what is a single linear regression
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Single Linear Regression
Conceptual Explanation
• Welcome to this explanation of Single Linear Regression.
• Welcome to this explanation of Single Linear Regression.• Single linear regression is an extension of
correlation.
• Welcome to this explanation of Single Linear Regression.• Single linear regression is an extension of
correlation.
Correlation Single Linear Regressionextends to
• Correlation is designed to render a single coefficient that represents the degree of coherence between two variables
• Correlation is designed to render a single coefficient that represents the degree of coherence between two variables
• Correlation is designed to render a single coefficient that represents the degree of coherence between two variables
• Correlation is designed to render a single coefficient that represents the degree of coherence between two variables
+.99
As one variable
increases the other
increases
• Correlation is designed to render a single coefficient that represents the degree of coherence between two variables
+.99
As one variable
increases the other
increases
This coefficient represents an almost perfect positive
correlation or relationship between these two variables.
• Correlation is designed to render a single coefficient that represents the degree of coherence between two variables
Ave Daily Temp
500
600
700
800
900
• Correlation is designed to render a single coefficient that represents the degree of coherence between two variables
Ave Daily Temp
500
600
700
800
900
As one variable
decreases the other
increases
• Correlation is designed to render a single coefficient that represents the degree of coherence between two variables
Ave Daily Temp
500
600
700
800
900
-.99
As one variable
decreases the other
increases
• Correlation is designed to render a single coefficient that represents the degree of coherence between two variables
Ave Daily Temp
500
600
700
800
900
-.99
As one variable
decreases the other
increases
Almost a perfect negative correlation or relationship
between these two variables.
• Single linear regression uses that information to predict the value of one variable based on the given value of the other variable.
• Single linear regression uses that information to predict the value of one variable based on the given value of the other variable.
• Single linear regression uses that information to predict the value of one variable based on the given value of the other variable.
• For example:
• For example:If the following data set were real, what would you predict ice cream sales would be when the temperature reaches 1000?
• For example:If the following data set were real, what would you predict ice cream sales would be when the temperature reaches 1000?
Ave Daily Ice Cream Sales
?560
480
350
320
230
Ave Daily Temp
1000
900
800
700
600
500
• Single linear regression uses that information to predict the value of one variable (ice cream) based on the given value of the other variable (temperature).
• Single linear regression uses that information to predict the value of one variable (ice cream) based on the given value of the other variable (temperature).
If the following data set were real, what would you predict ice cream sales would be when the temperature reaches 1000?
• Rather than simply examining the relationship between the variables (as is the case with the Pearson Product Moment Correlation), one variable will be used as the predictor (temperature) and the other value will be used as the outcome or predicted (ice cream sales).
Ave Daily Ice Cream Sales
630?560
480
350
320
230
Ave Daily Temp
1000
900
800
700
600
500
If the following data set were real, what would you predict ice cream sales would be when the temperature reaches 1000?
• Rather than simply examining the relationship between the variables (as is the case with the Pearson Product Moment Correlation), one variable will be used as the predictor (temperature) and the other value will be used as the outcome or predicted (ice cream sales). • Linear Regression makes it possible to estimate a value like
630
Ave Daily Ice Cream Sales
630?560
480
350
320
230
Ave Daily Temp
1000
900
800
700
600
500
• In some cases which variable is considered predictor or outcome is arbitrary.
• In some cases which variable is considered predictor or outcome is arbitrary.• Like measures of depression and anxiety
• In some cases which variable is considered predictor or outcome is arbitrary.• Like measures of depression and anxiety
Composite Depression Score
33
26
22
14
12
6
Composite Anxiety Score
103
100
92
74
52
26
• In some cases which variable is considered predictor or outcome is arbitrary.• Like measures of depression and anxiety
• It’s not clear which influences which. Most likely depression and anxiety mutually influence one another.
Composite Depression Score
33
26
22
14
12
6
Composite Anxiety Score
103
100
92
74
52
26
• In some cases, either by theory or by the nature of the research design, one variable will be rationally defined as the predictor and the other as the outcome.
• In some cases, either by theory or by the nature of the research design, one variable will be rationally defined as the predictor and the other as the outcome.
Ave Daily Exposure to Sunlight
3.3 hrs
2.6 hrs
2.2 hrs
1.4 hrs
1.2 hrs
0.6 hrs
• In some cases, either by theory or by the nature of the research design, one variable will be rationally defined as the predictor and the other as the outcome.
Ave Daily Exposure to Sunlight
3.3 hrs
2.6 hrs
2.2 hrs
1.4 hrs
1.2 hrs
0.6 hrs
Levels of Vitamin E after two months
10.3 units
8.1 units
7.3 units
7.0 units
6.8 units
5.7 units
• In some cases, either by theory or by the nature of the research design, one variable will be rationally defined as the predictor and the other as the outcome.
Ave Daily Exposure to Sunlight
3.3 hrs
2.6 hrs
2.2 hrs
1.4 hrs
1.2 hrs
0.6 hrs
Levels of Vitamin E after two months
10.3 units
8.1 units
7.3 units
7.0 units
6.8 units
5.7 units
In this example, exposure to sunlight may impact levels of
Vitamin E.
But, levels of Vitamin E would not impact the amount of sunlight
one gets.
• An easy way to conceptualize single linear regression is to create a scatterplot in Cartesian space.
• An easy way to conceptualize single linear regression is to create a scatterplot in Cartesian space.
Let’s plot the following data set:
• An easy way to conceptualize single linear regression is to create a scatterplot in Cartesian space.
Let’s plot the following data set:Composite
Depression Score33
26
22
14
12
6
Composite Anxiety Score
103
100
92
74
52
26
• First, we assign the predictor variable along the X axis, which in this case we’ll arbitrarily say is depression.
• First, we assign the predictor variable along the X axis, which in this case we’ll arbitrarily say is depression.
0 5 10 15 20 25 30 350
20
40
60
80
100
120
Relationship between Depression & Anxiety
Depression
Anx
iety
• ... and the outcome variable along the Y axis we’ll arbitrarily say is Anxiety.
• ... and the outcome variable along the Y axis we’ll arbitrarily say is Anxiety.
0 5 10 15 20 25 30 350
20
40
60
80
100
120
Relationship between Depression & Anxiety
Depression
Anx
iety
• Now, let’s identify or plot each point or dot
• Now, let’s identify or plot each point or dotDepression
33262214126
Anxiety
10310092745226
• Now, let’s identify or plot each point or dotDepression
33262214126
Anxiety
10310092745226
0 5 10 15 20 25 30 350
20
40
60
80
100
120
Relationship between Depression & Anxiety
Depression
Anx
iety
• Now, let’s identify or plot each point or dotDepression
33262214126
Anxiety
10310092745226
0 5 10 15 20 25 30 350
20
40
60
80
100
120
Relationship between Depression & Anxiety
Depression
Anx
iety
(33, 103)
• Now, let’s identify or plot each point or dotDepression
33262214126
Anxiety
10310092745226
• Now, let’s identify or plot each point or dotDepression
33262214126
Anxiety
10310092745226
0 5 10 15 20 25 30 350
20
40
60
80
100
120
Relationship between Depression & Anxiety
Depression
Anx
iety
(26, 100)
• Now, let’s identify or plot each point or dotDepression
33262214126
Anxiety
10310092745226
• Now, let’s identify or plot each point or dotDepression
33262214126
Anxiety
10310092745226
0 5 10 15 20 25 30 350
20
40
60
80
100
120
Relationship between Depression & Anxiety
Depression
Anx
iety
(22, 92)
• Now, let’s identify or plot each point or dotDepression
33262214126
Anxiety
10310092745226
• Now, let’s identify or plot each point or dotDepression
33262214126
Anxiety
10310092745226
0 5 10 15 20 25 30 350
20
40
60
80
100
120
Relationship between Depression & Anxiety
Depression
Anx
iety
(14, 74)
• Now, let’s identify or plot each point or dotDepression
33262214126
Anxiety
10310092745226
• Now, let’s identify or plot each point or dotDepression
33262214126
Anxiety
10310092745226
0 5 10 15 20 25 30 350
20
40
60
80
100
120
Relationship between Depression & Anxiety
Depression
Anx
iety
(12, 52)
• Now, let’s identify or plot each point or dotDepression
33262214126
Anxiety
10310092745226
• Now, let’s identify or plot each point or dotDepression
33262214126
Anxiety
10310092745226
0 5 10 15 20 25 30 350
20
40
60
80
100
120
Relationship between Depression & Anxiety
Depression
Anx
iety
(6, 26)
• Visually, one can see in the plotted space whether there is a tendency for the variables to be related and in what direction they are related.
