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==[[AP_Statistics_Curriculum_2007 | General Advance-Placement (AP) Statistics Curriculum]] - Regression ==
 
==[[AP_Statistics_Curriculum_2007 | General Advance-Placement (AP) Statistics Curriculum]] - Regression ==
  
As we discussed in the [[AP_Statistics_Curriculum_2007_GLM_Corr |Correlation section]], many applications involve the analysis of relationships between two, or more, variables involved in the process of interest. Suppose we have bivariate data (''X'' and ''Y'') of a process and we are interested on determining the linear relation between X and Y (e.g., determining a straight line that best fits the pairs of data (''X,Y'')). A linear relationship between ''X'' and ''Y'' will give us the power to make predictions - i.e., given a value of ''X'' predict a corresponding ''Y'' response. Note that in this design, data consists of paired observations (''X,Y'') - for example, the [[SOCR_Data_Dinov_021708_Earthquakes | Longitude and Latitude of the SOCR Earthquake dataset]].
+
As we discussed in the [[AP_Statistics_Curriculum_2007_GLM_Corr |Correlation section]], many applications involve the analysis of relationships between two, or more, variables involved in the process of interest. Suppose we have bivariate data (''X'' and ''Y'') of a process and we are interested in determining the linear relation between X and Y (e.g., determining a straight line that best fits the pairs of data (''X,Y'')). A linear relationship between ''X'' and ''Y'' will give us the power to make predictions - i.e., given a value of ''X'' predict a corresponding ''Y'' response. Note that in this design, data consists of paired observations (''X,Y'') - for example, the [[SOCR_Data_Dinov_021708_Earthquakes | Longitude and Latitude of the SOCR Earthquake dataset]].
  
 
===Lines in 2D===
 
===Lines in 2D===
Line 20: Line 20:
 
* Y is an observed variable and X is specified by the researcher - e.g., Y is hair growth after X months, for individuals at certain dose levels of hair growth cream.
 
* Y is an observed variable and X is specified by the researcher - e.g., Y is hair growth after X months, for individuals at certain dose levels of hair growth cream.
  
* X and Y are both observed variables - e.g., [[SOCR_Data_Dinov_020108_HeightsWeights | Height (Y) and weight (X)]] for 20 randomly selected individuals from the population.
+
* X and Y are both observed variables - e.g., [[SOCR_Data_Dinov_020108_HeightsWeights | height (Y) and weight (X)]] for 20 randomly selected individuals from the population.
  
 
Suppose we have ''n'' pairs ''(X,Y)'', {<math>X_1, X_2, X_3, \cdots, X_n</math>} and {<math>Y_1, Y_2, Y_3, \cdots, Y_n</math>}, of observations of the same process. If a [[SOCR_EduMaterials_Activities_ScatterChart |scatterplot]] of the data suggests a general linear trend, it would be reasonable to fit a line to the data. The main question is how to determine the best line?
 
Suppose we have ''n'' pairs ''(X,Y)'', {<math>X_1, X_2, X_3, \cdots, X_n</math>} and {<math>Y_1, Y_2, Y_3, \cdots, Y_n</math>}, of observations of the same process. If a [[SOCR_EduMaterials_Activities_ScatterChart |scatterplot]] of the data suggests a general linear trend, it would be reasonable to fit a line to the data. The main question is how to determine the best line?
  
 
====[[AP_Statistics_Curriculum_2007_GLM_Corr#Airfare_Example |Airfare Example]]====
 
====[[AP_Statistics_Curriculum_2007_GLM_Corr#Airfare_Example |Airfare Example]]====
We can see from the [[SOCR_EduMaterials_Activities_ScatterChart |scatterplot]] that greater distance is associated with higher airfare. In other words airports that tend to be further from Baltimore tend to be more expensive airfare. To decide on the best fitting line, we use the '''least-squares method''' to fit the least squares (regression) line.
+
We can see from the [[SOCR_EduMaterials_Activities_ScatterChart |scatterplot]] that [[AP_Statistics_Curriculum_2007_GLM_Corr#Airfare_Example | greater distance is associated with higher airfare]]. In other words airports that tend to be further from Baltimore tend to have more expensive airfare. To decide on the best fitting line, we use the '''least-squares method''' to fit the least squares (regression) line.
  
