Regression Models

Simple liner regression is a sloped line fitted to the values with two parameters.
On the other hand an intercept only model shows a fitted value, the mean of y. It's a straight line only measuring one parameter.

General Equation of a Straight Line

Generally the equation of slope is

where:
$Y=Responsevariable$
$X=Explanatoryvariable$
${\beta }_0=Intercept$ (y when x=0)
${\beta }_1=Slope$ (Increase in y when x increases by 1)
$ϵ=error$ which is assumed in a normal distribution $({ϵ}_{i}N(O,{\sigma }^2))$

If ${\beta }_0=0$ then...

• x and y aren't related
• there is no slope
• the mean of y doesn't depend on x
• the mean of y is constant

The problem is that we don't observe ${\beta }_1$ (population parameter), we only observe ${\beta }_1$ (estimate of the slope based on sample). Sometimes ${\beta }_1$ is small so the question becomes how far from 0 does the slope have to be before we accept x and y aren't related?

Null and Alternative Hypotheses

Null Hypothesis $(H_0)$ is what we believe unless we find evidence, $(H_0)$ is presumed true until evidence is found that it is false.
$H_0=({\beta }_1=0)$ or in other words, there is no association between x and y

Alternative Hypothesis$(H_1)$ is what is believed if evidence is found to reject the null hypothesis.
$H_1=({\beta }_1\ne 0)$, there's an association between x and y

P-Values

• Evidence against the null hypothesis
• Small p-values are more evidence against the null hypothesis

Slope

• If $H_0$ is true then $({\beta }_1=0)$ is a flat line $\therefore$ intercept only is suitable unless there is evidence otherwise
• If $H_1$ is true then $({\beta }_1\ne 0)$ then a slope line is better if we have evidence
• P value is the strength of said evidence

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