SMHS SLR - Revision history

Dinov: /* Inference on Correlation */

2026-02-22T01:29:46Z

‎Inference on Correlation

Dinov: /* References */

2026-02-10T21:01:16Z

‎References

Dinov: /* Inference on Correlation */

2025-12-05T21:24:20Z

‎Inference on Correlation

Dinov at 21:23, 5 December 2025

2025-12-05T21:23:06Z

Show changes

Dinov: /* Problems */

2014-11-21T15:15:23Z

‎Problems

Dinov: /* Problems */

2014-10-05T15:53:33Z

‎Problems

Dinov: /* In R */

2014-09-29T21:55:56Z

‎In R

Dinov: /* Simple linear regression (SLR) */

2014-09-29T21:53:49Z

‎Simple linear regression (SLR)

Dinov: /* SOCR Analyses */

2014-09-29T21:48:55Z

‎SOCR Analyses

Dinov: /* SOCR Analyses */

2014-09-29T21:46:05Z

‎SOCR Analyses

@@ Line 48: / Line 48: @@
 Because the sampling distribution of <math>r</math> is skewed when <math>\rho \ne 0,</math> we use Fisher’s transformation to compare correlations from two independent samples (<math>r_1</math> and <math>r_2</math>):
-<math>z' = \frac{1}{2} \ln \left( \frac{1+r}{1-r} \right).</math>
+<math>z' = \frac{1}{2} \ln \left( \frac{1+r}{1-r} \right)\equiv atanh(r),</math>
 The transformed value <math>z'</math> is approximately normally distributed with variance <math>\frac{1}{n - 3}</math>.

@@ Line 233: / Line 233: @@
 ===References===
 *[[Probability_and_statistics_EBook#Chapter_X:_Correlation_and_Regression | SOCR Probability and Statistics EBook, Correlation and Regression Chapter]]
 * Altman DG. (1991). *Practical Statistics for Medical Research*.
 * Dunn, G. (1989). *Design and Analysis of Reliability Studies*.
@@ Line 239: / Line 241: @@
 <hr>
-* SOCR Home page: http://www.socr.umich.edu
+* SOCR Home page: https://socr.umich.edu
-{{translate|pageName=http://wiki.socr.umich.edu/index.php?title=SMHS_SLR}}
+{{translate|pageName=https://wiki.socr.umich.edu/index.php?title=SMHS_SLR}}

@@ Line 39: / Line 39: @@
 ==== Inference on Correlation ====
 To assess whether an observed correlation reflects a true association in the population, we test:
-<math>H_0: \rho = 0 \quad \text{vs.} \quad H_a: \rho \ne 0</math>.
+<math>H_0: \rho = 0 \quad \text{vs.} \quad H_a: \rho \ne 0.</math>
 * Test statistic:
-<math>t = \frac{r\sqrt{n-2}}{\sqrt{1-r^2}}</math>,
+<math>t = \frac{r\sqrt{n-2}}{\sqrt{1-r^2}},</math>
 which follows a Student’s <math>t</math>-distribution with <math>n - 2</math> degrees of freedom.
 Comparing Two Independent Correlations (Fisher’s Z-Transformation):
-Because the sampling distribution of <math>r</math> is skewed when <math>\rho \ne 0</math>, we use Fisher’s transformation to compare correlations from two independent samples (<math>r_1</math> and <math>r_2</math>):
+Because the sampling distribution of <math>r</math> is skewed when <math>\rho \ne 0,</math> we use Fisher’s transformation to compare correlations from two independent samples (<math>r_1</math> and <math>r_2</math>):
-<math>z' = \frac{1}{2} \ln \left( \frac{1+r}{1-r} \right)</math>.
+<math>z' = \frac{1}{2} \ln \left( \frac{1+r}{1-r} \right).</math>
-The transformed value <math>z'</math> is approximately normally distributed with variance <math>\frac{1}{n - 3</math>.
+The transformed value <math>z'</math> is approximately normally distributed with variance <math>\frac{1}{n - 3}</math>.
 To test <math>H_0: \rho_1 = \rho_2</math>, compute:
-<math>Z = \frac{z'_1 - z'_2}{\sqrt{\frac{1}{n_1-3} + \frac{1}{n_2-3}}}</math>.
+<math>Z = \frac{z'_1 - z'_2}{\sqrt{\frac{1}{n_1-3} + \frac{1}{n_2-3}}}.</math>
-Under <math>H_0</math>, <math>Z \sim N(0,1)</math>.
+Under <math>H_0</math>, <math>Z \sim N(0,1).</math>
 === Simple Linear Regression (SLR) ===

