Multivariate Linear Model - MultivariateLS

This notebooks illustrates some features for the multivariate linear model estimated by least squares. The example is based on the UCLA stats example in https://stats.oarc.ucla.edu/stata/dae/multivariate-regression-analysis/ .

The model assumes that a multivariate dependent variable depends linearly on the same set of explanatory variables.

Y = X * B + u

where
- the dependent variable (endog) Y has shape (nobs, k_endog),
- the matrix of explanatory variables including constant (exog) X has shape (nobs, k_exog), and - the parameter matrix B has shape (k_exog, k_endog), i.e. coefficients for explanatory variables in rows and dependent variables in columns. - the disturbance term u has the same shape as Y, (nobs, k_endog), and is assumed to have mean zero and to be uncorrelated with the exog X.

Estimation is by least squares. The parameter estimates with common explanatory variables for each dependent variables corresponds to separate OLS estimates for each endog. The main advantage of the multivariate model is that we can make inference

[1]:

import os
import numpy as np

import pandas as pd

from statsmodels.base.model import LikelihoodModel
from statsmodels.regression.linear_model import OLS
from statsmodels.multivariate.manova import MANOVA
from statsmodels.multivariate.multivariate_ols import MultivariateLS

import statsmodels.multivariate.tests.results as path
dir_path = os.path.dirname(os.path.abspath(path.__file__))
csv_path = os.path.join(dir_path, 'mvreg.csv')
data_mvreg = pd.read_csv(csv_path)

[2]:

data_mvreg.head()

[2]:
locus_of_control self_concept motivation read write science prog
0 -1.143955 0.722641 0.368973 37.405548 39.032845 33.532822 academic
1 0.504134 0.111364 0.520319 52.760784 51.995041 65.225044 academic
2 1.628546 0.629934 0.436838 59.771915 54.651653 64.604500 academic
3 0.368096 -0.138528 -0.004324 42.854324 41.121357 48.493809 vocational
4 -0.280190 -0.452226 1.256924 54.756279 49.947208 50.381657 academic 
[3]:

formula = "locus_of_control + self_concept + motivation ~ read + write + science + prog"
mod = MultivariateLS.from_formula(formula, data=data_mvreg)
res = mod.fit()

Multivariate hypothesis tests mv_test

The mv_test method by default performs the hypothesis tests that each term in the formula has no effect on any of the dependent variables (endog). This is the same as the MANOVA test.
Note, MANOVA in statsmodels is implemented as test on coefficients in the multivariate model and is not restricted to categorical variables. In the current example, we have three continuous and one categorical explanatory variables, in addition to the constant.

Consequently, using mv_test in MultivariateLS and in MANOVA produces the same results. However. MANOVA only provides the hypothesis tests as feature, while MultivariateLS provide the usual model results.

More general versions of the mv_test are for hypothesis in the form L B M = C. Here L are restrictions corresponding to explanatory variables, M are restrictions corresponding to dependent (endog) variables and C is a matrix of constants for affine restrictions. See docstrings for details.

[4]:

mvt = res.mv_test()
mvt.summary_frame

[4]:
Value Num DF Den DF F Value Pr > F
Effect Statistic
Intercept Wilks' lambda 0.848467 3 592.0 35.242876 0.0
Pillai's trace 0.151533 3.0 592.0 35.242876 0.0
Hotelling-Lawley trace 0.178596 3 592.0 35.242876 0.0
Roy's greatest root 0.178596 3 592 35.242876 0.0
prog Wilks' lambda 0.891438 6 1184.0 11.670765 0.0
Pillai's trace 0.108649 6.0 1186.0 11.354963 0.0
Hotelling-Lawley trace 0.121685 6 787.558061 11.996155 0.0
Roy's greatest root 0.120878 3 593 23.893456 0.0
read Wilks' lambda 0.976425 3 592.0 4.764416 0.002727
Pillai's trace 0.023575 3.0 592.0 4.764416 0.002727
Hotelling-Lawley trace 0.024144 3 592.0 4.764416 0.002727
Roy's greatest root 0.024144 3 592 4.764416 0.002727
write Wilks' lambda 0.947394 3 592.0 10.957338 0.000001
Pillai's trace 0.052606 3.0 592.0 10.957338 0.000001
Hotelling-Lawley trace 0.055527 3 592.0 10.957338 0.000001
Roy's greatest root 0.055527 3 592 10.957338 0.000001
science Wilks' lambda 0.983405 3 592.0 3.329911 0.019305
Pillai's trace 0.016595 3.0 592.0 3.329911 0.019305
Hotelling-Lawley trace 0.016875 3 592.0 3.329911 0.019305
Roy's greatest root 0.016875 3 592 3.329911 0.019305 
[5]:

