Statistics and R Programming Basics

In R, the * symbol represents multiplication. You can use it to multiply numbers or variables together, for example, 3 * 5.

In R, the / symbol represents division. You can use it to divide one number or variable by another, for example, 10 / 2.

In R, the ^ symbol represents exponentiation. This means raising a number or variable to a power. For example, 2 ^ 3 is the same as 2 * 2 * 2, which equals 8.

In R, the sqrt function provides the square root of a number. For example, sqrt(9) returns 3, because 3 * 3 = 9.

In R, the log(x, base) function calculates the logarithm of 'x' with the specified 'base'. For example, log(8, 2) returns 3, because 2 raised to the power of 3 equals 8.

A numeric variable in R can store numerical values, such as integers, decimals, or fractions. Examples include 10, 3.14, or 0.5.

A character variable in R can store text or strings, for example, 'hello', 'world', or 'male'. They are usually enclosed in quotes.

A logical variable in R can represent true or false values. It is typically represented by T or F. For example, 'T' for 'true' and 'F' for 'false'.

The likelihood function is the probability function of the observed data (x), viewed as a function of the unknown parameter (θ). It helps determine how likely different values of θ are, given the observed data.

The maximum likelihood estimate (MLE) is the value of the parameter (θ) that maximizes the likelihood function. It's like finding the peak of a mountain - the value that makes the data most probable.

A confidence interval is a range of plausible values for an unknown parameter. It's calculated at a specific confidence level, representing the proportion of intervals that would contain the true value in the long run.

Regression models analyze how one variable (response, y) depends on other variables (explanatory/covariates, x). The simplest form uses a linear relationship between the response and a single covariate.

Simple linear regression models the relationship between a response variable (y) and a single predictor variable (x) using a linear equation. This is a basic model assuming a straight-line relationship.

The least squares approach is a commonly used method for fitting linear regression models. It minimizes the sum of squared differences between the observed values and the values predicted by the model.

"≈" means "is approximately modeled as". It signifies that the model is an approximation of the real-world relationship and not a perfect representation.

"We are regressing Y on X (or Y onto X)" indicates that we're studying how the response variable (Y) changes based on the predictor variable (X).

The standard error of the estimated mean (ˆµ) measures how much the estimated mean is likely to vary from the true population mean. It quantifies the uncertainty in our estimate.

The errors (ϵi) in a linear regression model represent the deviations of observed values (yi) from the predicted values (yˆi). Each error term has a constant variance (σ2) and is uncorrelated with other errors.

The residual standard error (RSE) is an estimate of the standard deviation of the errors (σ) in a linear regression model. It's obtained from the data and reflects how well the model fits the observed values.

A confidence interval is a range of values that is likely to contain the true value of an unknown parameter. A 95% confidence interval indicates that we are 95% certain that the true value falls within that range.

The coefficient of determination (R2) measures the proportion of variability in the dependent variable (y) that is explained by the independent variable (x) in a linear regression model. It ranges from 0 to 1, with higher values indicating a better fit.

The explained sum of squares (ESS) represents the squared deviations of the predicted values (yˆi) from their mean. It quantifies the variability in the data that is explained by the regression model.

The residual sum of squares (RSS) represents the squared deviations of the observed values (yi) from the predicted values (yˆi). It measures the unexplained variability or the 'error' in the regression model.

The total sum of squares (TSS) represents the total variability in the dependent variable (y). It is the sum of the ESS and RSS, reflecting the total variability in the data.

A mathematical equation that describes a relationship between a response variable and one or more predictor variables.

The process of predicting a qualitative (categorical) response variable, assigning observations to specific categories.

A technique used in classification to predict the probability of an observation belonging to each category of a qualitative variable.

A statistical method used to estimate the parameters of a linear regression model by minimizing the sum of squared differences between the observed and predicted responses.

The predicted values of the response variable based on the estimated regression model.

A specific classification method that predicts a binary (two-category) qualitative response, often used for yes/no scenarios.

The differences between the observed response values and the fitted values, representing the model's prediction errors.

Observations used to build and train a classifier, providing examples of the relationship between predictor variables and the response variable.

A test used to determine whether at least one of the predictor variables in a multiple linear regression model has a significant effect on the response variable.

Observations used to assess the performance of a trained classifier, evaluating its ability to predict outcomes on unseen data.

In multiple regression, the process of identifying which predictors contribute significantly to the variation in the response variable.

A set of training observations used to build a classifier, providing examples of the relationship between predictor variables and the response variable.

Evaluating how well the regression model fits the observed data, considering the overall relationship between the predictors and the response.

A set of observations used to assess the performance of a trained classifier, evaluating its ability to predict outcomes on unseen data.

The process of creating a model based on a set of training data, aiming to capture the underlying relationships between variables.

Predicting a specific response value based on a given set of predictor values, incorporating the uncertainty in the prediction.

The p-value is a measure of evidence against the null hypothesis. A small p-value suggests that the observed data is unlikely to have occurred by chance, making it more plausible to reject the null hypothesis. A larger p-value implies that the observed data is more consistent with the null hypothesis.

The null hypothesis is a statement about the population that is assumed to be true until proven otherwise. It's often a statement of 'no effect' or 'no difference'.

Regression analysis is a statistical technique used to model the relationship between a dependent variable (response) and one or more independent variables (predictors). It aims to understand how changes in the independent variable(s) affect the dependent variable.

Linear regression assumes that the relationship between the predictor and response variables is linear, meaning the relationship can be represented with a straight line.

Simple linear regression involves studying the relationship between one predictor variable and a response variable. It helps understand how changes in the predictor affect the response. Examples could include predicting house prices based on house size, or predicting student performance based on study time.

Multiple linear regression involves analyzing the relationship between multiple predictor variables and a response variable. It provides a more complete picture of the factors influencing the response. For instance, predicting house prices based on both house size and location.

The concept of homoscedasticity, also known as constant variance, refers to the assumption in regression analysis that the variance of the error term is consistent across all values of the predictor variable. It essentially means that the spread of the data points around the regression line is equal for all values of the predictor variable.

R-squared (R²) is a statistical measure in regression analysis that represents the proportion of the variance in the dependent variable that is explained by the independent variable(s). It's a value between 0 and 1, where a higher R² indicates a better fit of the regression model to the data.