#!/usr/bin/env python
# coding: utf-8

# DO NOT EDIT
# Autogenerated from the notebook mixed_lm_example.ipynb.
# Edit the notebook and then sync the output with this file.
#
# flake8: noqa
# DO NOT EDIT

# # Linear Mixed Effects Models

import numpy as np
import pandas as pd
import statsmodels.api as sm
import statsmodels.formula.api as smf
from statsmodels.tools.sm_exceptions import ConvergenceWarning

# **Note**: The R code and the results in this notebook has been converted
# to markdown so that R is not required to build the documents. The R
# results in the notebook were computed using R 3.5.1 and lme4 1.1.

# ```ipython
# %load_ext rpy2.ipython
# ```

# ```ipython
# %R library(lme4)
# ```

# ```
# array(['lme4', 'Matrix', 'tools', 'stats', 'graphics', 'grDevices',
#        'utils', 'datasets', 'methods', 'base'], dtype='<U9')
# ```

# Comparing R lmer to statsmodels MixedLM
# =======================================
#
# The statsmodels imputation of linear mixed models (MixedLM) closely
# follows the approach outlined in Lindstrom and Bates (JASA 1988).  This is
# also the approach followed in the  R package LME4.  Other packages such as
# Stata, SAS, etc. should also be consistent with this approach, as the
# basic techniques in this area are mostly mature.
#
# Here we show how linear mixed models can be fit using the MixedLM
# procedure in statsmodels.  Results from R (LME4) are included for
# comparison.
#
# Here are our import statements:

# ## Growth curves of pigs
#
# These are longitudinal data from a factorial experiment. The outcome
# variable is the weight of each pig, and the only predictor variable we
# will use here is "time".  First we fit a model that expresses the mean
# weight as a linear function of time, with a random intercept for each pig.
# The model is specified using formulas. Since the random effects structure
# is not specified, the default random effects structure (a random intercept
# for each group) is automatically used.

data = sm.datasets.get_rdataset("dietox", "geepack").data
md = smf.mixedlm("Weight ~ Time", data, groups=data["Pig"])
mdf = md.fit(method=["lbfgs"])
print(mdf.summary())

# Here is the same model fit in R using LMER:

# ```ipython
# %%R
# data(dietox, package='geepack')
# ```

# ```ipython
# %R print(summary(lmer('Weight ~ Time + (1|Pig)', data=dietox)))
# ```

# ```
# Linear mixed model fit by REML ['lmerMod']
# Formula: Weight ~ Time + (1 | Pig)
#    Data: dietox
#
# REML criterion at convergence: 4809.6
#
# Scaled residuals:
#     Min      1Q  Median      3Q     Max
# -4.7118 -0.5696 -0.0943  0.4877  4.7732
#
# Random effects:
#  Groups   Name        Variance Std.Dev.
#  Pig      (Intercept) 40.39    6.356
#  Residual             11.37    3.371
# Number of obs: 861, groups:  Pig, 72
#
# Fixed effects:
#             Estimate Std. Error t value
# (Intercept) 15.72352    0.78805   19.95
# Time         6.94251    0.03339  207.94
#
# Correlation of Fixed Effects:
#      (Intr)
# Time -0.275
# ```

# Note that in the statsmodels summary of results, the fixed effects and
# random effects parameter estimates are shown in a single table.  The
# random effect for animal is labeled "Intercept RE" in the statsmodels
# output above.  In the LME4 output, this effect is the pig intercept under
# the random effects section.
#
# There has been a lot of debate about whether the standard errors for
# random effect variance and covariance parameters are useful.  In LME4,
# these standard errors are not displayed, because the authors of the
# package believe they are not very informative.  While there is good reason
# to question their utility, we elected to include the standard errors in
# the summary table, but do not show the corresponding Wald confidence
# intervals.
#
# Next we fit a model with two random effects for each animal: a random
# intercept, and a random slope (with respect to time).  This means that
# each pig may have a different baseline weight, as well as growing at a
# different rate. The formula specifies that "Time" is a covariate with a
# random coefficient.  By default, formulas always include an intercept
# (which could be suppressed here using "0 + Time" as the formula).