• Visually, one can see in the plotted space whether there is a tendency for the variables to be related and in what direction they are related.
0 5 10 15 20 25 30 350
20
40
60
80
100
120
Relationship between Depression & Anxiety
Depression
Anx
iety
• Visually, one can see in the plotted space whether there is a tendency for the variables to be related and in what direction they are related.
0 5 10 15 20 25 30 350
20
40
60
80
100
120
Relationship between Depression & Anxiety
Depression
Anx
iety
In this case there is a strong tendency
to relate and the relationship is
positive
• With this data set the tendency for the variables to relate is strong and the direction is negative:
• With this data set the tendency for the variables to relate is strong and the direction is negative:
Depression
61214222633
Anxiety
10310092745226
• With this data set the tendency for the variables to relate is strong and the direction is negative:
Depression
61214222633
Anxiety
10310092745226
0 5 10 15 20 25 30 350
20
40
60
80
100
120
Relationship between Depression & Anxiety
Depression
Anx
iety
• With this data set the tendency for the variables to relate is strong and the direction is negative:
Depression
61214222633
Anxiety
10310092745226
0 5 10 15 20 25 30 350
20
40
60
80
100
120
Relationship between Depression & Anxiety
Depression
Anx
iety
Strong and Negative
• When no relationship exists the scatter plot tends to look like a big circle.
• When no relationship exists the scatter plot tends to look like a big circle.
Depression
2233126
1426
Anxiety
10310092745226
• When no relationship exists the scatter plot tends to look like a big circle.
Depression
2233126
1426
Anxiety
10310092745226
0 5 10 15 20 25 30 350
20
40
60
80
100
120
Relationship between Depression & Anxiety
DepressionA
nxie
ty
• When no relationship exists the scatter plot tends to look like a big circle.
Depression
2233126
1426
Anxiety
10310092745226
0 5 10 15 20 25 30 350
20
40
60
80
100
120
Relationship between Depression & Anxiety
DepressionA
nxie
ty
• When no relationship exists the scatter plot tends to look like a big circle.
Depression
226
33261412
Anxiety
10310092745226
0 5 10 15 20 25 30 350
20
40
60
80
100
120
Relationship between Depression & Anxiety
DepressionA
nxie
ty
• When no relationship exists the scatter plot tends to look like a big circle.
Depression
226
33261412
Anxiety
10310092745226
0 5 10 15 20 25 30 350
20
40
60
80
100
120
Relationship between Depression & Anxiety
DepressionA
nxie
ty
• When no relationship exists the scatter plot tends to look like a big circle.
Depression
226
33261412
Anxiety
10310092745226
Weak and Positive
• When no relationship exists the scatter plot tends to look like a big circle.
Depression
61433261222
Anxiety
10310074925226
0 5 10 15 20 25 30 350
20
40
60
80
100
120
Relationship between Depression & Anxiety
DepressionA
nxie
ty
• When no relationship exists the scatter plot tends to look like a big circle.
Depression
61433261222
Anxiety
10310074925226
0 5 10 15 20 25 30 350
20
40
60
80
100
120
Relationship between Depression & Anxiety
DepressionA
nxie
ty
• When no relationship exists the scatter plot tends to look like a big circle.
Depression
61433261222
Anxiety
10310074925226
Weak and Negative
• You might have noticed that as the variables are related either positively or negatively, the plot looks more like an oval tilted one way or the other.
• You might have noticed that as the variables are related either positively or negatively, the plot looks more like an oval tilted one way or the other.
0 5 10 15 20 25 30 350
20
40
60
80
100
120
Relationship between Depression & Anxiety
Depression
Anx
iety
0 5 10 15 20 25 30 350
20
40
60
80
100
120
Relationship between Depression & Anxiety
DepressionA
nxie
ty
• You might have noticed that as the variables are related either positively or negatively, the plot looks more like an oval tilted one way or the other.
Weak and Negative
0 5 10 15 20 25 30 350
20
40
60
80
100
120
Relationship between Depression & Anxiety
Depression
Anx
iety
0 5 10 15 20 25 30 350
20
40
60
80
100
120
Relationship between Depression & Anxiety
DepressionA
nxie
ty
Weak and Positive
• As mentioned before, Linear Regression is used to predict one variable (ice cream sales) from another related variable (temperature).
• As mentioned before, Linear Regression is used to predict one variable (ice cream sales) from another related variable (temperature). • The stronger the relationship (e.g., +.99 or -.99) the more
accurate the prediction.
• As mentioned before, Linear Regression is used to predict one variable (ice cream sales) from another related variable (temperature). • The stronger the relationship (e.g., +.99 or -.99) the more
accurate the prediction.
• The weaker the relationship (e.g., +.14 or -.03) the less accurate the prediction.
• As mentioned before, Linear Regression is used to predict one variable (ice cream sales) from another related variable (temperature). • The stronger the relationship (e.g., +.99 or -.99) the more
accurate the prediction.
• The weaker the relationship (e.g., +.14 or -.03) the less accurate the prediction.
• As mentioned before, Linear Regression is used to predict one variable (ice cream sales) from another related variable (temperature). • The stronger the relationship (e.g., +.99 or -.99) the more
accurate the prediction.
• The weaker the relationship (e.g., +.14 or -.03) the less accurate the prediction.
• One of the ways to represent those relationships is of course with the coefficients (e.g., +.99, +.14, -.03, -.99).
• As mentioned before, Linear Regression is used to predict one variable (ice cream sales) from another related variable (temperature). • The stronger the relationship (e.g., +.99 or -.99) the more
accurate the prediction.
• The weaker the relationship (e.g., +.14 or -.03) the less accurate the prediction.
• One of the ways to represent those relationships is of course with the coefficients (e.g., +.99, +.14, -.03, -.99).• Another way to represent it is by graphing the relationship.
• Recall that a line in Cartesian space is defined by its slope and its Y intercept (the value of Y when X equals 0).
• Recall that a line in Cartesian space is defined by its slope and its Y intercept (the value of Y when X equals 0).
[Y= intercept + (slope X)]∙
• Recall that a line in Cartesian space is defined by its slope and its Y intercept (the value of Y when X equals 0).
[Y= intercept + (slope X)]∙
0 1 2 3 4 5 60
1
2
3
4
5
6
• In this case the slope would be 1. You may remember that this value is derived by taking what is called the “rise” over the “run”.
0 1 2 3 4 5 60
1
2
3
4
5
6
rise1
• In this case the slope would be 1. You may remember that this value is derived by taking what is called the “rise” over the “run”.
run1
0 1 2 3 4 5 60
1
2
3
4
5
6
rise1
• In this case the slope would be 1. You may remember that this value is derived by taking what is called the “rise” over the “run”.
• So the equation for this line so far would look like this:
run1
0 1 2 3 4 5 60
1
2
3
4
5
6
rise1
• In this case the slope would be 1. You may remember that this value is derived by taking what is called the “rise” over the “run”.
• So the equation for this line so far would look like this:
run1
𝒚=0+11𝒙
0 1 2 3 4 5 60
1
2
3
4
5
6
rise1
run1
𝒚=0+11𝒙
0 1 2 3 4 5 60
1
2
3
4
5
6
rise1
run1
𝒚=0+11𝒙
This is where the line crosses the
Y axis.
0 1 2 3 4 5 60
1
2
3
4
5
6
rise1
run1
𝒚=0+11𝒙
This is the slope which is the rise
over the run.
• A line represents the functional relationship between variable X and variable Y, therefore, that line can be used to predict a Y value from any given X value.
• A line represents the functional relationship between variable X and variable Y, therefore, that line can be used to predict a Y value from any given X value.
Feb
Mar
Apr
May
Jun
Ave Monthly Temperature
500
600
700
800
900
Ave Monthly Ice Cream Sales
239
320
400
480
560
• In this case the two variables (temperature and ice cream sales) have a perfect linear relationship. This is rarely ever seen among variables such as these in the real world, but for illustrative purposes we have created a perfect relationship.
