## Uncomment and run this cell to install the packages
# !pip install pandas numpy statsmodelsStatistical Models
Statistical models are mathematical representations of relationships between variables in a dataset. Python provides a wide range of tools for statistical modeling and inference.
Keywords
statistical models, linear regression, logistic regression, statsmodels.api, statistics
Implementing Statistical Models in Python
Statistical models
Statistical models are mathematical representations of relationships between variables in a dataset. These models are used to make predictions, infer causal relationships, and understand patterns in data. Statistical modeling involves formulating hypotheses about the data generating process, estimating model parameters from observed data, and evaluating the fit of the model to the data.
The statsmodels.api library in Python provides a wide range of tools for statistical modeling and inference. It allows users to build, estimate, and analyze various statistical models using a simple and intuitive interface.
import statsmodels.api as sm
import numpy as np- Linear Regression:
- Linear regression is used to model the relationship between one or more independent variables and a continuous dependent variable.
# Generate example data
np.random.seed(0)
X = np.random.rand(100, 2) # Two independent variables# Dependent variable with noise
y = 2 * X[:, 0] + 3 * X[:, 1] + np.random.normal(0, 1, 100)# Add constant term for intercept
X = sm.add_constant(X)# Fit linear regression model
model = sm.OLS(y, X).fit()# Print model summary
print(model.summary()) OLS Regression Results
==============================================================================
Dep. Variable: y R-squared: 0.440
Model: OLS Adj. R-squared: 0.428
Method: Least Squares F-statistic: 38.06
Date: Wed, 17 Apr 2024 Prob (F-statistic): 6.31e-13
Time: 12:22:26 Log-Likelihood: -140.25
No. Observations: 100 AIC: 286.5
Df Residuals: 97 BIC: 294.3
Df Model: 2
Covariance Type: nonrobust
==============================================================================
coef std err t P>|t| [0.025 0.975]
------------------------------------------------------------------------------
const 0.2554 0.277 0.922 0.359 -0.294 0.805
x1 1.4260 0.356 4.011 0.000 0.720 2.132
x2 2.8054 0.351 8.004 0.000 2.110 3.501
==============================================================================
Omnibus: 1.210 Durbin-Watson: 2.349
Prob(Omnibus): 0.546 Jarque-Bera (JB): 0.703
Skew: 0.122 Prob(JB): 0.704
Kurtosis: 3.330 Cond. No. 5.58
==============================================================================
Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.0.2554/0.2770.9220216606498195- R-squared measures how well the independant variables explain the variability of the dependant variable
- F-Statistic measures the significance of the regression model
- t-statistic for each coefficient measures the significance level of each independent variable
- Logistic Regression:
- Logistic regression is used when the dependent variable is binary (e.g., 0 or 1, True or False).
# Generate example data for logistic regression
np.random.seed(0)
X = np.random.rand(100, 2) # Two independent variables
# Generate binary outcome variable based on a threshold
threshold = 0.6
y = (2 * X[:, 0] + 3 * X[:, 1] > threshold).astype(int)
# Add constant term for intercept
X = sm.add_constant(X)
# Fit logistic regression model
logit_model = sm.Logit(y, X).fit()
# Print model summary
print(logit_model.summary())Warning: Maximum number of iterations has been exceeded.
Current function value: 0.000000
Iterations: 35
Logit Regression Results
==============================================================================
Dep. Variable: y No. Observations: 100
Model: Logit Df Residuals: 97
Method: MLE Df Model: 2
Date: Wed, 17 Apr 2024 Pseudo R-squ.: 1.000
Time: 13:05:57 Log-Likelihood: -1.1958e-06
converged: False LL-Null: -9.8039
Covariance Type: nonrobust LLR p-value: 5.524e-05
==============================================================================
coef std err z P>|z| [0.025 0.975]
------------------------------------------------------------------------------
const -70.8342 4.95e+04 -0.001 0.999 -9.71e+04 9.7e+04
x1 196.7613 1.92e+05 0.001 0.999 -3.76e+05 3.76e+05
x2 488.7691 7.23e+05 0.001 0.999 -1.42e+06 1.42e+06
==============================================================================
Complete Separation: The results show that there iscomplete separation or perfect prediction.
In this case the Maximum Likelihood Estimator does not exist and the parameters
are not identified./home/jumashafara/venvs/dataanalysis/lib/python3.10/site-packages/statsmodels/base/model.py:607: ConvergenceWarning: Maximum Likelihood optimization failed to converge. Check mle_retvals
warnings.warn("Maximum Likelihood optimization failed to "These examples demonstrate how to implement linear regression and logistic regression using statsmodels.api. The summary output provides detailed information about the model parameters, goodness-of-fit measures, and statistical significance of predictors. This can be useful for interpreting the results and assessing the performance of the models.