Old School T, Chi Squared, and F Charts¶
Introduction¶
I had an idle whim while reading a very old introduction to statistics book (Facts from Figures, M. J. Moroney), "I wonder if I can create the various charts shown in the book" - some are hand drawn!. This from the days when reading a value off a chart was the preferred way to get a function value.
It actually taught me a lot about Matplotlib, and how to manage the appearence of a graphic!
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%load_ext watermark
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%reload_ext lab_black
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%matplotlib inline
All imports go here.
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import pandas as pd
import numpy as np
import seaborn as sn
import matplotlib.pyplot as plt
from matplotlib.ticker import FormatStrFormatter
from scipy import stats
F Charts¶
I decided to make a chart, that (given the Degrees of Freedom of Denominator and Numerator) you can use to look up the F statistic at a chosen confidence level (say N%). That is, given the Null Hypothesis of an equal variance estimate in the Denumerator and Numerator, you would expect a value equal to or smaller to the charted value N% of the time.
Thanks to scipy, the statistics part is easy; the charting, no so easy.
Lets walk through the code below in detail.
- I define a set of steps 1 to 10 by steps of 1, and then 10 to 50 by steps of 10.
- I create a matplotlib Figure abd Axes object, at a respectable size
- I then loop through all the Denominator Degrees of Freedom
- I create a list of F values for the Numerator Degrees of Freedom
- I plot it on the Axes object as a line
- at the right end of the line, I add text showing the Denominator Degree of Freedom
- At this stage all the real plotting is done, all the rest is putting lipstick on the pig. We set the Y axis scale to be
log, which in Matplotlib land meanslog10(fair enough, I suppose, but it makes me flinch) - I ask for minor tick marks to be displayed (to make it easier to read off values). This seems to be the crucial call to get the minor divisions of the x axis shown in the grid
- I ask for a grid to be drawn on both x and y axis, showing both major and minor axis divisions
- I have to set the minor tick labels on the y axis to be small, so they don't overlap each other, or the major tick labels
- By default, log scale axis tick labels are in scientific style notation (i.e. 10, 10^2, 10^3). Because my range is so small, I am choosing to go with just normal integer formats for the Y axis (both major and minor)
- Now we move onto the X axis. I specify where I want my minor tick (and grid) divisions to go, which all the integers between 1 and 50 inclusive. I then extract out the multiple of 10, because that is where the major tick marks will go, and we don't want overlap.
- We tell matplotlib where our minor divisions go on the X axis
- We tell the X axis we want small labels for the minor tick labels
- Just to be consistent, we set the same formatter for the X axis tick labels, as we set for the Y axis
- We are now in familiar territory: we set the X and Y axis labels
- We specify the style of the major grid: black, on both X and Y axis, quite dark. Sadly
alphahas two meanings in this code: one (the parameter call) for input into the F function, one (in matplotlib calls, being the opacity of the lines) - For minor gridlines, we draw dashed lines, less dark, on both X and Y axis directions
- Finally, we set the Axes Object title (note: Not the Figure title)
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def show_f_chart(alpha=0.95):
# 1
steps = list(range(1, 10)) + list(range(10, 51, 10))
# 2
fig, ax = plt.subplots(figsize=(15, 15))
# 3
for denom in steps:
f95 = [stats.f.ppf(alpha, i, denom) for i in steps]
ax.plot(
steps, f95, 'b-', label='F Denom ' + str(denom)
