AP Statistics Curriculum 2007 GLM Corr - Revision history

Tdlee: Text replacement - "{{translate|pageName=http://wiki.stat.ucla.edu/socr/" to ""{{translate|pageName=http://wiki.socr.umich.edu/"

2020-03-03T16:43:52Z

Text replacement - "{{translate|pageName=http://wiki.stat.ucla.edu/socr/" to ""{{translate|pageName=http://wiki.socr.umich.edu/"

Dinov: /* Lines in 2D */

2017-02-26T16:36:36Z

‎Lines in 2D

Dinov: /* Brain volume and age example */

2014-09-05T13:54:36Z

‎Brain volume and age example

Dinov: /* Statistical inference on correlation coefficients */

2014-09-05T13:49:09Z

‎Statistical inference on correlation coefficients

Dinov: /* Statistical inference on correlation coefficients */

2014-09-05T13:48:43Z

‎Statistical inference on correlation coefficients

Dinov: /* Statistical inference on correlation coefficients */

2014-09-05T13:43:46Z

‎Statistical inference on correlation coefficients

Dinov: /* Statistical inference on correlation coefficients */

2014-09-05T13:42:59Z

‎Statistical inference on correlation coefficients

IvoDinov: /* Lines in 2D */ typo

2013-05-29T15:44:06Z

‎Lines in 2D: typo

IvoDinov: minor typo in translation code at bottom of page

2013-02-22T17:47:29Z

minor typo in translation code at bottom of page

IvoDinov: /* Statistical inference on correlation coefficients */

2012-11-01T00:42:04Z

‎Statistical inference on correlation coefficients

← Older revision		Revision as of 16:43, 3 March 2020
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	* SOCR Home page: http://www.socr.ucla.edu		* SOCR Home page: http://www.socr.ucla.edu

−	{{translate\|pageName=http://wiki.~~stat~~.~~ucla~~.edu~~/socr~~/index.php?title=AP_Statistics_Curriculum_2007_GLM_Corr}}	+	"{{translate\|pageName=http://wiki.socr.umich.edu/index.php?title=AP_Statistics_Curriculum_2007_GLM_Corr}}

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 ===Lines in 2D===
-There are 3 types of lines in 2D planes - Vertical Lines, Horizontal Lines and Oblique Lines. In general, the mathematical representation of lines in 2D is given by equations like <math>aX + bY=c</math>, most frequently expressed as <math>Y=aX + b</math>, provided the line is not vertical.
+You may want to start by trying the [http://www.socr.ucla.edu/htmls/game/Bivariate_Game.html Interactive SOCR Java Scatterplot Game] to see how the selection of points in the plan effect the least squares estimates for the slope, linear model and correlation. There are 3 types of lines in 2D planes - Vertical Lines, Horizontal Lines and Oblique Lines. In general, the mathematical representation of lines in 2D is given by equations like <math>aX + bY=c</math>, most frequently expressed as <math>Y=aX + b</math>, provided the line is not vertical.
 Recall that there is a one-to-one correspondence between any line in 2D and (linear) equations of the form
-: If the line is '''vertical''' (<math>X_1 =X_2</math>): <math>X=X_1</math>;
+: If the line is '''vertical''' (<math>X_1 =X_2</math>): <math>X=X_1</math>
-: If the line is '''horizontal''' (<math>Y_1 =Y_2</math>): <math>Y=Y_1</math>;
+: If the line is '''horizontal''' (<math>Y_1 =Y_2</math>): <math>Y=Y_1</math>
 : Otherwise ('''oblique''' line): <math>{Y-Y_1 \over Y_2-Y_1}= {X-X_1 \over X_2-X_1}</math>, (for <math>X_1\not=X_2</math> and <math>Y_1\not=Y_2</math>)
 where <math>(X_1,Y_1)</math> and <math>(X_2, Y_2)</math> are two points on the line of interest (2-distinct points in 2D determine a unique line).

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 : We have 2 independent groups (Group1 and Group2) and Y=Volume1 (response) and X=Age1 (predictor); V=Volume2 and U=Age2, n1=27, n2=27. We can also [http://www.socr.ucla.edu/htmls/ana/SimpleRegression_Analysis.html compute the 2 correlation coefficients]:
-:: <math>r1=corr(X,Y)=-0.75339</math>, and
+:: <math>r1=corr(X,Y)=-0.75339,</math> and
 :: <math>r2=corr(U,V) = -0.49491</math>
 : Using the Fisher's transformation we obtain:
-::<math>r11= \frac{1}{2} \log_e \| \frac{1+r1}{1-r1} \|=-0.980749</math>, and
+::<math>r11= \frac{1}{2} \log_e \| \frac{1+r1}{1-r1} \|=-0.980749,</math> and
-::<math>r22= \frac{1}{2} \log_e \| \frac{1+r22}{1-r22} \|=-0.5425423</math>
+::<math>r22= \frac{1}{2} \log_e \| \frac{1+r2}{1-r2} \|=-0.5425423</math>
 : And thus, <math>Z_o=\frac{r11-r22}{\sqrt{\frac{1}{24} +\frac{1}{24}}}= -1.517993</math> and a [http://socr.ucla.edu/htmls/dist/Normal_Distribution.html corresponding (1-sided) p-value] = 0.064508 (double-sided p-value = 0.129016).

