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Think Stats by Allen B. Downey Think Stats is an introduction to Probability and Statistics for Python programmers.

This is the accompanying code for this book.

Website: http://greenteapress.com/wp/think-stats-2e/

Project: Support and Testing

Path: think-stats-code / chap11ex.ipynb

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License: GPL3

Kernel: Python 3

Examples and Exercises from Think Stats, 2nd Edition

http://thinkstats2.com

MIT License: https://opensource.org/licenses/MIT

In [1]:

from __future__ import print_function, division

%matplotlib inline

import numpy as np
import pandas as pd

import random

import thinkstats2
import thinkplot

Multiple regression

Let's load up the NSFG data again.

In [2]:

import first

live, firsts, others = first.MakeFrames()

Here's birth weight as a function of mother's age (which we saw in the previous chapter).

In [3]:

import statsmodels.formula.api as smf

formula = 'totalwgt_lb ~ agepreg'
model = smf.ols(formula, data=live)
results = model.fit()
results.summary()

OLS Regression Results
Dep. Variable:	totalwgt_lb	R-squared:	0.005
Model:	OLS	Adj. R-squared:	0.005
Method:	Least Squares	F-statistic:	43.02
Date:	Mon, 25 Feb 2019	Prob (F-statistic):	5.72e-11
Time:	16:34:15	Log-Likelihood:	-15897.
No. Observations:	9038	AIC:	3.180e+04
Df Residuals:	9036	BIC:	3.181e+04
Df Model:	1
Covariance Type:	nonrobust

	coef	std err	t	P>\|t\|	[0.025	0.975]
Intercept	6.8304	0.068	100.470	0.000	6.697	6.964
agepreg	0.0175	0.003	6.559	0.000	0.012	0.023

Omnibus:	1024.052	Durbin-Watson:	1.618
Prob(Omnibus):	0.000	Jarque-Bera (JB):	3081.833
Skew:	-0.601	Prob(JB):	0.00
Kurtosis:	5.596	Cond. No.	118.

Warnings:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

We can extract the parameters.

In [4]:

inter = results.params['Intercept']
slope = results.params['agepreg']
inter, slope

(6.830396973311047, 0.017453851471802877)

And the p-value of the slope estimate.

In [5]:

slope_pvalue = results.pvalues['agepreg']
slope_pvalue

5.722947107312786e-11

And the coefficient of determination.

In [6]:

results.rsquared

0.004738115474710369

The difference in birth weight between first babies and others.

In [7]:

diff_weight = firsts.totalwgt_lb.mean() - others.totalwgt_lb.mean()
diff_weight

-0.12476118453549034

The difference in age between mothers of first babies and others.

In [8]:

diff_age = firsts.agepreg.mean() - others.agepreg.mean()
diff_age

-3.5864347661500275

The age difference plausibly explains about half of the difference in weight.

In [9]:

slope * diff_age

-0.06259709972169267

Running a single regression with a categorical variable, isfirst:

In [10]:

live['isfirst'] = live.birthord == 1
formula = 'totalwgt_lb ~ isfirst'
results = smf.ols(formula, data=live).fit()
results.summary()

OLS Regression Results
Dep. Variable:	totalwgt_lb	R-squared:	0.002
Model:	OLS	Adj. R-squared:	0.002
Method:	Least Squares	F-statistic:	17.74
Date:	Mon, 25 Feb 2019	Prob (F-statistic):	2.55e-05
Time:	16:34:16	Log-Likelihood:	-15909.
No. Observations:	9038	AIC:	3.182e+04
Df Residuals:	9036	BIC:	3.184e+04
Df Model:	1
Covariance Type:	nonrobust

	coef	std err	t	P>\|t\|	[0.025	0.975]
Intercept	7.3259	0.021	356.007	0.000	7.286	7.366
isfirst[T.True]	-0.1248	0.030	-4.212	0.000	-0.183	-0.067

