Example: Simple Linear Model

In the following, we will go through a simple example of how to solve a linear mixed effects model. Consider the following setup,

\[y_{ij} = (\beta_0 + u_{0i}) + x \beta_1 + \epsilon_{ij}\]

where \(y\) is the measurement, \(x\) is the covariate, \(\beta_0\) and \(\beta_1\) is the fixed effects, \(u_0\) is the random intercept and \(\epsilon\) is the measurement error. And \(i\) is index for study, \(j\) is index for observation within study.

Assume our data frame looks like,

y	x	y_se	study_id
0.20	0.0	0.1	A
0.29	0.1	0.1	A
0.09	0.2	0.1	B
0.14	0.3	0.1	C
0.40	0.4	0.1	D

and our goal is to obtain the fixed effects and random effects for each study.

Create Data Object

The first step is to create a MRData object to carry the data information.

from mrtool import MRData

data = MRData()
data.load_df(
    df,
    col_obs='y',
    col_covs=['x'],
    col_obs_se='y_se',
    col_study_id='study_id'
)

Notice that the MRData will automatically create an intercept in the covariate list.

Configure Covariate Models

The second step is to create covariate models.

from mrtool import LinearCovModel

cov_intercept = LinearCovModel('intercept', use_re=True)
cov_x = LinearCovModel('x')

Create Model and Fit Model

The third step is to create the model to group data and covariate models. And use the optimization routine to find result.

from mrtool import MRBRT

model = MRBRT(
    data,
    [cov_intercept, cov_x]
)
model.fit_model()

You could get the fixed effects and random effects by calling model.beta_soln and model.re_soln.

Predict and Create Draws

The last step is to predict and create draws.

# first create data object used for predict
# the new data frame has to provide the same covariates as in the fitting
data_pred = MRData()
data_pred.load_df(
    df_pred,
    col_covs=['x']
)

# create point prediction
y_pred = model.predict(data_pred)

# sampling solutions
beta_samples, gamma_samples = model.sample_soln(sample_size=1000)

# create draws
y_draws = model.create_draws(
    data_pred,
    beta_samples,
    gamma_samples
)

Here y_pred is the point prediction and y_draws contains 1000 draws of the outcome.