Prove That the Regression $R ^ { 2 }$ Is Identical to the Square of the Correlation Coefficient Between

question 28

Essay

Prove that the regression $R ^ { 2 }$ is identical to the square of the correlation coefficient between two variables Y and X . Regression functions are written in a form that suggests causation running from X to Y . Given your proof, does a high regression $R ^ { 2 }$ present supportive evidence of a causal relationship? Can you think of some regression examples where the direction of causality is not clear? Is without a doubt?

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Definitions:

Inhalants

Solvents, such as gasoline, glue, or paint thinner, that one inhales to produce a high and that can cause permanent central nervous system damage as well as liver and kidney disease.

Inhalant Use

The act of inhaling substances, often household or industrial chemicals, to achieve a psychoactive or mind-altering effect.

Phencyclidine (PCP)

A hallucinogenic drug that can induce dissociation, euphoria, and sometimes violent behavior, commonly known as Angel Dust.

Hallucinogen

A class of drug that causes hallucinations—perceptions or experiences that do not correspond with reality.