Omnibus:	988.919	Durbin-Watson:	1.613
Prob(Omnibus):	0.000	Jarque-Bera (JB):	2897.107
Skew:	-0.589	Prob(JB):	0.00
Kurtosis:	5.511	Cond. No.	2.58

Warnings:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

Now finally running a multiple regression:

In [11]:

formula = 'totalwgt_lb ~ isfirst + agepreg'
results = smf.ols(formula, data=live).fit()
results.summary()

OLS Regression Results
Dep. Variable:	totalwgt_lb	R-squared:	0.005
Model:	OLS	Adj. R-squared:	0.005
Method:	Least Squares	F-statistic:	24.02
Date:	Mon, 25 Feb 2019	Prob (F-statistic):	3.95e-11
Time:	16:34:16	Log-Likelihood:	-15894.
No. Observations:	9038	AIC:	3.179e+04
Df Residuals:	9035	BIC:	3.182e+04
Df Model:	2
Covariance Type:	nonrobust

	coef	std err	t	P>\|t\|	[0.025	0.975]
Intercept	6.9142	0.078	89.073	0.000	6.762	7.066
isfirst[T.True]	-0.0698	0.031	-2.236	0.025	-0.131	-0.009
agepreg	0.0154	0.003	5.499	0.000	0.010	0.021

Omnibus:	1019.945	Durbin-Watson:	1.618
Prob(Omnibus):	0.000	Jarque-Bera (JB):	3063.682
Skew:	-0.599	Prob(JB):	0.00
Kurtosis:	5.588	Cond. No.	137.

Warnings:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

As expected, when we control for mother's age, the apparent difference due to isfirst is cut in half.

If we add age squared, we can control for a quadratic relationship between age and weight.

In [12]:

live['agepreg2'] = live.agepreg**2
formula = 'totalwgt_lb ~ isfirst + agepreg + agepreg2'
results = smf.ols(formula, data=live).fit()
results.summary()

OLS Regression Results
Dep. Variable:	totalwgt_lb	R-squared:	0.007
Model:	OLS	Adj. R-squared:	0.007
Method:	Least Squares	F-statistic:	22.64
Date:	Mon, 25 Feb 2019	Prob (F-statistic):	1.35e-14
Time:	16:34:16	Log-Likelihood:	-15884.
No. Observations:	9038	AIC:	3.178e+04
Df Residuals:	9034	BIC:	3.181e+04
Df Model:	3
Covariance Type:	nonrobust

	coef	std err	t	P>\|t\|	[0.025	0.975]
Intercept	5.6923	0.286	19.937	0.000	5.133	6.252
isfirst[T.True]	-0.0504	0.031	-1.602	0.109	-0.112	0.011
agepreg	0.1124	0.022	5.113	0.000	0.069	0.155
agepreg2	-0.0018	0.000	-4.447	0.000	-0.003	-0.001

Omnibus:	1007.149	Durbin-Watson:	1.616
Prob(Omnibus):	0.000	Jarque-Bera (JB):	3003.343
Skew:	-0.594	Prob(JB):	0.00
Kurtosis:	5.562	Cond. No.	1.39e+04

Warnings:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.
[2] The condition number is large, 1.39e+04. This might indicate that there are
strong multicollinearity or other numerical problems.

When we do that, the apparent effect of isfirst gets even smaller, and is no longer statistically significant.

These results suggest that the apparent difference in weight between first babies and others might be explained by difference in mothers' ages, at least in part.

Data Mining

We can use join to combine variables from the preganancy and respondent tables.

In [13]:

import nsfg

live = live[live.prglngth>30]
resp = nsfg.ReadFemResp()
resp.index = resp.caseid
join = live.join(resp, on='caseid', rsuffix='_r')

And we can search for variables with explanatory power.

Because we don't clean most of the variables, we are probably missing some good ones.

