Bivariate - Revision history

Rice: /* Correlation */

2024-04-05T17:38:28Z

Correlation

← Older revision		Revision as of 17:38, 5 April 2024
Line 21:		Line 21:

	== Correlation ==		== Correlation ==
	The '''correlation''' of two random variables measures the '''line'''		The '''Pearson correlation''' of two random variables measures the '''line'''
	dependent''' between <math>X</math> and <math>Y</math>'''		dependent''' between <math>X</math> and <math>Y</math>'''

Line 29:		Line 29:

	Correlation is ''normalized''. It has no units, and is always between -1 and 1. When r = 1, the relationship between X and Y is '''perfect positive linear'''. When r = -1, it is '''perfect negative linear'''. If it is 0, there is no linear relationship. This doesn't mean that there is no relationship. Notably, any symmetric scatter plot has a correlation of 0.		Correlation is ''normalized''. It has no units, and is always between -1 and 1. When r = 1, the relationship between X and Y is '''perfect positive linear'''. When r = -1, it is '''perfect negative linear'''. If it is 0, there is no linear relationship. This doesn't mean that there is no relationship. Notably, any symmetric scatter plot has a correlation of 0.

			When the relationship between teh two is non-linear, we use the '''Spearman's correlation'''.

			<math>\rho = 1 - \frac{6 \sum d^2_i}{n (n^2 - 1)}</math>

			where d<sub>i</sub> is the difference between the two ranks of each observation. For the two variables of each observation, we ''rank'' them by sorting from smallest to largest.

	= Bivariate Normal =		= Bivariate Normal =

2024-04-05T17:26:34Z

Correlation

← Older revision		Revision as of 17:26, 5 April 2024
Line 28:		Line 28:
	</math>		</math>

	Correlation is always between -1 and 1. When r = 1, the relationship between X and Y is '''perfect positive linear'''. When r = -1, it is '''perfect negative linear'''. If it is 0, there is no linear relationship. This doesn't mean that there is no relationship. Notably, any symmetric scatter plot has a correlation of 0.		Correlation is ''normalized''. It has no units, and is always between -1 and 1. When r = 1, the relationship between X and Y is '''perfect positive linear'''. When r = -1, it is '''perfect negative linear'''. If it is 0, there is no linear relationship. This doesn't mean that there is no relationship. Notably, any symmetric scatter plot has a correlation of 0.

	= Bivariate Normal =		= Bivariate Normal =

2024-03-19T19:02:51Z

Regression Effect

← Older revision		Revision as of 19:02, 19 March 2024
Line 83:		Line 83:

	== Regression Effect ==		== Regression Effect ==
	~~[[File:Regression Effect Scatter Plot.png\|thumb\|Regression effect demonstrated by SD line and Regression line]]~~
	The '''regression effect''' is the phenomenon that the best prediction		The '''regression effect''' is the phenomenon that the best prediction
	of <math>Y</math> given <math>X = x</math> is less rare for		of <math>Y</math> given <math>X = x</math> is less rare for
	<math>Y</math> than <math>x</math>; Future predictions regress to		<math>Y</math> than <math>x</math>; Future predictions regress to
	mediocrity.		mediocrity.

			If the sample is random (or at least somewhat random), it is unlikely that a subject that has an unlikely score in ''x'' got another unlikely score in ''y''. Therefore, the expectation of ''Y'' given ''X'' is close to the mean. It's barely elaborated on in class, go look at [[wikipedia:Regression_toward_the_mean\|Wikipedia]]

	When you plot all the predicted <math>E(Y\|X = x)</math>, you get the		When you plot all the predicted <math>E(Y\|X = x)</math>, you get the

2024-03-19T06:24:03Z

← Older revision		Revision as of 06:24, 19 March 2024
Line 1:		Line 1:
	[[Category:Distribution (Statistics)]]		[[Category:Distribution (Statistics)]][[Category:Statistics]]
	'''Bivariate''' data consider two variables instead of the usual one; each value of one of the variables is paired with a value of the other variable. We will be using <math>X, Y</math> to denote the two random variables throughout this page.		'''Bivariate''' data consider two variables instead of the usual one; each value of one of the variables is paired with a value of the other variable. We will be using <math>X, Y</math> to denote the two random variables throughout this page.

2024-03-19T06:23:32Z

← Older revision		Revision as of 06:23, 19 March 2024
Line 1:		Line 1:
			[[Category:Distribution (Statistics)]]
	'''Bivariate''' data consider two variables instead of the usual one; each value of one of the variables is paired with a value of the other variable. We will be using <math>X, Y</math> to denote the two random variables throughout this page.		'''Bivariate''' data consider two variables instead of the usual one; each value of one of the variables is paired with a value of the other variable. We will be using <math>X, Y</math> to denote the two random variables throughout this page.

2024-03-19T05:37:43Z

← Older revision		Revision as of 05:37, 19 March 2024
Line 27:		Line 27:
	</math>		</math>

	Correlation is always between -1 and 1		Correlation is always between -1 and 1. When r = 1, the relationship between X and Y is '''perfect positive linear'''. When r = -1, it is '''perfect negative linear'''. If it is 0, there is no linear relationship. This doesn't mean that there is no relationship. Notably, any symmetric scatter plot has a correlation of 0.

	= Bivariate Normal =		= Bivariate Normal =

2024-03-19T01:09:53Z

@@ Line 1: / Line 1: @@
-Consider two numerica random variables <math>X</math> and
+'''Bivariate''' data consider two variables instead of the usual one; each value of one of the variables is paired with a value of the other variable. We will be using <math>X, Y</math> to denote the two random variables throughout this page.
-<math>Cov(X, Y)</math>
+= Summary Statistics =
-The '''correlation''' of two random variables measures the '''line
+== Covariance ==
-dependent''' between <math>X</math> and <math>Y</math>
+The '''covariance''' measures the total variation of two RVs and their centers. It indicates the relationship of two variables whenever one changes, measuring how much the two vary together.
 <math>

2024-03-19T00:33:13Z

← Older revision		Revision as of 00:33, 19 March 2024
(No difference)

2024-03-18T19:02:06Z

@@ Line 77: / Line 77: @@
 = Linear Regression =
 <math>
@@ Line 92: / Line 108: @@
 <math>
- \hat{y_i} = \hat{\beta_0} + \hat{\beta_1} x_i
+\hat{y_i} = \hat{\beta_0} + \hat{\beta_1} x_i
 </math>

2024-03-18T18:53:07Z