Solved – How to get regression equation from statsmodel (python)

multiple regressionpythonregressionstatsmodels

I am poor in statistics. I did a multiple linear regression in python by using statsmodels. I know that I can transform a non-linear relation into a linear relation in different ways using this model. Combinations of different independent variables are shown in below:

smf.ols(formula = 'y ~ a + I(a**2) + log(b) + c + d', data=data).fit()

where $y$ is a dependent variable depends on second degree polynomial of $a$, log of $b$ and linear in $c$ and $d$.

My question is in such a case what will my equation be?

Best Answer

If I understand your question, you're asking for the model form:

$$y = \alpha + \beta_0a + \beta_1a^2 + \beta_2\log(b) + \beta_3c + \beta_4d + \epsilon$$

where:

$\epsilon \sim N(0,\sigma^2)$ is the OLS assumption
$a,b,c,d$ are in your data
$\beta_0,...\beta_4$ are your coefficients. $\alpha$ is the intercept which is added automatically when you use formulas, unless specified otherwise (by including a -1 in the formula).

See here for more on how Python formulas work.

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There are 2 issues that you need to understand. First is that your equations do not take into account the correlation between the coefficient estimates and the second is the difference between a confidence interval and a prediction interval.

The confidence interval tells where you think the mean response will be for a given set of x-values. Much fewer than 95% of your observations (and future observations) will fall within the confidence bands. What you are asking for is a prediction interval, which tells where a future single value is likely to fall.

There are many resources that discuss prediction and confidence intervals. Regression textbooks will give the formulas to use along with more detailed explanation. One online resource (there are many, this is just one I found to point to) is https://onlinecourses.science.psu.edu/stat501/node/315 which refers to a formula on the previous page (section 7.1) that you may need to click on to understand the full formula.

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There are situations where it would make sense to include the predicted random effects (BLUPs) in a prediction. If you have observations on a group (e.g. a person) and you want to make predictions about additional responses from the same group, then you would generally want to include the BLUP.

This is possible in statsmodels, as illustrated below. We should add an option to make this easier. For now, you have to do it manually, as shown below.

import pandas as pd
import statsmodels.api as sm
import numpy as np

# Simulate data for illustration
group_size = 5
n_groups = 100
x1 = np.random.normal(size=group_size*n_groups)
x2 = np.random.normal(size=group_size*n_groups)
u = np.kron(np.random.normal(size=n_groups), np.ones(group_size))
g = np.kron(np.arange(n_groups), np.ones(group_size))
e = np.random.normal(size=group_size*n_groups)
y = x1 - x2 + u + e

# Fit a multilevel model
df = pd.DataFrame({"y":y, "x1":x1, "x2":x2, "g":g})
model = sm.MixedLM.from_formula("y ~ x1 + x2", groups="g", data=df)
result = model.fit()

# The BLUPs
re = result.random_effects

# Multiply each BLUP by the random effects design matrix for one group
rex = [np.dot(model.exog_re_li[j], re[k]) for (j, k) in enumerate(model.group_labels)]

# Add the fixed and random terms to get the overall prediction
rex = np.concatenate(rex)
yp = result.fittedvalues + rex

Best Answer

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Solved – Equation for Confidence Interval of Linear Regression

Linear Mixed Models – Including Random Effects in Prediction with LMM

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