manova = MANOVA.from_formula(formula, data=data_mvreg)
manova.mv_test().summary_frame

[5]:
Value Num DF Den DF F Value Pr > F
Effect Statistic
Intercept Wilks' lambda 0.848467 3 592.0 35.242876 0.0
Pillai's trace 0.151533 3.0 592.0 35.242876 0.0
Hotelling-Lawley trace 0.178596 3 592.0 35.242876 0.0
Roy's greatest root 0.178596 3 592 35.242876 0.0
prog Wilks' lambda 0.891438 6 1184.0 11.670765 0.0
Pillai's trace 0.108649 6.0 1186.0 11.354963 0.0
Hotelling-Lawley trace 0.121685 6 787.558061 11.996155 0.0
Roy's greatest root 0.120878 3 593 23.893456 0.0
read Wilks' lambda 0.976425 3 592.0 4.764416 0.002727
Pillai's trace 0.023575 3.0 592.0 4.764416 0.002727
Hotelling-Lawley trace 0.024144 3 592.0 4.764416 0.002727
Roy's greatest root 0.024144 3 592 4.764416 0.002727
write Wilks' lambda 0.947394 3 592.0 10.957338 0.000001
Pillai's trace 0.052606 3.0 592.0 10.957338 0.000001
Hotelling-Lawley trace 0.055527 3 592.0 10.957338 0.000001
Roy's greatest root 0.055527 3 592 10.957338 0.000001
science Wilks' lambda 0.983405 3 592.0 3.329911 0.019305
Pillai's trace 0.016595 3.0 592.0 3.329911 0.019305
Hotelling-Lawley trace 0.016875 3 592.0 3.329911 0.019305
Roy's greatest root 0.016875 3 592 3.329911 0.019305

The core multivariate regression results are displayed by the summary method.

[6]:

print(res.summary())

                                      MultivariateLS Regression Results
==============================================================================================================
Dep. Variable:     ['locus_of_control', 'self_concept', 'motivation']   No. Observations:                  600
Model:                                                 MultivariateLS   Df Residuals:                      594
Method:                                                 Least Squares   Df Model:                           15
Date:                                                Thu, 14 Nov 2024
Time:                                                        16:49:50
======================================================================================
  locus_of_control       coef    std err          t      P>|t|      [0.025      0.975]
--------------------------------------------------------------------------------------
Intercept             -1.4970      0.157     -9.505      0.000      -1.806      -1.188
prog[T.general]       -0.1278      0.064     -1.998      0.046      -0.253      -0.002
prog[T.vocational]     0.1239      0.058      2.150      0.032       0.011       0.237
read                   0.0125      0.004      3.363      0.001       0.005       0.020
write                  0.0121      0.003      3.581      0.000       0.005       0.019
science                0.0058      0.004      1.582      0.114      -0.001       0.013
--------------------------------------------------------------------------------------
      self_concept       coef    std err          t      P>|t|      [0.025      0.975]
--------------------------------------------------------------------------------------
Intercept             -0.0959      0.179     -0.536      0.592      -0.447       0.255
prog[T.general]       -0.2765      0.073     -3.808      0.000      -0.419      -0.134
prog[T.vocational]     0.1469      0.065      2.246      0.025       0.018       0.275
read                   0.0013      0.004      0.310      0.757      -0.007       0.010
write                 -0.0043      0.004     -1.115      0.265      -0.012       0.003
science                0.0053      0.004      1.284      0.200      -0.003       0.013
--------------------------------------------------------------------------------------
        motivation       coef    std err          t      P>|t|      [0.025      0.975]
--------------------------------------------------------------------------------------
Intercept             -0.9505      0.198     -4.811      0.000      -1.339      -0.563
prog[T.general]       -0.3603      0.080     -4.492      0.000      -0.518      -0.203
prog[T.vocational]     0.2594      0.072      3.589      0.000       0.117       0.401
read                   0.0097      0.005      2.074      0.038       0.001       0.019
write                  0.0175      0.004      4.122      0.000       0.009       0.026
science               -0.0090      0.005     -1.971      0.049      -0.018   -3.13e-05
======================================================================================

