Variance Component Analysis¶
This notebook illustrates variance components analysis for two-level nested and crossed designs.
[1]:
import numpy as np
import statsmodels.api as sm
from statsmodels.regression.mixed_linear_model import VCSpec
import pandas as pd
Make the notebook reproducible
[2]:
np.random.seed(3123)
Nested analysis¶
In our discussion below, “Group 2” is nested within “Group 1”. As a concrete example, “Group 1” might be school districts, with “Group 2” being individual schools. The function below generates data from such a population. In a nested analysis, the group 2 labels that are nested within different group 1 labels are treated as independent groups, even if they have the same label. For example, two schools labeled “school 1” that are in two different school districts are treated as independent schools, even though they have the same label.
[3]:
def generate_nested(n_group1=200, n_group2=20, n_rep=10, group1_sd=2,
group2_sd=3, unexplained_sd=4):
# Group 1 indicators
group1 = np.kron(np.arange(n_group1), np.ones(n_group2 * n_rep))
# Group 1 effects
u = group1_sd * np.random.normal(size=n_group1)
effects1 = np.kron(u, np.ones(n_group2 * n_rep))
# Group 2 indicators
group2 = np.kron(np.ones(n_group1), np.kron(np.arange(n_group2), np.ones(n_rep)))
# Group 2 effects
u = group2_sd * np.random.normal(size=n_group1*n_group2)
effects2 = np.kron(u, np.ones(n_rep))
e = unexplained_sd * np.random.normal(size=n_group1 * n_group2 * n_rep)
y = effects1 + effects2 + e
df = pd.DataFrame({"y":y, "group1": group1, "group2": group2})
return df
Generate a data set to analyze.
[4]:
df = generate_nested()
Using all the default arguments for generate_nested, the population values of “group 1 Var” and “group 2 Var” are 2^2=4 and 3^2=9, respectively. The unexplained variance, listed as “scale” at the top of the summary table, has population value 4^2=16.
[5]:
model1 = sm.MixedLM.from_formula("y ~ 1", re_formula="1", vc_formula={"group2": "0 + C(group2)"},
groups="group1", data=df)
result1 = model1.fit()
print(result1.summary())
Mixed Linear Model Regression Results
==========================================================
Model: MixedLM Dependent Variable: y
No. Observations: 40000 Method: REML
No. Groups: 200 Scale: 15.8825
Min. group size: 200 Log-Likelihood: -116022.3805
Max. group size: 200 Converged: Yes
Mean group size: 200.0
-----------------------------------------------------------
Coef. Std.Err. z P>|z| [0.025 0.975]
-----------------------------------------------------------
Intercept -0.035 0.149 -0.232 0.817 -0.326 0.257
group1 Var 3.917 0.112
group2 Var 8.742 0.063
==========================================================
If we wish to avoid the formula interface, we can fit the same model by building the design matrices manually.
[6]:
def f(x):
n = x.shape[0]
g2 = x.group2
u = g2.unique()
u.sort()
uv = {v: k for k, v in enumerate(u)}
mat = np.zeros((n, len(u)))
for i in range(n):
mat[i, uv[g2.iloc[i]]] = 1
colnames = ["%d" % z for z in u]
return mat, colnames
Then we set up the variance components using the VCSpec class.
[7]:
vcm = df.groupby("group1").apply(f).to_list()
mats = [x[0] for x in vcm]
colnames = [x[1] for x in vcm]
names = ["group2"]
vcs = VCSpec(names, [colnames], [mats])
Finally we fit the model. It can be seen that the results of the two fits are identical.
[8]:
oo = np.ones(df.shape[0])
model2 = sm.MixedLM(df.y, oo, exog_re=oo, groups=df.group1, exog_vc=vcs)
result2 = model2.fit()
print(result2.summary())
Mixed Linear Model Regression Results
==========================================================
Model: MixedLM Dependent Variable: y
No. Observations: 40000 Method: REML
No. Groups: 200 Scale: 15.8825
Min. group size: 200 Log-Likelihood: -116022.3805
Max. group size: 200 Converged: Yes
Mean group size: 200.0
-----------------------------------------------------------
Coef. Std.Err. z P>|z| [0.025 0.975]
-----------------------------------------------------------
const -0.035 0.149 -0.232 0.817 -0.326 0.257
x_re1 Var 3.917 0.112
group2 Var 8.742 0.063
==========================================================
Crossed analysis¶
In a crossed analysis, the levels of one group can occur in any combination with the levels of the another group. The groups in Statsmodels MixedLM are always nested, but it is possible to fit a crossed model by having only one group, and specifying all random effects as variance components. Many, but not all crossed models can be fit in this way. The function below generates a crossed data set with two levels of random structure.
[9]:
def generate_crossed(n_group1=100, n_group2=100, n_rep=4, group1_sd=2,
group2_sd=3, unexplained_sd=4):
# Group 1 indicators
group1 = np.kron(np.arange(n_group1, dtype=np.int),
np.ones(n_group2 * n_rep, dtype=np.int))
group1 = group1[np.random.permutation(len(group1))]
# Group 1 effects
u = group1_sd * np.random.normal(size=n_group1)
effects1 = u[group1]
# Group 2 indicators
group2 = np.kron(np.arange(n_group2, dtype=np.int),
np.ones(n_group2 * n_rep, dtype=np.int))
group2 = group2[np.random.permutation(len(group2))]
# Group 2 effects
u = group2_sd * np.random.normal(size=n_group2)
effects2 = u[group2]
e = unexplained_sd * np.random.normal(size=n_group1 * n_group2 * n_rep)
y = effects1 + effects2 + e
df = pd.DataFrame({"y":y, "group1": group1, "group2": group2})
return df
Generate a data set to analyze.
