing essential features.
• A system refers to the processes, elements, or entities being studied or analyzed.
• Assumptions are the basic premises on which the model is built, potentially simplifying complexities.
• Structure describes the relationships between variables or components within the system.

Variable Connections

• Describes how variables are connected, evolve over time, or their dependencies.

Types of Models

• Deterministic models have outcomes completely determined by input values, with no randomness.
• Stochastic models describe systems with outcomes influenced by unpredictable variations.

Types of Data

• Time-series data involves observations of a variable's values over time.
• Cross-sectional data captures data on one or more variables at the same time.

Steps in the Model-Building Process

• Planning involves defining the problem, identifying variables, and establishing goals.
• Model development includes collecting data, specifying the model, fitting the model, validating assumptions, and addressing regression problems.

Verification and Maintenance

• Includes checking model adequacy, the sign of coefficients, parameter stability, forecasting ability, and updating parameters.

Lecture 2

• Regression analysis is a statistical tool that utilizes the relation between a dependent variable and one or more independent variables so that the dependent variable can be predicted using the independent variable/s.
• Francis Galton's idea is the foundation of linear regression analysis, where we predict the value of one variable based on another, showing how values tend to "regress” towards the mean of the dependent variable

Uses of Regression Analysis

• Describing relationships explains the relationship between dependent and independent variables. Aids in understanding how variables change relative to each other.
• Estimation uses observed values of independents in order to estimate the the value of the dependent variable.
• Prediction forecasts the expected value of a variable of interest based on the values of other variables.
• Controlling is assessing the effect of one+ independent variables, while investigating the relationship of one independent variable with a dependent variable.
• Structural analysis uses an estimated model for quantitatively measuring variable relationships, facilitating comparison of theories.

Types of Regression Analysis

• Linear regression examines the relationship between one independent variable and one dependent variable. With a continuous dependent variable.
• Logistic regression calculates the likelihood of an event with binary outcomes, where the dependent variable is categorical
• Multiple regression extends simple linear regression to examine relationships between multiple independent and dependent variables simultaneously.

The Linear Model

• Two points can be represented by a straight line using Y = a + mX, where a is the y-intercept, and m is the slope.
• Accounts for randomness with an error term ∈ to create a stochastic linear model.

Justification of the Error Term

• Represents factors not within the scope of the model that also affect the dependant variable
• There is an unpredictable element of randomness in responses modeled by the error term
• Accounts for errors in recording
• Errors can be positive or negative but they are expected to be small
• Matrix notations

Linear in Matrix Form

• The k-variable, n-observations linear model can be written as Y = Xβ + ε

Lecture 3

• Is a procedure using using the method of least squares to determine the best for line to data.