 
<center>[[Image:SOCR_EBook_Dinov_GLM_Regr_021708_Fig1.jpg|500px]]</center>
 
<center>[[Image:SOCR_EBook_Dinov_GLM_Regr_021708_Fig1.jpg|500px]]</center>
  
 
====Estimating the Best Linear Fit====
 
====Estimating the Best Linear Fit====
The parameters of the linear regression line, <math>Y = a + bX</math>, can be estimated using [http://en.wikipedia.org/wiki/Ordinary_Least_Squares  Least Squares]. This method finds the line that minimizes the sum of the squares of the regression '''residuals''', <math> \sum_{i=1}^N {\hat{\varepsilon_i}^2} = \sum_{i=1}^N (\hat{y}_{i}-y_i)^2 </math>, where <math>y_i</math> and <math>\hat{y}_{i}=a+bx_i</math> are the observed and the predicted values of ''Y'' for <math>x_i</math>, respectfully.
+
The parameters of the linear regression line, <math>Y = a + bX</math>, can be estimated using [http://en.wikipedia.org/wiki/Ordinary_Least_Squares  Least Squares]. The least squares technique finds the line that minimizes the sum of the squares of the regression '''residuals''', <math>\hat{\varepsilon_i}=\hat{y}_{i}-y_i</math>, <math> \sum_{i=1}^N {\hat{\varepsilon_i}^2} = \sum_{i=1}^N (\hat{y}_{i}-y_i)^2 </math>, where <math>y_i</math> and <math>\hat{y}_{i}=a+bx_i</math> are the observed and the predicted values of ''Y'' for <math>x_i</math>.
  
 
The minimization problem can be solved using calculus, by finding the first order partial derivatives and setting them equal to zero. The solution gives the slope and y-intercept of the regressions line:
 
The minimization problem can be solved using calculus, by finding the first order partial derivatives and setting them equal to zero. The solution gives the slope and y-intercept of the regressions line:
Line 50: Line 50:
 
If the error terms are Normally distributed, the estimate of the slope coefficient has a normal distribution with mean equal to '''b''' and ''standard error'' given by:
 
If the error terms are Normally distributed, the estimate of the slope coefficient has a normal distribution with mean equal to '''b''' and ''standard error'' given by:
  
: <math> s_ \hat{b} = \sqrt { \frac {\sum_{i=1}^N \hat{\varepsilon_i}^2 /(N-2)} {\sum_{i=1}^N (x_i - \bar{x})^2} }</math>.
+
:: <math> SE(\hat{b}) = s_ \hat{b} = \sqrt { {1\over (N-2)} \frac {\sum_{i=1}^N \hat{\varepsilon_i}^2} {\sum_{i=1}^N (x_i - \bar{x})^2} }</math>.
  
A confidence interval for ''b'' can be created using a [[AP_Statistics_Curriculum_2007_StudentsT | T-distribution with N-2 degrees of freedom]]:
+
* A confidence interval for ''b'' can be created using a [[AP_Statistics_Curriculum_2007_StudentsT | T-distribution with N-2 degrees of freedom]]:
  
:<math> [ \hat{b} - s_ \hat{b} t_{N-2}^*,\hat{b} + s_ \hat{b} t_{N-2}^*] </math>
+
::<math> [ \hat{b} - s_ \hat{b} t_{(\alpha/2, N-2)},\hat{b} + s_ \hat{b} t_{(\alpha/2, N-2)}] </math>
 +
 
 +
* The corresponding test for the regression slope coefficient ''b'' is analogously computed (<math>H_o: b=b_o</math> vs. <math>H_a: b \not= b_0</math>, frequently <math>b_0 = 0</math>)
 +
 
 +
::<math> t_o = {\hat{b} - b_o \over s_ \hat{b} } \sim T_{df=N-2}. </math>
  
 
===Hands-on Example===
 
===Hands-on Example===
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===Earthquake Example===
 
===Earthquake Example===
  
Use the [[SOCR_Data_Dinov_021708_Earthquakes | SOCR Earthquake Dataset]] to determine the best-leaner fit between the [http://nationalatlas.gov/articles/mapping/a_latlong.html Longitude and the [http://nationalatlas.gov/articles/mapping/a_latlong.html Latitude] of the California Earthquakes since 1900. What is the interpretation of this regression line ([http://en.wikipedia.org/wiki/San_Andreas_Fault San Andreas fault]?). You can see the [http://socr.ucla.edu/docs/resources/SOCR_Data/SOCR_Earthquake5Data_GoogleMap.html SOCR Geomap of these Earthquakes].
+
Use the [[SOCR_Data_Dinov_021708_Earthquakes | SOCR Earthquake Dataset]] to determine the best-linear fit between the [http://nationalatlas.gov/articles/mapping/a_latlong.html Longitude] and the [http://nationalatlas.gov/articles/mapping/a_latlong.html Latitude] of the California Earthquakes since 1900. What is the interpretation of this regression line ([http://en.wikipedia.org/wiki/San_Andreas_Fault San Andreas fault]?). You can see the [http://socr.ucla.edu/docs/resources/SOCR_Data/SOCR_Earthquake5Data_GoogleMap.html SOCR Geomap of these Earthquakes]. The image below shows how to use the [[SOCR_EduMaterials_AnalysisActivities_SLR |Simple Linear regression]] in [http://socr.ucla.edu/htmls/SOCR_Analyses.html SOCR Analyses] to calculate the regression line:
 +
: <math>Latitude = -40.149 -0.6457\times Longitude</math>
  