[10]:
df = generate_crossed()
Next we fit the model, note that the groups vector is constant. Using the default parameters for generate_crossed, the level 1 variance should be 2^2=4, the level 2 variance should be 3^2=9, and the unexplained variance should be 4^2=16.
[11]:
vc = {"g1": "0 + C(group1)", "g2": "0 + C(group2)"}
oo = np.ones(df.shape[0])
model3 = sm.MixedLM.from_formula("y ~ 1", groups=oo, vc_formula=vc, data=df)
result3 = model3.fit()
print(result3.summary())
Mixed Linear Model Regression Results
==========================================================
Model: MixedLM Dependent Variable: y
No. Observations: 40000 Method: REML
No. Groups: 1 Scale: 15.9824
Min. group size: 40000 Log-Likelihood: -112684.9688
Max. group size: 40000 Converged: Yes
Mean group size: 40000.0
-----------------------------------------------------------
Coef. Std.Err. z P>|z| [0.025 0.975]
-----------------------------------------------------------
Intercept -0.251 0.353 -0.710 0.478 -0.943 0.442
g1 Var 4.282 0.154
g2 Var 8.150 0.291
==========================================================
If we wish to avoid the formula interface, we can fit the same model by building the design matrices manually.
[12]:
def f(g):
n = len(g)
u = g.unique()
u.sort()
uv = {v: k for k, v in enumerate(u)}
mat = np.zeros((n, len(u)))
for i in range(n):
mat[i, uv[g.iloc[i]]] = 1
colnames = ["%d" % z for z in u]
return [mat], [colnames]
vcm = [f(df.group1), f(df.group2)]
print(vcm)
mats = [x[0] for x in vcm]
colnames = [x[1] for x in vcm]
names = ["group1", "group2"]
vcs = VCSpec(names, colnames, mats)
[([array([[0., 0., 0., ..., 0., 0., 0.],
[0., 0., 0., ..., 0., 0., 0.],
[0., 0., 0., ..., 0., 0., 0.],
...,
[1., 0., 0., ..., 0., 0., 0.],
[0., 0., 0., ..., 0., 0., 0.],
[0., 0., 0., ..., 0., 0., 0.]])], [['0', '1', '2', '3', '4', '5', '6', '7', '8', '9', '10', '11', '12', '13', '14', '15', '16', '17', '18', '19', '20', '21', '22', '23', '24', '25', '26', '27', '28', '29', '30', '31', '32', '33', '34', '35', '36', '37', '38', '39', '40', '41', '42', '43', '44', '45', '46', '47', '48', '49', '50', '51', '52', '53', '54', '55', '56', '57', '58', '59', '60', '61', '62', '63', '64', '65', '66', '67', '68', '69', '70', '71', '72', '73', '74', '75', '76', '77', '78', '79', '80', '81', '82', '83', '84', '85', '86', '87', '88', '89', '90', '91', '92', '93', '94', '95', '96', '97', '98', '99']]), ([array([[0., 0., 0., ..., 0., 0., 0.],
[0., 0., 0., ..., 0., 0., 0.],
[0., 0., 0., ..., 0., 0., 0.],
...,
[0., 0., 0., ..., 0., 0., 0.],
[0., 0., 0., ..., 0., 0., 0.],
[0., 0., 0., ..., 0., 0., 0.]])], [['0', '1', '2', '3', '4', '5', '6', '7', '8', '9', '10', '11', '12', '13', '14', '15', '16', '17', '18', '19', '20', '21', '22', '23', '24', '25', '26', '27', '28', '29', '30', '31', '32', '33', '34', '35', '36', '37', '38', '39', '40', '41', '42', '43', '44', '45', '46', '47', '48', '49', '50', '51', '52', '53', '54', '55', '56', '57', '58', '59', '60', '61', '62', '63', '64', '65', '66', '67', '68', '69', '70', '71', '72', '73', '74', '75', '76', '77', '78', '79', '80', '81', '82', '83', '84', '85', '86', '87', '88', '89', '90', '91', '92', '93', '94', '95', '96', '97', '98', '99']])]
Here we fit the model without using formulas, it is simple to check that the results for models 3 and 4 are identical.
[13]:
oo = np.ones(df.shape[0])
model4 = sm.MixedLM(df.y, oo[:, None], exog_re=None, groups=oo, exog_vc=vcs)
result4 = model4.fit()
print(result4.summary())
Mixed Linear Model Regression Results
==========================================================
Model: MixedLM Dependent Variable: y
No. Observations: 40000 Method: REML
No. Groups: 1 Scale: 15.9824
Min. group size: 40000 Log-Likelihood: -112684.9688
Max. group size: 40000 Converged: Yes
Mean group size: 40000.0
-----------------------------------------------------------
Coef. Std.Err. z P>|z| [0.025 0.975]
-----------------------------------------------------------
const -0.251 0.353 -0.710 0.478 -0.943 0.442
group1 Var 4.282 0.154
group2 Var 8.150 0.291
==========================================================