Regressions

This notebook demonstrates various regressions, and accompanies the videos in Week 8.

Setup

We’re going to import our modules. Of particular note is statsmodels.formula.api, which gives us the regression APIs.

import pandas as pd
import numpy as np
import matplotlib.pyplot as plt
import seaborn as sns
import scipy.stats as sps
import statsmodels.api as sm
import statsmodels.formula.api as smf

We’re going to use the Penguins data:

penguins = pd.read_csv('../data/penguins.csv')
penguins.head()
species island bill_length_mm bill_depth_mm flipper_length_mm body_mass_g sex year
0 Adelie Torgersen 39.1 18.7 181.0 3750.0 male 2007
1 Adelie Torgersen 39.5 17.4 186.0 3800.0 female 2007
2 Adelie Torgersen 40.3 18.0 195.0 3250.0 female 2007
3 Adelie Torgersen NaN NaN NaN NaN NaN 2007
4 Adelie Torgersen 36.7 19.3 193.0 3450.0 female 2007

I’m also going to create a utility function for plotting lines:

def plot_line(intercept, slope, xmin, xmax, **kwargs):
    xs = np.linspace(xmin, xmax, 100)
    ys = intercept + slope * xs
    plt.plot(xs, ys, **kwargs)

Explore Correlations

The Seaborn pairplot function shows a scatterplot matrix, allowing us to view possible correlations:

sns.pairplot(penguins[['species', 'sex', 'bill_length_mm', 'bill_depth_mm', 'flipper_length_mm', 'body_mass_g']], hue='species')
plt.show()
../../../_images/Regressions_9_0.png

Let’s also look at the distribution of sex and species, to see if we have sex imbalances between species:

sns.countplot(x='species', hue='sex', data=penguins)
<AxesSubplot:xlabel='species', ylabel='count'>
../../../_images/Regressions_11_1.png

Nope!

Now a correlation matrix between our various numeric variables:

penguins.drop(columns=['year']).corr()
bill_length_mm bill_depth_mm flipper_length_mm body_mass_g
bill_length_mm 1.000000 -0.235053 0.656181 0.595110
bill_depth_mm -0.235053 1.000000 -0.583851 -0.471916
flipper_length_mm 0.656181 -0.583851 1.000000 0.871202
body_mass_g 0.595110 -0.471916 0.871202 1.000000

The .corr method on a data frame will compute correlations between all pairs of numeric variables, and show them in a table. The diagonal is 1, because a variable is always correlated with itself.

Dummy Variable Demo

Here’s the code that demonstrates dummy-coding from the Categorical Predictors lecture:

pd.get_dummies(penguins['species'])
Adelie Chinstrap Gentoo
0 1 0 0
1 1 0 0
2 1 0 0
3 1 0 0
4 1 0 0
... ... ... ...
339 0 1 0
340 0 1 0
341 0 1 0
342 0 1 0
343 0 1 0

344 rows × 3 columns

And dropping the first level:

pd.get_dummies(penguins['species'], drop_first=True)
Chinstrap Gentoo
0 0 0
1 0 0
2 0 0
3 0 0
4 0 0
... ... ...
339 1 0
340 1 0
341 1 0
342 1 0
343 1 0

344 rows × 2 columns

Flipper Regression

From the pair plot, the flipper length and body mass look like good candidates for regression. Let’s show them with a regplot:

sns.regplot(x='flipper_length_mm', y='body_mass_g', data=penguins, line_kws={'color': 'orange'})
plt.title('Body mass by flipper length')
plt.show()
../../../_images/Regressions_21_0.png

Now we’re going to fit a linear regression.

The first step is to create the OLS model. This has two inputs:

  1. The formula specifying the regression model (in this case, predicting body mass with flipper length)

  2. The data to train the model on

Let’s do it:

bm_mod = smf.ols('body_mass_g ~ flipper_length_mm', data=penguins)

Once we have created the model, we need to fit it. The fit method will return an object containing the results of fitting the model, such as its paramters; it can be summarized with summary:

bmf = bm_mod.fit()
bmf.summary()
OLS Regression Results
Dep. Variable: body_mass_g R-squared: 0.759
Model: OLS Adj. R-squared: 0.758
Method: Least Squares F-statistic: 1071.
Date: Fri, 22 Oct 2021 Prob (F-statistic): 4.37e-107
Time: 15:29:50 Log-Likelihood: -2528.4
No. Observations: 342 AIC: 5061.
Df Residuals: 340 BIC: 5069.
Df Model: 1
Covariance Type: nonrobust
coef std err t P>|t| [0.025 0.975]
Intercept -5780.8314 305.815 -18.903 0.000 -6382.358 -5179.305
flipper_length_mm 49.6856 1.518 32.722 0.000 46.699 52.672
Omnibus: 5.634 Durbin-Watson: 2.190
Prob(Omnibus): 0.060 Jarque-Bera (JB): 5.585
Skew: 0.313 Prob(JB): 0.0613
Kurtosis: 3.019 Cond. No. 2.89e+03


Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.
[2] The condition number is large, 2.89e+03. This might indicate that there are
strong multicollinearity or other numerical problems.

Now we need to check our residual assumptions. There are two plots to create:

  • Residuals vs. Fitted

  • Residual Q-Q

I’m going to create a function that plots them both:

def plot_lm_diag(fit):
    "Plot linear fit diagnostics"
    sns.regplot(x=fit.fittedvalues, y=fit.resid)
    plt.xlabel('Fitted')
    plt.ylabel('Residuals')
    plt.title('Residuals vs. Fitted')
    plt.show()

    sm.qqplot(fit.resid, fit=True, line='45')
    plt.title('Residuals')
    plt.show()

And now use it:

plot_lm_diag(bmf)
../../../_images/Regressions_29_0.png ../../../_images/Regressions_29_1.png

Standardizing Variables

We can also standardize our variables before fitting a model. This converts variable values into z-scores, such that

\[z_i = \frac{x_i - \bar{x}}{s}\]

Z-standardized (or z-normalized) variables have a mean of 0 and a standard deviation of 1 (and since \(1^2 = 1\), the variance is also 1). The result is that model coefficients are in units of standard deviations.

Let’s write a function to convert a series to its z-scores:

def zscore(xs):
    xbar = xs.mean()
    s = xs.std()
    return (xs - xbar) / s

And create standardized versions of our variables:

penguin_std = pd.DataFrame({
    'species': penguins['species'],
    'sex': penguins['sex'],
    'mass': zscore(penguins['body_mass_g']),
    'flipper': zscore(penguins['flipper_length_mm']),
    'bill_len': zscore(penguins['bill_length_mm']),
    'bill_depth': zscore(penguins['bill_depth_mm'])
})
penguin_std
species sex mass flipper bill_len bill_depth
0 Adelie male -0.563317 -1.416272 -0.883205 0.784300
1 Adelie female -0.500969 -1.060696 -0.809939 0.126003
2 Adelie female -1.186793 -0.420660 -0.663408 0.429833
3 Adelie NaN NaN NaN NaN NaN
4 Adelie female -0.937403 -0.562890 -1.322799 1.088129
... ... ... ... ... ... ...
339 Chinstrap male -0.251578 0.432721 2.175637 1.341320
340 Chinstrap female -0.999750 0.077145 -0.077282 0.480471
341 Chinstrap male -0.532143 -0.562890 1.040019 0.531109
342 Chinstrap male -0.126883 0.646066 1.259816 0.936215
343 Chinstrap female -0.532143 -0.207315 1.149917 0.784300

344 rows × 6 columns

And now we’ll fit and describe a model using these:

bm_mod = smf.ols('mass ~ flipper', data=penguin_std)
bmf = bm_mod.fit()
bmf.summary()
OLS Regression Results
Dep. Variable: mass R-squared: 0.759
Model: OLS Adj. R-squared: 0.758
Method: Least Squares F-statistic: 1071.
Date: Fri, 22 Oct 2021 Prob (F-statistic): 4.37e-107
Time: 15:29:51 Log-Likelihood: -241.46
No. Observations: 342 AIC: 486.9
Df Residuals: 340 BIC: 494.6
Df Model: 1
Covariance Type: nonrobust
coef std err t P>|t| [0.025 0.975]
Intercept 1.75e-16 0.027 6.58e-15 1.000 -0.052 0.052
flipper 0.8712 0.027 32.722 0.000 0.819 0.924
Omnibus: 5.634 Durbin-Watson: 2.190
Prob(Omnibus): 0.060 Jarque-Bera (JB): 5.585
Skew: 0.313 Prob(JB): 0.0613
Kurtosis: 3.019 Cond. No. 1.00


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

The residual plots will be identical to the unstandardized model, except for the scales of the axes, because all we did was rescale and recenter the variables:

plot_lm_diag(bmf)
../../../_images/Regressions_37_0.png ../../../_images/Regressions_37_1.png

Mass and Bill Length

Let’s do a regression we don’t expect to work well: mass vs. bill length.

