Chapter 2.3- 2.6

2.3 Regression

- explains the relationship between two variables only when one variable helps explain or

predicts the other

Regression line: a straight line which shows a response variable y changes as the explanatory x

varable changes

-Used to predict the value of y for a given value of x

2.12

Increase in energy use is an explanatory variable and fat gain is a response variable

Fitting a line = line of best fit

y = a + bx

-where b is the slope and a is the intercept (when x=0)

-b is the rate of change; as x changes by one unit, y changes by the slope value

-y = prediction where as y = actual observation

Extrapolation: use of the regression line for prediction far outside of the range of values

-not so accurate because cannot say whether the graph continues to increase or decrease or

whether the relationship remains linear at extreme values

Least-squares Regression

-y = a + bx

-method of fitting a line to a scatter plot

-it is not resistant to outliers

-a line that is close as possible to all the points in the vertical direction

-error = observed – predicted

-e > 0 when observed is greater than prediction and vice versa

-makes prediction errors as small as possible

-least-squares regression line of y on x is the line that makes the sum of the errors = 0

-sum of error: e1 +e2+e3 = 0

-sum of sq of errors = to the smallest value

To calculate (do not round values that are further used in calculations):

b = r (sy / sx)

a = meany – b(meanx)

Interpreting the regression

-when changing units, correlation does not change however the least-squares line does

change

-least sq regression line always passes through the point: (x mean, y mean)

-when mean = 0 and s = 1, the regression line passes through the origin and has a slope = r

www.notesolution.com

-correlation of r is the slope of the least sq regression line when in standardized unit

-if b >0 then r >0

-R2 = (r)2 r = + sq root of R2

Correlation and regression

-correlation is the relationship between two quantitative variables whereas regression is the

relationship of explanatory and response

Square of correlation (r2): is the proportion of the variation in the y that is explained by x

-r2 as the measure of how successful the regression explains the response

ex: if r = +/-.7 r2 = .49 which is 50% of the variation is accounted for the linear

relationship

r = -/+1, r 2 = 1 which indicates that all the variation in one variable is accounted by the

linear relationship of the other variable

Another way to calculate r2

r2 = variance of predicted value y / variance of observed value y

Transforming relationships

2.18

- if relationship is not linear, take the logarithm to get a linear relationship

2.4 Cautions about correlation and regression

Residuals

-shows how far the data fall from the regression line

-observed y – predicted y(regression line)

-mean of the least sq residual is always ZERO

Residual Plot

-is a scatter plot of regression residual vs. explanatory variable

-if scatter plot with a regression line has a pattern then the residual plot should have NO

pattern in the residuals.

-If scatter plot is curved rather than linear, the residuals will follow a curve pattern

2.20

- regression line does not catch the important fact that the variability of field measurements

increases with defect depth increases

Outliers and Influential Observations

-When a point is within a range of a scatter plot, the residual is large

-Influential: a point that is outside the range and can make a huge difference to the

regression line when removed

-If influential has a small residual then the difference in regression line is small

-If the influential has a big residual then the difference in the regression line is huge

www.notesolution.com

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