 
<center>[[Image:SOCR_EBook_Dinov_GLM_Regr_021708_Fig2.jpg|500px]]</center>
 
<center>[[Image:SOCR_EBook_Dinov_GLM_Regr_021708_Fig2.jpg|500px]]</center>
 +
 +
===Hands-on Example===
 +
Compute by hand the regression line between X and Y and validate your results using [http://socr.ucla.edu/htmls/SOCR_Analyses.html the SOCR Regression Analysis].
 +
<center>
 +
{| class="wikitable" style="text-align:center; width:75%" border="1"
 +
|-
 +
| X  || Y in cm || <math>x_i-\bar{x}</math> ||  <math>y_i-\bar{y}</math> || <math>(x_i-\bar{x})^2</math> || <math>(y_i-\bar{y})^2</math> || <math>(x_i-\bar{x})(y_i-\bar{y})</math>
 +
|-
 +
| -1 || 0 || -3 || -0.5 || 9 || 0.25 || 1.5
 +
|-
 +
| 2 || -1 || 0 || -1.5 || 0 || 2.25 || 0
 +
|-
 +
| 3 || 1 || 1 || 0.5 || 1 || 0.25 || 0.5
 +
|-
 +
| 4 || 2 || 2 || 1.5 || 4 || 2.25 || 3
 +
|-
 +
| <math>\bar{x}</math>  || <math>\bar{y}</math> ||  ||  ||  ||  ||
 +
|}
 +
</center>
  
 
<hr>
 
<hr>
===References===
+
 
 +
===[[EBook_Problems_GLM_Regress|Problems]]===
 +
 
 +
===See also===
 +
* [[SOCR_EduMaterials_AnalysisActivities_SLR| SOCR Linear Regression Activity]]
  
 
<hr>
 
<hr>
 
* 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_Regress}}
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"{{translate|pageName=http://wiki.socr.umich.edu/index.php?title=AP_Statistics_Curriculum_2007_GML_Regress}}

Latest revision as of 13:48, 3 March 2020

General Advance-Placement (AP) Statistics Curriculum - Regression

As we discussed in the Correlation section, many applications involve the analysis of relationships between two, or more, variables involved in the process of interest. Suppose we have bivariate data (X and Y) of a process and we are interested in determining the linear relation between X and Y (e.g., determining a straight line that best fits the pairs of data (X,Y)). A linear relationship between X and Y will give us the power to make predictions - i.e., given a value of X predict a corresponding Y response. Note that in this design, data consists of paired observations (X,Y) - for example, the Longitude and Latitude of the SOCR Earthquake dataset.

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 \(aX + bY=c\), most frequently expressed as \(Y=aX + b\), provides 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 (\(X_1 =X_2\))\[X=X_1\];
If the line is horizontal (\(Y_1 =Y_2\))\[Y=Y_1\];
Otherwise (oblique line)\[{Y-Y_1 \over Y_2-Y_1}= {X-X_1 \over X_2-X_1}\], (for \(X_1\not=X_2\) and \(Y_1\not=Y_2\))

where \((X_1,Y_1)\) and \((X_2, Y_2)\) are two points on the line of interest (2-distinct points in 2D determine a unique line).

Y=2X+1
Y=-3X-5

Linear Modeling - Regression

There are two contexts for regression:

  • Y is an observed variable and X is specified by the researcher - e.g., Y is hair growth after X months, for individuals at certain dose levels of hair growth cream.

Suppose we have n pairs (X,Y), {\(X_1, X_2, X_3, \cdots, X_n\)} and {\(Y_1, Y_2, Y_3, \cdots, Y_n\)}, of observations of the same process. If a scatterplot of the data suggests a general linear trend, it would be reasonable to fit a line to the data. The main question is how to determine the best line?