First, we’ll fit the regression:

bm_mod = smf.ols('mass ~ bill_len', data=penguin_std)
bmf = bm_mod.fit()
bmf.summary()
OLS Regression Results
Dep. Variable: mass R-squared: 0.354
Model: OLS Adj. R-squared: 0.352
Method: Least Squares F-statistic: 186.4
Date: Fri, 22 Oct 2021 Prob (F-statistic): 3.81e-34
Time: 15:29:51 Log-Likelihood: -410.02
No. Observations: 342 AIC: 824.0
Df Residuals: 340 BIC: 831.7
Df Model: 1
Covariance Type: nonrobust
coef std err t P>|t| [0.025 0.975]
Intercept 1.75e-16 0.044 4.02e-15 1.000 -0.086 0.086
bill_len 0.5951 0.044 13.654 0.000 0.509 0.681
Omnibus: 5.671 Durbin-Watson: 0.866
Prob(Omnibus): 0.059 Jarque-Bera (JB): 4.857
Skew: -0.211 Prob(JB): 0.0882
Kurtosis: 2.597 Cond. No. 1.00


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

\(R^2\) is much lower than the previous models, indicating this model explains much less of the variance.

Let’s look at our diagnostic plots:

plot_lm_diag(bmf)
../../../_images/Regressions_41_0.png ../../../_images/Regressions_41_1.png

There’s some noticable non-normality at the tails of the residuals, but it isn’t bad. The residuals, however, exhibit notable heteroskedasticity,.

Multivariate Regression

This section goes with the Multiple Regression video.

Per-Species Intercepts

Let’s add the species as a predictor, so we have per-species intercepts for the variable:

bm_mod = smf.ols('mass ~ flipper + species', data=penguin_std)
bmf = bm_mod.fit()
bmf.summary()
OLS Regression Results
Dep. Variable: mass R-squared: 0.783
Model: OLS Adj. R-squared: 0.781
Method: Least Squares F-statistic: 405.7
Date: Fri, 22 Oct 2021 Prob (F-statistic): 1.25e-111
Time: 15:29:51 Log-Likelihood: -223.79
No. Observations: 342 AIC: 455.6
Df Residuals: 338 BIC: 470.9
Df Model: 3
Covariance Type: nonrobust
coef std err t P>|t| [0.025 0.975]
Intercept -0.0685 0.057 -1.207 0.228 -0.180 0.043
species[T.Chinstrap] -0.2575 0.072 -3.577 0.000 -0.399 -0.116
species[T.Gentoo] 0.3327 0.119 2.801 0.005 0.099 0.566
flipper 0.7137 0.054 13.255 0.000 0.608 0.820
Omnibus: 3.390 Durbin-Watson: 2.556
Prob(Omnibus): 0.184 Jarque-Bera (JB): 3.397
Skew: 0.242 Prob(JB): 0.183
Kurtosis: 2.934 Cond. No. 6.47


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

Species is automatically dummy-coded with the first level dropped. We therefore get the following intercepts:

  • Adelie: Intercept

  • Chinstrap: Intercept + species[T.Chinstrap]

  • Gentoo: Intercept + species[T.Gentoo]

Let’s check fit:

plot_lm_diag(bmf)
../../../_images/Regressions_47_0.png ../../../_images/Regressions_47_1.png

And we’ll plot the different lines on top of a scatterplot:

sns.scatterplot(x='flipper', y='mass', hue='species', data=penguin_std)
plot_line(bmf.params['Intercept'], bmf.params['flipper'], -2, 2, label='Adelie')
plot_line(bmf.params['Intercept'] + bmf.params['species[T.Gentoo]'],
          bmf.params['flipper'],
          -2, 2, label='Gentoo')
plot_line(bmf.params['Intercept'] + bmf.params['species[T.Chinstrap]'],
          bmf.params['flipper'],
          -2, 2, label='Chinstrap')
plt.title('Species-specific intercepts')
plt.show()
../../../_images/Regressions_49_0.png

Per-Species Slopes and Intercepts

Now let’s add an interaction term between flipper and species.

