multiple linear regression
Post on 20-Jun-2015
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What is a Multiple Linear Regression?
Welcome to this learning module onMultiple Linear Regression
In this presentation we will cover the following aspects of Multiple Regression:
In this presentation we will cover the following aspects of Multiple Regression:- Connection to Partial Correlation and ANCOVA
In this presentation we will cover the following aspects of Multiple Regression:- Connection to Partial Correlation and ANCOVA- Unique contribution of each variable
In this presentation we will cover the following aspects of Multiple Regression:- Connection to Partial Correlation and ANCOVA- Unique contribution of each variable- Contribution of all variables at the same time
In this presentation we will cover the following aspects of Multiple Regression:- Connection to Partial Correlation and ANCOVA- Unique contribution of each variable- Contribution of all variables at the same time- Type of data multiple regression can handle
In this presentation we will cover the following aspects of Multiple Regression:- Connection to Partial Correlation and ANCOVA- Unique contribution of each variable- Contribution of all variables at the same time- Type of data multiple regression can handle- Types of relationships multiple regression
describes
In this presentation we will cover the following aspects of Multiple Regression:- Connection to Partial Correlation and ANCOVA- Unique contribution of each variable- Contribution of all variables at the same time- Type of data multiple regression can handle- Types of relationships multiple regression
describes
In this presentation we will cover the following aspects of Multiple Regression:- Connection to Partial Correlation and ANCOVA- Unique contribution of each variable- Contribution of all variables at the same time- Type of data multiple regression can handle- Types of relationships multiple regression
describes
In this presentation we will cover the concept of Partial Correlation.
In this presentation we will cover the following aspects of Multiple Regression:- Connection to Partial Correlation and ANCOVA- Unique contribution of each variable- Contribution of all variables at the same time- Type of data multiple regression can handle- Types of relationships multiple regression
describes
After going through this presentation look at the presentation on Analysis of Covariance and consider what multiple regression and ANCOVA have in common.
In this presentation we will cover the following aspects of Multiple Regression:- Connection to Partial Correlation and ANCOVA- Unique contribution of each variable- Contribution of all variables at the same time- Type of data multiple regression can handle- Types of relationships multiple regression
describes
What is a Partial Correlation?
Partial correlation estimates the relationship between two variables while removing the influence of a third variable from the relationship.
Like in the example that follows,
Like in the example that follows, a Pearson Correlation between height and weight would yield a .825 correlation.
Like in the example that follows, a Pearson Correlation between height and weight would yield a .825 correlation. We might then control for gender (because we think being female or male has an effect on the relationship between height and weight).
However, when controlling for gender the correlation between height and weight drops to .770.
However, when controlling for gender the correlation between height and weight drops to .770.
Individual Height (inches) Weight (pounds) Sex (1 – male, 2 – female)A 73 240 1B 70 210 1C 69 180 1D 68 160 1E 70 150 2F 68 140 2G 67 135 2H 62 120 2
However, when controlling for gender the correlation between height and weight drops to .770.
Individual Height (inches) Weight (pounds) Sex (1 – male, 2 – female)A 73 240 1B 70 210 1C 69 180 1D 68 160 1E 70 150 2F 68 140 2G 67 135 2H 62 120 2
However, when controlling for gender the correlation between height and weight drops to .770.
Individual Height (inches) Weight (pounds) Sex (1 – male, 2 – female)A 73 240 1B 70 210 1C 69 180 1D 68 160 1E 70 150 2F 68 140 2G 67 135 2H 62 120 2
&
However, when controlling for gender the correlation between height and weight drops to .770.
Individual Height (inches) Weight (pounds) Sex (1 – male, 2 – female)A 73 240 1B 70 210 1C 69 180 1D 68 160 1E 70 150 2F 68 140 2G 67 135 2H 62 120 2
&
However, when controlling for gender the correlation between height and weight drops to .770.
Individual Height (inches) Weight (pounds) Sex (1 – male, 2 – female)A 73 240 1B 70 210 1C 69 180 1D 68 160 1E 70 150 2F 68 140 2G 67 135 2H 62 120 2
& controlling for
However, when controlling for gender the correlation between height and weight drops to .770.
Individual Height (inches) Weight (pounds) Sex (1 – male, 2 – female)A 73 240 1B 70 210 1C 69 180 1D 68 160 1E 70 150 2F 68 140 2G 67 135 2H 62 120 2
& controlling for
However, when controlling for gender the correlation between height and weight drops to .770.
Individual Height (inches) Weight (pounds) Sex (1 – male, 2 – female)A 73 240 1B 70 210 1C 69 180 1D 68 160 1E 70 150 2F 68 140 2G 67 135 2H 62 120 2
& controlling for = .770
This is very helpful because we may think two variables (height and weight) are highly correlated but we can determine if that correlation holds when we take out the effect of a third variable (gender).
