AP Statistics Curriculum 2007 GLM Corr

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General Advance-Placement (AP) Statistics Curriculum - Correlation

Many biomedical, social, engineering and science applications involve the analysis of relationships, if any, between two or more variables involved in the process of interest. We begin with the simplest of all situations where bivariate data (X and Y) are measured for a process and we are interested in determining the association, relation or an appropriate model for these observations (e.g., fitting a straight line to the pairs of (X,Y) data). If we are successful determining a relationship between X and Y, we can use this model 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 height and weight of individuals.

Lines in 2D

You may want to start by trying the 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 \(aX + bY=c\), most frequently expressed as \(Y=aX + b\), 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 (\(X_1 =X_2\))\[X=X_1\]
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

The Correlation Coefficient

Correlation coefficient (\(-1 \leq \rho \leq 1\)) is a measure of linear association, or clustering around a line of multivariate data. The main relationship between two variables (X, Y) can be summarized by\[(\mu_X, \sigma_X)\], \((\mu_Y, \sigma_Y)\) and the correlation coefficient, denoted by \(\rho=\rho_{(X,Y)}=R(X,Y)\).

  • If \(\rho=1\), we have a perfect positive correlation (straight line relationship between the two variables)
  • If \(\rho=0\), there is no correlation (random cloud scatter), i.e., no linear relation between X and Y.
  • If \(\rho = -1\), there is a perfect negative correlation between the variables.

Computing \(\rho=R(X,Y)\)

The protocol for computing the correlation involves standardizing, multiplication and averaging.

\[\rho_{X,Y}={\mathrm{COV}(X,Y) \over \sigma_X \sigma_Y} ={E((X-\mu_X)(Y-\mu_Y)) \over \sigma_X\sigma_Y},\] where E is the expected value operator and COV means covariance. Since μX = E(X), \(\sigma_X^2 = E(X^2)-E^2(X)\) and similarly for Y, we may also write

\[\rho_{X,Y}=\frac{E(XY)-E(X)E(Y)}{\sqrt{E(X^2)-E^2(X)}~\sqrt{E(Y^2)-E^2(Y)}}.\]

  • Sample correlation - we only have sampled data - we replace the (unknown) expectations and standard deviations by their sample analogues (sample-mean and sample-standard deviation) to compute the sample correlation:
Suppose {\(X_1, X_2, X_3, \cdots, X_n\)} and {\(Y_1, Y_2, Y_3, \cdots, Y_n\)} are bivariate observations of the same process and \((\mu_X, \sigma_X)\) and \((\mu_Y, \sigma_Y)\) are the means and standard deviations for the X and Y measurements, respectively.

\[ \rho_{x,y}=\frac{\sum x_iy_i-n \bar{x} \bar{y}}{(n-1) s_x s_y}=\frac{n\sum x_iy_i-\sum x_i\sum y_i} {\sqrt{n\sum x_i^2-(\sum x_i)^2}~\sqrt{n\sum y_i^2-(\sum y_i)^2}}. \]

\[ \rho_{x,y}=\frac{\sum (x_i-\bar{x})(y_i-\bar{y})}{(n-1) s_x s_y} = {1 \over n-1} \sum_{i=1}^n { {x_i-\bar{x} \over s_x} {y_i-\bar{y}\over s_y}}, \]

where \(\bar{x}\) and \(\bar{y}\) are the sample means of X  and Y , sx  and sy  are the sample standard deviations of X  and Y  and the sum is from i = 1 to n. We may rewrite this as

\[ \rho_{x,y}=\frac{\sum x_iy_i-n \bar{x} \bar{y}}{(n-1) s_x s_y}=\frac{n\sum x_iy_i-\sum x_i\sum y_i} {\sqrt{n\sum x_i^2-(\sum x_i)^2}~\sqrt{n\sum y_i^2-(\sum y_i)^2}}. \]

  • Note: The correlation is defined only if both of the standard deviations are finite and are nonzero. It is a corollary of the Cauchy-Schwarz inequality that the correlation is always bounded \(-1 \leq \rho \leq 1\).

Examples

Human weight and height

Suppose we took only 6 of the over 25,000 observations of human weight and height included in this SOCR Dataset.

Subject Index Height(\(x_i\)) in cm Weight (\(y_i\)) in kg \(x_i-\bar{x}\) \(y_i-\bar{y}\) \((x_i-\bar{x})^2\) \((y_i-\bar{y})^2\) \((x_i-\bar{x})(y_i-\bar{y})\)
1 167 60 6 4.67 36 21,82 28.02
2 170 64 9 8.67 81 75.17 78.03
3 160 57 -1 1.67 1 2.79 -1.67
4 152 46 -9 -9.33 81 87.05 83.97
5 157 55 -4 -0.33 16 0.11 1.32
6 160 50 -1 -5.33 1 28.41 5.33
Total 966 332 0 0 216 215.33 195.0

We can easily now compute by hand \(\bar{x}=966/6=161\) (cm), \(\bar{y}=332/6=55\) (kg), \(s_x=\sqrt{216/5}=6.57\) and \(s_y=\sqrt{215.3/5}=6.56\).

Therefore, \( \rho_{x,y}= {1 \over n-1} \sum_{i=1}^n { {x_i-\bar{x} \over s_x} {y_i-\bar{y}\over s_y}} = 0.904. \)

Of course, these calculations become difficult for more than a few paired observations and that is why we use the Simple Linear Regression in SOCR Analyses to compute the correlation and other linear associations in the bivariate case. The image below shows the calculations for the same data shown above in SOCR.

