SADC Course in Statistics

Simple Linear Regression

(Session 02)

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Learning ObjectivesAt the end of this session, you will be able to

• understand the meaning of a simple linear regression model, its aims and terminology

• determine the best fitting line describing the relationship between a quantitative response (y) and a quantitative explanatory variable (x)

• Interpret the unknown parameters of the regression line

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An illustrative example

Data on the next slide shows the average number of cigarettes smoked per adult in 1930 and the death rate per million in 1952 for sixteen countries.

The question of interest is whether there is a relationship between the death rate (y) and level of smoking (x). Here both y and x are quantitative measurements.

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The DataCountry Cig. Smoked (x) Death rate (y)England and Wales 1378 461Finland 1662 433

Austria 960 380Nethelands 632 276Belgium 1066 254Switzerland 706 236New Zealand 478 216U.S.A. 1296 202Denmark 465 179Australia 504 177Canada 760 176France 585 140Italy 455 110Sweden 388 89Norway 359 77Japan 723 40

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Start by plotting - shows pattern

-a straight line relationship seems plausible here.

010

020

030

040

050

0D

eat

h ra

te (

y)

0 500 1000 1500 2000Cigarettes smoked (x)

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Recall reasons for modelling

• To determine which of (often) several factors explain variability in the key response of interest;

• To summarise the relationship(s);

• For predictive purposes, e.g. predicting y for given x’s, or identifying x’s that optimise y in some way;

Note: Presence of an association betweenvariables does not necessarily implycausation.

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Describe variation in response (here death rate) in terms of its relationship with the explanatory variable (here cig. numbers).

Model : Model : data = pattern + residual

–can describe pattern as: a + bx , if straight line relationship seems

reasonable

–residual is unexplained variation - assumed to be random.

Describing the Regression Model

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If there is only one explanatory variable, we have a Simple Linear Regression Model.

Here data = pattern + residual becomes:

y = + x +

where + x =pattern and = residual.• is called the intercept• is called the slope• the ’s represent the departure of the true line from the observed values.

Simple Linear Regression Model

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A Diagrammatic Representation

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x

y

y x

i

x

y

××

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×

×

i

i

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and are the unknown parameters in the model. They are estimated from the data

• The random error, , is assumed to have a– normal distribution– with constant variance (whatever the

value of x)

We shall return to these assumptions later.

Parameters of Model & Assumptions

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Results of model fitting------------------------------------------------------ deathrate|Coef. Std.Err. t P>|t| [95% Conf.Int.]---------+--------------------------------------------Cigars | .2410 .0544 4.43 0.001 .1245 .3577Const. | 28.31 46.92 0.60 0.556 -72.34 128.95------------------------------------------------------

These are estimates of coefficients of the regression equation since this is a sample of data - precision quantified by standard errors

Estimated equation is: y = 28.31 + 0.241 * x

Note: The t and P>|t| columns will be discussed in the next session.

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The fitted line0

100

200

300

400

500

0 500 1000 1500 2000Cigarettes smoked (x)

Death rate (y) Fitted values

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Interpreting model parameters

• Slope (regression coefficient): If cigarettes smoked increases by 1 unit per year, death rate will increase by 0.24 units. In other words, if cigarettes smoked increases by 100 units, death rate will increase by 24 units.

• Intercept of 28.31 only has meaning if the range of x values (cigarettes smoked) under study includes the value of zero. Here zero cigarettes smoked still gives an estimated death rate of 28.3

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Predictions from the lineThe model equation can also be used to

predict y at a given value of x

Thus from y = 28.31 + 0.241 x, predicted death rate ( ) in a country where

number of cigarettes smoked is x=1000, is given by

= 28.31 + 0.241 (1000)= 269.3

Note: Predictions will be discussed in greater detail in Session 9.

ˆˆy x

y

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Computation of model estimates (for reference only)

i iˆy x ˆˆ y xn

i i i i

2 2i i

x y ( x )( y ) / n Sxyˆx ( x ) / n Sxx

Note: Can also write i i

2i

(x x)(y y)Sxy

Sxx (x x)

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Practical work follows to ensure learning objectives are

achieved…