While in partial correlation, only two variables are correlated while holding a third variable constant, in multiple regression several variables are grouped together to predict an outcome variable.
While in partial correlation, only two variables are correlated while holding a third variable constant, in multiple regression several variables are grouped together to predict an outcome variable
Independent or Predictor Variables
While in partial correlation, only two variables are correlated while holding a third variable constant, in multiple regression several variables are grouped together to predict an outcome variable
Height
Independent or Predictor Variables
While in partial correlation, only two variables are correlated while holding a third variable constant, in multiple regression several variables are grouped together to predict an outcome variable
Height
Gender
Independent or Predictor Variables
While in partial correlation, only two variables are correlated while holding a third variable constant, in multiple regression several variables are grouped together to predict an outcome variable
Height
Gender
Age
Independent or Predictor Variables
While in partial correlation, only two variables are correlated while holding a third variable constant, in multiple regression several variables are grouped together to predict an outcome variable
Height
Gender
Age
Soda Drinking
Independent or Predictor Variables
While in partial correlation, only two variables are correlated while holding a third variable constant, in multiple regression several variables are grouped together to predict an outcome variable
Height
Gender
Age
Soda Drinking
Exercise
Independent or Predictor Variables
While in partial correlation, only two variables are correlated while holding a third variable constant, in multiple regression several variables are grouped together to predict an outcome variable
Height
Gender
Age
Soda Drinking
Exercise
Independent or Predictor Variables
Dependent, Response or Outcome Variable
While in partial correlation, only two variables are correlated while holding a third variable constant, in multiple regression several variables are grouped together to predict an outcome variable
Height
Gender
Age
Soda Drinking
Exercise
Independent or Predictor Variables
Dependent, Response or Outcome Variable
Weight
While in partial correlation, only two variables are correlated while holding a third variable constant, in multiple regression several variables are grouped together to predict an outcome variable
Height
Gender
Age
Soda Drinking
Exercise
Independent or Predictor Variables
Dependent, Response or Outcome Variable
Weight
all have an influence on . . .
While in partial correlation, only two variables are correlated while holding a third variable constant, in multiple regression several variables are grouped together to predict an outcome variable
Height
Gender
Age
Soda Drinking
Exercise
Independent or Predictor Variables
Dependent, Response or Outcome Variable
Weight
all have an influence on . . .
While in partial correlation, only two variables are correlated while holding a third variable constant, in multiple regression several variables are grouped together to predict an outcome variable
Height
Gender
Age
Soda Drinking
Exercise
Independent or Predictor Variables
Dependent, Response or Outcome Variable
Weight
all have an influence on . . .
While in partial correlation, only two variables are correlated while holding a third variable constant, in multiple regression several variables are grouped together to predict an outcome variable
Height
Gender
Age
Soda Drinking
Exercise
Independent or Predictor Variables
Dependent, Response or Outcome Variable
Weight
all have an influence on . . .
While in partial correlation, only two variables are correlated while holding a third variable constant, in multiple regression several variables are grouped together to predict an outcome variable
Height
Gender
Age
Soda Drinking
Exercise
Independent or Predictor Variables
Dependent, Response or Outcome Variable
Weight
all have an influence on . . .
While in partial correlation, only two variables are correlated while holding a third variable constant, in multiple regression several variables are grouped together to predict an outcome variable
Height
Weight
Gender
Age
Soda Drinking
Exercise
Independent or Predictor Variables
Dependent, Response or Outcome Variable
all have an influence on . . .
In this presentation we will cover the following aspects of Multiple Regression:- Connection to Partial Correlation and ANCOVA- Unique contribution of each variable- Contribution of all variables at the same time- Type of data multiple regression can handle- Types of relationships multiple regression
describes
Essentially, the group of predictors are all covariates to each other.
Height
Weight
Gender
Age
Soda Drinking
Exercise
Independent or Predictor Variables
Dependent, Response or Outcome Variable
all have an influence on . . .
Meaning,
Height
Weight
Gender
Age
Soda Drinking
Exercise
Independent or Predictor Variables
Dependent, Response or Outcome Variable
all have an influence on . . .
Meaning, for example,
Height
Weight
Gender
Age
Soda Drinking
Exercise
Independent or Predictor Variables
Dependent, Response or Outcome Variable
all have an influence on . . .
Meaning, for example, that it is possible to identify the unique prediction power of height on weight
Height
Weight
Gender
Age
Soda Drinking
Exercise
Independent or Predictor Variables
Dependent, Response or Outcome Variable
all have an influence on . . .
Meaning, for example, that it is possible to identify the unique prediction power of height on weight
Gender
Age
Soda Drinking
Exercise
Independent or Predictor Variables
Dependent, Response or Outcome Variable
all have an influence on . . . Height
Weight
Meaning, for example, that it is possible to identify the unique prediction power of height on weight after you’ve taken out the influence of all of the other predictors.
