Regression Plots ================== .. _regression_plots_notebook: `Link to Notebook GitHub `_ .. raw:: html
In [ ]:
from __future__ import print_function
   from statsmodels.compat import lzip
   import numpy as np
   import pandas as pd
   import matplotlib.pyplot as plt
   import statsmodels.api as sm
   from statsmodels.formula.api import ols

Duncan's Prestige Dataset

Load the Data

We can use a utility function to load any R dataset available from the great Rdatasets package.

In [ ]:
prestige = sm.datasets.get_rdataset("Duncan", "car", cache=True).data
In [ ]:
prestige.head()
In [ ]:
prestige_model = ols("prestige ~ income + education", data=prestige).fit()
In [ ]:
print(prestige_model.summary())

Influence plots

Influence plots show the (externally) studentized residuals vs. the leverage of each observation as measured by the hat matrix.

Externally studentized residuals are residuals that are scaled by their standard deviation where

$$var(\hat{\epsilon}_i)=\hat{\sigma}^2_i(1-h_{ii})$$

with

$$\hat{\sigma}^2_i=\frac{1}{n - p - 1 \;\;}\sum_{j}^{n}\;\;\;\forall \;\;\; j \neq i$$

$n$ is the number of observations and $p$ is the number of regressors. $h_{ii}$ is the $i$-th diagonal element of the hat matrix

$$H=X(X^{\;\prime}X)^{-1}X^{\;\prime}$$

The influence of each point can be visualized by the criterion keyword argument. Options are Cook's distance and DFFITS, two measures of influence.

In [ ]:
fig, ax = plt.subplots(figsize=(12,8))
   fig = sm.graphics.influence_plot(prestige_model, ax=ax, criterion="cooks")
   
                            OLS Regression Results                            
   ==============================================================================
   Dep. Variable:               prestige   R-squared:                       0.828
   Model:                            OLS   Adj. R-squared:                  0.820
   Method:                 Least Squares   F-statistic:                     101.2
   Date:                Mon, 20 Jul 2015   Prob (F-statistic):           8.65e-17
   Time:                        17:44:26   Log-Likelihood:                -178.98
   No. Observations:                  45   AIC:                             364.0
   Df Residuals:                      42   BIC:                             369.4
   Df Model:                           2                                         
   Covariance Type:            nonrobust                                         
   ==============================================================================
                    coef    std err          t      P>|t|      [95.0% Conf. Int.]
   ------------------------------------------------------------------------------
   Intercept     -6.0647      4.272     -1.420      0.163       -14.686     2.556
   income         0.5987      0.120      5.003      0.000         0.357     0.840
   education      0.5458      0.098      5.555      0.000         0.348     0.744
   ==============================================================================
   Omnibus:                        1.279   Durbin-Watson:                   1.458
   Prob(Omnibus):                  0.528   Jarque-Bera (JB):                0.520
   Skew:                           0.155   Prob(JB):                        0.771
   Kurtosis:                       3.426   Cond. No.                         163.
   ==============================================================================
   
   Warnings:
   [1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

As you can see there are a few worrisome observations. Both contractor and reporter have low leverage but a large residual.
RR.engineer has small residual and large leverage. Conductor and minister have both high leverage and large residuals, and,
therefore, large influence.

Partial Regression Plots

Since we are doing multivariate regressions, we cannot just look at individual bivariate plots to discern relationships.
Instead, we want to look at the relationship of the dependent variable and independent variables conditional on the other
independent variables. We can do this through using partial regression plots, otherwise known as added variable plots.

In a partial regression plot, to discern the relationship between the response variable and the $k$-th variabe, we compute
the residuals by regressing the response variable versus the independent variables excluding $X_k$. We can denote this by
$X_{\sim k}$. We then compute the residuals by regressing $X_k$ on $X_{\sim k}$. The partial regression plot is the plot
of the former versus the latter residuals.

The notable points of this plot are that the fitted line has slope $\beta_k$ and intercept zero. The residuals of this plot
are the same as those of the least squares fit of the original model with full $X$. You can discern the effects of the
individual data values on the estimation of a coefficient easily. If obs_labels is True, then these points are annotated
with their observation label. You can also see the violation of underlying assumptions such as homooskedasticity and
linearity.

In [ ]:
fig, ax = plt.subplots(figsize=(12,8))
   fig = sm.graphics.plot_partregress("prestige", "income", ["income", "education"], data=prestige, ax=ax)
In [ ]:
fix, ax = plt.subplots(figsize=(12,14))
   fig = sm.graphics.plot_partregress("prestige", "income", ["education"], data=prestige, ax=ax)

As you can see the partial regression plot confirms the influence of conductor, minister, and RR.engineer on the partial relationship between income and prestige. The cases greatly decrease the effect of income on prestige. Dropping these cases confirms this.

In [ ]:
subset = ~prestige.index.isin(["conductor", "RR.engineer", "minister"])
   prestige_model2 = ols("prestige ~ income + education", data=prestige, subset=subset).fit()
   print(prestige_model2.summary())

For a quick check of all the regressors, you can use plot_partregress_grid. These plots will not label the
points, but you can use them to identify problems and then use plot_partregress to get more information.