We do this with flipper * species, which expands to flipper + species + flipper:species:

bm_mod = smf.ols('mass ~ flipper * species', data=penguin_std)
bmf = bm_mod.fit()
bmf.summary()
OLS Regression Results
Dep. Variable: mass R-squared: 0.790
Model: OLS Adj. R-squared: 0.786
Method: Least Squares F-statistic: 252.2
Date: Fri, 22 Oct 2021 Prob (F-statistic): 2.22e-111
Time: 15:29:52 Log-Likelihood: -218.25
No. Observations: 342 AIC: 448.5
Df Residuals: 336 BIC: 471.5
Df Model: 5
Covariance Type: nonrobust
coef std err t P>|t| [0.025 0.975]
Intercept -0.1761 0.074 -2.393 0.017 -0.321 -0.031
species[T.Chinstrap] -0.1888 0.101 -1.871 0.062 -0.387 0.010
species[T.Gentoo] 0.1579 0.135 1.172 0.242 -0.107 0.423
flipper 0.5757 0.081 7.095 0.000 0.416 0.735
flipper:species[T.Chinstrap] 0.0305 0.138 0.222 0.825 -0.240 0.301
flipper:species[T.Gentoo] 0.3821 0.122 3.139 0.002 0.143 0.621
Omnibus: 5.822 Durbin-Watson: 2.534
Prob(Omnibus): 0.054 Jarque-Bera (JB): 5.758
Skew: 0.317 Prob(JB): 0.0562
Kurtosis: 3.034 Cond. No. 10.8


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

AIC is improving slightly, not a lot.

Look at assumption checks:

plot_lm_diag(bmf)
../../../_images/Regressions_53_0.png ../../../_images/Regressions_53_1.png

Getting better, but still some curve on the Q-Q and doesn’t quite look homoskedastic. (Watch the video for more explanation.)

Let’s plot these different lines:

sns.scatterplot(x='flipper', y='mass', hue='species', data=penguin_std)
plot_line(bmf.params['Intercept'], bmf.params['flipper'], -2, 2, label='Adelie')
plot_line(bmf.params['Intercept'] + bmf.params['species[T.Gentoo]'],
          bmf.params['flipper'] + bmf.params['flipper:species[T.Gentoo]'],
          -2, 2, label='Gentoo')
plot_line(bmf.params['Intercept'] + bmf.params['species[T.Chinstrap]'],
          bmf.params['flipper'] + bmf.params['flipper:species[T.Chinstrap]'],
          -2, 2, label='Chinstrap')
plt.title('Models with per-species slopes and intercepts')
plt.show()
../../../_images/Regressions_56_0.png

Sexual Dimorphism

Now let’s look at penguin sex as a predictor, and we’ll keep the flipper:species interaction:

bm_mod = smf.ols('mass ~ flipper + species * sex + flipper:species', data=penguin_std)
bmf = bm_mod.fit()
bmf.summary()
OLS Regression Results
Dep. Variable: mass R-squared: 0.876
Model: OLS Adj. R-squared: 0.873
Method: Least Squares F-statistic: 285.9
Date: Fri, 22 Oct 2021 Prob (F-statistic): 7.40e-142
Time: 15:29:52 Log-Likelihood: -125.89
No. Observations: 333 AIC: 269.8
Df Residuals: 324 BIC: 304.0
Df Model: 8
Covariance Type: nonrobust
coef std err t P>|t| [0.025 0.975]
Intercept -0.7717 0.076 -10.090 0.000 -0.922 -0.621
species[T.Chinstrap] 0.2359 0.120 1.968 0.050 3.11e-05 0.472
species[T.Gentoo] 1.0222 0.120 8.509 0.000 0.786 1.259
sex[T.male] 0.7474 0.063 11.790 0.000 0.623 0.872
species[T.Chinstrap]:sex[T.male] -0.5059 0.124 -4.086 0.000 -0.749 -0.262
species[T.Gentoo]:sex[T.male] -0.0023 0.109 -0.021 0.983 -0.217 0.212
flipper 0.2861 0.069 4.172 0.000 0.151 0.421
flipper:species[T.Chinstrap] 0.1817 0.126 1.442 0.150 -0.066 0.429
flipper:species[T.Gentoo] 0.1260 0.117 1.074 0.284 -0.105 0.357
Omnibus: 0.719 Durbin-Watson: 2.050
Prob(Omnibus): 0.698 Jarque-Bera (JB): 0.836
Skew: 0.083 Prob(JB): 0.658
Kurtosis: 2.819 Cond. No. 16.2


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

This is good, but some non-significant relationships. In particular, the flipper/species interaction disappeared.

Assumption checks:

plot_lm_diag(bmf)
../../../_images/Regressions_60_0.png ../../../_images/Regressions_60_1.png

That is looking pretty good.

Drop interaction, due to non-significant coefficients:

bm_mod = smf.ols('mass ~ flipper + species * sex', data=penguin_std)
bmf = bm_mod.fit()
bmf.summary()
OLS Regression Results
Dep. Variable: mass R-squared: 0.875
Model: OLS Adj. R-squared: 0.873
Method: Least Squares F-statistic: 380.2
Date: Fri, 22 Oct 2021 Prob (F-statistic): 6.74e-144
Time: 15:29:53 Log-Likelihood: -127.17
No. Observations: 333 AIC: 268.3
Df Residuals: 326 BIC: 295.0
Df Model: 6
Covariance Type: nonrobust
coef std err t P>|t| [0.025 0.975]
Intercept -0.7034 0.062 -11.305 0.000 -0.826 -0.581
species[T.Chinstrap] 0.0968 0.076 1.280 0.202 -0.052 0.246
species[T.Gentoo] 0.9982 0.108 9.273 0.000 0.786 1.210
sex[T.male] 0.7233 0.061 11.767 0.000 0.602 0.844
species[T.Chinstrap]:sex[T.male] -0.4188 0.106 -3.953 0.000 -0.627 -0.210
species[T.Gentoo]:sex[T.male] 0.0549 0.090 0.612 0.541 -0.122 0.231
flipper 0.3592 0.049 7.291 0.000 0.262 0.456
Omnibus: 0.476 Durbin-Watson: 2.046
Prob(Omnibus): 0.788 Jarque-Bera (JB): 0.596
Skew: 0.069 Prob(JB): 0.742
Kurtosis: 2.845 Cond. No. 9.89


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

This is definitely the best AIC we’ve seen. Let’s look at those assumption check plots:

plot_lm_diag(bmf)
../../../_images/Regressions_65_0.png ../../../_images/Regressions_65_1.png

That looks quite good.

Bootstrap

Now let’s bootstrap some of these model parameters. We’ll start with a function that will compute bootstrap samples of the rows of a data frame, and pass them to a function to get a statistic:

def boot_rows(df, stat, nboot=10000):
    n = len(df)
    obs = stat(df)
    boots = [stat(df.sample(n=n, replace=True)) for i in range(nboot)]
    return obs, np.quantile(boots, [0.025, 0.975])

Now we’ll write a function to train our full OLS model, and return the flipper length coefficient:

def flipper_coefficient(df):
    bm_mod = smf.ols('mass ~ flipper + species * sex', data=df)
    bmf = bm_mod.fit()
    return bmf.params['flipper']

And we’ll bootstrap the flipper length coefficient:

boot_rows(penguin_std, flipper_coefficient, nboot=1000)
(0.3592089725886189, array([0.26818419, 0.45622605]))

That matches the CIs from the model itself pretty closely!

Simultaneous Bootstrap of Multiple Parameters

In the video, I said you could do this. Here’s the code.

First, we need a boot_rows function that can deal with the stat function returning a series of multiple statistics:

def boot_rows(df, stat, nboot=10000):
    n = len(df)
    obs = stat(df)
    boots = [stat(df.sample(n=n, replace=True)) for i in range(nboot)]
    if isinstance(obs, pd.Series):
        # we have multiple statistics - stack them into a frame
        boots = pd.DataFrame.from_records(boots)
        # compute confidence intervals - columns will be params, and rows CI bounds
        conf = boots.quantile([0.025, 0.975])
        # transpose will put the params on rows
        # join with frame-ified version of the original observations
        return obs.to_frame('value').join(conf.transpose())
    else:
        return obs, np.quantile(boots, [0.025, 0.975])

And a function to return the parameters from a trained model:

def flipper_params(df):
    bm_mod = smf.ols('mass ~ flipper + species * sex', data=df)
    bmf = bm_mod.fit()
    return bmf.params

And let’s bootstrap all of this:

boot_rows(penguin_std, flipper_params, nboot=1000)
value 0.025 0.975
Intercept -0.703441 -0.816331 -0.593793
species[T.Chinstrap] 0.096813 -0.040241 0.229157
species[T.Gentoo] 0.998247 0.808552 1.206571
sex[T.male] 0.723339 0.613021 0.847359
species[T.Chinstrap]:sex[T.male] -0.418757 -0.625197 -0.216513
species[T.Gentoo]:sex[T.male] 0.054908 -0.113788 0.221861
flipper 0.359209 0.265790 0.448324