import pandas as pd
import statsmodels.api as sm
# Importing CSV
htwt = pd.read_csv("../data/htwt.csv")Linear Regression
To demonstrate the use of linear regression we examine a dataset that illustrates the relationship between Height and Weight in a group of 237 teen-aged boys and girls. The dataset is available here and is imported to the workspace.
Descriptive Statistics
The first step is to obtain the simple descriptive statistics for the numeric variables of htwt data, and one-way frequencies for categorical variables. This is accomplished by employing summary function. There are 237 participants who are from 13.9 to 25 years old. It is a cross-sectional study, with each participant having one observation. We can use this data set to examine the relationship of participants’ height to their age and sex.
In order to create a regression model to demonstrate the relationship between age and height for females, we first need to create a flag variable identifying females and an interaction variable between age and female gender flag.
htwt['female'] = (htwt['SEX'] == 'f').astype(int)
htwt['fem_age'] = htwt['AGE'] * htwt['female']
htwt.head()| ROW | SEX | AGE | HEIGHT | WEIGHT | female | fem_age | |
|---|---|---|---|---|---|---|---|
| 0 | 1 | f | 14.3 | 56.3 | 85.0 | 1 | 14.3 |
| 1 | 2 | f | 15.5 | 62.3 | 105.0 | 1 | 15.5 |
| 2 | 3 | f | 15.3 | 63.3 | 108.0 | 1 | 15.3 |
| 3 | 4 | f | 16.1 | 59.0 | 92.0 | 1 | 16.1 |
| 4 | 5 | f | 19.1 | 62.5 | 112.5 | 1 | 19.1 |
Regression Analysis
Next, we fit a regression model, representing the relationships between gender, age, height and the interaction variable created in the datastep above. We again use a where statement to restrict the analysis to those who are less than or equal to 19 years old. We use the clb option to get a 95% confidence interval for each of the parameters in the model. The model that we are fitting is height = b0 + b1 x female + b2 x age + b3 x fem_age + e
X = htwt[['female', 'AGE', 'fem_age']][htwt['AGE'] <= 19]
X = sm.add_constant(X)
Y = htwt['HEIGHT'][htwt['AGE'] <= 19]
model = sm.OLS(Y, X).fit()
model.summary()| Dep. Variable: | HEIGHT | R-squared: | 0.460 |
| Model: | OLS | Adj. R-squared: | 0.452 |
| Method: | Least Squares | F-statistic: | 60.93 |
| Date: | Fri, 25 Oct 2024 | Prob (F-statistic): | 1.50e-28 |
| Time: | 08:31:55 | Log-Likelihood: | -534.17 |
| No. Observations: | 219 | AIC: | 1076. |
| Df Residuals: | 215 | BIC: | 1090. |
| Df Model: | 3 | ||
| Covariance Type: | nonrobust |
| coef | std err | t | P>|t| | [0.025 | 0.975] | |
| const | 28.8828 | 2.873 | 10.052 | 0.000 | 23.219 | 34.547 |
| female | 13.6123 | 4.019 | 3.387 | 0.001 | 5.690 | 21.534 |
| AGE | 2.0313 | 0.178 | 11.435 | 0.000 | 1.681 | 2.381 |
| fem_age | -0.9294 | 0.248 | -3.750 | 0.000 | -1.418 | -0.441 |
| Omnibus: | 1.300 | Durbin-Watson: | 2.284 |
| Prob(Omnibus): | 0.522 | Jarque-Bera (JB): | 0.981 |
| Skew: | -0.133 | Prob(JB): | 0.612 |
| Kurtosis: | 3.191 | Cond. No. | 450. |
Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.
From the coefficients table b0,b1,b2,b3 are estimated as b0=28.88 b1=13.61 b2=2.03 b3=-0.92942
The resulting regression model for height, age and gender based on the available data is height=28.8828 + 13.6123 x female + 2.0313 x age -0.9294 x fem_age