In [14]:

import patsy

def GoMining(df):
    """Searches for variables that predict birth weight.

    df: DataFrame of pregnancy records

    returns: list of (rsquared, variable name) pairs
    """
    variables = []
    for name in df.columns:
        try:
            if df[name].var() < 1e-7:
                continue

            formula = 'totalwgt_lb ~ agepreg + ' + name
            
            # The following seems to be required in some environments
            # formula = formula.encode('ascii')

            model = smf.ols(formula, data=df)
            if model.nobs < len(df)/2:
                continue

            results = model.fit()
        except (ValueError, TypeError):
            continue

        variables.append((results.rsquared, name))

    return variables

In [15]:

variables = GoMining(join)

The following functions report the variables with the highest values of $R^2$ .

In [16]:

import re

def ReadVariables():
    """Reads Stata dictionary files for NSFG data.

    returns: DataFrame that maps variables names to descriptions
    """
    vars1 = thinkstats2.ReadStataDct('2002FemPreg.dct').variables
    vars2 = thinkstats2.ReadStataDct('2002FemResp.dct').variables

    all_vars = vars1.append(vars2)
    all_vars.index = all_vars.name
    return all_vars

def MiningReport(variables, n=30):
    """Prints variables with the highest R^2.

    t: list of (R^2, variable name) pairs
    n: number of pairs to print
    """
    all_vars = ReadVariables()

    variables.sort(reverse=True)
    for r2, name in variables[:n]:
        key = re.sub('_r$', '', name)
        try:
            desc = all_vars.loc[key].desc
            if isinstance(desc, pd.Series):
                desc = desc[0]
            print(name, r2, desc)
        except (KeyError, IndexError):
            print(name, r2)

Some of the variables that do well are not useful for prediction because they are not known ahead of time.

In [17]:

MiningReport(variables)

totalwgt_lb 1.0
birthwgt_lb 0.9498127305978009 BD-3 BIRTHWEIGHT IN POUNDS - 1ST BABY FROM THIS PREGNANCY
lbw1 0.30082407844707704 LOW BIRTHWEIGHT - BABY 1
prglngth 0.13012519488625052 DURATION OF COMPLETED PREGNANCY IN WEEKS
wksgest 0.12340041363361054 GESTATIONAL LENGTH OF COMPLETED PREGNANCY (IN WEEKS)
agecon 0.10203149928155986 AGE AT TIME OF CONCEPTION
mosgest 0.027144274639579802 GESTATIONAL LENGTH OF COMPLETED PREGNANCY (IN MONTHS)
babysex 0.01855092529394209 BD-2 SEX OF 1ST LIVEBORN BABY FROM THIS PREGNANCY
race_r 0.016199503586253217
race 0.016199503586253217
nbrnaliv 0.016017752709788113 BC-2 NUMBER OF BABIES BORN ALIVE FROM THIS PREGNANCY
paydu 0.014003795578114597 IB-10 CURRENT LIVING QUARTERS OWNED/RENTED, ETC
rmarout03 0.013430066465713209 INFORMAL MARITAL STATUS WHEN PREGNANCY ENDED - 3RD
birthwgt_oz 0.013102457615706053 BD-3 BIRTHWEIGHT IN OUNCES - 1ST BABY FROM THIS PREGNANCY
anynurse 0.012529022541810764 BH-1 WHETHER R BREASTFED THIS CHILD AT ALL - 1ST FROM THIS PREG
bfeedwks 0.012193688404495862 DURATION OF BREASTFEEDING IN WEEKS
totincr 0.011870069031173491 TOTAL INCOME OF R'S FAMILY
marout03 0.011807801994374811 FORMAL MARITAL STATUS WHEN PREGNANCY ENDED - 3RD
marcon03 0.011752599354395654 FORMAL MARITAL STATUS WHEN PREGNANCY BEGAN - 3RD
cebow 0.011437770919637047 NUMBER OF CHILDREN BORN OUT OF WEDLOCK
rmarout01 0.011407737138640073 INFORMAL MARITAL STATUS WHEN PREGNANCY ENDED - 1ST
rmarout6 0.011354138472805642 INFORMAL MARITAL STATUS AT PREGNANCY OUTCOME - 6 CATEGORIES
marout01 0.011269357246806555 FORMAL MARITAL STATUS WHEN PREGNANCY ENDED - 1ST
hisprace_r 0.011238349302030715
hisprace 0.011238349302030715
mar1diss 0.010961563590751733 MONTHS BTW/1ST MARRIAGE & DISSOLUTION (OR INTERVIEW)
fmarcon5 0.0106049646842995 FORMAL MARITAL STATUS AT CONCEPTION - 5 CATEGORIES
rmarout02 0.0105469132065652 INFORMAL MARITAL STATUS WHEN PREGNANCY ENDED - 2ND
marcon02 0.010481401795534362 FORMAL MARITAL STATUS WHEN PREGNANCY BEGAN - 2ND
fmarout5 0.010461691367377068 FORMAL MARITAL STATUS AT PREGNANCY OUTCOME

