Question

Estimate a simple linear regression model and present the estimated linear equation. Display the regression summary table and interpret the intercept and slope coefficient estimates of the linear model.                                                           Estimate a simple linear regression model and present the estimated linear equation. Display the regression summary table and interpret the intercept and slope coefficient estimates of the linear model.                                                           

Answer 1

Simple linear regression - It is a linear regression model where the response variable is influenced or dependent on a single predictor variable and the relationship by which the dependent variable (usually denoted by y) is related to the independent variable (usually denoted by x) is given as follows.

Note that there are two kinds of relationships :

1. Deterministic : where y and x are exactly related. for example : Circumference = 2 (radius), where there is no error term, and a value of radius will give the exact value of circumference. Or,
2. Statistical/ probabilistic : where there is a relationship between (x,y) but the relationship isn't perfect.

In a simple linear regression problem we study the relationship between one response variable y and one predictor variable x.

To estimate the model parameters, we use the technique of ordinary least square estimation where the model error sum of square is minimised with respect to the model parameters.

Define,

minimising SSE with respect to we get.

and we get the following estimates:

,

.

Then the regression summary table can be presented as:

Sources of variation	Degrees of freedom	SS	MS	F
Regression	2-1=1	SSR	MSR=SSR/1	F= MSR/ MSE
Residual	n-2	SSE	MSE=SSE/(n-2)
Total	n-1	TSS

Interpretation of intercept term:

is the predicted mean response when x = 0. Though it is nonsensical at times, but has to be included in the model.

Interpretation of slope term:

IIt can be explained by our model that the mean response to increase or decrease by   units for every one unit increase in x.