Gender
Age
Soda Drinking
Exercise
Independent or Predictor Variables
Dependent, Response or Outcome Variable
all have an influence on . . . Height
Weight
Meaning, for example, that it is possible to identify the unique prediction power of height on weight after you’ve taken out the influence of all of the other predictors.
Gender
Age
Soda Drinking
Exercise
Independent or Predictor Variables
Dependent, Response or Outcome Variable
all have an influence on . . . Height
Weight
For example,
Height
Weight
Gender
Age
Soda Drinking
Exercise
Independent or Predictor Variables
Dependent, Response or Outcome Variable
all have an influence on . . .
For example, here is the correlation between Height and Weight without controlling for all of the other variables.
Gender
Age
Soda Drinking
Exercise
Independent or Predictor Variables
Dependent, Response or Outcome Variable
all have an influence on . . . Height
Weight
For example, here is the correlation between Height and Weight without controlling for all of the other variables.
Gender
Age
Soda Drinking
Exercise
Independent or Predictor Variables
Dependent, Response or Outcome Variable
all have an influence on . . . Height
Weight
Correlation = .825
However,
Height
Weight
Gender
Age
Soda Drinking
Exercise
Independent or Predictor Variables
Dependent, Response or Outcome Variable
all have an influence on . . .
However, here is the correlation between Height and Weight after taking out the effect of all of the other variables.
Height
Weight
Gender
Age
Soda Drinking
Exercise
Independent or Predictor Variables
Dependent, Response or Outcome Variable
all have an influence on . . .
However, here is the correlation between Height and Weight after taking out the effect of all of the other variables.
Gender
Age
Soda Drinking
Exercise
Independent or Predictor Variables
Dependent, Response or Outcome Variable
all have an influence on . . .
Weight
Height
However, here is the correlation between Height and Weight after taking out the effect of all of the other variables.
Gender
Age
Soda Drinking
Exercise
Independent or Predictor Variables
Dependent, Response or Outcome Variable
all have an influence on . . . Height
Weight
Correlation = .601
However, here is the correlation between Height and Weight after taking out the effect of all of the other variables.
Gender
Age
Soda Drinking
Exercise
Independent or Predictor Variables
Dependent, Response or Outcome Variable
all have an influence on . . .
Weight
Correlation = .601
So, after eliminating the effect of gender, age, soda, and exercise on weight, the
unique correlation that height shares with weight is .601.
Height
Even though we were only correlating height and weight when we computed a correlation of .825,
Even though we were only correlating height and weight when we computed a correlation of .825, the other four variables still had an influence on weight.
Even though we were only correlating height and weight when we computed a correlation of .825, the other four variables still had an influence on weight. However, that influence was not accounted for and remained hidden.
With multiple regression we can control for these four variables and account for their influence
With multiple regression we can control for these four variables and account for their influence thus calculating the unique contribution height makes on weight without their influence being present.
We can do the same for any of these other variables. Like the relationship between Gender and Weight.
We can do the same for any of these other variables. Like the relationship between Gender and Weight.
Height
Age
Soda Drinking
Exercise
Independent or Predictor Variables
Dependent, Response or Outcome Variable
all have an influence on . . .
Weight
Gender
We can do the same for any of these other variables. Like the relationship between Gender and Weight.
Height
Age
Soda Drinking
Exercise
Independent or Predictor Variables
Dependent, Response or Outcome Variable
all have an influence on . . .
Weight
Gender Correlation = .701
But when you take out the influence of the other variables the correlation drops from .701 to .582.
BEFORE
Height
Age
Soda Drinking
Exercise
Independent or Predictor Variables
Dependent, Response or Outcome Variable
all have an influence on . . .
Weight
Gender Correlation = .701
AFTER
AFTER
Height
Age
Soda Drinking
Exercise
Independent or Predictor Variables
Dependent, Response or Outcome Variable
all have an influence on . . .
Weight
Gender
AFTER
Height
Age
Soda Drinking
Exercise
Independent or Predictor Variables
Dependent, Response or Outcome Variable
all have an influence on . . .
Weight
Gender Correlation = .582
Here is the correlation between age and weight before you take out the effect of the other variables:
Here is the correlation between age and weight before you take out the effect of the other variables:
Height
Gender
Soda Drinking
Exercise
Independent or Predictor Variables
Dependent, Response or Outcome Variable
all have an influence on . . .
Weight Age
Here is the correlation between age and weight before you take out the effect of the other variables:
Height
Gender
Soda Drinking
Exercise
Independent or Predictor Variables
Dependent, Response or Outcome Variable
all have an influence on . . .