Combining the variables that seem to have the most explanatory power.

In [18]:

formula = ('totalwgt_lb ~ agepreg + C(race) + babysex==1 + '
               'nbrnaliv>1 + paydu==1 + totincr')
results = smf.ols(formula, data=join).fit()
results.summary()

OLS Regression Results
Dep. Variable:	totalwgt_lb	R-squared:	0.060
Model:	OLS	Adj. R-squared:	0.059
Method:	Least Squares	F-statistic:	79.98
Date:	Mon, 25 Feb 2019	Prob (F-statistic):	4.86e-113
Time:	16:35:03	Log-Likelihood:	-14295.
No. Observations:	8781	AIC:	2.861e+04
Df Residuals:	8773	BIC:	2.866e+04
Df Model:	7
Covariance Type:	nonrobust

	coef	std err	t	P>\|t\|	[0.025	0.975]
Intercept	6.6303	0.065	102.223	0.000	6.503	6.757
C(race)[T.2]	0.3570	0.032	11.215	0.000	0.295	0.419
C(race)[T.3]	0.2665	0.051	5.175	0.000	0.166	0.367
babysex == 1[T.True]	0.2952	0.026	11.216	0.000	0.244	0.347
nbrnaliv > 1[T.True]	-1.3783	0.108	-12.771	0.000	-1.590	-1.167
paydu == 1[T.True]	0.1196	0.031	3.861	0.000	0.059	0.180
agepreg	0.0074	0.003	2.921	0.004	0.002	0.012
totincr	0.0122	0.004	3.110	0.002	0.005	0.020

Omnibus:	398.813	Durbin-Watson:	1.604
Prob(Omnibus):	0.000	Jarque-Bera (JB):	1388.362
Skew:	-0.037	Prob(JB):	3.32e-302
Kurtosis:	4.947	Cond. No.	221.

Warnings:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

Logistic regression

Example: suppose we are trying to predict y using explanatory variables x1 and x2.

In [19]:

y = np.array([0, 1, 0, 1])
x1 = np.array([0, 0, 0, 1])
x2 = np.array([0, 1, 1, 1])

According to the logit model the log odds for the $i$ th element of $y$ is

$\log o = \beta_0 + \beta_1 x_1 + \beta_2 x_2$

So let's start with an arbitrary guess about the elements of $\beta$ :

In [20]:

beta = [-1.5, 2.8, 1.1]

Plugging in the model, we get log odds.

In [21]:

log_o = beta[0] + beta[1] * x1 + beta[2] * x2
log_o

array([-1.5, -0.4, -0.4,  2.4])

Which we can convert to odds.

In [22]:

o = np.exp(log_o)
o

array([ 0.22313016,  0.67032005,  0.67032005, 11.02317638])

And then convert to probabilities.

In [23]:

p = o / (o+1)
p

array([0.18242552, 0.40131234, 0.40131234, 0.9168273 ])

The likelihoods of the actual outcomes are $p$ where $y$ is 1 and $1-p$ where $y$ is 0.

