Lecture notes
Linear regression
- Institution
- The University Of Essex (UoE)
This document is full explanation of linear regression
[Show more]
Seller
Follow
Content preview
MA317 (2020/21) 4 MODELLING EXPERIMENTAL DATA2 Lecture: Revision simple linear regression I
The example life expectancy (years) and average income (USD) of EU countries is used to revise simple linear
regression. Least square estimation and maximum-likelihood-estimation of the intercept and slope parameter are
derived. The distribution of the parameters is discussed.
2.1 Introduction
Lets assume we have a data set of length n with pairs (xi , yi ), where x is a vector containing all the xi values
(predictor variables) and y is a vector containing all the yi values (response variables) and want to fit a regression
line-of-best fit to the data e.g.
y = a + bx
Then we can use a statistical model of a simple linear regression (for example cf. Ross, 2009, pages 353-378) to find
the ‘best’ values for a and b (â and b̂).
Here, we use the model that proposes
Yi ∼ N (a + bxi , σ 2 ), i = 1, 2, . . . n,
where the random variables Y1 , Y2 , . . . , Yn are independently distributed, σ > 0, a ∈ R and b ∈ R are unknown
parameters.
NOTE:
A random variable X is said to have a normal distribution with parameters µ and σ 2 , if its pdf f is given by:
(x − µ)2
2 1
f (µ, σ ; x) = √ exp − ,
2π σ 2σ 2
x ∈ R, µ ∈ R, σ > 0.
The normal distribution with parameters µ and σ 2 is denoted by N (µ, σ 2 ).
X ∼ N (µ, σ 2 ) is short form of: A random variable X is distributed by N (µ, σ 2 ).
Therefore, our model assumes the individual response variable, yi , can be modelled by a normal distribution with
mean a + bxi and standard deviation σ.
Using this model we can calculate the least-squares estimators (LSE) as
Sxy
â = ȳ − b̂ x̄, b̂ =
Sxx
.
Where the LSE minimise the sum of squared differences, defined by
n
X
SSR = (yi − ŷi )2
i=1
In this case as yˆi = â + b̂xi , so we have
n
X 2
SSR = yi − â − b̂xi
i=1
, MA317 (2020/21) 5 MODELLING EXPERIMENTAL DATA
With,
n n
1X 1X
x̄ = xi , ȳ = yi
n n
i=1 i=1
n
X n
X
Sxx = (xi − x̄)2 = x2i − n (x̄)2
i=1 i=1
n
X X n
Sxy = (xi − x̄)yi = xi yi − nx̄ȳ
i=1 i=1
n
X n
X
Syy = (yi − ȳ)2 = yi2 − n (ȳ)2
i=1 i=1
Example 2.1. Life expectancy (years) and average income (USD) of EU countries (further details see example 3.2
later on).
> euyear2008
............. income (USD) lifespan (years)
Austria.......... 49525.06 80.45
Belgium.......... 47148.85 80.11
Bulgaria.......... 6546.31 73.32
Cyprus........... 31409.84 79.66
Czech Republic... 20728.85 77.21
Denmark.......... 62035.78 78.70
Estonia.......... 17541.30 73.97
Finland.......... 50775.44 79.79
France........... 44471.50 81.52
Germany.......... 44524.52 80.09
Greece........... 31173.57 79.96
Hungary.......... 15408.01 74.01
Ireland.......... 60178.22 79.86
Italy............ 38384.51 81.95
Latvia........... 14937.07 72.24
Lithuania........ 14034.31 71.82
Luxembourg...... 117954.68 80.52
Netherlands, The. 53075.91 80.40
Poland........... 13857.40 75.53
Portugal......... 22955.13 79.25
Romania........... 9299.74 73.37
Slovak Republic.. 18211.64 74.81
Slovenia......... 26910.67 78.97
Spain............ 35000.35 81.09
Sweden........... 52884.46 81.24
United Kingdom... 43360.77 79.90
We use R to compute the least square estimates of the intercept a and the slope b of a simple linear regression for the
response variable lifespan (years) y and the predictor variable average income (USD) x:
y = a + b x,
The benefits of buying summaries with Stuvia:
Guaranteed quality through customer reviews
Stuvia customers have reviewed more than 700,000 summaries. This how you know that you are buying the best documents.
Quick and easy check-out
You can quickly pay through credit card for the summaries. There is no membership needed.
Focus on what matters
Your fellow students write the study notes themselves, which is why the documents are always reliable and up-to-date. This ensures you quickly get to the core!
Frequently asked questions
What do I get when I buy this document?
You get a PDF, available immediately after your purchase. The purchased document is accessible anytime, anywhere and indefinitely through your profile.
Satisfaction guarantee: how does it work?
Our satisfaction guarantee ensures that you always find a study document that suits you well. You fill out a form, and our customer service team takes care of the rest.
Who am I buying these notes from?
Stuvia is a marketplace, so you are not buying this document from us, but from seller vandanaakbari. Stuvia facilitates payment to the seller.
Will I be stuck with a subscription?
No, you only buy these notes for £7.49. You're not tied to anything after your purchase.
Can Stuvia be trusted?
4.6 stars on Google & Trustpilot (+1000 reviews)
68175 documents were sold in the last 30 days
Founded in 2010, the go-to place to buy revision notes and other study material for 15 years now