Introduction to Statistics: General Linear Model

A statistical model that uses a straight line to represent the relationship between two continuous variables.

The value of the dependent variable (Y) when the independent variable (X) is equal to zero. It's the point where the line intercepts the Y-axis on the scatter plot.

The rate of change in the dependent variable (Y) for every unit change in the independent variable (X). It's the slope of the line on the scatter plot.

The predicted value of the dependent variable (Y) for a given value of the independent variable (X).

The difference between the actual value of Y and the predicted value of Y. This represents how far off the prediction is from the actual value.

A method used in the general linear model to find the line of best fit by minimizing the sum of squared errors.

The process of getting the estimated values for the intercept (b0) and slope (b1) in the general linear model.

The use of the original units of measurement for variables when estimating the coefficients in the general linear model.

All of the following are simply various versions of the General Linear Model: t-tests, ANOVA, ANCOVA, MANOVA, MANCOVA, correlation (Pearson and Spearman), linear regression, multiple regression, chi-square test, some machine learning.

The General Linear Model equation uses the intercept, slope, and a residual error term to estimate the dependent variable.

A standardized measure of the linear relationship between two variables, with values ranging from -1.00 to +1.00.

The closer the correlation coefficient (r) is to -1 or +1, the stronger the linear relationship between the two variables.

In a scatterplot, points that form a straight line with a positive slope represent a positive correlation, while points forming a line with a negative slope represent a negative correlation.

The cor.test() function in R can be used to calculate the correlation coefficient and perform hypothesis tests.

A correlation coefficient of r=1 represents a perfect positive linear association, while r=-1 represents a perfect negative linear association.

The difference between the observed value of the outcome variable (Yi) and the predicted value of the outcome variable (Ŷi). In other words, it's how far off the prediction is from the actual value.

The estimated regression coefficients, b0 and b1, are the values that minimize the sum of the squared residuals. This means the line defined by these coefficients is the best fit for the data.

In the general linear model equation, Y refers to the outcome variable, an estimate of the ith observation of it. This can be your predicted value of Y given the predictor variable X.

In the general linear model equation, X refers to the predictor variable, the ith observation of it. The observation can be a single value, or a value from a vector/set of predictors.

In the general linear model equation, b0 refers to the intercept of the regression line, which is the predicted value of Y when the predictor variable (X) is zero.

In the general linear model equation, b1 refers to the slope of the regression line, which indicates the change in Y for every one unit change in X. It shows the relationship between X and Y.

Using data from one dataset to make predictions about another dataset.

The extent to which a predictor (independent variable) explains the variation in the outcome (dependent variable).

A statistical measure that considers both the magnitude and direction of the relationship between variables.

The process of adding more predictors (independent variables) to a model in an attempt to improve its predictive power.

A statistical model that uses multiple independent variables to predict a dependent variable.

A directed acyclic graph (DAG) represents relationships between variables, where arrows show causal direction. It helps visualize potential confounders, mediators, and other factors influencing outcomes.

Confounding variables are factors that influence both the exposure (X) and outcome (Y) of interest, potentially distorting the observed relationship between them.

Mediators are variables that explain the path from exposure (X) to outcome (Y). They are caused by the exposure and then influence the outcome.

The normality assumption in regression states that the residuals (differences between predicted and actual values) should follow a normal distribution.

The linearity assumption in regression assumes the relationship between variables is linear (straight line).

The homogeneity of variance assumption in regression states that the spread of the residuals should be similar across all values of the independent variable.

The uncorrelated predictors assumption in regression requires that the independent variables are not highly correlated with each other.

Process of including multiple variables in your regression model, to examine the relationship between a target variable (Y) and a predictor variable (Z) while holding another variable (X) constant. Essentially, it asks "What would the relationship between Y and Z be if everyone had the average level of X?"

A statistical model that aims to represent the relationship between multiple dependent variables and independent variables using a straight line, minimizing the difference between the predicted and actual values.

While controlling for variables can help isolate the effect of one variable on another, they cannot magically establish causality. Causal conclusions are only justifiable if the data comes from a randomized experiment, where the treatment groups are assigned randomly.

They can be used to predict future values or outcomes, but conclusions drawn from them may not always represent true causal relationships. The validity of causal inferences heavily depends on the data source.