In [24]:

likes = np.where(y, p, 1-p)
likes

array([0.81757448, 0.40131234, 0.59868766, 0.9168273 ])

The likelihood of $y$ given $\beta$ is the product of likes:

In [25]:

like = np.prod(likes)
like

0.1800933529673034

Logistic regression works by searching for the values in $\beta$ that maximize like.

Here's an example using variables in the NSFG respondent file to predict whether a baby will be a boy or a girl.

In [26]:

import first
live, firsts, others = first.MakeFrames()
live = live[live.prglngth>30]
live['boy'] = (live.babysex==1).astype(int)

The mother's age seems to have a small effect.

In [27]:

model = smf.logit('boy ~ agepreg', data=live)
results = model.fit()
results.summary()

Optimization terminated successfully.
         Current function value: 0.693015
         Iterations 3

Logit Regression Results
Dep. Variable:	boy	No. Observations:	8884
Model:	Logit	Df Residuals:	8882
Method:	MLE	Df Model:	1
Date:	Mon, 25 Feb 2019	Pseudo R-squ.:	6.144e-06
Time:	16:35:05	Log-Likelihood:	-6156.7
converged:	True	LL-Null:	-6156.8
		LLR p-value:	0.7833

	coef	std err	z	P>\|z\|	[0.025	0.975]
Intercept	0.0058	0.098	0.059	0.953	-0.185	0.197
agepreg	0.0010	0.004	0.275	0.783	-0.006	0.009

Here are the variables that seemed most promising.

In [28]:

formula = 'boy ~ agepreg + hpagelb + birthord + C(race)'
model = smf.logit(formula, data=live)
results = model.fit()
results.summary()

Optimization terminated successfully.
         Current function value: 0.692944
         Iterations 3

Logit Regression Results
Dep. Variable:	boy	No. Observations:	8782
Model:	Logit	Df Residuals:	8776
Method:	MLE	Df Model:	5
Date:	Mon, 25 Feb 2019	Pseudo R-squ.:	0.0001440
Time:	16:35:05	Log-Likelihood:	-6085.4
converged:	True	LL-Null:	-6086.3
		LLR p-value:	0.8822

	coef	std err	z	P>\|z\|	[0.025	0.975]
Intercept	-0.0301	0.104	-0.290	0.772	-0.234	0.173
C(race)[T.2]	-0.0224	0.051	-0.439	0.660	-0.122	0.077
C(race)[T.3]	-0.0005	0.083	-0.005	0.996	-0.163	0.162
agepreg	-0.0027	0.006	-0.484	0.629	-0.014	0.008
hpagelb	0.0047	0.004	1.112	0.266	-0.004	0.013
birthord	0.0050	0.022	0.227	0.821	-0.038	0.048

To make a prediction, we have to extract the exogenous and endogenous variables.

In [29]:

endog = pd.DataFrame(model.endog, columns=[model.endog_names])
exog = pd.DataFrame(model.exog, columns=model.exog_names)

The baseline prediction strategy is to guess "boy". In that case, we're right almost 51% of the time.

In [30]:

actual = endog['boy']
baseline = actual.mean()
baseline

0.507173764518333

If we use the previous model, we can compute the number of predictions we get right.

In [31]:

predict = (results.predict() >= 0.5)
true_pos = predict * actual
true_neg = (1 - predict) * (1 - actual)
sum(true_pos), sum(true_neg)

(3944.0, 548.0)

And the accuracy, which is slightly higher than the baseline.

In [32]:

acc = (sum(true_pos) + sum(true_neg)) / len(actual)
acc

0.5115007970849464

To make a prediction for an individual, we have to get their information into a DataFrame.

In [33]:

columns = ['agepreg', 'hpagelb', 'birthord', 'race']
new = pd.DataFrame([[35, 39, 3, 2]], columns=columns)
y = results.predict(new)
y

0    0.513091
dtype: float64

This person has a 51% chance of having a boy (according to the model).