In [ ]:
fig = plt.figure(figsize=(12,8))
   fig = sm.graphics.plot_partregress_grid(prestige_model, fig=fig)
   
                            OLS Regression Results                            
   ==============================================================================
   Dep. Variable:               prestige   R-squared:                       0.876
   Model:                            OLS   Adj. R-squared:                  0.870
   Method:                 Least Squares   F-statistic:                     138.1
   Date:                Mon, 20 Jul 2015   Prob (F-statistic):           2.02e-18
   Time:                        17:44:28   Log-Likelihood:                -160.59
   No. Observations:                  42   AIC:                             327.2
   Df Residuals:                      39   BIC:                             332.4
   Df Model:                           2                                         
   Covariance Type:            nonrobust                                         
   ==============================================================================
                    coef    std err          t      P>|t|      [95.0% Conf. Int.]
   ------------------------------------------------------------------------------
   Intercept     -6.3174      3.680     -1.717      0.094       -13.760     1.125
   income         0.9307      0.154      6.053      0.000         0.620     1.242
   education      0.2846      0.121      2.345      0.024         0.039     0.530
   ==============================================================================
   Omnibus:                        3.811   Durbin-Watson:                   1.468
   Prob(Omnibus):                  0.149   Jarque-Bera (JB):                2.802
   Skew:                          -0.614   Prob(JB):                        0.246
   Kurtosis:                       3.303   Cond. No.                         158.
   ==============================================================================
   
   Warnings:
   [1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

Component-Component plus Residual (CCPR) Plots

The CCPR plot provides a way to judge the effect of one regressor on the
response variable by taking into account the effects of the other
independent variables. The partial residuals plot is defined as
$\text{Residuals} + B_iX_i \text{ }\text{ }$ versus $X_i$. The component adds $B_iX_i$ versus
$X_i$ to show where the fitted line would lie. Care should be taken if $X_i$
is highly correlated with any of the other independent variables. If this
is the case, the variance evident in the plot will be an underestimate of
the true variance.

In [ ]:
fig, ax = plt.subplots(figsize=(12, 8))
   fig = sm.graphics.plot_ccpr(prestige_model, "education", ax=ax)

As you can see the relationship between the variation in prestige explained by education conditional on income seems to be linear, though you can see there are some observations that are exerting considerable influence on the relationship. We can quickly look at more than one variable by using plot_ccpr_grid.

In [ ]:
fig = plt.figure(figsize=(12, 8))
   fig = sm.graphics.plot_ccpr_grid(prestige_model, fig=fig)

Regression Plots

The plot_regress_exog function is a convenience function that gives a 2x2 plot containing the dependent variable and fitted values with confidence intervals vs. the independent variable chosen, the residuals of the model vs. the chosen independent variable, a partial regression plot, and a CCPR plot. This function can be used for quickly checking modeling assumptions with respect to a single regressor.

In [ ]:
fig = plt.figure(figsize=(12,8))
   fig = sm.graphics.plot_regress_exog(prestige_model, "education", fig=fig)

Fit Plot

The plot_fit function plots the fitted values versus a chosen independent variable. It includes prediction confidence intervals and optionally plots the true dependent variable.

In [ ]:
fig, ax = plt.subplots(figsize=(12, 8))
   fig = sm.graphics.plot_fit(prestige_model, "education", ax=ax)

Statewide Crime 2009 Dataset

Compare the following to http://www.ats.ucla.edu/stat/stata/webbooks/reg/chapter4/statareg_self_assessment_answers4.htm

Though the data here is not the same as in that example. You could run that example by uncommenting the necessary cells below.

In [ ]:
#dta = pd.read_csv("http://www.stat.ufl.edu/~aa/social/csv_files/statewide-crime-2.csv")
   #dta = dta.set_index("State", inplace=True).dropna()
   #dta.rename(columns={"VR" : "crime",
   #                    "MR" : "murder",
   #                    "M"  : "pctmetro",
   #                    "W"  : "pctwhite",
   #                    "H"  : "pcths",
   #                    "P"  : "poverty",
   #                    "S"  : "single"
   #                    }, inplace=True)
   #
   #crime_model = ols("murder ~ pctmetro + poverty + pcths + single", data=dta).fit()
In [ ]:
dta = sm.datasets.statecrime.load_pandas().data
In [ ]:
crime_model = ols("murder ~ urban + poverty + hs_grad + single", data=dta).fit()
   print(crime_model.summary())

Partial Regression Plots

In [ ]:
fig = plt.figure(figsize=(12,8))
   fig = sm.graphics.plot_partregress_grid(crime_model, fig=fig)
   