)
ax.text(steps[-1], f95[-1], 'D=' + str(denom))
# end for
# 4
ax.set_yscale('log')
# 5
ax.minorticks_on()
# 6
ax.grid(True, axis='both', which='both')
# 7
ax.tick_params(axis='y', which='minor', labelsize=8)
# 8
ax.yaxis.set_minor_formatter(FormatStrFormatter('%d'))
ax.yaxis.set_major_formatter(FormatStrFormatter('%d'))
# 9
xminor = list(range(1, 51))
for i in range(10, 51, 10):
xminor.remove(i)
# endfor
# 10
ax.set_xticks(xminor, minor=True)
# 11
ax.tick_params(axis='x', which='minor', labelsize=8)
# 12
ax.xaxis.set_minor_formatter(FormatStrFormatter('%d'))
ax.xaxis.set_major_formatter(FormatStrFormatter('%d'))
# 13
ax.set_xlabel('Numerator DoF (N)')
ax.set_ylabel(f'F(N,D) {alpha} value')
# 14
ax.grid(
which='major', alpha=0.9, color='k', axis='both'
)
# 15
ax.grid(
which='minor',
linestyle='dashed',
alpha=0.5,
color='k',
axis='both',
)
# 16
ax.set_title(
f'F Test ({alpha})\n[D is Denominator DoF]'
)
return ax
# end show_f_chart
_ = show_f_chart(0.95)
Same again, except for 99% confidence level
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_ = show_f_chart(0.99)
T Statistic¶
We repeat the same steps for the T statistic, except that I was happy with the default minor divisions on the X axis
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steps = list(range(1, 11)) + list(range(20, 101, 10))
fig, ax = plt.subplots(figsize=(15, 15))
# plt.ylim(bottom=1, top=100)
t99 = [stats.t.ppf(0.995, i) for i in steps]
t95 = [stats.t.ppf(0.975, i) for i in steps]
t999 = [stats.t.ppf(0.9995, i) for i in steps]
ax.plot(steps, t99, 'b-')
ax.plot(steps, t95, 'b-')
ax.plot(steps, t999, 'b-')
ax.text(steps[-1], t99[-1], '99.5%')
ax.text(steps[-1], t95[-1], '97.5%')
ax.text(steps[-1], t999[-1], '99.9%')
ax.set_xlim(left=1)
# 4
ax.set_yscale('log')
ax.set_xscale('log')
# 5
ax.minorticks_on()
# 6
ax.grid(True, axis='both', which='both')
# 7
ax.tick_params(axis='y', which='minor', labelsize=8)
# 8
ax.yaxis.set_minor_formatter(FormatStrFormatter('%d'))
ax.yaxis.set_major_formatter(FormatStrFormatter('%d'))
# 12
ax.xaxis.set_minor_formatter(FormatStrFormatter('%d'))
ax.xaxis.set_major_formatter(FormatStrFormatter('\n%d'))
# 13
ax.set_xlabel('DoF (N)')
ax.set_ylabel(f'T(N)')
# 14
ax.grid(which='major', alpha=0.9, color='k', axis='both')
# 15
ax.grid(
which='minor',
linestyle='dashed',
alpha=0.2,
color='k',
axis='both',
)
ax.set_title('T Test')
CHI Squared Charts¶
This time, we use Tex in the Y axis labels to add a little bit of class. As a bit of hackery, we label both ends of all lines, as some lines end below our Y axis limit
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steps = list(range(1, 30))
fig, ax = plt.subplots(figsize=(15, 15))
ci_limits = [0.95, 0.99, 0.999, 0.05, 0.01]
for ci in ci_limits:
chi2 = [stats.chi2.ppf(ci, i) for i in steps]
ax.plot(steps, chi2, 'b-')
if chi2[0] > 1:
ax.text(
steps[0],
chi2[0],
f'\n {1-ci:3.1%} Level',
verticalalignment='top',
)
# end if
ax.text(
steps[-1],
chi2[-1],
f' {1-ci:3.1%} Level',
verticalalignment='top',
)
# end for
ax.set_xlim(left=1)
ax.set_ylim(bottom=1)
# 4
ax.set_yscale('log')
ax.set_xscale('log')
# 5
ax.minorticks_on()
# 6
ax.grid(True, axis='both', which='both')
# 7
ax.tick_params(axis='y', which='minor', labelsize=8)
# 8
ax.yaxis.set_minor_formatter(FormatStrFormatter('%d'))
ax.yaxis.set_major_formatter(FormatStrFormatter('%d'))
# 12
ax.xaxis.set_minor_formatter(FormatStrFormatter('%d'))
ax.xaxis.set_major_formatter(FormatStrFormatter('\n%d'))
# 13
ax.set_xlabel('DoF (N)', fontsize=30)
ax.set_ylabel('$\chi^2(N)$', fontsize=30)
# 14
ax.grid(which='major', alpha=0.9, color='k', axis='both')
# 15
ax.grid(
which='minor',
linestyle='dashed',
alpha=0.2,
color='k',
axis='both',
)
ax.set_title('$\chi^2 $Test', fontsize=40)
Environment¶
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%watermark -h -iv
%watermark
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