@@ Line 374: / Line 374: @@
 ===Problems===
 Example 1: Simple linear correlation and regression in R:
-  > library(MASS)
+  library(MASS)
-  > data(cats)
+  data(cats)
-  > str(cats)
+  str(cats)
   'data.frame':	144 obs. of  3 variables:
@@ Line 383: / Line 383: @@
   Hwt: num  7 7.4 9.5 7.2 7.3 7.6 8.1 8.2 8.3 8.5 ...
-  > summary(cats)
+  summary(cats)
   Sex         Bwt             Hwt
@@ Line 392: / Line 392: @@
 rd Qu.:3.025   3rd Qu.:12.12
          Max.   :3.900   Max.   :20.50
 [[Image:SMHS SLR Fig10.png|300]]

@@ Line 379: / Line 379: @@
   'data.frame':	144 obs. of  3 variables:
-  $ Sex: Factor w/ 2 levels "F","M": 1 1 1 1 1 1 1 1 1 1 ...
+  Sex: Factor with 2 levels "F","M": 1 1 1 1 1 1 1 1 1 1 ...
-  $ Bwt: num  2 2 2 2.1 2.1 2.1 2.1 2.1 2.1 2.1 ...
+  Bwt: num  2 2 2 2.1 2.1 2.1 2.1 2.1 2.1 2.1 ...
-  $ Hwt: num  7 7.4 9.5 7.2 7.3 7.6 8.1 8.2 8.3 8.5 ...
+  Hwt: num  7 7.4 9.5 7.2 7.3 7.6 8.1 8.2 8.3 8.5 ...
   > summary(cats)

@@ Line 198: / Line 198: @@
   cor(x,y)
   [1] 0.8654923
 The correlation between x and y is 0.8655 which is pretty strong positive correlation. So it would be reasonable to make the assumption of a linear regression model of number of runs scored and the average batting.

@@ Line 340: / Line 340: @@
 *Estimating the best linear fit: simple linear regression model $Y=a+bX+\varepsilon $ can be estimated using least square, which fits a line minimizing the sums of $ \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 $ \hat y_{i} = a+bx_{i}$ are observed and predicted values of $Y$ for $x_{i}$.
-*Solving for the minimization problem:
+*Solving for the minimization problem $ f(a,b)=\min_{a,b} {\big (\sum_{i=1}^{N}((a+bx_{i})-y_{i} )^{2} \big )}$:
 : $\hat{b} = \frac {\sum_{i=1}^{N}  (x_{i} - \bar{x})(y_{i} - \bar{y}) }  {\sum_{i=1}^{N} (x_{i} - \bar{x}) ^2} = \frac {\sum_{i=1}^{N} {(x_{i}y_{i})} - N \bar{x} \bar{y}}  {\sum_{i=1}^{N} (x_{i})^2 - N \bar{x}^2}  = \rho_{X,Y} \frac {s_y}{s_x},$
 : where [[AP_Statistics_Curriculum_2007_GLM_Corr | $\rho_{X,Y}$ is the correlation coefficient]].

@@ Line 281: / Line 281: @@
 [[Image:SMHS_SLR_Fig9.png|600px]]
-[[Image:SMHS_SLR_Fig10.png|600px]]
+[[Image:SMHS_SLR_Fig11.png|600px]]
 ====Statistical inference on correlation coefficient====

@@ Line 275: / Line 275: @@
   P-Value:            .00001
-[[Image:SMHS SLR Fig7.png|600px]]
+[[Image:SMHS_SLR_Fig7.png|600px]]
-[[Image:SMHS SLR Fig8.png|600px]]
+[[Image:SMHS_SLR_Fig8.png|600px]]
-[[Image:SMHS SLR Fig9.png|600px]]
+[[Image:SMHS_SLR_Fig9.png|600px]]
 ====Statistical inference on correlation coefficient====