• In this case the two variables (temperature and ice cream sales) have a perfect linear relationship. This is rarely ever seen among variables such as these in the real world, but for illustrative purposes we have created a perfect relationship.
40 60 80 100 1200
100
200
300
400
500
600
700
Average Monthly Ice Cream Sales
Ave
Mon
thly
Tem
pera
ture
• Now let’s say we have data for the average temperature during the month of July. But, we don’t have the data for the average ice cream sales for July
• Now let’s say we have data for the average temperature during the month of July. But, we don’t have the data for the average ice cream sales for July
Feb
Mar
Apr
May
Jun
JUL
Ave Monthly Temperature
500
600
700
800
900
1000
Ave Monthly Ice Cream Sales
239
320
400
480
560
?
• Now let’s say we have data for the average temperature during the month of July. But, we don’t have the data for the average ice cream sales for July
• Using single linear regression we can predict the average ice cream sales for July. Here is the formula we will use for the prediction:
Feb
Mar
Apr
May
Jun
JUL
Ave Monthly Temperature
500
600
700
800
900
1000
Ave Monthly Ice Cream Sales
239
320
400
480
560
?
• Now let’s say we have data for the average temperature during the month of July. But, we don’t have the data for the average ice cream sales for July
• Using single linear regression we can predict the average ice cream sales for July. Here is the formula we will use for the prediction:
Feb
Mar
Apr
May
Jun
JUL
Ave Monthly Temperature
500
600
700
800
900
1000
Ave Monthly Ice Cream Sales
239
320
400
480
560
?
¿
• Now let’s say we have data for the average temperature during the month of July. But, we don’t have the data for the average ice cream sales for July
• Using single linear regression we can predict the average ice cream sales for July. Here is the formula we will use for the prediction:
• There are many ways to write this equation. Here is one way:
Feb
Mar
Apr
May
Jun
JUL
Ave Monthly Temperature
500
600
700
800
900
1000
Ave Monthly Ice Cream Sales
239
320
400
480
560
?
¿
• Now let’s say we have data for the average temperature during the month of July. But, we don’t have the data for the average ice cream sales for July
• Using single linear regression we can predict the average ice cream sales for July. Here is the formula we will use for the prediction:
• There are many ways to write this equation. Here is one way:
Feb
Mar
Apr
May
Jun
JUL
Ave Monthly Temperature
500
600
700
800
900
1000
Ave Monthly Ice Cream Sales
239
320
400
480
560
?
¿
¿
• Using this data set we can create a formula for a straight line that represents that relationship:
• Using this data set we can create a formula for a straight line that represents that relationship:
Feb
Mar
Apr
May
Jun
Ave Monthly Temperature
500
600
700
800
900
Ave Monthly Ice Cream Sales
239
320
400
480
560
• Using this data set we can create a formula for a straight line that represents that relationship:
Feb
Mar
Apr
May
Jun
Ave Monthly Temperature
500
600
700
800
900
Ave Monthly Ice Cream Sales
239
320
400
480
560
40 60 80 100 1200
100
200
300
400
500
600
700
Average Monthly Ice Cream Sales
Ave
Mon
thly
Tem
pera
ture
𝑦= -162+8( )𝑥
• With this equation we can now plug in the average temperature for July (1000) and see what the predicted average ice cream sales would be:
• With this equation we can now plug in the average temperature for July (1000) and see what the predicted average ice cream sales would be:
Feb
Mar
Apr
May
Jun
Jul
Ave Monthly Temperature
500
600
700
800
900
1000
Ave Monthly Ice Cream Sales
239
320
400
480
560
• With this equation we can now plug in the average temperature for July (1000) and see what the predicted average ice cream sales would be:
Feb
Mar
Apr
May
Jun
Jul
Ave Monthly Temperature
500
600
700
800
900
1000
Ave Monthly Ice Cream Sales
239
320
400
480
560
40 60 80 100 1200
100
200
300
400
500
600
700
Average Monthly Ice Cream Sales
Ave
Mon
thly
Tem
pera
ture
𝑦 ̂� = -162 + 8(100)
• With this equation we can now plug in the average temperature for July (1000) and see what the predicted average ice cream sales would be:
Feb
Mar
Apr
May
Jun
Jul
Ave Monthly Temperature
500
600
700
800
900
1000
Ave Monthly Ice Cream Sales
239
320
400
480
560
40 60 80 100 1200
100
200
300
400
500
600
700
Average Monthly Ice Cream Sales
Ave
Mon
thly
Tem
pera
ture
𝑦 ̂� = -162 + 8(100)
• With this equation we can now plug in the average temperature for July (1000) and see what the predicted average ice cream sales would be:
Feb
Mar
Apr
May
Jun
Jul
Ave Monthly Temperature
500
600
700
800
900
1000
Ave Monthly Ice Cream Sales
239
320
400
480
560
40 60 80 100 1200
100
200
300
400
500
600
700
Average Monthly Ice Cream Sales
Ave
Mon
thly
Tem
pera
ture
𝑦 ̂� = -162 + 800
• With this equation we can now plug in the average temperature for July (1000) and see what the predicted average ice cream sales would be:
Feb
Mar
Apr
May
Jun
Jul
Ave Monthly Temperature
500
600
700
800
900
1000
Ave Monthly Ice Cream Sales
239
320
400
480
560
40 60 80 100 1200
100
200
300
400
500
600
700
Average Monthly Ice Cream Sales
Ave
Mon
thly
Tem
pera
ture
𝑦 ̂� = 638
• With this equation we can now plug in the average temperature for July (1000) and see what the predicted average ice cream sales would be:
Feb
Mar
Apr
May
Jun
Jul
Ave Monthly Temperature
500
600
700
800
900
1000
Ave Monthly Ice Cream Sales
239
320
400
480
560
638
40 60 80 100 1200
100
200
300
400
500
600
700
Average Monthly Ice Cream Sales
Ave
Mon
thly
Tem
pera
ture
𝑦 ̂� = 638
• So, based on our single linear regression analysis we would predict that in the month of July that the average monthly ice cream sales will be 638.
Feb
Mar
Apr
May
Jun
Jul
Ave Monthly Temperature
500
600
700
800
900
1000
Ave Monthly Ice Cream Sales
239
320
400
480
560
638
40 60 80 100 1200
100
200
300
400
500
600
700
Average Monthly Ice Cream Sales
Ave
Mon
thly
Tem
pera
ture
𝑦 ̂� = 638
• So, based on our single linear regression analysis we would predict that in the month of July that the average monthly ice cream sales will be 638.
• This is a simple demonstration of how regression works.
Feb
Mar
Apr
May
Jun
Jul
Ave Monthly Temperature
500
600
700
800
900
1000
Ave Monthly Ice Cream Sales
239
320
400
480
560
638
40 60 80 100 1200
100
200
300
400
500
600
700
Average Monthly Ice Cream Sales
Ave
Mon
thly
Tem
pera
ture
𝑦 ̂� = 638
• So, based on our single linear regression analysis we would predict that in the month of July that the average monthly ice cream sales will be 638.
• This is a simple demonstration of how regression works.• In reality, however, most variables will not correlate so
perfectly like this did:
Feb
Mar
Apr
May
Jun
Jul
Ave Monthly Temperature
500
600
700
800
900
1000
Ave Monthly Ice Cream Sales
239
320
400
480
560
638
40 60 80 100 1200
100
200
300
400
500
600
700
Average Monthly Ice Cream Sales
Ave
Mon
thly
Tem
pera
ture
𝑦 ̂� = 638
• Most will look like this:
• Most will look like this:
• Most will look like this:
• This line is called the best fitting line because it minimizes the distance between the line and all of the points. You will notice again that we have a linear equation for that line:
• Most will look like this:
• This line is called the best fitting line because it minimizes the distance between the line and all of the points. You will notice again that we have a linear equation for that line:𝑦= -
50.93+7.21(x)
• Most will look like this:
• This equation is calculated by using the standard deviations and means of the two variables. For brevity sake we will not go into this here. 𝑦= -
50.93+7.21(x)
• Given the infinite number of positive linear fitting through a scatterplot, the one closer to represent the functional relationship between X and Y is the line that results in the cumulative least squared error between the predicted values of Y and the true observed values of Y for each given X.