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Use the Simple Linear Regression to compute the correlation between the height and weight in the first 200 measurements in the human weight and height included in this SOCR Dataset.

Hot-dogs dataset

Use the Simple Linear Regression to compute the correlation between the calories and sodium in the Hot-dogs dataset.

Airfare Example

Suppose we have the following bivariate X={airfare} and Y={distance traveled from Baltimore, MD} measurements:

Destination Distance Airfare
Atlanta 576 178
Boston 370 138
Chicago 612 94
Dallas 1216 278
Detroit 409 158
Denver 1502 258
Miami 946 198
New_Orleans 998 188
New_York 189 98
Orlando 787 179
Pittsburgh 210 138
St._Louis 737 98

Use the Simple Linear Regression to find the correlation between ticket fare and the distance traveled by passengers. Explain your findings.

Properties of the Correlation Coefficient

  • The correlation is associative operation\[\rho_{(X,Y)} = \rho_{(Y,X)}\]
  • The correlation is (almost) linearly invariant\[\rho_{(aX+b,Y)} = \sgn(a)\times \rho_{(X,Y)}\]. If \(a>0\), then \(\rho_{(aX+b,Y)} = \rho_{(X,Y)}\). If \(a<0\), then \(\rho_{(aX+b,Y)} = -\rho_{(X,Y)}\).
  • A trivial correlation, \(\rho_{X,Y}=0\) only implies that there is no linear relation between X and Y, but there may be other relations (e.g., quadratic). Therefore, statistical independence of X and Y does imply that \(\rho_{X,Y}=0\). However the converse is false, \(\rho_{X,Y}=0\) does not imply independence!
  • A high correlation between X and Y does not imply causality (i.e., does not mean that one of the variables causes the observed behavior in the other. For example, consider X={math scores} and Y={shoe size) for all K-12 students. X and Y are very highly positively correlated, yet higher shoe sizes do not imply better math skills.
  • The complete properties of the Correlation coefficients may be found here.

Statistical inference on correlation coefficients

  • 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 \(rho\). If our bivariate sample is of size N and the observed sample correlation is r, then the test statistics is:
\[t_o = \frac{r-\rho}{\sqrt{\frac{1-r^2}{N-2}}},\] which has T-distribution with df=N-2.
  • Comparing two correlation coefficients: The Fisher's transformation provides a mechanism to test for comparing two correlation coefficients using Normal distribution. Suppose we have 2 independent paired samples \(\{(X_i,Y_i)\}_{i=1}^{n_1}\) and \(\{(U_j,V_j)\}_{j=1}^{n_2}\), 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:
\[\hat{r}= \frac{1}{2} \log_e \| \frac{1+r}{1-r} \|.\]
Transforming the two correlation coefficients separately yields:
\[r11= \frac{1}{2} \log_e \| \frac{1+r1}{1-r1} \|,\] and
\[r22= \frac{1}{2} \log_e \| \frac{1+r22}{1-r22} \|.\]
Then the test statistics \(Z_o= \frac{r11-r22}{\sqrt{\frac{1}{n1-3}+ \frac{1}{n2-3}}}\) 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 (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

The following dataset represents the brain volume (responses) measurements and ages (predictors) for 2 cohorts of subjects (Group1 and Group2):

Group1 Age1 Volume1 Group2 Age2 Volume2
1 64 0.245517 2 30 0.29165
1 30 0.308443 2 38 0.297111
1 37 0.294692 2 46 0.275401
1 62 0.279998 2 55 0.284923
1 59 0.256802 2 37 0.287809
1 51 0.293875 2 41 0.291287
1 52 0.28895 2 57 0.26833
1 59 0.29262 2 69 0.268375
1 33 0.283666 2 69 0.253352
1 47 0.27458 2 41 0.292208
1 62 0.27269 2 60 0.276306
1 58 0.269609 2 59 0.27905
1 55 0.277243 2 50 0.262916
1 61 0.236264 2 58 0.290697
1 70 0.218015 2 58 0.269361
1 38 0.287205 2 61 0.268247
1 41 0.307387 2 57 0.294204
1 40 0.271063 2 50 0.292699
1 25 0.307688 2 38 0.273969
1 70 0.237811 2 57 0.29049
1 49 0.293371 2 64 0.286564
1 56 0.252592 2 71 0.257386
1 56 0.251349 2 34 0.314958
1 40 0.29616 2 53 0.298022
1 50 0.249596 2 53 0.269229
1 55 0.282721 2 25 0.270634
1 69 0.247565 2 61 0.266905
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 compute the 2 correlation coefficients:
\[r1=corr(X,Y)=-0.75339,\] and
\[r2=corr(U,V) = -0.49491\]
Using the Fisher's transformation we obtain:
\[r11= \frac{1}{2} \log_e \| \frac{1+r1}{1-r1} \|=-0.980749,\] and
\[r22= \frac{1}{2} \log_e \| \frac{1+r2}{1-r2} \|=-0.5425423\]
And thus, \(Z_o=\frac{r11-r22}{\sqrt{\frac{1}{24} +\frac{1}{24}}}= -1.517993\) and a corresponding (1-sided) p-value = 0.064508 (double-sided p-value = 0.129016).

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