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 * Testing a single correlation coefficient \(H_o: r=\rho \) vs. \(H_a: r\not= \rho \)):
 : There is a simple statistical test for the correlation coefficient. Suppose we want to test if the correlation between X and Y is equal to <math>rho</math>. If our bivariate sample is of size N and the observed sample correlation is ''r'', then the test statistics is:
-:: <math>t_o = \frac{r-\rho}{\sqrt{\frac{1-r^2}{N-2}}},</math> which has [[AP_Statistics_Curriculum_2007_StudentsT |T-distribution with df=N—2]].
+:: <math>t_o = \frac{r-\rho}{\sqrt{\frac{1-r^2}{N-2}}},</math> which has [[AP_Statistics_Curriculum_2007_StudentsT |T-distribution with df=N-2]].
 * Comparing two correlation coefficients: The [http://en.wikipedia.org/wiki/Fisher_transformation Fisher's transformation] provides a mechanism to test for comparing two correlation coefficients using [[AP_Statistics_Curriculum_2007#Chapter_V:_Normal_Probability_Distribution |Normal distribution]]. Suppose we have 2 independent paired samples <math>\{(X_i,Y_i)\}_{i=1}^{n_1}</math> and <math>\{(U_j,V_j)\}_{j=1}^{n_2}</math>, and the r1=corr(X,Y) and r2=corr(U,V) and we are testing \(H_o: r1=r2\) vs. \(H_a: r1\not= r2\). The Fisher's transformation for the 2 correlations is defined by:

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 : Then the test statistics <math>Z_o= \frac{r11-r22}{\sqrt{\frac{1}{n1-3}+ \frac{1}{n2-3}}}</math> is Standard Normally distributed.
-Note that the hypotheses for the single and double sample inference are \( H_o: r=0\) vs. \(H_a: r \not= 0 \), and \( H_o: r1-r2=0\) vs. \(H_a: r1-r2 \not= 0 \), respectively. And an estimate of the standard deviation of the correlation is \(SD ( \hat{r}) =  \sqrt{\frac{1}{n-3}}\). Thus, \( r \sim N(0, \sqrt{\frac{1}{n-3}}) \).
+Note that the hypotheses for the single and double sample inference are \( H_o: r=0\) vs. \(H_a: r \not= 0 \), and \( H_o: r1-r2=0\) vs. \(H_a: r1-r2 \not= 0 \), respectively. And an estimate of the standard deviation of the (Fisher transformed!) correlation is \(SD ( \hat{r}) =  \sqrt{\frac{1}{n-3}}\). Thus, \( r \sim N(0, \sqrt{\frac{1}{n-3}}) \).
 ====Brain volume and age example====

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 * Testing a single correlation coefficient \(H_o: r=\rho \) vs. \(H_a: r\not= \rho \)):
 : There is a simple statistical test for the correlation coefficient. Suppose we want to test if the correlation between X and Y is equal to <math>rho</math>. If our bivariate sample is of size N and the observed sample correlation is ''r'', then the test statistics is:
-:: <math>t_o = \frac{r-\rho}{\sqrt{\frac{1-r^2}{N-2}}}</math>, which has [[AP_Statistics_Curriculum_2007_StudentsT |T-distribution with df=N—2]].
+:: <math>t_o = \frac{r-\rho}{\sqrt{\frac{1-r^2}{N-2}}},</math> which has [[AP_Statistics_Curriculum_2007_StudentsT |T-distribution with df=N—2]].
 * Comparing two correlation coefficients: The [http://en.wikipedia.org/wiki/Fisher_transformation Fisher's transformation] provides a mechanism to test for comparing two correlation coefficients using [[AP_Statistics_Curriculum_2007#Chapter_V:_Normal_Probability_Distribution |Normal distribution]]. Suppose we have 2 independent paired samples <math>\{(X_i,Y_i)\}_{i=1}^{n_1}</math> and <math>\{(U_j,V_j)\}_{j=1}^{n_2}</math>, and the r1=corr(X,Y) and r2=corr(U,V) and we are testing \(H_o: r1=r2\) vs. \(H_a: r1\not= r2\). The Fisher's transformation for the 2 correlations is defined by:
-:: <math>\hat{r}= \frac{1}{2} \log_e \| \frac{1+r}{1-r} \|</math>
+:: <math>\hat{r}= \frac{1}{2} \log_e \| \frac{1+r}{1-r} \|.</math>
 : Transforming the two correlation coefficients separately yields:
-::<math>r11= \frac{1}{2} \log_e \| \frac{1+r1}{1-r1} \|</math>, and
+::<math>r11= \frac{1}{2} \log_e \| \frac{1+r1}{1-r1} \|,</math> and
-::<math>r22= \frac{1}{2} \log_e \| \frac{1+r22}{1-r22} \|</math>
+::<math>r22= \frac{1}{2} \log_e \| \frac{1+r22}{1-r22} \|.</math>
 : Then the test statistics <math>Z_o= \frac{r11-r22}{\sqrt{\frac{1}{n1-3}+ \frac{1}{n2-3}}}</math> is Standard Normally distributed.

← Older revision		Revision as of 17:47, 22 February 2013
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	* SOCR Home page: http://www.socr.ucla.edu		* SOCR Home page: http://www.socr.ucla.edu

−	{{translate\|pageName=http://wiki.stat.ucla.edu/socr/index.php?title=~~AP_Statistics_Curriculum_2007_GML_Corr~~}}	+	{{translate\|pageName=http://wiki.stat.ucla.edu/socr/index.php?title=AP_Statistics_Curriculum_2007_GLM_Corr}}