• Given the infinite number of positive linear fitting through a scatterplot, the one closer to represent the functional relationship between X and Y is the line that results in the cumulative least squared error between the predicted values of Y and the true observed values of Y for each given X.
• Given the infinite number of positive linear fitting through a scatterplot, the one closer to represent the functional relationship between X and Y is the line that results in the cumulative least squared error between the predicted values of Y and the true observed values of Y for each given X.
This line is the predicted values of Y calculated from the
equation
• Given the infinite number of positive linear fitting through a scatterplot, the one closer to represent the functional relationship between X and Y is the line that results in the cumulative least squared error between the predicted values of Y and the true observed values of Y for each given X.
These dots represent the actual data
This line is the predicted values of Y calculated from the
equation
• We don’t have to actually plot the coordinates and lines. We can operate solely on the equations to generate predicted values and errors in prediction. In this way we can determine if temperature is a statistically significant predictor of ice cream sales.
• So here are the actual data we plotted the data from:
• So here are the actual data we plotted the data from:
Jan 40 300
Feb 50 320
Mar 60 370
Apr 70 480
May 80 560
Jun 90 640
Jul 100 720
Aug 90 600
Sep 80 400
Oct 60 300
Nov 40 200
Dec 20 122
(X) Ave Monthly
Temp
(y) Actual Ave Monthly Ice Cream Sales
• So here are the actual data we plotted the data from:
Jan 40 300
Feb 50 320
Mar 60 370
Apr 70 480
May 80 560
Jun 90 640
Jul 100 720
Aug 90 600
Sep 80 400
Oct 60 300
Nov 40 200
Dec 20 122
(X) Ave Monthly
Temp
(y) Actual Ave Monthly Ice Cream Sales
• So here are the actual data we plotted the data from:
Jan 40 300
Feb 50 320
Mar 60 370
Apr 70 480
May 80 560
Jun 90 640
Jul 100 720
Aug 90 600
Sep 80 400
Oct 60 300
Nov 40 200
Dec 20 122
(X) Ave Monthly
Temp
(y) Actual Ave Monthly Ice Cream Sales
• We can now plot the predicted Y using the equation:
• So here are the actual data we plotted the data from:
Jan 40 300
Feb 50 320
Mar 60 370
Apr 70 480
May 80 560
Jun 90 640
Jul 100 720
Aug 90 600
Sep 80 400
Oct 60 300
Nov 40 200
Dec 20 122
(X) Ave Monthly
Temp
(y) Actual Ave Monthly Ice Cream Sales
• We can now plot the predicted Y using the equation: = -50.93+7.21(x)
• So here are the actual data we plotted the data from:
Jan 40 300
Feb 50 320
Mar 60 370
Apr 70 480
May 80 560
Jun 90 640
Jul 100 720
Aug 90 600
Sep 80 400
Oct 60 300
Nov 40 200
Dec 20 122
(X) Ave Monthly
Temp
(y) Actual Ave Monthly Ice Cream Sales
• We can now plot the predicted Y using the equation:
• Which is the equation for the best fitting line between these two variables:
= -50.93+7.21(x)
• We can now plot the predicted Y using the equation:
• We can now plot the predicted Y using the equation:= -50.93+7.21(x)
• We can now plot the predicted Y using the equation:
Jan 40 300
Feb 50 320
Mar 60 370
Apr 70 480
May 80 560
Jun 90 640
Jul 100 720
Aug 90 600
Sep 80 400
Oct 60 300
Nov 40 200
Dec 20 122
(X) Ave Monthly
Temp
(y) Actual Ave Monthly Ice Cream Sales
= -50.93+7.21(x)
• We can now plot the predicted Y using the equation:
Jan 40 300
Feb 50 320
Mar 60 370
Apr 70 480
May 80 560
Jun 90 640
Jul 100 720
Aug 90 600
Sep 80 400
Oct 60 300
Nov 40 200
Dec 20 122
(X) Ave Monthly
Temp
(y) Actual Ave Monthly Ice Cream Sales
= -50.93+7.21(x)
= -50.93+7.21(300) == -50.93+7.21(320) =
= -50.93+7.21(480) =
= -50.93+7.21(370) =
= -50.93+7.21(560) =
= -50.93+7.21(640) =
= -50.93+7.21(720) =
= -50.93+7.21(600) =
= -50.93+7.21(400) == -50.93+7.21(300) =
= -50.93+7.21(200) == -50.93+7.21(122) =
• We can now plot the predicted Y using the equation:
Jan 40 300
Feb 50 320
Mar 60 370
Apr 70 480
May 80 560
Jun 90 640
Jul 100 720
Aug 90 600
Sep 80 400
Oct 60 300
Nov 40 200
Dec 20 122
(X) Ave Monthly
Temp
(y) Actual Ave Monthly Ice Cream Sales
= -50.93+7.21(x)
= -50.93+7.21(300) == -50.93+7.21(320) =
= -50.93+7.21(480) =
= -50.93+7.21(370) =
= -50.93+7.21(560) =
= -50.93+7.21(640) =
= -50.93+7.21(720) =
= -50.93+7.21(600) =
= -50.93+7.21(400) == -50.93+7.21(300) =
= -50.93+7.21(200) == -50.93+7.21(122) =
Predicted Ave Monthly Ice Cream
Sales
237.47
309.57
381.67
453.77
525.87
597.97
670.07
597.97
525.87
381.67
237.47
93.27
• We can now plot the predicted Y using the equation:
• With this information we can now determine if x (temperature) is a statistically significant predictor of “y” (ice cream sales).
Jan 40 300
Feb 50 320
Mar 60 370
Apr 70 480
May 80 560
Jun 90 640
Jul 100 720
Aug 90 600
Sep 80 400
Oct 60 300
Nov 40 200
Dec 20 122
(X) Ave Monthly
Temp
(y) Actual Ave Monthly Ice Cream Sales
= -50.93+7.21(x)
= -50.93+7.21(300) == -50.93+7.21(320) =
= -50.93+7.21(480) =
= -50.93+7.21(370) =
= -50.93+7.21(560) =
= -50.93+7.21(640) =
= -50.93+7.21(720) =
= -50.93+7.21(600) =
= -50.93+7.21(400) == -50.93+7.21(300) =
= -50.93+7.21(200) == -50.93+7.21(122) =
Predicted Ave Monthly Ice Cream
Sales
237.47
309.57
381.67
453.77
525.87
597.97
670.07
597.97
525.87
381.67
237.47
93.27
• To begin we need to determine the total sum of squares just like we would do with analysis of variance.
• To begin we need to determine the total sum of squares just like we would do with analysis of variance.
• This is done by subtracting the actual “Y” (ice cream sales) values from the average or mean ice cream sales for the whole year.
• To begin we need to determine the total sum of squares just like we would do with analysis of variance.
• This is done by subtracting the actual “Y” (ice cream sales) values from the average or mean ice cream sales for the whole year.
• The mean is calculated by adding up the values and divided them by how many there are.
• To begin we need to determine the total sum of squares just like we would do with analysis of variance.
• This is done by subtracting the actual “Y” (ice cream sales) values from the average or mean ice cream sales for the whole year.
• The mean is calculated by adding up the values and divided them by how many there are.
• (300+320+370+480+560+640+720+600+400+300+200+122) / 12 = 417 average ice cream sales
• We then subtract each y value from the mean
• We then subtract each y value from the mean
(y) Actual Ave Monthly Ice Cream Sales
300
320
370
480
560
640
720
600
400
300
200
122
Predicted Ave Monthly Ice Cream
Sales
237.47
309.57
381.67
453.77
525.87
597.97
670.07
597.97
525.87
381.67
237.47
93.27
Difference
-117
-97
-47
63
143
223
303
183
-17
-117
-217
-295
------------
============
• We then subtract each y value from the mean
• Note - if we did not know the functional relationship between X and Y, our best prediction of any one person’s Y value would be the mean of Y.
