regression regression relationship = trend + scatter observed value = predicted value + prediction...
TRANSCRIPT
Regression
Regression relationship = trend + scatter
Observed value = predicted value + prediction error
8
y = 5 + 2x
data point(8, 25)
25
21
prediction error
Regression is about fitting a line or curve to bivariate data to predict the value of a variable y based on the value of an independent variable x.
Residual
A line of best fit will be used to predict a value of y for a given value of x. The difference between the measured value y and the predicted value ŷ is called the residual.
Residual = y-ŷ Residual = observed value – predicted
value
Regression Line
Obviously, we would like all theseresiduals to be as small as possible.
A technique is least squares regression minimises the sum of the squares of the residuals, the line found by this technique is therefore called the least squares regression line of y on x, or simply the regression line.
Complete the table below
xy 25ˆ xy 25ˆ xy 25ˆ xbby 10ˆ
y
y - y
Data Point (8, 25) (3, 7) (-2, -3) (x, y)
Observed y-value 25 y
Fitted line
Predicted value / fitted value 21
Prediction error / residual 4
Complete the table below
xy 25ˆ xy 25ˆ xy 25ˆ xbby 10ˆ
y
y - y
Data Point (8, 25) (3, 7) (-2, -3) (x, y)
Observed y-value 25 7 -3 y
Fitted line
Predicted value / fitted value 21 19 -1
Prediction error / residual 4 -12 -2
The Least Squares Regression Line Choose the line with smallest sum of
squared prediction errors.
Minimise the sum of squared prediction errors
Minimise 2 errors prediction
Which line?
The Least Squares Regression Line
There is one and only one least squares regression line for every linear regression
for the least squares line but it is also true for many other lines
is on the least squares line Calculator or computer gives the
equation of the least squares line
0errorsprediction
),( yx
Residuals Plot
The pattern of residuals allows you to see if your regression line is a good fit for the data and how reliable interpolation and extrapolation will be.
If the model is a good fit, the residuals will oscillate closely above and below the zero line.
Temperature oF
Chirps per second
69.4 15.4
69.7 14.7
71.6 16.0
75.2 15.5
76.3 14.4
79.6 15.0
80.6 17.1
80.6 16.0
82.0 17.1
82.6 17.2
83.3 16.2
83.5 17.0
84.3 18.4
88.6 20.0
93.3 19.8
Crickets: Temperature vs Chirps
y = 0.2119x - 0.3091R2 = 0.6975
10.0
12.0
14.0
16.0
18.0
20.0
22.0
60 70 80 90 100
Temperature oF
Nu
mb
er o
f ch
irp
s p
er s
eco
nd
Correlation coefficient = 0.8352This is √0.6975
Predicted chirps per
second
Observed chirps per
second Residuals
14.4 15.4 1.0
14.5 14.7 0.2
14.9 16.0 1.1
15.6 15.5 -0.1
15.9 14.4 -1.5
16.6 15.0 -1.6
16.8 17.1 0.3
16.8 16.0 -0.8
17.1 17.1 0.0
17.2 17.2 0.0
17.3 16.2 -1.1
17.4 17.0 -0.4
17.6 18.4 0.8
18.5 20.0 1.5
19.5 19.8 0.3
The regression line is:y = 0.2119x - 0.3091which is what we use to get the predicted value of y.Eg. x = 69.4 oFy = 0.2119(69.4) – 0.3091
= 14.4 chirps per second
Residual = Observed – Predicted Value Value
Temperature oF
Chirps per second
69.4 15.4
69.7 14.7
71.6 16.0
75.2 15.5
76.3 14.4
79.6 15.0
80.6 17.1
80.6 16.0
82.0 17.1
82.6 17.2
83.3 16.2
83.5 17.0
84.3 18.4
88.6 20.0
93.3 19.8
ResidualsResiduals
Predicted chirps per
second
Observed chirps per
second Residuals
14.4 15.4 1.0
14.5 14.7 0.2
14.9 16.0 1.1
15.6 15.5 -0.1
15.9 14.4 -1.5
16.6 15.0 -1.6
16.8 17.1 0.3
16.8 16.0 -0.8
17.1 17.1 0.0
17.2 17.2 0.0
17.3 16.2 -1.1
17.4 17.0 -0.4
17.6 18.4 0.8
18.5 20.0 1.5
19.5 19.8 0.3
-2.0
-1.5
-1.0
-0.5
0.0
0.5
1.0
1.5
2.0
60 70 80 90 100
Temperature
Re
sid
ua
ls
ResidualsResiduals
The plot of the residuals shows that they are randomly scattered, so in this case a linear model is appropriate.