md = smf.mixedlm("Weight ~ Time", data, groups=data["Pig"], re_formula="~Time")
mdf = md.fit(method=["lbfgs"])
print(mdf.summary())

# Here is the same model fit using LMER in R:

# ```ipython
# %R print(summary(lmer("Weight ~ Time + (1 + Time | Pig)", data=dietox)))
# ```

# ```
# Linear mixed model fit by REML ['lmerMod']
# Formula: Weight ~ Time + (1 + Time | Pig)
#    Data: dietox
#
# REML criterion at convergence: 4434.1
#
# Scaled residuals:
#     Min      1Q  Median      3Q     Max
# -6.4286 -0.5529 -0.0416  0.4841  3.5624
#
# Random effects:
#  Groups   Name        Variance Std.Dev. Corr
#  Pig      (Intercept) 19.493   4.415
#           Time         0.416   0.645    0.10
#  Residual              6.038   2.457
# Number of obs: 861, groups:  Pig, 72
#
# Fixed effects:
#             Estimate Std. Error t value
# (Intercept) 15.73865    0.55012   28.61
# Time         6.93901    0.07982   86.93
#
# Correlation of Fixed Effects:
#      (Intr)
# Time 0.006
# ```

# The random intercept and random slope are only weakly correlated $(0.294
# / \sqrt{19.493 * 0.416} \approx 0.1)$.  So next we fit a model in which
# the two random effects are constrained to be uncorrelated:

0.294 / (19.493 * 0.416)**0.5

md = smf.mixedlm("Weight ~ Time", data, groups=data["Pig"], re_formula="~Time")
free = sm.regression.mixed_linear_model.MixedLMParams.from_components(
    np.ones(2), np.eye(2))

mdf = md.fit(free=free, method=["lbfgs"])
print(mdf.summary())

# The likelihood drops by 0.3 when we fix the correlation parameter to 0.
# Comparing 2 x 0.3 = 0.6 to the chi^2 1 df reference distribution suggests
# that the data are very consistent with a model in which this parameter is
# equal to 0.
#
# Here is the same model fit using LMER in R (note that here R is
# reporting the REML criterion instead of the likelihood, where the REML
# criterion is twice the log likelihood):

# ```ipython
# %R print(summary(lmer("Weight ~ Time + (1 | Pig) + (0 + Time | Pig)",
# data=dietox)))
# ```

# ```
# Linear mixed model fit by REML ['lmerMod']
# Formula: Weight ~ Time + (1 | Pig) + (0 + Time | Pig)
#    Data: dietox
#
# REML criterion at convergence: 4434.7
#
# Scaled residuals:
#     Min      1Q  Median      3Q     Max
# -6.4281 -0.5527 -0.0405  0.4840  3.5661
#
# Random effects:
#  Groups   Name        Variance Std.Dev.
#  Pig      (Intercept) 19.8404  4.4543
#  Pig.1    Time         0.4234  0.6507
#  Residual              6.0282  2.4552
# Number of obs: 861, groups:  Pig, 72
#
# Fixed effects:
#             Estimate Std. Error t value
# (Intercept) 15.73875    0.55444   28.39
# Time         6.93899    0.08045   86.25
#
# Correlation of Fixed Effects:
#      (Intr)
# Time -0.086
# ```
#

# ## Sitka growth data
#
# This is one of the example data sets provided in the LMER R library.
# The outcome variable is the size of the tree, and the covariate used here
# is a time value.  The data are grouped by tree.

data = sm.datasets.get_rdataset("Sitka", "MASS").data
endog = data["size"]
data["Intercept"] = 1
exog = data[["Intercept", "Time"]]

# Here is the statsmodels LME fit for a basic model with a random
# intercept.  We are passing the endog and exog data directly to the LME
# init function as arrays.  Also note that endog_re is specified explicitly
# in argument 4 as a random intercept (although this would also be the
# default if it were not specified).

md = sm.MixedLM(endog, exog, groups=data["tree"], exog_re=exog["Intercept"])
mdf = md.fit()
print(mdf.summary())