(y) Actual Ave Monthly Ice Cream Sales
300
320
370
480
560
640
720
600
400
300
200
122
Predicted Ave Monthly Ice Cream
Sales
237.47
309.57
381.67
453.77
525.87
597.97
670.07
597.97
525.87
381.67
237.47
93.27
Difference
-117
-97
-47
63
143
223
303
183
-17
-117
-217
-295
------------
============
• Because we are calculating the total sum of squares we will need to square the results and then take the average of the sum of squares. This is the same as the variance of all of the scores.
• Because we are calculating the total sum of squares we will need to square the results and then take the average of the sum of squares. This is the same as the variance of all of the scores.
(y) Actual Ave Monthly Ice Cream Sales
300
320
370
480
560
640
720
600
400
300
200
122
Predicted Ave Monthly Ice Cream
Sales
237.47
309.57
381.67
453.77
525.87
597.97
670.07
597.97
525.87
381.67
237.47
93.27
Difference
-117
-97
-47
63
143
223
303
183
-17
-117
-217
-295
------------
============
Squared
13689
9409
2209
3969
20449
49729
91809
33489
289
13689
47089
87025
• Because we are calculating the total sum of squares we will need to square the results and then sum up the results
• Because we are calculating the total sum of squares we will need to square the results and then sum up the result
(y) Actual Ave Monthly Ice Cream Sales
300
320
370
480
560
640
720
600
400
300
200
122
Predicted Ave Monthly Ice Cream
Sales
237.47
309.57
381.67
453.77
525.87
597.97
670.07
597.97
525.87
381.67
237.47
93.27
Difference
-117
-97
-47
63
143
223
303
183
-17
-117
-217
-295
------------
============
Squared
13689
9409
2209
3969
20449
49729
91809
33489
289
13689
47089
87025
Sum up
SUM 372844
• Now we find regression (good) and residual (bad). To have better prediction power we want the regression sums of squares to be large and the residual or error sums of squares to be small.
• Now we find regression (good) and residual (bad). To have better prediction power we want the regression sums of squares to be large and the residual or error sums of squares to be small.• Let’s see if the residual or the regression is greater.
• Now we find regression (good) and residual (bad). To have better prediction power we want the regression sums of squares to be large and the residual or error sums of squares to be small.• Let’s see if the residual or the regression is greater.• We know that the total sums of squares is 31,070.
• Now we find regression (good) and residual (bad). To have better prediction power we want the regression sums of squares to be large and the residual or error sums of squares to be small.• Let’s see if the residual or the regression is greater.• We know that the total sums of squares is 31,070.
Sum of Squares df Mean Square F-ratio Significance Total 372,844
• Now we find regression (good) and residual (bad). To have better prediction power we want the regression sums of squares to be large and the residual or error sums of squares to be small.• Let’s see if the residual or the regression is greater.• We know that the total sums of squares is 31,070. • Now we will calculate the residual (error) and the
regression sums of squares which will add up to 372,844. Sum of Squares df Mean Square F-ratio Significance Total 372,844
• Now we find regression (good) and residual (bad). To have better prediction power we want the regression sums of squares to be large and the residual or error sums of squares to be small.• Let’s see if the residual or the regression is greater.• We know that the total sums of squares is 31,070. • Now we will calculate the residual (error) and the
regression sums of squares which will add up to 372,844. Sum of Squares df Mean Square F-ratio SignificanceRegression ? Residual (error) ? Total 372,844
• Before we calculate residual and regression let’s see visually how we calculated the total sums of squares -372,844.
• Before we calculate residual and regression let’s see visually how we calculated the total sums of squares -372,844.• Once again we subtract the actual Y values from the mean
of the actual Y values
• Before we calculate residual and regression let’s see visually how we calculated the total sums of squares -372,844.• Once again we subtract the actual Y values from the mean
of the actual Y values(y) Actual Ave Monthly Ice Cream Sales
300
320
370
480
560
640
720
600
400
300
200
122
Predicted Ave Monthly Ice Cream
Sales
417
417
417
417
417
417
417
417
417
417
417
417
------------
• Before we calculate residual and regression let’s see visually how we calculated the total sums of squares -372,844.• Once again we subtract the actual Y values from the mean
of the actual Y values(y) Actual Ave Monthly Ice Cream Sales
300
320
370
480
560
640
720
600
400
300
200
122
Predicted Ave Monthly Ice Cream
Sales
417
417
417
417
417
417
417
417
417
417
417
417
------------
10 20 30 40 50 60 70 80 90 100 1100
100
200
300
400
500
600
700
800
Final Exam
Mid
term
Exa
m
• Before we calculate residual and regression let’s see visually how we calculated the total sums of squares -372,844.• Once again we subtract the actual Y values from the mean
of the actual Y values(y) Actual Ave Monthly Ice Cream Sales
300
320
370
480
560
640
720
600
400
300
200
122
Predicted Ave Monthly Ice Cream
Sales
417
417
417
417
417
417
417
417
417
417
417
417
------------
10 20 30 40 50 60 70 80 90 100 1100
100
200
300
400
500
600
700
800
Final Exam
Mid
term
Exa
m
• Before we calculate residual and regression let’s see visually how we calculated the total sums of squares -372,844.• Once again we subtract the actual Y values from the mean
of the actual Y values(y) Actual Ave Monthly Ice Cream Sales
300
320
370
480
560
640
720
600
400
300
200
122
Predicted Ave Monthly Ice Cream
Sales
417
417
417
417
417
417
417
417
417
417
417
417
------------
10 20 30 40 50 60 70 80 90 100 1100
100
200
300
400
500
600
700
800
Final Exam
Mid
term
Exa
m
• Before we calculate residual and regression let’s see visually how we calculated the total sums of squares -372,844.• Once again we subtract the actual Y values from the mean
of the actual Y values(y) Actual Ave Monthly Ice Cream Sales
300
320
370
480
560
640
720
600
400
300
200
122
Predicted Ave Monthly Ice Cream
Sales
417
417
417
417
417
417
417
417
417
417
417
417
------------
10 20 30 40 50 60 70 80 90 100 1100
100
200
300
400
500
600
700
800
Final Exam
Mid
term
Exa
m
• The first data set are the actual Y values. We subtract them from the mean (417) which would be our best prediction if we did not know the relationship between X (temperature) and Y (ice cream sales)
• The first data set are the actual Y values. We subtract them from the mean (417) which would be our best prediction if we did not know the relationship between X (temperature) and Y (ice cream sales)
(y) Actual Ave Monthly Ice Cream Sales
300
320
370
480
560
640
720
600
400
300
200
122
Predicted Ave Monthly Ice Cream
Sales
417
417
417
417
417
417
417
417
417
417
417
417
------------
10 20 30 40 50 60 70 80 90 100 1100
100
200
300
400
500
600
700
800
Final Exam
Mid
term
Exa
m
• Here is the graphic depiction of our subtracting each data point from the mean (417):
• Here is the graphic depiction of our subtracting each data point from the mean (417):
10 20 30 40 50 60 70 80 90 100 1100
100
200
300
400
500
600
700
800
122
Final Exam
Mid
term
Exa
m
122-417
= -295
417
• Here is the graphic depiction of our subtracting each data point from the mean (417):
10 20 30 40 50 60 70 80 90 100 1100
100
200
300
400
500
600
700
800
122
Final Exam
Mid
term
Exa
m
122-417
= -295
417
(y) Actual Ave Monthly Ice Cream Sales
300
320
370
480
560
640
720
600
400
300
200
122
Predicted Ave Monthly Ice Cream
Sales
417
417
417
417
417
417
417
417
417
417
417
417
Difference
-117
-97
-47
63
143
223
303
183
-17
-117
-217
-295
------------
============
• Here is the graphic depiction of our subtracting each data point from the mean (417):
10 20 30 40 50 60 70 80 90 100 1100
100
200
300
400
500
600
700
800
122
Final Exam
Mid
term
Exa
m
122-417
= -295
417
(y) Actual Ave Monthly Ice Cream Sales
300
320
370
480
560
640
720
600
400
300
200
122
Predicted Ave Monthly Ice Cream
Sales
417
417
417
417
417
417
417
417
417
417
417
417
Difference
-117
-97
-47
63
143
223
303
183
-17
-117
-217
-295
------------
============
• Here is the graphic depiction of our subtracting each data point from the mean (417):
10 20 30 40 50 60 70 80 90 100 1100
100
200
300
400
500
600
700
800
200
Final Exam
Mid
term
Exa
m
200-417
= -217
417