Weight Age Correlation = .435
The correlation drops from .435 to .385 after taking out the influence of the other variables:
The correlation drops from .435 to .385 after taking out the influence of the other variables:
Height
Gender
Soda Drinking
Exercise
Independent or Predictor Variables
Dependent, Response or Outcome Variable
all have an influence on . . .
Weight Age
The correlation drops from .435 to .385 after taking out the influence of the other variables:
Height
Gender
Soda Drinking
Exercise
Independent or Predictor Variables
Dependent, Response or Outcome Variable
all have an influence on . . .
Weight Age Correlation = .385
In this presentation we will cover the following aspects of Multiple Regression:- Connection to Partial Correlation and ANCOVA- Unique contribution of each variable- Contribution of all variables at the same time- Type of data multiple regression can handle- Types of relationships multiple regression
describes
Beyond estimating the unique power of each predictor,
Beyond estimating the unique power of each predictor, multiple regression also estimates the combined power of the group of predictors.
Beyond estimating the unique power of each predictor, multiple regression also estimates the combined power of the group of predictors.
Beyond estimating the unique power of each predictor, multiple regression also estimates the combined power of the group of predictors.
Height
Weight
Gender
Age
Soda Drinking
Exercise
Independent or Predictor Variables
Dependent, Response or Outcome Variable
Beyond estimating the unique power of each predictor, multiple regression also estimates the combined power of the group of predictors.
Height
Weight
Gender
Age
Soda Drinking
Exercise
Independent or Predictor Variables
Dependent, Response or Outcome Variable
Combined Correlation
= .982
In this presentation we will cover the following aspects of Multiple Regression:- Connection to Partial Correlation and ANCOVA- Unique contribution of each variable- Contribution of all variables at the same time- Type of data multiple regression can handle- Types of relationships multiple regression
describes
Multiple regression can estimate the effects of continuous and categorical variables in the same model.
Multiple regression can estimate the effects of continuous and categorical variables in the same model.
Height
Independent or Predictor Variables
Dependent, Response or Outcome Variable
Age
Soda Drinking
Exercise
Gender
Weight
Multiple regression can estimate the effects of continuous and categorical variables in the same model.
Height
Independent or Predictor Variables
Dependent, Response or Outcome Variable
Age
Soda Drinking
Exercise
Gender
Weight
Height is represented by continuous data – because height can take on any value between two points in inches or centimeters.
Multiple regression can estimate the effects of continuous and categorical variables in the same model.
Multiple regression can estimate the effects of continuous and categorical variables in the same model.
Height
Age
Soda Drinking
Exercise
Independent or Predictor Variables
Dependent, Response or Outcome Variable
all have an influence on . . .
Weight
Gender Gender is a represented by categorical data – because gender can take on two values (female or male)
In this presentation we will cover the following aspects of Multiple Regression:- Connection to Partial Correlation and ANCOVA- Unique contribution of each variable- Contribution of all variables at the same time- Type of data multiple regression can handle- Types of relationships multiple regression
describes
Multiple regression can describe or estimate linear relationships like average monthly temperature and ice cream sales:
Multiple regression can describe or estimate linear relationships like average monthly temperature and ice cream sales:
0 20 40 60 80 100 1200
100
200
300
400
500
600
700
Average Monthly Ice Cream Sales
Ave
Mon
thly
Tem
pera
ture
Multiple regression can describe or estimate linear relationships like average monthly temperature and ice cream sales:
0 20 40 60 80 100 1200
100
200
300
400
500
600
700
Average Monthly Ice Cream Sales
Ave
Mon
thly
Tem
pera
ture
Linea
r Rela
tionsh
ip
It can also describe or estimate curvilinear relationships.
For example,
For example, what if in our fantasy world the temperature reached 100 degrees and then 120 degrees. Let’s say with such extreme temperatures ice cream sales actually dip as consumers seek out products like electrolyte-enhanced drinks or slushies.
Then the relationship might look like this:
Then the relationship might look like this:
0 20 40 60 80 100 1200
100
200
300
400
500
600
700
Average Monthly Ice Cream Sales
Ave
Mon
thly
Tem
pera
ture
Then the relationship might look like this:
0 20 40 60 80 100 1200
100
200
300
400
500
600
700
Average Monthly Ice Cream Sales
Ave
Mon
thly
Tem
pera
ture
This is an example of a Curvilinear
Relationship
In summary,
In summary, Multiple Regression is like single linear regression but instead of determining the predictive power of one variable (temperature) on another variable (ice cream sales) we consider the predictive power of other variables (such as socio-economic status or age).
With multiple regression you can estimate the predictive power of many variables on a certain outcome,
With multiple regression you can estimate the predictive power of many variables on a certain outcome, as well as the unique influence each single variable makes on that outcome after taking out the influence of all of the other variables.
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