The the standard results attributes for the parameter estimates like params, bse, tvalues and pvalues, are two dimensional arrays or dataframes with explanatory variables (exog) in rows and dependend (endog) variables in columns.

[7]:

res.params

[7]:
0 1 2
Intercept -1.496970 -0.095858 -0.950513
prog[T.general] -0.127795 -0.276483 -0.360329
prog[T.vocational] 0.123875 0.146876 0.259367
read 0.012505 0.001308 0.009674
write 0.012145 -0.004293 0.017535
science 0.005761 0.005306 -0.009001 
[8]:

res.bse

[8]:
0 1 2
Intercept 0.157499 0.178794 0.197563
prog[T.general] 0.063955 0.072602 0.080224
prog[T.vocational] 0.057607 0.065396 0.072261
read 0.003718 0.004220 0.004664
write 0.003391 0.003850 0.004254
science 0.003641 0.004133 0.004567 
[9]:

res.pvalues

[9]:
0 1 2
Intercept 4.887129e-20 0.592066 0.000002
prog[T.general] 4.615006e-02 0.000155 0.000008
prog[T.vocational] 3.193055e-02 0.025075 0.000359
read 8.192738e-04 0.756801 0.038481
write 3.700449e-04 0.265214 0.000043
science 1.141093e-01 0.199765 0.049209

General MV and Wald tests

The multivariate linear model allows for multivariate test in the L B M form as well as standard wald tests on linear combination of parameters.

The multivariate tests are based on eigenvalues or trace of the matrices. Wald tests are standard test base on the flattened (stacked) parameter array and their covariance, hypothesis are of the form R b = c where b is the column stacked parameter array. The tests are asymptotically equivalent under the model assumptions but differ in small samples.

The linear restriction can be defined either as hypothesis matrices (numpy arrays) or as strings or list of strings.

[10]:

yn = res.model.endog_names
xn = res.model.exog_names
yn, xn

[10]:

(['locus_of_control', 'self_concept', 'motivation'],
 ['Intercept',
  'prog[T.general]',
  'prog[T.vocational]',
  'read',
  'write',
  'science'])

[11]:

# test for an individual coefficient

mvt = res.mv_test(hypotheses=[("coef", ["science"], ["locus_of_control"])])
mvt.summary_frame

[11]:
Value Num DF Den DF F Value Pr > F
Effect Statistic
coef Wilks' lambda 0.995803 1 594.0 2.50373 0.114109
Pillai's trace 0.004197 1.0 594.0 2.50373 0.114109
Hotelling-Lawley trace 0.004215 1 594.0 2.50373 0.114109
Roy's greatest root 0.004215 1 594 2.50373 0.114109 
[12]:

tt = res.t_test("ylocus_of_control_science")
tt, tt.pvalue

[12]:

(<class 'statsmodels.stats.contrast.ContrastResults'>
                              Test for Constraints
 ==============================================================================
                  coef    std err          t      P>|t|      [0.025      0.975]
 ------------------------------------------------------------------------------
 c0             0.0058      0.004      1.582      0.114      -0.001       0.013
 ==============================================================================,
 array(0.11410929))

We can use either mv_test or wald_test for the joint hypothesis that an explanatory variable has no effect on any of the dependent variables, that is all coefficient for the explanatory variable are zero.

