# Difference between revisions of "Eleisha's Segment 19: The Chi-Square Statistic"

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Since <math> \mathbf x </math> is a random draw from the mulitvariate normal, x can be written as <math> \mathbf {x = Ly + \mu} </math>, where y is filled with | Since <math> \mathbf x </math> is a random draw from the mulitvariate normal, x can be written as <math> \mathbf {x = Ly + \mu} </math>, where y is filled with | ||

− | <math> y_i \text{'s} </math> drawn from a normal with a | + | <math> y_i \text{'s} </math> drawn from a normal with a \mu zero and \sigma one. (Shown in lecture 17) |

If you manipulate <math> \mathbf {x = Ly + \mu} </math> you get <math> \mathbf {Ly = x - \mu} </math> | If you manipulate <math> \mathbf {x = Ly + \mu} </math> you get <math> \mathbf {Ly = x - \mu} </math> | ||

Line 15: | Line 15: | ||

= (y^TL^T)(LL^T)^{-1}(Ly) = y^Ty} = \sum y_i \text{'s} </math> | = (y^TL^T)(LL^T)^{-1}(Ly) = y^Ty} = \sum y_i \text{'s} </math> | ||

− | Since the <math> y_i | + | Since the <math> y_i\text{'s} </math> are drawn for the normal distributions with mean \mu zero and \sigma one, \sum y_i \text{'s} is the sum of squared t-values. |

+ | |||

+ | This means that t <math> ({\mathbf x-\mathbf\mu})^T{\mathbf\Sigma}^{-1}({\mathbf x-\mathbf\mu}) </math> is <math> \chi^2 </math> distributed. | ||

<b>To Think About </b> | <b>To Think About </b> |

## Revision as of 09:29, 30 April 2014

** To Calculate **

1. Prove the assertion on lecture slide 5, namely that, for a multivariate normal distribution, the quantity **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ({\mathbf x-\mathbf\mu})^T{\mathbf\Sigma}^{-1}({\mathbf x-\mathbf\mu}), }**
where **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf x }**
is a random draw from the multivariate normal, is **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \chi^2 }**
distributed.

Must prove that **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ({\mathbf x-\mathbf\mu})^T{\mathbf\Sigma}^{-1}({\mathbf x-\mathbf\mu}) }**
is **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \chi^2 }**
distributed.

Since **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf x }**
is a random draw from the mulitvariate normal, x can be written as **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf {x = Ly + \mu} }**
, where y is filled with
**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y_i \text{'s} }**
drawn from a normal with a \mu zero and \sigma one. (Shown in lecture 17)

If you manipulate **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf {x = Ly + \mu} }**
you get **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf {Ly = x - \mu} }**

Therefore

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ({\mathbf x-\mathbf\mu})^T{\mathbf\Sigma}^{-1}({\mathbf x-\mathbf\mu}) = \mathbf{(Ly)^T \Sigma^{-1} (Ly) = (y^TL^T)\Sigma^{-1}(Ly) = (y^TL^T)(LL^T)^{-1}(Ly) = y^Ty} = \sum y_i \text{'s} }**

Since the **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y_i\text{'s} }**
are drawn for the normal distributions with mean \mu zero and \sigma one, \sum y_i \text{'s} is the sum of squared t-values.

This means that t **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ({\mathbf x-\mathbf\mu})^T{\mathbf\Sigma}^{-1}({\mathbf x-\mathbf\mu}) }**
is **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \chi^2 }**
distributed.

**To Think About **

1. Why are we so interested in t-values? Why do we square them?

2. Suppose you measure a bunch of quantities **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_i }**
, each of which is measured with a measurement accuracy **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sigma_i }**
and has a theoretically expected value **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mu_i }**
. Describe in detail how you might use a chi-square test statistic as a p-value test to see if your theory is viable? Should your test be 1 or 2 tailed?

**Back To: ** Eleisha Jackson