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📚 Statistical Inference for Everyone
Chapters ▾
1 Introduction to Probability2 Applications of Probability3 Random Sequences and Visualization4 Introduction to Model Comparison5 Applications of Model Comparison6 Introduction to Parameter Estimation7 Priors, Likelihoods, and Posteriors8 Common Statistical Significance Tests9 Applications of Parameter Estimation and Inference10 Multi-parameter Models11 Introduction to MCMC12 Concluding Thoughts13 Appendix A: Computational Analysis14 Appendix B: Notation and Standards15 Appendix C: Common Distributions and Their Properties16 Appendix D: Tables

10.5 Computer Examples

from sie import *
data=load_data('data/shoesize.xls')
data.head()
Show expected output
   Index Gender  Size  Height
0      1      F   5.5      60
1      2      F   6.0      60
2      3      F   7.0      60
3      4      F   8.0      60
4      5      F   8.0      60
import random
random.seed(102)
rows = random.sample(data.index, 10)
newdata=data.ix[rows]
data=newdata
data
Show expected output
     Index Gender  Size  Height
60      61      F   7.0      64
251    252      M   9.0      70
69      70      F   8.0      64
290    291      M  11.0      71
247    248      M  12.0      69
156    157      F   9.5      68
231    232      M  10.0      69
17      18      F   6.5      61
216    217      M  10.0      68
252    253      M   9.0      70
plot(data['Height'],data['Size'],'o')
gca().set_xlim([60,72])
gca().set_ylim([4,14])
xlabel('Height [inches]')
ylabel('Shoe Size')
Show expected output
<matplotlib.text.Text at 0x10adae0d0>
Scatter of the nine-person shoe-size subset from the sample code: shoe sizes 6.5 to 12 against heights 61 to 71 inches, trending upward.
result=regression('Size ~ Height',data)
Show expected output
<matplotlib.figure.Figure at 0x10d200710>
Posterior curve for the regression intercept with dashed percentile lines: 1% at -40.27, 50% at -19.26 and 99% at 1.76.
Show expected output
<matplotlib.figure.Figure at 0x10d702610>
Posterior curve for the height coefficient β with dashed percentile lines: 1% at 0.11, 50% at 0.42 and 99% at 0.73; the entire curve sits above zero.
plot(data['Height'],data['Size'],'o')

h=linspace(60,72,10)
plot(h,result['_Predict'](Height=h),'-')

gca().set_xlim([60,72])
gca().set_ylim([4,14])
xlabel('Height [inches]')
ylabel('Shoe Size')

b=result.Intercept.mean()
m=result.Height.mean()

if b>0:
    text(62,12,'$y=%.3f x + %.3f$' % (m,b),fontsize=30)
else:
    text(62,12,'$y=%.3f x %.3f$' % (m,b),fontsize=30)
The shoe-size scatter with the fitted line y=0.422x-19.256 overlaid, running from about (60, 6) to (72, 11.2).
data=load_data('data/sat.csv')
result=regression('total ~ expenditure',data)
Show expected output
<matplotlib.figure.Figure at 0x1109156d0>
Posterior for the SAT regression intercept: a bell centered at 1089.29 with percentile lines from 982.47 (1%) to 1196.12 (99%).
Show expected output
<matplotlib.figure.Figure at 0x110953110>
Posterior for the expenditure coefficient in the SAT fit: a bell centered at -20.89 with percentile lines from -38.53 (1%) to -3.26 (99%) — entirely below zero.
plot(data['expenditure'],data['total'],'o')
xlabel('Expenditure [per pupil, thousands]')
ylabel('SAT Total')
h=linspace(3,10,10)
plot(h,result['_Predict'](expenditure=h),'-')

b=result.Intercept.mean()
m=result.expenditure.mean()

if b>0:
    text(4.5,1125,'$y=%.3f x + %.3f$' % (m,b),fontsize=30)
else:
    text(4.5,1125,'$y=%.3f x %.3f$' % (m,b),fontsize=30)
Scatter of total SAT score (about 840-1110) against per-pupil expenditure in thousands, with the downward fitted line y=-20.892x+1089.294. Spending more appears naively associated with LOWER scores.
result=regression('percent_taking ~ expenditure',data)
Show expected output
<matplotlib.figure.Figure at 0x111cfff10>
Posterior for the intercept of the percent-taking fit: a bell centered at -33.48 with percentile lines from -66.77 (1%) to -0.20 (99%).
Show expected output
<matplotlib.figure.Figure at 0x111867750>
Posterior for the expenditure coefficient in the percent-taking fit: a bell centered at 11.64 with percentile lines from 6.14 (1%) to 17.13 (99%) — entirely above zero.
plot(data['expenditure'],data['percent_taking'],'o')
xlabel('Expenditure [per pupil, thousands]')
ylabel('SAT Total')
h=linspace(3,10,10)
plot(h,result['_Predict'](expenditure=h),'-')

b=result.Intercept.mean()
m=result.expenditure.mean()

if b>0:
    text(4.5,85,'$y=%.3f x + %.3f$' % (m,b),fontsize=30)
else:
    text(4.5,85,'$y=%.3f x %.3f$' % (m,b),fontsize=30)
Scatter of the percentage of students taking the SAT against per-pupil expenditure with the rising fitted line y=11.638x-33.485; higher-spending states have far more of their students take the test. (The plot's y-axis label mistakenly reads 'SAT Total'.)
result=regression('total ~ expenditure + percent_taking',data)
Show expected output
<matplotlib.figure.Figure at 0x1107f5fd0>
Posterior for the intercept of the two-variable SAT model: a bell centered at 993.83 with percentile lines from 941.25 (1%) to 1046.41 (99%).
Show expected output
<matplotlib.figure.Figure at 0x10d70a690>
Posterior for the expenditure coefficient once percent-taking is included: a bell centered at 12.29 with percentile lines from 2.11 (1%) to 22.46 (99%) — now entirely positive.
Show expected output
<matplotlib.figure.Figure at 0x110fbcf90>
Posterior for the percent-taking coefficient in the two-variable SAT model: a bell centered at -2.85 with percentile lines from -3.37 (1%) to -2.33 (99%).

Adapted from Statistical Inference for Everyone, by Brian Blais (Bryant University), licensed under CC BY-SA 4.0 (dual-licensed under the GNU FDL 1.2 or later; this adaptation uses the CC BY-SA grant). Changes were made; this adaptation is distributed under the same license. License: CC-BY-SA-4.0.