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 \mu zero and \sigma one.  (Shown in lecture 17)
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<math> y_i \text{'s}  </math> drawn from a normal with a <math>\mu </math> zero and <math>\sigma</math> 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>
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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.
 
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.  
+
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 10:30, 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 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 } zero and 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} 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?

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