                            OLS Regression Results                            
   ==============================================================================
   Dep. Variable:                 murder   R-squared:                       0.813
   Model:                            OLS   Adj. R-squared:                  0.797
   Method:                 Least Squares   F-statistic:                     50.08
   Date:                Mon, 20 Jul 2015   Prob (F-statistic):           3.42e-16
   Time:                        17:44:31   Log-Likelihood:                -95.050
   No. Observations:                  51   AIC:                             200.1
   Df Residuals:                      46   BIC:                             209.8
   Df Model:                           4                                         
   Covariance Type:            nonrobust                                         
   ==============================================================================
                    coef    std err          t      P>|t|      [95.0% Conf. Int.]
   ------------------------------------------------------------------------------
   Intercept    -44.1024     12.086     -3.649      0.001       -68.430   -19.774
   urban          0.0109      0.015      0.707      0.483        -0.020     0.042
   poverty        0.4121      0.140      2.939      0.005         0.130     0.694
   hs_grad        0.3059      0.117      2.611      0.012         0.070     0.542
   single         0.6374      0.070      9.065      0.000         0.496     0.779
   ==============================================================================
   Omnibus:                        1.618   Durbin-Watson:                   2.507
   Prob(Omnibus):                  0.445   Jarque-Bera (JB):                0.831
   Skew:                          -0.220   Prob(JB):                        0.660
   Kurtosis:                       3.445   Cond. No.                     5.80e+03
   ==============================================================================
   
   Warnings:
   [1] Standard Errors assume that the covariance matrix of the errors is correctly specified.
   [2] The condition number is large, 5.8e+03. This might indicate that there are
   strong multicollinearity or other numerical problems.
In [ ]:
fig, ax = plt.subplots(figsize=(12,8))
   fig = sm.graphics.plot_partregress("murder", "hs_grad", ["urban", "poverty", "single"],  ax=ax, data=dta)

Leverage-Resid2 Plot

Closely related to the influence_plot is the leverage-resid2 plot.

In [ ]:
fig, ax = plt.subplots(figsize=(8,6))
   fig = sm.graphics.plot_leverage_resid2(crime_model, ax=ax)

Influence Plot

In [ ]:
fig, ax = plt.subplots(figsize=(8,6))
   fig = sm.graphics.influence_plot(crime_model, ax=ax)

Using robust regression to correct for outliers.

Part of the problem here in recreating the Stata results is that M-estimators are not robust to leverage points. MM-estimators should do better with this examples.

In [ ]:
from statsmodels.formula.api import rlm
In [ ]:
rob_crime_model = rlm("murder ~ urban + poverty + hs_grad + single", data=dta, 
                         M=sm.robust.norms.TukeyBiweight(3)).fit(conv="weights")
   print(rob_crime_model.summary())
In [ ]:
#rob_crime_model = rlm("murder ~ pctmetro + poverty + pcths + single", data=dta, M=sm.robust.norms.TukeyBiweight()).fit(conv="weights")
   #print(rob_crime_model.summary())
   
                    Robust linear Model Regression Results                    
   ==============================================================================
   Dep. Variable:                 murder   No. Observations:                   51
   Model:                            RLM   Df Residuals:                       46
   Method:                          IRLS   Df Model:                            4
   Norm:                   TukeyBiweight                                         
   Scale Est.:                       mad                                         
   Cov Type:                          H1                                         
   Date:                Mon, 20 Jul 2015                                         
   Time:                        17:44:33                                         
   No. Iterations:                    50                                         
   ==============================================================================
                    coef    std err          z      P>|z|      [95.0% Conf. Int.]
   ------------------------------------------------------------------------------
   Intercept     -4.2986      9.494     -0.453      0.651       -22.907    14.310
   urban          0.0029      0.012      0.241      0.809        -0.021     0.027
   poverty        0.2753      0.110      2.499      0.012         0.059     0.491
   hs_grad       -0.0302      0.092     -0.328      0.743        -0.211     0.150
   single         0.2902      0.055      5.253      0.000         0.182     0.398
   ==============================================================================
   
   If the model instance has been used for another fit with different fit
   parameters, then the fit options might not be the correct ones anymore .

There isn't yet an influence diagnostics method as part of RLM, but we can recreate them. (This depends on the status of issue #888)

In [ ]:
weights = rob_crime_model.weights
   idx = weights > 0
   X = rob_crime_model.model.exog[idx.values]
   ww = weights[idx] / weights[idx].mean()
   hat_matrix_diag = ww*(X*np.linalg.pinv(X).T).sum(1)
   resid = rob_crime_model.resid
   resid2 = resid**2
   resid2 /= resid2.sum()
   nobs = int(idx.sum())
   hm = hat_matrix_diag.mean()
   rm = resid2.mean()
In [ ]:
from statsmodels.graphics import utils
   fig, ax = plt.subplots(figsize=(12,8))
   ax.plot(resid2[idx], hat_matrix_diag, 'o')
   ax = utils.annotate_axes(range(nobs), labels=rob_crime_model.model.data.row_labels[idx], 
                       points=lzip(resid2[idx], hat_matrix_diag), offset_points=[(-5,5)]*nobs,
                       size="large", ax=ax)
   ax.set_xlabel("resid2")
   ax.set_ylabel("leverage")
   ylim = ax.get_ylim()
   ax.vlines(rm, *ylim)
   xlim = ax.get_xlim()
   ax.hlines(hm, *xlim)
   ax.margins(0,0)