(y) Actual Ave Monthly Ice Cream Sales
300
320
370
480
560
640
720
600
400
300
200
122
Predicted Ave Monthly Ice Cream
Sales
417
417
417
417
417
417
417
417
417
417
417
417
Difference
-117
-97
-47
63
143
223
303
183
-17
-117
-217
-295
------------
============
• Here is the graphic depiction of our subtracting each data point from the mean (417):
10 20 30 40 50 60 70 80 90 100 1100
100
200
300
400
500
600
700
800
200
Final Exam
Mid
term
Exa
m
200-417
= -217
417
(y) Actual Ave Monthly Ice Cream Sales
300
320
370
480
560
640
720
600
400
300
200
122
Predicted Ave Monthly Ice Cream
Sales
417
417
417
417
417
417
417
417
417
417
417
417
Difference
-117
-97
-47
63
143
223
303
183
-17
-117
-217
-295
------------
============
• Here is the graphic depiction of our subtracting each data point from the mean (417):
10 20 30 40 50 60 70 80 90 100 1100
100
200
300
400
500
600
700
800
Final Exam
Mid
term
Exa
m
200-417
= +303
417
(y) Actual Ave Monthly Ice Cream Sales
300
320
370
480
560
640
720
600
400
300
200
122
Predicted Ave Monthly Ice Cream
Sales
417
417
417
417
417
417
417
417
417
417
417
417
Difference
-117
-97
-47
63
143
223
303
183
-17
-117
-217
-295
------------
============
• Here is the graphic depiction of our subtracting each data point from the mean (417):
10 20 30 40 50 60 70 80 90 100 1100
100
200
300
400
500
600
700
800
Final Exam
Mid
term
Exa
m
200-417
= +303
417
(y) Actual Ave Monthly Ice Cream Sales
300
320
370
480
560
640
720
600
400
300
200
122
Predicted Ave Monthly Ice Cream
Sales
417
417
417
417
417
417
417
417
417
417
417
417
Difference
-117
-97
-47
63
143
223
303
183
-17
-117
-217
-295
------------
============
• Now we have the difference between the actual values for Y (ice cream sales) and the mean of the values for Y (417)
10 20 30 40 50 60 70 80 90 100 1100
100
200
300
400
500
600
700
800
Final Exam
Mid
term
Exa
m
(y) Actual Ave Monthly Ice Cream Sales
300
320
370
480
560
640
720
600
400
300
200
122
Predicted Ave Monthly Ice Cream
Sales
417
417
417
417
417
417
417
417
417
417
417
417
Difference
-117
-97
-47
63
143
223
303
183
-17
-117
-217
-295
------------
============
• Now we have the difference between the actual values for Y (ice cream sales) and the mean of the values for Y (417)
10 20 30 40 50 60 70 80 90 100 1100
100
200
300
400
500
600
700
800
Final Exam
Mid
term
Exa
m
(y) Actual Ave Monthly Ice Cream Sales
300
320
370
480
560
640
720
600
400
300
200
122
Predicted Ave Monthly Ice Cream
Sales
417
417
417
417
417
417
417
417
417
417
417
417
Difference
-117
-97
-47
63
143
223
303
183
-17
-117
-217
-295
------------
============
• As we showed previously we have to square this value because if we don’t when we sum the differences they will come to zero.
• As we showed previously we have to square this value because if we don’t when we sum the differences they will come to zero.
Difference
-117
-97
-47
63
143
223
303
183
-17
-117
-217
-295
Squared
13689
9409
2209
3969
20449
49729
91809
33489
289
13689
47089
87025
SUM
= 0
SUM
= 372,844
• As we showed previously we have to square this value because if we don’t when we sum the differences they will come to zero.
Difference
-117
-97
-47
63
143
223
303
183
-17
-117
-217
-295
Squared
13689
9409
2209
3969
20449
49729
91809
33489
289
13689
47089
87025
SUM
= 0
SUM
= 372,844
• We are doing all this once again to show a visual depiction of what the total sums of squares are:
• As we showed previously we have to square this value because if we don’t when we sum the differences they will come to zero.
Difference
-117
-97
-47
63
143
223
303
183
-17
-117
-217
-295
Squared
13689
9409
2209
3969
20449
49729
91809
33489
289
13689
47089
87025
SUM
= 0
SUM
= 372,844
• We are doing all this once again to show a visual depiction of what the total sums of squares are:
Sum of Squares
df Mean Square F-ratio Significance
Total 372,844
• Now that we’ve seen a visual depiction of how we calculated total sums of squares we compare the sums of squares that are associated with error (residual) and those associated with regression.
• Now that we’ve seen a visual depiction of how we calculated total sums of squares we compare the sums of squares that are associated with error (residual) and those associated with regression.
Sum of Squares
df Mean Square F-ratio Significance
Regression Residual Total 372,844
• Now that we’ve seen a visual depiction of how we calculated total sums of squares we compare the sums of squares that are associated with error (residual) and those associated with regression.
• Let’s calculate the error or residual sums of squares now.
Sum of Squares
df Mean Square F-ratio Significance
Regression Residual Total 372,844
• The error or residual sums of squares are computed by subtracting each actual Y value from each Y predicted value.
• The error or residual sums of squares are computed by subtracting each actual Y value from each Y predicted value.• Here are the actual Y values
• The error or residual sums of squares are computed by subtracting each actual Y value from each Y predicted value.• Here are the actual Y values
10 20 30 40 50 60 70 80 90 100 1100
100
200
300
400
500
600
700
800
Final Exam
Mid
term
Exa
m
These are the actual Y values or average ice cream sales
aver
age
ice
crea
m s
ales
• The error or residual sums of squares are computed by subtracting each actual Y value from each Y predicted value.• Here are the actual Y values
10 20 30 40 50 60 70 80 90 100 1100
100
200
300
400
500
600
700
800
Final Exam
Mid
term
Exa
m
These are the actual Y values or average ice cream sales
aver
age
ice
crea
m s
ales
(y) Actual Ave Monthly Ice Cream Sales
300
320
370
480
560
640
720
600
400
300
200
122
• Here are the predicted values using the linear regression formula:
• Here are the predicted values using the linear regression formula:
10 20 30 40 50 60 70 80 90 100 1100
100
200
300
400
500
600
700
800
Final Exam
Mid
term
Exa
m
These are the ac-tual Y values or average ice cream sales
aver
age
ice
crea
m s
ales
300
320
370
480
560
640
720
600
400
300
200
122
(y) Actual Ave Monthly Ice Cream Sales
= -50.93+7.21(300) == -50.93+7.21(320) =
= -50.93+7.21(480) =
= -50.93+7.21(370) =
= -50.93+7.21(560) =
= -50.93+7.21(640) =
= -50.93+7.21(720) =
= -50.93+7.21(600) =
= -50.93+7.21(400) == -50.93+7.21(300) =
= -50.93+7.21(200) == -50.93+7.21(122) =
Predicted Ave Monthly Ice Cream
Sales
237.47
309.57
381.67
453.77
525.87
597.97
670.07
597.97
525.87
381.67
237.47
93.27
• Here are the predicted values using the linear regression formula:
300
320
370
480
560
640
720
600
400
300
200
122
(y) Actual Ave Monthly Ice Cream Sales
= -50.93+7.21(300) == -50.93+7.21(320) =
= -50.93+7.21(480) =
= -50.93+7.21(370) =
= -50.93+7.21(560) =
= -50.93+7.21(640) =
= -50.93+7.21(720) =
= -50.93+7.21(600) =
= -50.93+7.21(400) == -50.93+7.21(300) =
= -50.93+7.21(200) == -50.93+7.21(122) =
Predicted Ave Monthly Ice Cream
Sales
237.47
309.57
381.67
453.77
525.87
597.97
670.07
597.97
525.87
381.67
237.47
93.27
10 20 30 40 50 60 70 80 90 100 1100
100
200
300
400
500
600
700
800
Final Exam
Mid
term
Exa
m
aver
age
ice
crea
m s
ales
• Here are the predicted values using the linear regression formula:
300
320
370
480
560
640
720
600
400
300
200
122
(y) Actual Ave Monthly Ice Cream Sales
= -50.93+7.21(300) == -50.93+7.21(320) =
= -50.93+7.21(480) =
= -50.93+7.21(370) =
= -50.93+7.21(560) =
= -50.93+7.21(640) =
= -50.93+7.21(720) =
= -50.93+7.21(600) =
= -50.93+7.21(400) == -50.93+7.21(300) =
= -50.93+7.21(200) == -50.93+7.21(122) =
Predicted Ave Monthly Ice Cream
Sales
237.47
309.57
381.67
453.77
525.87
597.97
670.07
597.97
525.87
381.67
237.47
93.27
10 20 30 40 50 60 70 80 90 100 1100
100
200
300
400
500
600
700
800
Final Exam
Mid
term
Exa
m
aver
age
ice
crea
m s
ales
• From these points and the linear regression formula a line can be drawn
• From these points and the linear regression formula a line can be drawn
10 20 30 40 50 60 70 80 90 100 1100
100
200
300
400
500
600
700
800
Final Exam
Mid
term
Exa
m
aver
age
ice
crea
m s
ales
• From these points and the linear regression formula a line can be drawn
10 20 30 40 50 60 70 80 90 100 1100
100
200
300
400
500
600
700
800
Final Exam
Mid
term
Exa
m
aver
age
ice
crea
m s
ales
• The difference between each actual value (orange) and the predicted value (green line) is what is called error or residual. The closer these two values are to each other the smaller the error. The farther these two values are from each other the larger the error and the weaker the predictive power of the regression line.
