# AP Statistics Curriculum 2007 GLM MultLin

(Difference between revisions)
 Revision as of 19:09, 14 June 2007 (view source)IvoDinov (Talk | contribs)← Older edit Revision as of 00:24, 19 February 2008 (view source)IvoDinov (Talk | contribs) Newer edit → Line 1: Line 1: ==[[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 == + + 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 === - Example on how to attach images to Wiki documents in included below (this needs to be replaced by an appropriate figure for this section)! + 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 -
[[Image:AP_Statistics_Curriculum_2007_IntroVar_Dinov_061407_Fig1.png|500px]]
+ + : $Y = \beta_0 + \beta_1 X_1 + \beta_2 X_2 + \cdots +\beta_p X_p + \varepsilon$, 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[/itex]) to the independent variables is assumed to be a [http://en.wikipedia.org/wiki/Linear_function linear function] of the parameters $\beta_i. Note that multilinear regression is a linear modeling technique '''not''' because is that the graph of [itex]Y = \beta_{0}+\beta x$ 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=== ===Approach===

## 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 Xi, 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 β0 is the intercept ("constant" term) and βis 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 βi. Note that multilinear regression is a linear modeling technique not because is that the graph of Y = β0 + βx 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 ( βi), 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 x2 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

• TBD

### Examples

Computer simulations and real observed data.

• TBD

### Hands-on activities

Step-by-step practice problems.

• TBD

• TBD