Regression Analysis Quiz

The process of estimating the relationship between a response variable (dependent) and one or more explanatory variables (independent) using data.

The distribution of the response variable (Y) given the values of the explanatory variables (X). It represents the probability of different values of Y for each specific combination of X values.

The goal of regression analysis is to estimate the unknown conditional density, meaning we try to understand how Y changes for different values of X.

Data points that include both the response variable and explanatory variable values. These points represent individual observations from the population.

The response variable is the outcome we are trying to predict or understand. Example: House prices

The explanatory variable is the factor that we believe influences the response variable. Example: Number of bedrooms in a house

In linear regression, we assume the response variable can be represented as a linear combination of the explanatory variables. This means the relationship is modeled as a straight line.

The parameters in the linear regression model that determine the slope and intercept of the line, representing the strength and direction of the relationship between the variables. These are estimated from the data to best fit the relationship.

The conditional mean function is a way to describe how the expected value of a response variable changes with different values of explanatory variables.

In mean regression, we're primarily interested in finding the relationship between the average response and the explanatory variables.

This is the transformed version of the conditional mean function, where it is assumed to be a linear function of the explanatory variables.

It's a known smooth function that connects the conditional mean to the linear predictor.

It denotes how much each explanatory variable contributes to the linear predictor.

This is the process of estimating the unknown parameters in our model.

This refers to the estimation of the conditional mean function based on data. It aims to understand the relationship between the explanatory variables and the expected value of the response variable.

In generalized linear mean regression, we assume that the response variables are independent and that after transformation, the conditional mean function can be expressed as a linear function of the explanatory variables.

The difference between the actual value of a dependent variable and its predicted value in a regression model.

A set of parameters that defines the distribution of the residuals. It's like a blueprint for the 'errors' in the model.

A vector of parameters that represents the relationship between the independent and dependent variables in a model.

Combined, the two parameter vectors 𝜽(1) and 𝜽(2) represent the overall parameters of the model.

The median of the conditional distribution of 𝒴𝑖 given 𝒳𝑖. It represents the central value of the dependent variable for a given value of the independent variable.

The expected value of 𝒴𝑖 given 𝒳𝑖. It represents the average value of the dependent variable for a given value of the independent variable.

A statistical model where the dependent variable has a distribution that depends on both a location parameter (𝜇) and a scale parameter (𝜈).

A model that describes the relationship between a dependent variable and a set of independent variables by assuming a linear relationship with a constant term and coefficients for each independent variable.

Knowing the conditional mean alone is not enough to fully understand the distribution of a response variable (Y) given explanatory variables (X).

A conditional mean needs additional characteristics to be uniquely determined. These characteristics are like extra pieces of information that help define its shape and spread.

Mathematical function that describes the probability of different values of the response variable (Y) for each specific combination of explanatory variables (X).

A set of possible conditional density functions, defined by a specific mathematical form with adjustable parameters that control its shape.

A function used to model the conditional mean, often taking specific forms like exponential or sigmoid, to ensure the mean falls within a desired range.

The parameters of the conditional mean function, which control the relationship between explanatory variables (X) and the response variable (Y). They need to be estimated from the data.

The objective in regression analysis is to estimate the unknown parameters of the conditional density function. By doing so, we can better understand the relationship between the explanatory variables and the response.

The specific chosen form for the conditional mean function ensures that the predicted mean values are within the appropriate range for the response variable. For example, a logarithmic function can be chosen to predict positive values.

The process of finding the best values for the parameters of a statistical model, based on the observed data, where "best" means maximizing the likelihood of observing the observed data.

A function that connects the expected value of the response variable (Y) to a linear combination of the explanatory variables (X). It's like a bridge between the response and linear predictor.

The process of estimating the conditional mean function, which is the expected value of the response variable for given values of the explanatory variables, based on the collected data.

A way to describe the relationship between the response variable and the explanatory variables by assuming the conditional mean function is a linear combination of the explanatory variables, after a suitable transformation.

The function that represents the negative loglikelihood function, often minimized in parameter estimation. Minimizing this function means maximizing the likelihood function.

The natural logarithm of the odds of an event with probability 𝑢 occurring.

A statistical term for a family of probability distributions where the density function has a specific form with parameters like 𝜂 (canonical parameter) and 𝛿. Used in modeling a variety of data types like continuous and discrete.

The parameter that determines the mean and variance of a distribution in an exponential family. It plays a crucial role in linking the mean and variance.

The parameter that controls the spread (variance) of a distribution in an exponential family. It's a constant value for certain distributions (like Poisson), but can vary for others.

A function that determines the mean and variance of a distribution in an exponential family. Its derivative determines the mean, and its second derivative is related to the variance.

A link function transforms the linear combination of explanatory variables into the expected value of the response variable. Each link function relates the conditional mean to an appropriate linear predictor for a specific distribution.

A statistical model where the relationship between the response variable and explanatory variables is expressed through a linear combination of the explanatory variables. The expected value of the response variable is a linear function of the explanatory variables.