In this example, the pvalues agree at 3 decimals.

[13]:

wt = res.wald_test(["ylocus_of_control_science", "yself_concept_science", "ymotivation_science"], scalar=True)
wt

[13]:

<class 'statsmodels.stats.contrast.ContrastResults'>
<F test: F=3.3411603250015216, p=0.01901163430173511, df_denom=594, df_num=3>

[14]:

mvt = res.mv_test(hypotheses=[("science", ["science"], yn)])
mvt.summary_frame

[14]:
Value Num DF Den DF F Value Pr > F
Effect Statistic
science Wilks' lambda 0.983405 3 592.0 3.329911 0.019305
Pillai's trace 0.016595 3.0 592.0 3.329911 0.019305
Hotelling-Lawley trace 0.016875 3 592.0 3.329911 0.019305
Roy's greatest root 0.016875 3 592 3.329911 0.019305 
[15]:

# t_test provides a vectorized results for each of the simple hypotheses

tt = res.t_test(["ylocus_of_control_science", "yself_concept_science", "ymotivation_science"])
tt, tt.pvalue

[15]:

(<class 'statsmodels.stats.contrast.ContrastResults'>
                              Test for Constraints
 ==============================================================================
                  coef    std err          t      P>|t|      [0.025      0.975]
 ------------------------------------------------------------------------------
 c0             0.0058      0.004      1.582      0.114      -0.001       0.013
 c1             0.0053      0.004      1.284      0.200      -0.003       0.013
 c2            -0.0090      0.005     -1.971      0.049      -0.018   -3.13e-05
 ==============================================================================,
 array([0.11410929, 0.19976543, 0.0492095 ]))

Warning: the naming pattern for the flattened parameter names used in t_test and wald_test might still change.

The current pattern is "y{endog_name}_{exog_name}".

examples:

[16]:

[f"y{endog_name}_{exog_name}" for endog_name in yn for exog_name in ["science"]]

[16]:

['ylocus_of_control_science', 'yself_concept_science', 'ymotivation_science']

[17]:

c = [f"y{endog_name}_{exog_name}" for endog_name in yn for exog_name in ["prog[T.general]", "prog[T.vocational]"]]
c

[17]:

['ylocus_of_control_prog[T.general]',
 'ylocus_of_control_prog[T.vocational]',
 'yself_concept_prog[T.general]',
 'yself_concept_prog[T.vocational]',
 'ymotivation_prog[T.general]',
 'ymotivation_prog[T.vocational]']

The previous restriction corresponds to the MANOVA type test that the categorical variable “prog” has no effect.

[18]:

mant = manova.mv_test().summary_frame
mant.loc["prog"] #["Pr > F"].to_numpy()

[18]:
Value Num DF Den DF F Value Pr > F
Statistic
Wilks' lambda 0.891438 6 1184.0 11.670765 0.0
Pillai's trace 0.108649 6.0 1186.0 11.354963 0.0
Hotelling-Lawley trace 0.121685 6 787.558061 11.996155 0.0
Roy's greatest root 0.120878 3 593 23.893456 0.0 
[19]:

res.wald_test(c, scalar=True)

[19]:

<class 'statsmodels.stats.contrast.ContrastResults'>
<F test: F=12.046814522691747, p=8.548081236477388e-13, df_denom=594, df_num=6>

Note: The degrees of freedom differ across hypothesis test methods. The model can be considered as a multivariate model with nobs=600 in this case, or as a stacked model with nobs_total = nobs * k_endog = 1800.

For within endog restriction, inference is based on the same covariance of the parameter estimates in MultivariateLS and OLS. The degrees of freedom in a single output OLS are df_resid = 600 - 6 = 594. Using the same degrees of freedom in MultivariateLS preserves the equivalence for the analysis of each endog. Using the total df_resid for hypothesis tests would make them more liberal.

Asymptotic inference based on normal and chisquare distribution (use_t=False) is not affected by how df_resid are defined.

It is not yet decided whether there will be additional options to choose different degrees of freedom in the Wald tests.