Exercises

Exercise: Suppose one of your co-workers is expecting a baby and you are participating in an office pool to predict the date of birth. Assuming that bets are placed during the 30th week of pregnancy, what variables could you use to make the best prediction? You should limit yourself to variables that are known before the birth, and likely to be available to the people in the pool.

In [34]:

import first
live, firsts, others = first.MakeFrames()
live = live[live.prglngth>30]

The following are the only variables I found that have a statistically significant effect on pregnancy length.

In [35]:

import statsmodels.formula.api as smf
model = smf.ols('prglngth ~ birthord==1 + race==2 + nbrnaliv>1', data=live)
results = model.fit()
results.summary()

OLS Regression Results
Dep. Variable:	prglngth	R-squared:	0.011
Model:	OLS	Adj. R-squared:	0.011
Method:	Least Squares	F-statistic:	34.28
Date:	Mon, 25 Feb 2019	Prob (F-statistic):	5.09e-22
Time:	16:35:08	Log-Likelihood:	-18247.
No. Observations:	8884	AIC:	3.650e+04
Df Residuals:	8880	BIC:	3.653e+04
Df Model:	3
Covariance Type:	nonrobust

	coef	std err	t	P>\|t\|	[0.025	0.975]
Intercept	38.7617	0.039	1006.410	0.000	38.686	38.837
birthord == 1[T.True]	0.1015	0.040	2.528	0.011	0.023	0.180
race == 2[T.True]	0.1390	0.042	3.311	0.001	0.057	0.221
nbrnaliv > 1[T.True]	-1.4944	0.164	-9.086	0.000	-1.817	-1.172

Omnibus:	1587.470	Durbin-Watson:	1.619
Prob(Omnibus):	0.000	Jarque-Bera (JB):	6160.751
Skew:	-0.852	Prob(JB):	0.00
Kurtosis:	6.707	Cond. No.	10.9

Warnings:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

Exercise: The Trivers-Willard hypothesis suggests that for many mammals the sex ratio depends on “maternal condition”; that is, factors like the mother’s age, size, health, and social status. See https://en.wikipedia.org/wiki/Trivers-Willard_hypothesis

Some studies have shown this effect among humans, but results are mixed. In this chapter we tested some variables related to these factors, but didn’t find any with a statistically significant effect on sex ratio.

As an exercise, use a data mining approach to test the other variables in the pregnancy and respondent files. Can you find any factors with a substantial effect?

In [36]:

import regression
join = regression.JoinFemResp(live)

In [37]:

# Solution goes here

In [38]:

# Solution goes here

In [39]:

# Solution goes here

Exercise: If the quantity you want to predict is a count, you can use Poisson regression, which is implemented in StatsModels with a function called poisson. It works the same way as ols and logit. As an exercise, let’s use it to predict how many children a woman has born; in the NSFG dataset, this variable is called numbabes.

Suppose you meet a woman who is 35 years old, black, and a college graduate whose annual household income exceeds $75,000. How many children would you predict she has born?

In [40]:

# Solution goes here

In [41]:

# Solution goes here

Now we can predict the number of children for a woman who is 35 years old, black, and a college graduate whose annual household income exceeds $75,000

In [42]:

# Solution goes here

Exercise: If the quantity you want to predict is categorical, you can use multinomial logistic regression, which is implemented in StatsModels with a function called mnlogit. As an exercise, let’s use it to guess whether a woman is married, cohabitating, widowed, divorced, separated, or never married; in the NSFG dataset, marital status is encoded in a variable called rmarital.

Suppose you meet a woman who is 25 years old, white, and a high school graduate whose annual household income is about $45,000. What is the probability that she is married, cohabitating, etc?

In [43]:

# Solution goes here

Make a prediction for a woman who is 25 years old, white, and a high school graduate whose annual household income is about $45,000.

In [44]:

# Solution goes here

In [ ]: