# Difference between revisions of "AP Statistics Curriculum 2007 GLM MultLin"

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==[[AP_Statistics_Curriculum_2007 | General Advance-Placement (AP) Statistics Curriculum]] - Multiple Linear Regression == | ==[[AP_Statistics_Curriculum_2007 | General Advance-Placement (AP) Statistics Curriculum]] - Multiple Linear Regression == | ||

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+ | In the previous sections we saw how to study the relations in bivariate designs. Now we extend that to any finite number of varaibles (mulitvariate case). | ||

=== Multiple Linear Regression === | === Multiple Linear Regression === | ||

− | + | We are interested in determining the '''linear regression''', as a model, of the relationship between one '''dependent''' variable ''Y'' and many '''independent''' variables ''X''<sub>''i''</sub>, ''i'' = 1, ..., ''p''. The multilinear regression model can be written as | |

− | < | + | |

+ | : <math>Y = \beta_0 + \beta_1 X_1 + \beta_2 X_2 + \cdots +\beta_p X_p + \varepsilon</math>, where <math>\varepsilon</math> is the error term. | ||

+ | |||

+ | The coefficient <math>\beta_0</math> is the intercept ("constant" term) and <math>\beta_i</math>s are the respective parameters of the '' p'' independent variables. There are ''p+1'' parameters to be estimated in the multilinear regression. | ||

+ | |||

+ | * Multilinear vs. non-linear regression: This multilinear regression method is "linear" because the relation of the response (the dependent variable <math>Y</math>) to the independent variables is assumed to be a [http://en.wikipedia.org/wiki/Linear_function linear function] of the parameters <math>\beta_i</math>. Note that multilinear regression is a linear modeling technique '''not''' because is that the graph of <math>Y = \beta_{0}+\beta x </math> is a straight line '''nor''' because <math>Y</math> is a linear function of the ''X'' variables. But the "linear" terms refers to the fact that <math>Y</math> can be considered a linear function of the parameters ( <math>\beta_i</math>), even though it is not a linear function of <math>X</math>. Thus, any model like | ||

+ | |||

+ | : <math>Y = \beta_o + \beta_1 x + \beta_2 x^2 + \varepsilon</math> | ||

+ | |||

+ | is still one of '''linear''' regression, that is, linear in <math>x</math> and <math>x^2</math> respectively, even though the graph on <math>x</math> by itself is not a straight line. | ||

===Approach=== | ===Approach=== |

## Revision as of 20:30, 18 February 2008

## Contents

## General Advance-Placement (AP) Statistics Curriculum - Multiple Linear Regression

In the previous sections we saw how to study the relations in bivariate designs. Now we extend that to any finite number of varaibles (mulitvariate case).

### Multiple Linear Regression

We are interested in determining the **linear regression**, as a model, of the relationship between one **dependent** variable *Y* and many **independent** variables *X*_{i}, *i* = 1, ..., *p*. The multilinear regression model can be written as

\[Y = \beta_0 + \beta_1 X_1 + \beta_2 X_2 + \cdots +\beta_p X_p + \varepsilon\], where \(\varepsilon\) is the error term.

The coefficient \(\beta_0\) is the intercept ("constant" term) and \(\beta_i\)s are the respective parameters of the * p* independent variables. There are *p+1* parameters to be estimated in the multilinear regression.

- Multilinear vs. non-linear regression: This multilinear regression method is "linear" because the relation of the response (the dependent variable \(Y\)) to the independent variables is assumed to be a linear function of the parameters \(\beta_i\). Note that multilinear regression is a linear modeling technique
**not**because is that the graph of \(Y = \beta_{0}+\beta x \) is a straight line**nor**because \(Y\) is a linear function of the*X*variables. But the "linear" terms refers to the fact that \(Y\) can be considered a linear function of the parameters ( \(\beta_i\)), even though it is not a linear function of \(X\). Thus, any model like

\[Y = \beta_o + \beta_1 x + \beta_2 x^2 + \varepsilon\]

is still one of **linear** regression, that is, linear in \(x\) and \(x^2\) respectively, even though the graph on \(x\) by itself is not a straight line.

### Approach

Models & strategies for solving the problem, data understanding & inference.

- TBD

### Model Validation

Checking/affirming underlying assumptions.

- TBD

### Computational Resources: Internet-based SOCR Tools

- TBD

### Examples

Computer simulations and real observed data.

- TBD

### Hands-on activities

Step-by-step practice problems.

- TBD

### References

- TBD

- SOCR Home page: http://www.socr.ucla.edu

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