• The difference between each actual value (orange) and the predicted value (green line) is what is called error or residual. The closer these two values are to each other the smaller the error. The farther these two values are from each other the larger the error and the weaker the predictive power of the regression line.
10 20 30 40 50 60 70 80 90 100 1100
100
200
300
400
500
600
700
800
Final Exam
Mid
term
Exa
m
aver
age
ice
crea
m s
ales
Difference
Difference
• Let’s subtract the orange actual values and the green line predicted values:
• Let’s subtract the orange actual values and the green line predicted values:
10 20 30 40 50 60 70 80 90 100 1100
100
200
300
400
500
600
700
800
Final Exam
Mid
term
Exa
m
aver
age
ice
crea
m s
ales
(y) Actual Ave Monthly Ice Cream Sales
300
320
370
480
560
640
720
600
400
300
200
122
Predicted Ave Monthly Ice Cream
Sales
237.47
309.57
381.67
453.77
525.87
597.97
670.07
597.97
525.87
381.67
237.47
93.27
Difference
62.53
10.43
-11.67
26.23
34.13
42.03
49.93
2.03
-125.87
-81.67
-37.47
28.73
------------
============
+28.73122
93
• Let’s subtract the orange actual values and the green line predicted values:
10 20 30 40 50 60 70 80 90 100 1100
100
200
300
400
500
600
700
800
Final Exam
Mid
term
Exa
m
aver
age
ice
crea
m s
ales
(y) Actual Ave Monthly Ice Cream Sales
300
320
370
480
560
640
720
600
400
300
200
122
Predicted Ave Monthly Ice Cream
Sales
237.47
309.57
381.67
453.77
525.87
597.97
670.07
597.97
525.87
381.67
237.47
93.27
Difference
62.53
10.43
-11.67
26.23
34.13
42.03
49.93
2.03
-125.87
-81.67
-37.47
28.73
------------
============
-125.87
525
400
• Let’s subtract the orange actual values and the green line predicted values:
• And so on…
10 20 30 40 50 60 70 80 90 100 1100
100
200
300
400
500
600
700
800
Final Exam
Mid
term
Exa
m
aver
age
ice
crea
m s
ales
(y) Actual Ave Monthly Ice Cream Sales
300
320
370
480
560
640
720
600
400
300
200
122
Predicted Ave Monthly Ice Cream
Sales
237.47
309.57
381.67
453.77
525.87
597.97
670.07
597.97
525.87
381.67
237.47
93.27
Difference
62.53
10.43
-11.67
26.23
34.13
42.03
49.93
2.03
-125.87
-81.67
-37.47
28.73
------------
============
-125.87
525
400
• We then square those difference (deviations)
• We then square those difference (deviations)
(y) Actual Ave Monthly Ice Cream Sales
300
320
370
480
560
640
720
600
400
300
200
122
Predicted Ave Monthly Ice Cream
Sales
237.47
309.57
381.67
453.77
525.87
597.97
670.07
597.97
525.87
381.67
237.47
93.27
Difference
62.53
10.43
-11.67
26.23
34.13
42.03
49.93
2.03
-125.87
-81.67
-37.47
28.73
------------
============
Squared
3910.00
108.78
136.19
688.01
1164.86
1766.52
2493.00
4.12
15843.26
6669.99
1404.00
825.41
• We then square those difference (deviations) and sum them up
(y) Actual Ave Monthly Ice Cream Sales
300
320
370
480
560
640
720
600
400
300
200
122
Predicted Ave Monthly Ice Cream
Sales
237.47
309.57
381.67
453.77
525.87
597.97
670.07
597.97
525.87
381.67
237.47
93.27
Difference
62.53
10.43
-11.67
26.23
34.13
42.03
49.93
2.03
-125.87
-81.67
-37.47
28.73
------------
============
Squared
3910.00
108.78
136.19
688.01
1164.86
1766.52
2493.00
4.12
15843.26
6669.99
1404.00
825.41
Sum up
= 35,014
Sum of Squares
df Mean Square F-ratio Significance
Regression Residual 35,014 Total 372,844
• We will now calculate the regression sums of squares.
• We will now calculate the regression sums of squares.
• Our hope is that this value will be much bigger than the residual (35,014).
Sum of Squares
df Mean Square F-ratio Significance
Regression Residual 35,014 Total 372,844
• The regression sums of squares is calculated by subtracting the predicted values from the mean.
• The regression sums of squares is calculated by subtracting the predicted values from the mean.• Let’s see what this looks like visually. The green line is the
predicted values for Y or the regression line.
• The regression sums of squares is calculated by subtracting the predicted values from the mean.• Let’s see what this looks like visually. The green line is the
predicted values for Y or the regression line.
10 20 30 40 50 60 70 80 90 100 1100
100
200
300
400
500
600
700
800
Final Exam
Mid
term
Exa
m
aver
age
ice
crea
m s
ales
• The regression sums of squares is calculated by subtracting the predicted values from the mean.• Let’s see what this looks like visually. The green line is the
predicted values for Y or the regression line.
10 20 30 40 50 60 70 80 90 100 1100
100
200
300
400
500
600
700
800
Final Exam
Mid
term
Exa
m
aver
age
ice
crea
m s
ales
• The regression sums of squares is calculated by subtracting the predicted values from the mean.• Let’s see what this looks like visually. The green line is the
predicted values for Y or the regression line.
10 20 30 40 50 60 70 80 90 100 1100
100
200
300
400
500
600
700
800
Final Exam
Mid
term
Exa
m
aver
age
ice
crea
m s
ales
The blue line is the mean (417)
which is the best predictor absent
anything else.
• You can probably already tell that it will be bigger because a simple way to calculate it is to subtract the residual (35,014) from the total (372,844).
10 20 30 40 50 60 70 80 90 100 1100
100
200
300
400
500
600
700
800
Final Exam
Mid
term
Exa
m
aver
age
ice
crea
m s
ales
• You can probably already tell that it will be bigger because a simple way to calculate it is to subtract the residual (35,014) from the total (372,844).• However, we will calculate it the long way so you can see what
is happening.