Airfare Example

We can see from the scatterplot that greater distance is associated with higher airfare. In other words airports that tend to be further from Baltimore tend to have more expensive airfare. To decide on the best fitting line, we use the least-squares method to fit the least squares (regression) line.

SOCR EBook Dinov GLM Regr 021708 Fig1.jpg

Estimating the Best Linear Fit

The parameters of the linear regression line, \(Y = a + bX\), can be estimated using Least Squares. The least squares technique finds the line that minimizes the sum of the squares of the regression residuals, \(\hat{\varepsilon_i}=\hat{y}_{i}-y_i\), \( \sum_{i=1}^N {\hat{\varepsilon_i}^2} = \sum_{i=1}^N (\hat{y}_{i}-y_i)^2 \), where \(y_i\) and \(\hat{y}_{i}=a+bx_i\) are the observed and the predicted values of Y for \(x_i\).

The minimization problem can be solved using calculus, by finding the first order partial derivatives and setting them equal to zero. The solution gives the slope and y-intercept of the regressions line:

  • Regression line Slope:

\[ \hat{b} = \frac {\sum_{i=1}^{N} (x_{i} - \bar{x})(y_{i} - \bar{y}) } {\sum_{i=1}^{N} (x_{i} - \bar{x}) ^2} \] \[ \hat{b} = \frac {\sum_{i=1}^{N} {(x_{i}y_{i})} - N \bar{x} \bar{y}} {\sum_{i=1}^{N} (x_{i})^2 - N \bar{x}^2} = \rho_{X,Y} \frac {s_y}{s_x} \], where \(\rho_{X,Y}\) is the correlation coefficient.

  • Y-intercept:

\[ \hat{a} = \bar{y} - \hat{b} \bar{x} \]

  • Properties of the least square line:
    • The line goes through the point \( (\bar{X},\bar{Y}) \)
    • The sum of the residuals is equal to zero
    • The estimates are unbiased (their expected values are equal to the real slope and intercept values).

Regression Coefficients Inference

If the error terms are Normally distributed, the estimate of the slope coefficient has a normal distribution with mean equal to b and standard error given by:

\[ SE(\hat{b}) = s_ \hat{b} = \sqrt { {1\over (N-2)} \frac {\sum_{i=1}^N \hat{\varepsilon_i}^2} {\sum_{i=1}^N (x_i - \bar{x})^2} }\].
\[ [ \hat{b} - s_ \hat{b} t_{(\alpha/2, N-2)},\hat{b} + s_ \hat{b} t_{(\alpha/2, N-2)}] \]
  • The corresponding test for the regression slope coefficient b is analogously computed (\(H_o: b=b_o\) vs. \(H_a: b \not= b_0\), frequently \(b_0 = 0\))
\[ t_o = {\hat{b} - b_o \over s_ \hat{b} } \sim T_{df=N-2}. \]

Hands-on Example

Suppose we have 3 sample of points {(1,-1),(2,4),(6,3)}. The mean of X is 3 and the mean of Y is 2. The slope coefficient estimate is given by:

\[ \hat{b} = \frac {(1 - 3)((-1) - 2) + (2 - 3)(4 - 2) + (6 - 3)(3 - 2)} {(1 - 3)^2 + (2 - 3)^2 + (6 - 3)^2 } = 7/14 = 0.5 \]

The standard error of the slope coefficient (b) is 0.866. A 95% confidence interval is given by:

CI(b): [0.5 - 0.866 x 12.7062, 0.5 + 0.866 x 12.7062] = [-10.504, 11.504].

Earthquake Example

Use the SOCR Earthquake Dataset to determine the best-linear fit between the Longitude and the Latitude of the California Earthquakes since 1900. What is the interpretation of this regression line (San Andreas fault?). You can see the SOCR Geomap of these Earthquakes. The image below shows how to use the Simple Linear regression in SOCR Analyses to calculate the regression line: \[Latitude = -40.149 -0.6457\times Longitude\]

SOCR EBook Dinov GLM Regr 021708 Fig2.jpg

Hands-on Example

Compute by hand the regression line between X and Y and validate your results using the SOCR Regression Analysis.

X Y in cm \(x_i-\bar{x}\) \(y_i-\bar{y}\) \((x_i-\bar{x})^2\) \((y_i-\bar{y})^2\) \((x_i-\bar{x})(y_i-\bar{y})\)
-1 0 -3 -0.5 9 0.25 1.5
2 -1 0 -1.5 0 2.25 0
3 1 1 0.5 1 0.25 0.5
4 2 2 1.5 4 2.25 3
\(\bar{x}\) \(\bar{y}\)

Problems

See also


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