[20]:

res.df_resid

[20]:

594

Both mv_test and wald_test can be used to test hypothesis on contrasts between coefficients

[21]:

c = [f"y{endog_name}_prog[T.general] - y{endog_name}_prog[T.vocational]" for endog_name in yn]
c

[21]:

['ylocus_of_control_prog[T.general] - ylocus_of_control_prog[T.vocational]',
 'yself_concept_prog[T.general] - yself_concept_prog[T.vocational]',
 'ymotivation_prog[T.general] - ymotivation_prog[T.vocational]']

[22]:

res.wald_test(c, scalar=True)

[22]:

<class 'statsmodels.stats.contrast.ContrastResults'>
<F test: F=23.929409268979654, p=1.2456536486105104e-14, df_denom=594, df_num=3>

[23]:

mvt = res.mv_test(hypotheses=[("diff_prog", ["prog[T.general] - prog[T.vocational]"], yn)])
mvt.summary_frame

[23]:
Value Num DF Den DF F Value Pr > F
Effect Statistic
diff_prog Wilks' lambda 0.892176 3 592.0 23.848839 0.0
Pillai's trace 0.107824 3.0 592.0 23.848839 0.0
Hotelling-Lawley trace 0.120856 3 592.0 23.848839 0.0
Roy's greatest root 0.120856 3 592 23.848839 0.0

Example: hypothesis that coefficients are the same across endog equations.

We can test that the difference between the parameters of the later two equation with the first equation are zero.

[24]:

mvt = res.mv_test(hypotheses=[("diff_prog", xn, ["self_concept - locus_of_control", "motivation - locus_of_control"])])
mvt.summary_frame

[24]:
Value Num DF Den DF F Value Pr > F
Effect Statistic
diff_prog Wilks' lambda 0.867039 12 1186.0 7.307879 0.0
Pillai's trace 0.13714 12.0 1188.0 7.28819 0.0
Hotelling-Lawley trace 0.14853 12 919.36321 7.331042 0.0
Roy's greatest root 0.100625 6 594 9.961898 0.0

In a similar hypothesis test, we can test that equation have the same slope coefficients but can have different constants.

[25]:

xn[1:]

[25]:

['prog[T.general]', 'prog[T.vocational]', 'read', 'write', 'science']

[26]:

mvt = res.mv_test(hypotheses=[("diff_prog", xn[1:], ["self_concept - locus_of_control", "motivation - locus_of_control"])])
mvt.summary_frame

[26]:
Value Num DF Den DF F Value Pr > F
Effect Statistic
diff_prog Wilks' lambda 0.879133 10 1186.0 7.890322 0.0
Pillai's trace 0.124212 10.0 1188.0 7.866738 0.0
Hotelling-Lawley trace 0.133679 10 886.75443 7.918284 0.0
Roy's greatest root 0.092581 5 594 10.998679 0.0

Prediction

The regression model and its results instance have methods for prediction and residuals.

Note, because the parameter estimates are the same as in the OLS estimate for individual endog, the predictions will also be the same between the MultivariateLS model and the set of individual OLS models.

[27]:

res.resid

[27]:
locus_of_control self_concept motivation
0 -0.781981 0.759249 0.575027
1 0.334075 0.015388 0.635811
2 1.342126 0.539488 0.432337
3 0.426497 -0.326337 -0.012298
4 -0.364810 -0.480846 1.255411
... ... ... ...
595 -1.849566 0.920851 -0.318799
596 -1.278212 -1.080592 -0.031575
597 -1.060668 -1.065596 -1.577958
598 -0.661946 0.368192 0.132774
599 -0.129760 0.702698 1.020835

600 rows × 3 columns

[28]:

res.predict()

[28]:

array([[-0.36197321, -0.03660735, -0.2060539 ],
       [ 0.17005867,  0.09597616, -0.11549243],
       [ 0.28641963,  0.09044546,  0.00450077],
       ...,
       [ 0.6252098 , -0.23716973,  0.11864199],
       [-0.3024846 , -0.29586741, -0.47584179],
       [ 0.77574136,  0.2878978 ,  0.42480766]])