10 20 30 40 50 60 70 80 90 100 1100
100
200
300
400
500
600
700
800
Final Exam
Mid
term
Exa
m
aver
age
ice
crea
m s
ales
• We subtract each predicted value from the mean of the actual Y values
(y) Actual Ave Monthly Ice Cream Sales
237.47
309.57
381.67
453.77
525.87
597.97
670.07
597.97
525.87
381.67
237.47
93.27
Mean Monthly Ice Cream Sales
417.7
417.7
417.7
417.7
417.7
417.7
417.7
417.7
417.7
417.7
417.7
417.7
Difference
-180.2
-108.1
-36.0
36.1
108.2
180.3
252.4
180.3
108.2
-36.0
-180.2
-324.4
------------
============
• We subtract each predicted value from the mean of the actual Y values
10 20 30 40 50 60 70 80 90 100 1100
100
200
300
400
500
600
700
800
Final Exam
Mid
term
Exa
m
aver
age
ice
crea
m s
ales
• We subtract each predicted value from the mean of the actual Y values
10 20 30 40 50 60 70 80 90 100 1100
100
200
300
400
500
600
700
800
Final Exam
Mid
term
Exa
m
aver
age
ice
crea
m s
ales
(y) Actual Ave Monthly Ice Cream Sales
237.47
309.57
381.67
453.77
525.87
597.97
670.07
597.97
525.87
381.67
237.47
93.27
Mean Monthly Ice Cream Sales
417.7
417.7
417.7
417.7
417.7
417.7
417.7
417.7
417.7
417.7
417.7
417.7
Difference
-180.2
-108.1
-36.0
36.1
108.2
180.3
252.4
180.3
108.2
-36.0
-180.2
-324.4
------------
============
93- 417- 324
• We subtract each predicted value from the mean of the actual Y values
10 20 30 40 50 60 70 80 90 100 1100
100
200
300
400
500
600
700
800
Final Exam
Mid
term
Exa
m
aver
age
ice
crea
m s
ales
(y) Actual Ave Monthly Ice Cream Sales
237.47
309.57
381.67
453.77
525.87
597.97
670.07
597.97
525.87
381.67
237.47
93.27
Mean Monthly Ice Cream Sales
417.7
417.7
417.7
417.7
417.7
417.7
417.7
417.7
417.7
417.7
417.7
417.7
Difference
-180.2
-108.1
-36.0
36.1
108.2
180.3
252.4
180.3
108.2
-36.0
-180.2
-324.4
------------
============
670- 417+252
• Then we square the differences (or deviations)
• Then we square the differences (or deviations)
(y) Actual Ave Monthly Ice Cream Sales
237.47
309.57
381.67
453.77
525.87
597.97
670.07
597.97
525.87
381.67
237.47
93.27
Mean Monthly Ice Cream Sales
417.7
417.7
417.7
417.7
417.7
417.7
417.7
417.7
417.7
417.7
417.7
417.7
Difference
-180.2
-108.1
-36.0
36.1
108.2
180.3
252.4
180.3
108.2
-36.0
-180.2
-324.4
------------
============
Squared
32470.8
11684.9
1295.76
1303.45
11708
32509.3
63707.4
32509.3
11708
1295.76
32470.8
105233
• Then we square the differences (or deviations) and sum them up
(y) Actual Ave Monthly Ice Cream Sales
237.47
309.57
381.67
453.77
525.87
597.97
670.07
597.97
525.87
381.67
237.47
93.27
Mean Monthly Ice Cream Sales
417.7
417.7
417.7
417.7
417.7
417.7
417.7
417.7
417.7
417.7
417.7
417.7
Difference
-180.2
-108.1
-36.0
36.1
108.2
180.3
252.4
180.3
108.2
-36.0
-180.2
-324.4
------------
============
Squared
32470.8
11684.9
1295.76
1303.45
11708
32509.3
63707.4
32509.3
11708
1295.76
32470.8
105233
Sum up
= 337,830
• Then we square the differences (or deviations) and sum them up
Sum of Squares
df Mean Square F-ratio Significance
Regression 337,830 Residual 35,014 Total 372,844
• Now we have all of the information to test for significance
• Now we have all of the information to test for significance
Sum of Squares
df Mean Square F-ratio Significance
Regression 337,830 Residual 35,014 Total 372,844
• The degrees of freedom (df) for the regression are the number of parameters that are being estimated which in this case is the Y intercept and the slope in this equation minus
• The degrees of freedom (df) for the regression are the number of parameters that are being estimated which in this case is the Y intercept and the slope in this equation minus • 2 parameters -1 = 1
Sum of Squares
df Mean Square F-ratio Significance
Regression 337,830 1
Residual 35,014 Total 372,844
• The degrees of freedom for residual is the number of cases (12) minus the number of parameters (2)
• The degrees of freedom for residual is the number of cases (12) minus the number of parameters (2)• 12 months – 2 parameters (slope / y intercept) = 10
• The degrees of freedom for residual is the number of cases (12) minus the number of parameters (2)• 12 months – 2 parameters (slope / y intercept) = 10
Sum of Squares
df Mean Square F-ratio Significance
Regression 337,830 1
Residual 35,014 10 Total 372,844
• We now have the information we need to calculate the Mean Square values. They are calculated by dividing the sums of squares by the degrees of freedom.
• We now have the information we need to calculate the Mean Square values. They are calculated by dividing the sums of squares by the degrees of freedom.
Sum of Squares
df Mean Square F-ratio Significance
Regression 337,830 1 =337,830
Residual 35,014 10 =3,501
Total 372,844
• The F-ratio is computed by dividing the Regression Mean Square by the Residual Mean Square
• The F-ratio is computed by dividing the Regression Mean Square by the Residual Mean Square
• 337,830 / 3,501 = 96.5
• The F-ratio is computed by dividing the Regression Mean Square by the Residual Mean Square
• 337,830 / 3,501 = 96.5
Sum of Squares
df Mean Square F-ratio Significance
Regression 337,830 1 337,830 96.5
Residual 35,014 10 3,501
Total 372,844
• With this information we can turn to the F-distribution table to determine the significance value.
• With this information we can turn to the F-distribution table to determine the significance value.
Sum of Squares
df Mean Square F-ratio Significance
Regression 337,830 1 337,830 96.5
Residual 35,014 10 3,501
Total 372,844
Sum of Squares
df Mean Square F-ratio Significance
Regression 337,830 1 337,830 96.5
Residual 35,014 10 3,501
Total 372,844
• The regression degrees of freedom (1) is represented by the columns below:
Sum of Squares
df Mean Square F-ratio Significance
Regression 337,830 1 337,830 96.5
Residual 35,014 10 3,501
Total 372,844
• The regression degrees of freedom (1) is represented by the columns below:
Sum of Squares
df Mean Square F-ratio Significance
Regression 337,830 1 337,830 96.5
Residual 35,014 10 3,501
Total 372,844
Sum of Squares
df Mean Square F-ratio Significance
Regression 337,830 1 337,830 96.5
Residual 35,014 10 3,501
Total 372,844
• The residual degrees of freedom (10) is represented by the rows below:
Sum of Squares
df Mean Square F-ratio Significance
Regression 337,830 1 337,830 96.5
Residual 35,014 10 3,501
Total 372,844
• The residual degrees of freedom (10) is represented by the rows below:
Sum of Squares
df Mean Square F-ratio Significance
Regression 337,830 1 337,830 96.5
Residual 35,014 10 3,501
Total 372,844
Sum of Squares
df Mean Square F-ratio Significance
Regression 337,830 1 337,830 96.5
Residual 35,014 10 3,501
Total 372,844
• Put them together and we have found the critical F value at the .05 alpha level to be 4.96.
Sum of Squares
df Mean Square F-ratio Significance
Regression 337,830 1 337,830 96.5
Residual 35,014 10 3,501
Total 372,844
• Put them together and we have found the critical F value at the .05 alpha level to be 4.96.
Sum of Squares
df Mean Square F-ratio Significance
Regression 337,830 1 337,830 96.5
Residual 35,014 10 3,501
Total 372,844
• Because the F-ratio (96.5) exceeds the F-critical (4.96) we will reject the null hypothesis and indicate that temperature is a statistically significant predictor of ice cream sales
In Summary
In Summary
• The whole point of this demonstration was to
In Summary
• The whole point of this demonstration was to (1) explain that linear regression is used to predict the value of one variable (ice cream sales) based on another variable (temperature)
In Summary
• The whole point of this demonstration was to (1) explain that linear regression is used to predict the value of one variable (ice cream sales) based on another variable (temperature)(2) show that the total variance in Y can be partitioned into regression (prediction power) and residual (error)
In Summary
• The whole point of this demonstration was to (1) explain that linear regression is used to predict the value of one variable (ice cream sales) based on another variable (temperature)(2) show that the total variance in Y can be partitioned into regression (prediction power) and residual (error) (3) show how this can be used to test whether the prediction is better than by chance.
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