Python Programming for Biology: Bioinformatics and Beyond

Figure 22.9.  An example of simple linear regression applied to a linearly correlated

data set. The two random variables X and Y have been fitted by the minimisation of the

difference, at a specified X-value, between the Y-value of each point and the Y-value at the

line. The Y-intercept, α (where x = 0) and gradient, β were calculated as illustrated in the

Python example shown in the text.

Accordingly,  the  fitted  line  has  a  gradient  and  Y-intercept  close  to  the  artificial  test

values of −0.7 and 2.0. There is also a handy function in the scipy.stats function linregress

which  calculates  lots  of  useful  things  in  one  fell  swoop,  including  the  correlation

coefficient and the two-tailed probability of the hypothesis that the gradient is zero. Here

we do the linear regression and then plot the best-fit line on the same graph as the input

data (like

Figure 22.9

):

from scipy.stats import linregress

from matplotlib import pyplot

grad, yInt, corrCoeff, pValue, stdErr = linregress(xVals, yVals)

print('LR 2:', grad, yInt, corrCoeff, pValue, stdErr)

# Result for one run was: -0.711, 2.04, -0.949, 9.639e-51, 0.0240

xValsFit = [xVals.min(),xVals.max()]

yValsFit = [yInt + x*grad for x in xValsFit]

pyplot.plot(xVals, yVals, 'o')

pyplot.plot(xValsFit, yValsFit)

pyplot.show()

Tailed tests and confidence intervals may be calculated more generally for the gradient

using a T-test (with n−2 degrees of freedom). Here the T-statistic is calculated as follows,

where   is the estimated gradient and

is a y value estimated by the fit line:

---

1

The probability of a Type I error is usually denoted by α, and the probability of a Type

II  error  by  β.  We  sometimes  speak  of  the  power  of  a  statistical  analysis,  which  is  the

probability  that  the  null  hypothesis  has  been  rejected  when  it  really  is  false,  which  is

equivalent to 1 − β.