[29]:

res.predict(data_mvreg)

[29]:
0 1 2
0 -0.361973 -0.036607 -0.206054
1 0.170059 0.095976 -0.115492
2 0.286420 0.090445 0.004501
3 -0.058400 0.187809 0.007974
4 0.084621 0.028620 0.001513
... ... ... ...
595 0.185458 0.036897 0.034498
596 0.330408 0.097329 0.489407
597 0.625210 -0.237170 0.118642
598 -0.302485 -0.295867 -0.475842
599 0.775741 0.287898 0.424808

600 rows × 3 columns

[30]:

res.fittedvalues

[30]:
locus_of_control self_concept motivation
0 -0.361973 -0.036607 -0.206054
1 0.170059 0.095976 -0.115492
2 0.286420 0.090445 0.004501
3 -0.058400 0.187809 0.007974
4 0.084621 0.028620 0.001513
... ... ... ...
595 0.185458 0.036897 0.034498
596 0.330408 0.097329 0.489407
597 0.625210 -0.237170 0.118642
598 -0.302485 -0.295867 -0.475842
599 0.775741 0.287898 0.424808

600 rows × 3 columns

The predict methods can take user provided data for the explanatory variables, but currently are not able to automatically create sets of explanatory variables for interesting effects.

In the following, we construct at dataframe that we can use to predict the conditional expectation of the dependent variables for each level of the categorical variable “prog” at the sample means of the continuous variables.

[31]:

data_exog = data_mvreg[['read', 'write', 'science', 'prog']]

ex = pd.DataFrame(data_exog["prog"].unique(), columns=["prog"])
mean_ex = data_mvreg[['read', 'write', 'science']].mean()

ex.loc[:, ['read', 'write', 'science']] = mean_ex.values
ex

[31]:
prog read write science
0 academic 51.901833 52.384833 51.763333
1 vocational 51.901833 52.384833 51.763333
2 general 51.901833 52.384833 51.763333 
[32]:

pred = res.predict(ex)

pred.index = ex["prog"]
pred.columns = res.fittedvalues.columns
print("predicted mean by 'prog':")
pred

predicted mean by 'prog':

[32]:
locus_of_control self_concept motivation
prog
academic 0.086493 0.021752 0.004209
vocational 0.210368 0.168628 0.263575
general -0.041303 -0.254731 -0.356121

Outlier-Influence

This is currently in draft version.
resid_distance is a one dimensional residual measure based on Mahalanobis distance for each sample observation. The hat matrix in the MultivariateLS model is the same as in OLS, the diagonal of the hat matrix is temporarily attached as results._hat_matrix_diag.

Note, individual components of the multivariate dependent variable can be analyzed with OLS and are available in the corresponding post-estimation methods like OLSInfluence.

[33]:

res.resid_distance[:5]

[33]:

array([3.74332128, 0.95395412, 5.15221877, 0.82580531, 4.5260778 ])

[34]:

res.cov_resid

[34]:

array([[0.36844484, 0.05748939, 0.06050103],
       [0.05748939, 0.4748153 , 0.13103368],
       [0.06050103, 0.13103368, 0.57973305]])

[35]:

import matplotlib.pyplot as plt

[36]:

plt.plot(res.resid_distance)

[36]:

[<matplotlib.lines.Line2D at 0x7fb9bdcf45e0>]
../../../_images/examples_notebooks_generated_multivariate_ls_50_1.png 
[37]:

plt.plot(res._hat_matrix_diag)

[37]:

[<matplotlib.lines.Line2D at 0x7fb9bdbca530>]
../../../_images/examples_notebooks_generated_multivariate_ls_51_1.png 
[38]:

plt.plot(res._hat_matrix_diag, res.resid_distance, ".")

[38]:

[<matplotlib.lines.Line2D at 0x7fb9bda3ccd0>]
../../../_images/examples_notebooks_generated_multivariate_ls_52_1.png
Last update: Nov 14, 2024