Regression - Algebra vs Calculus and Probability

Regression with ALGEBRA ONLY

y = mx + b <or> Y_i = b₀ + b₁*X_i

Regression with CALCULUS and PROBABILITY

Y_i = b₀ + b₁*X_i + ε_i

Stochastic

The ε_i term is the difference, and is where probability and calculus are needed.

Regression with CALCULUS and PROBABILITY

Model summary:

Can use Least Squares, Maximum Likelihood, other

Includes:

Deterministic

Regression with ALGEBRA ONLY

Model summary:

Only includes a Least Squares fit

Might also sometimes include R-squared value.

Regression with ALGEBRA ONLY

Model assumptions regarding the ε_i term (model residuals).

Regression with CALCULUS and PROBABILITY

Does NOT include regression confidence bands.

Conclusions

Visually, it appears that the slope is greater than zero.

Includes regression confidence bands, which characterize the sampling uncertainty.

Conclusions

We cannot conclude that the slope is not zero.