Linear Regression

What are the three aspects of statistical relationships discussed in correlation?

Form, strength, and direction.

What does the linear form of a relationship mean?

A straight line (regression line) can represent how the variables relate.

What statistic quantifies the strength of a linear relationship?

Pearson product-moment correlation coefficient (r).

What does the sign (+/−) of the correlation coefficient indicate?

The direction of the relationship (positive or negative), not its strength.

What does a positive relationship look like on a scatterplot?

Points slope upward from left to right.

What does a negative relationship look like on a scatterplot?

Points slope downward from left to right.

What is regression used for?

Prediction — predicting values of one variable (Y) from another (X).

What are the two components of a regression line?

Slope (β₁) and intercept (β₀).

What does the slope (β₁) represent in regression?

The expected change in Y for a one-unit increase in X.

What does the intercept (β₀) represent?

The predicted value of Y when X = 0.

What is the formula for the simple linear regression model?

Yi​=β0​+β1​Xi​+εi​

What is the predicted value formula in regression?

Y^i​=β0​+β1​Xi​

What is a residual (εᵢ) in regression?

The difference between actual and predicted Y values: Yi−Y^iYi​−Y^i​

How do you calculate the slope (β₁)?

β1​=Scp ∑(Xi​−Xˉ)(Yi​−Yˉ)​

———- = ————————

SsX ∑(Xi​−Xˉ)2​​

How do you calculate the intercept (β₀)?

β0​=Yˉ−β1​Xˉ

What does model fit mean in regression?

How well the regression model represents the actual data.

What two methods are used to assess regression model fit?

ANOVA and R² (coefficient of determination).

What does R² represent?

The proportion of total variance in Y explained by the regression model.

How do you calculate R²?

R2=​SSREG​​

——————-

SSTOTAL

Or square the correlation r if only one predictor is used.

What is the formula for the F-ratio in regression ANOVA?

F=MSREG SSREG/dfREG

————— = ———————

MSRES SSRES/dfRES​​

How are the degrees of freedom partitioned in regression?

• dfTOTAL=N−1

• dfREG=K

• dfRES=N−K−1

Where K = number of predictors.