Simple Linear Regression
In many scientific investigations, one is interested to find how something
is related with something else. For example the distance traveled and the time
spent driving; one’s age and height. Generally, there are two types of
relationships between a pair of variable: deterministic relationship and
probabilistic relationship.
Deterministic relationship
s  s0  vt
S: distance travel
S0: initial distance
v: speed
t: traveled
distance
slope
S0
v
intercept
time
Probabilistic Relationship
In many occasions we are facing a different situation. One variable is
related to another variable as in the following.
height
age
Here we can not definitely predict one’s height from his age as we did
in
s  s0  vt
Linear Regression
Statistically, the way to characterize the relationship between two variables
as we shown before is to use a linear model as in the following:
y  a  bx  
Here, x is called independent variable
y is called dependent variable
 is the error term
y
a is intercept
b is slope
Error: 
b
a
x
Least Square Regression
Given some pairs of data for independent and dependent variables,
we may draw many lines through the scattered points
y
x
The least square regression finds a line passing through the points that minimize
the vertical distance between the points and the line. In other words, the least
square line minimizes the error term .
Least Square Method
For notational convenience, the line that fits through the
points is often written as
yˆ  a  bx
The linear model we wrote before is
y  a  bx  
If we use the value on the line, ŷ , to estimate y, the difference is (y- ŷ)
For points above the line, the difference is positive, while the difference
is negative for points below the line.
y
yˆ  a  bx
ŷ
(y- ŷ)
Error Sum of Squares
For some points, the values of (y- ŷ) are positive (points above the line) and for some
other points, the values of (y- ŷ) are negative (points below the line). If we add all
these up, the positive and negative values can get cancelled. Therefore, we take a
square for all these difference and sum them up. Such a sum is called the Sum
of Squares of Error (SSE)
n
SSE   ( y  yˆ ) 2
i 1
The constant a and b is estimated so that the error sum of squares is
minimized, therefore the name least square.
Estimating Regression Coefficients
If we solve the regression coefficients a and b from by minimizing SSE,
the following are the solutions.
n
b
 ( x  x )( y
i
i 1
n
i
 y)
2
(
x

x
)
 i

S xy
S xx
i 1
a  y  bx
Where xi is the ith independent variable value
yi is dependdent variable value corresponding to xi
x_bar and y_bar are the mean value of x and y.
Interpretation of a and b
The constant b is the slope, which gives the change in y (dependent variable) due to a
change of one unit in x (independent variable). If b> 0, x and y are positively correlated,
meaning y increases as x increases, vice versus. If b<0, x and y are negatively correlated.
y
y
a
a
b<0
b>0
x
x
Correlation Coefficient
Although now we have a regression line to describe the relationship between the
dependent variable and the independent variable, it is not enough to characterize
the relationship between x and y. We may see the situation in the following graphs.
y
(1)
y
x
(2)
x
Obviously the relationship between x and y in (1) is stronger than that in (2) even
though the line in (2) is the best fit line. The statistic that characterizes the strength
of the relationship is correlation coefficient or R2
How R2 is Calculated?
y
ŷ
y
y  y  ( y  yˆ )  ( yˆ  y )
If we use y_bar to represent y, the error is (y-y_bar). If we use ŷ to represents y, the
error is (y- ŷ ). Therefore the error is reduced to (y- ŷ ). Thus (ŷ- y_bar )
is the improvement over using y_bar. This is true for all points in the graph. To
account how much total improvement we get, we take a sum of all improvements, (ŷ
-y_bar). Again we face the same situation as we did while calculating variance. We
take the square of the difference and sum the squared difference for all points
R Square
Regression Sum of Squares
y
n
SSR   ( yˆ i  y )
2
ŷ
i 1
Total Sum of Squares
y
n
SST   ( yi  y ) 2
i 1
R2 
SSR
SST
R square indicates the percent variance in y explained by the regression.
We already calculated SSE (Error Sum of Squares) while estimating a and b. In fact,
the following relationship holds true:
SST=SSR+SSE
An Simple Linear Regression Example
The followings are some survey data showing how much a family spend on
food in relation to household income (x=income in thousand $, y=is percent of
income left after spending on food)
x
y
6.5
81
4
96
2.5
93
7.2
68
8.1
63
3.4
84
5.5
71
sum
37.2
556
mean
5.31429 79.4286
slope
-5.2071
intercept 107.101
SST
953.714
SSR
706.834
SSE
246.881
SST+SSR 953.715
R-square 0.74114
x-x_bar
1.185714
-1.31429
-2.81429
1.885714
2.785714
-1.91429
0.185714
y-y_bar
(x-x_bar)(y-y_bar)
1.571429
1.863265306
16.57143
-21.77959184
13.57143
-38.19387755
-11.4286
-21.55102041
-16.4286
-45.76530612
4.571429
-8.751020408
-8.42857
-1.565306122
-135.7428571
n
b
 ( x  x )( y
i
i 1
i
 y)
n
(x  x)
i 1
i
2
(x-x_bar)^2
1.40591837
1.72734694
7.92020408
3.55591837
7.76020408
3.6644898
0.0344898
26.0685714
y_hat
73.254325
86.2722
94.082925
69.60932
64.922885
89.39649
78.461475
(y-y_bar)^2 (y_hat-y_bar)^2 (y-y_hat)^2
2.46938776
38.12130132
59.99548121
274.612245
46.83527158
94.63009284
184.183673
214.7501205
1.172726556
130.612245
96.41767056
2.589910862
269.897959
210.4148973
3.697486723
20.8979592
99.35942913
29.12210432
71.0408163
0.935272739
55.67360918
953.714286
706.8339631
246.8814117