<SYSTEM_TASK:>
Given the following text description, write Python code to implement the functionality described below step by step
<END_TASK>
<USER_TASK:>
Description:
 4. Simulating Data
Step1:  Here are some checks you might do to make sure the data were correctly generated
Step2:  Check correlations between predictors
Step3:  Check coefficient recovery
Step4:  You have the option of being as general or specific as you like when generating data. Here's a simpler example that generates 100 observations with 5 predictors from a standard normal distribution, i.e. mean = 0, sd = 1 with random correlations between predictors. 
Step5:  Generating multi-level regression data
Generating data for a multi-level regression is just as simple and returns a pandas dataframe with outcome and predictor variables ready for use with 
Step6:  Again here are some checks you might do to make sure the data were correctly generated (by default lmm data will generally be a bit noisier due to within and across group/cluster variance; see the API for how to customize this)
Step7:  Check mean of predictors within each group
Step8:  Check correlations between predictors within each group
Step9:  Check coefficient recovery
<ASSISTANT_TASK:>
Python Code:
# Import the simulation function
from pymer4.simulate import simulate_lm
# Also fix the random number generator for reproducibility
import numpy as np
np.random.seed(10)
data, b = simulate_lm(
    500, 3, coef_vals=[100, 1.2, -40.1, 3], mus=[10, 30, 1], noise_params=(0, 5)
)
print(f"True coefficients:\n{b}\n")
print(f"Data:\n{data.head()}")
Explanation: 4. Simulating Data
:code:pymer4 comes with some easy-to-use functions for simulating data that can be modeled with :code:Lm and multi-level data that can be modeled with :code:Lmer or :code:Lm2. These functions can be found in the :code:pymer4.simulate module and are aptly named: :code:simulate_lm() and :code:simulate_lmm() respectively.
:code:pymer4 gives you a lot of control over what you want your data to look like by setting properties such as:
Number of data points and number of coefficients
Specific coefficient values
Means and standard deviations of predictors
Correlations between predictors
Amount of error (noise) in the data
Number of groups/clusters (multi-level data only)
Variance of random effects (multi-level data only)
Generating standard regression data
Generating data for a standard regression returns a pandas dataframe with outcome and predictor variables ready for use with :code:Lm(), along with an array of coefficients used to produce the data.
Let's generate 500 observations, with coefficient values of: 1.2, -40.1, and 3. We also have an intercept with a value of 100. The means of the columns of our design matrix (i.e. means of the predictors) will be: 10, 30, and 1. We'll also add noise from a normal distribution with mean = 0, and sd = 5. Any correlations between predictors are purely random.
End of explanation
print(data.iloc[:, 1:].mean(axis=0))
Explanation: Here are some checks you might do to make sure the data were correctly generated:
Check the means of predictors
End of explanation
print(data.iloc[:, 1:].corr())
Explanation: Check correlations between predictors
End of explanation
from pymer4.models import Lm
model = Lm("DV ~ IV1+IV2+IV3", data=data)
model.fit(summarize=False)
print(model.coefs.loc[:, "Estimate"])
Explanation: Check coefficient recovery
End of explanation
data, b = simulate_lm(100, 5)
print(f"True coefficients:\n{b}\n")
print(f"Data:\n{data.head()}")
Explanation: You have the option of being as general or specific as you like when generating data. Here's a simpler example that generates 100 observations with 5 predictors from a standard normal distribution, i.e. mean = 0, sd = 1 with random correlations between predictors. :code:pymer4 will randomly decide what to set the coefficient values to.
End of explanation
from pymer4.simulate import simulate_lmm
num_obs = 50
num_coef = 3
num_grps = 100
mus = [10.0, 30.0, 2.0]
coef_vals = [4.0, 1.8, -2, 10]
corrs = 0.15
data, blups, b = simulate_lmm(
    num_obs, num_coef, num_grps, coef_vals=coef_vals, mus=mus, corrs=corrs
)
print(f"True coefficients:\n{b}\n")
print(f"BLUPs:\n{blups.head()}\n")
print(f"Data:\n{data.head()}\n")
Explanation: Generating multi-level regression data
Generating data for a multi-level regression is just as simple and returns a pandas dataframe with outcome and predictor variables ready for use with :code:Lmer(), another dataframe with group/cluster level coefficients (i.e. BLUPs), and a vector of population-level coefficients.
Here's an example generating 5000 observations, organized as 100 groups with 50 observations each. We'll have three predictors with the coefficients: 1.8, -2, and 10. We also have an intercept with a coefficient of 4. The means of the columns of our design matrix (i.e. means of the predictors) will be: 10, 30, and 2. We'll also introduce correlations between our predictors of with a mean r of .15. We'll leave the default of standard normal noise i.e., mean = 0, and sd = 1.
End of explanation
# Group the data before running checks
group_data = data.groupby("Group")
Explanation: Again here are some checks you might do to make sure the data were correctly generated (by default lmm data will generally be a bit noisier due to within and across group/cluster variance; see the API for how to customize this):
End of explanation
print(group_data.apply(lambda grp: grp.iloc[:, 1:-1].mean(axis=0)))
Explanation: Check mean of predictors within each group
End of explanation
print(group_data.apply(lambda grp: grp.iloc[:, 1:-1].corr()))
Explanation: Check correlations between predictors within each group
End of explanation
from pymer4.models import Lmer
model = Lmer("DV ~ IV1+IV2+IV3 + (1|Group)", data=data)
model.fit(summarize=False)
print(model.coefs.loc[:, "Estimate"])
Explanation: Check coefficient recovery
End of explanation
<END_TASK>