9/9/2009

Pearson’s correlation coefficient

Fitting a line to data

• In general, no straight line will pass through all of the points.  We need a method for selecting the line with best fit.

Method of Least Squares

• Look at the deviations of the plotted points from the line in the vertical direction.  (Residual = observed - predicted)
• Compute the sum of the squares of the deviations.  (sum of squared residuals)
• Choose the line that minimizes the sum of squared residuals.  This line is called the least squares regression line

Computing the least squares regression line

• Exercise 5.44 on p. 226 (predicting telomere length from perceived stress)
• Body measurements (p:\data\math\stats\neckwast.mtw)

Connections between correlation and regression

• r-squared, the square of the correlation coefficient, represents the fraction of variation in the values of Y that is explained by using least squares regression with the explanatory variable X.
• The slope of the least squares regression line can be written as b = r * (sy/sx).

Class Exercises

• Exercise 5.9 on p. 199
• Exercise 5.66 on p. 260
• The federal government maintains a large facility in Hanford WA at which various activities related to nuclear reactors and bombs are carried out. The facility occupies land adjacent to the Columbia River. Over the years there has been leaks from open pit storage of radioactive wastes into the Columbia River. Data concerning cancer downstream from Hanford in Oregon has been collected from nine counties in Oregon, and can be compared with data on radioactivity levels. Some of the data from a study conducted in 1959-64 is contained in the minitab worksheet p:\data\math\stats\hanford.mtw. The index of exposure (C1) was based on several factors, including distance from Hanford, and average distance of the population from water frontage on the Columbia. Column C2 gives the Cancer Mortality per 100,000 person-years and column C3 gives the name of the county.

  a. Make a scatterplot of the data. Which variable is the explanatory variable?
  b. Is the association between the variables positive or negative?
  c. Find the least squares regression line for predicting cancer deaths from the index of exposure. For each of the exposure indexes, compute the predicted value of cancer mortality and the associated residual.
  d. What percentage of the variation in cancer deaths is explained by using the index of exposure?
  e. Interpret the value of the slope in the least squares line. i.e., explain what this slope says about the change in cancer death rates for different exposure indexes.
  f. Plot the residuals versus the index of exposure. What does the plot indicate about the adequacy of the linear fit?
  g. Make another scatterplot of the data and include the least squares line on the plot.
  h. Suppose you lived in a county with radioactive contamination index of exposure equal to 5. Use the least squares line to predict the cancer mortality in your home county.
  i. Compute the correlation coefficient r between index of exposure and cancer mortality.
  j. Create two new variables x^* = 10x and y^*=y/10. This can be done easily by using Calc > Calculator to create:

  c4=10*c1
  c5=c2/10

  k. Make a scatterplot of the transformed indexes and mortality rates. Does this plot have the same appearance as the plot you constructed in part a?
  l. Is the correlation coefficient for the transformed values the same as the correlation coefficient for the original values?
  m. Does the slope of the least squares line of y^* on x^* have the same slope as the regression line of y on x?

• Many libraries have gates that automatically count persons as they leave the building, thus making it easy to find out how many persons used the library in a given year. (Of course, someone who enters, leaves, and enters again is counted twice.) Jeff Witmer at Oberlin College conducted a survey of liberal arts college libraries and found the following descriptive statistics on enrollment and gate count.

  MTB > correlation 'enroll' 'gate'

  Correlation of Enroll and Gate = 0.701

  MTB > describe 'enroll' 'gate'
   

  	N	MEAN	MEDIAN	TRMEAN	STDEV	SEMEAN
  Enroll	17	2009	2007	2024	657	159
  Gate	17	247235	254116	247827	104807	25419

  
   

  	MIN	MAX	Q1	Q3
  Enroll	810	2980	1622	2642
  Gate	54738	430858	151011	347350

  a. Find the equation of the least squares line for predicting the gate count from enrollment.
  b. What percentage of the variation in the gate counts is explained by enrollments?
  c. Predict the number of persons that will use the library at a small liberal arts college with an enrollment of 1445.
  d. One of the reporting colleges has an enrollment of 2200 and a gate count of 130000. Find the value of the residual for this college.

Please read Sections 5.4 for class on Friday