Linear RegressionLinear Regression

Handbook Chapter Handbook Chapter

Experimental TestingExperimental Testing

Data are collected, in scientific experiments, Data are collected, in scientific experiments, to test the relationship between various to test the relationship between various measurable quantities that are predicted by a measurable quantities that are predicted by a hypothesis, either to support or invalidate the hypothesis, either to support or invalidate the hypothesis. hypothesis.

The dependent variable is measured under The dependent variable is measured under varying, but preset, values of the independent varying, but preset, values of the independent variable.variable.

Model FittingModel Fitting

Correlation = Correlation = If the variables have a correlation, it is If the variables have a correlation, it is

convenient to express the relationship convenient to express the relationship in the form of a mathematical equation, in the form of a mathematical equation, known as a model or natural law.known as a model or natural law.

The mathematical equation describes a The mathematical equation describes a functional relationship between the functional relationship between the variables.variables.

Model FittingModel Fitting

It may be relatively simple to It may be relatively simple to demonstrate the functional relationship demonstrate the functional relationship between the dependent and between the dependent and independent variables if the relationship independent variables if the relationship is direct.is direct.

Direct relationships (linear relationships) Direct relationships (linear relationships) can be represented by a straight line.can be represented by a straight line.

Linear ModelsLinear Models

A straight line is simplest statistically A straight line is simplest statistically analyzable function. analyzable function.

One way to express the linear One way to express the linear relationship between an independent relationship between an independent variable (x) and a dependent variable (x) and a dependent variable (y) is the slope –intercept variable (y) is the slope –intercept formula: y = m x + b.formula: y = m x + b.

Slope –Intercept FormulaSlope –Intercept Formula

y = m x + b y = m x + b y = dependent variabley = dependent variablex = independent variablex = independent variablem = slope of the linem = slope of the lineb = y-intercept of the lineb = y-intercept of the line

SlopeSlope

m = slope of the linem = slope of the line

• The slope represents the amount of tilt The slope represents the amount of tilt (slant) the straight line has relative to (slant) the straight line has relative to the x- and y- axes.the x- and y- axes.

y – Intercepty – Intercept

b = y-intercept of the lineb = y-intercept of the line

• The y-intercept is the point at which a The y-intercept is the point at which a straight line intersects the y-axis.straight line intersects the y-axis.

Slope –Intercept FormulaSlope –Intercept Formula

If the slope and y –intercept of a straight If the slope and y –intercept of a straight line is given, then the y value can be line is given, then the y value can be calculated, with confidence, for every calculated, with confidence, for every given value of x.given value of x.

The points of the straight line have the The points of the straight line have the coordinates corresponding to the (x, y) coordinates corresponding to the (x, y) pairs.pairs.

Linear RegressionLinear Regression

Since measurements inherently contain Since measurements inherently contain both systematic and random errors, both systematic and random errors, data points will not fit a given equation data points will not fit a given equation perfectly. perfectly.

Linear regression is a statistical method Linear regression is a statistical method for finding the best fitting straight line to for finding the best fitting straight line to a set of (x, y) pairs.a set of (x, y) pairs.

The expression giving the most probable The expression giving the most probable slope of the fitted straight line is:slope of the fitted straight line is:

NN NN

N N x xii––[[ x xi i ]]

i=1i=1 i=1 i=1

NN NN NN

N N y yiixxii–– y yii

xxii i=1i=1 i=1 i=1 i=1i=1

m =m =

The expression giving the most probable The expression giving the most probable y-intercept of the fitted straight line is:y-intercept of the fitted straight line is:

NN NN

N N x xii––[[ x xi i ]]

i=1i=1 i=1 i=1

NN NN NN N N

xxii22 y yii–– x xii

yyiixxiii=1i=1 i=1i=1 i=1i=1 i=1 i=1b =b =