# Here is the same model fit in R using LMER:

# ```ipython
# %R
# data(Sitka, package="MASS")
# print(summary(lmer("size ~ Time + (1 | tree)", data=Sitka)))
# ```

# ```
# Linear mixed model fit by REML ['lmerMod']
# Formula: size ~ Time + (1 | tree)
#    Data: Sitka
#
# REML criterion at convergence: 164.8
#
# Scaled residuals:
#     Min      1Q  Median      3Q     Max
# -2.9979 -0.5169  0.1576  0.5392  4.4012
#
# Random effects:
#  Groups   Name        Variance Std.Dev.
#  tree     (Intercept) 0.37451  0.612
#  Residual             0.03921  0.198
# Number of obs: 395, groups:  tree, 79
#
# Fixed effects:
#              Estimate Std. Error t value
# (Intercept) 2.2732443  0.0878955   25.86
# Time        0.0126855  0.0002654   47.80
#
# Correlation of Fixed Effects:
#      (Intr)
# Time -0.611
# ```

# We can now try to add a random slope.  We start with R this time.  From
# the code and output below we see that the REML estimate of the variance of
# the random slope is nearly zero.

# ```ipython
# %R print(summary(lmer("size ~ Time + (1 + Time | tree)", data=Sitka)))
# ```

# ```
# Linear mixed model fit by REML ['lmerMod']
# Formula: size ~ Time + (1 + Time | tree)
#    Data: Sitka
#
# REML criterion at convergence: 153.4
#
# Scaled residuals:
#     Min      1Q  Median      3Q     Max
# -2.7609 -0.5173  0.1188  0.5270  3.5466
#
# Random effects:
#  Groups   Name        Variance  Std.Dev. Corr
#  tree     (Intercept) 2.217e-01 0.470842
#           Time        3.288e-06 0.001813 -0.17
#  Residual             3.634e-02 0.190642
# Number of obs: 395, groups:  tree, 79
#
# Fixed effects:
#             Estimate Std. Error t value
# (Intercept) 2.273244   0.074655   30.45
# Time        0.012686   0.000327   38.80
#
# Correlation of Fixed Effects:
#      (Intr)
# Time -0.615
# convergence code: 0
# Model failed to converge with max|grad| = 0.793203 (tol = 0.002,
# component 1)
# Model is nearly unidentifiable: very large eigenvalue
#  - Rescale variables?
# ```

# If we run this in statsmodels LME with defaults, we see that the
# variance estimate is indeed very small, which leads to a warning about the
# solution being on the boundary of the parameter space.  The regression
# slopes agree very well with R, but the likelihood value is much higher
# than that returned by R.

exog_re = exog.copy()
md = sm.MixedLM(endog, exog, data["tree"], exog_re)
mdf = md.fit()
print(mdf.summary())

# We can further explore the random effects structure by constructing
# plots of the profile likelihoods. We start with the random intercept,
# generating a plot of the profile likelihood from 0.1 units below to 0.1
# units above the MLE. Since each optimization inside the profile likelihood
# generates a warning (due to the random slope variance being close to
# zero), we turn off the warnings here.

import warnings

with warnings.catch_warnings():
    warnings.filterwarnings("ignore")
    likev = mdf.profile_re(0, "re", dist_low=0.1, dist_high=0.1)

# Here is a plot of the profile likelihood function.  We multiply the log-
# likelihood difference by 2 to obtain the usual $\chi^2$ reference
# distribution with 1 degree of freedom.

import matplotlib.pyplot as plt

plt.figure(figsize=(10, 8))
plt.plot(likev[:, 0], 2 * likev[:, 1])
plt.xlabel("Variance of random intercept", size=17)
plt.ylabel("-2 times profile log likelihood", size=17)

# Here is a plot of the profile likelihood function. The profile
# likelihood plot shows that the MLE of the random slope variance parameter
# is a very small positive number, and that there is low uncertainty in this
# estimate.

re = mdf.cov_re.iloc[1, 1]
with warnings.catch_warnings():
    # Parameter is often on the boundary
    warnings.simplefilter("ignore", ConvergenceWarning)
    likev = mdf.profile_re(1, "re", dist_low=0.5 * re, dist_high=0.8 * re)

plt.figure(figsize=(10, 8))
plt.plot(likev[:, 0], 2 * likev[:, 1])
plt.xlabel("Variance of random slope", size=17)
lbl = plt.ylabel("-2 times profile log likelihood", size=17)
