## Introduction to Quantile Regression

Regression analysis is one of the most common statistical tools used in data analysis. It is used to understand the relationship between a dependent variable and one or more independent variables.

The traditional linear regression model assumes that the relationship between the dependent and independent variables is linear. However, in reality, this assumption may not always hold.

In such cases, quantile regression is a powerful alternative.

## Definition of Quantile Regression

Quantile regression is a type of regression analysis used to model the relationship between the dependent variable and one or more independent variables. Unlike traditional linear regression, which assumes a linear relationship between the dependent and independent variables, quantile regression allows us to model the relationship at different points of the distribution of the dependent variable.

This means that we can estimate the relationship between the independent variables and the dependent variable for different quantiles of the distribution.

## Differences between Linear Regression and Quantile Regression

Linear regression is a powerful tool used to estimate the relationship between a dependent variable and one or more independent variables. It assumes that the relationship is linear, which means that the slope of the line is constant across all points of the distribution.

In contrast, quantile regression allows us to estimate the relationship for different quantiles of the distribution. For example, suppose we want to estimate the effect of hours studied on exam scores.

We might be interested in knowing how the effect of hours studied differs at different points of the distribution of exam scores. In linear regression, we would estimate a single slope for the entire distribution of exam scores.

In contrast, in quantile regression, we could estimate different slopes for different quantiles of the distribution, such as the 10th, 25th, 50th, 75th, and 90th percentiles. This allows us to understand how the relationship between hours studied and exam scores changes at different levels of performance.

## Loading Packages and Creating Data

In order to demonstrate the use of quantile regression, we will use a simulated dataset. We will use Python and the following packages: NumPy, pandas, statsmodels, and Matplotlib.

First, we need to import the necessary packages and functions:

```
import numpy as np
import pandas as pd
import statsmodels.api as sm
import matplotlib.pyplot as plt
```

Next, we will create a dataset for the example. We will simulate a dataset with 100 observations, where the independent variable is the number of hours studied, and the dependent variable is the exam score.

We will also add some random noise to the exam scores to make the dataset more realistic. We will set a random seed to ensure reproducibility:

```
np.random.seed(123)
hours_studied = np.random.normal(5, 1, 100)
exam_score = 50 + 10 * hours_studied + np.random.normal(0, 5, 100)
df = pd.DataFrame({'hours_studied': hours_studied, 'exam_score': exam_score})
```

### We can now visualize the relationship between hours studied and exam score using a scatter plot:

```
plt.scatter(df['hours_studied'], df['exam_score'])
plt.xlabel('Hours Studied')
plt.ylabel('Exam Score')
plt.show()
```

## Conclusion

In conclusion, quantile regression is a powerful tool that allows us to model the relationship between a dependent variable and one or more independent variables at different points of the distribution. This is particularly useful when the relationship is non-linear or when we want to understand how the relationship varies at different levels of performance.

By using Python and popular packages such as NumPy, pandas, statsmodels, and Matplotlib, we can easily implement quantile regression in our data analysis workflows.

## Performing Quantile Regression

In this section, we will demonstrate how to perform quantile regression using Python and the previously created dataset. We will fit a quantile regression model to estimate the relationship between hours studied and exam scores.

## Fitting a Quantile Regression Model

We will use the Python package statsmodels to estimate the quantile regression model. The function we will use is called QuantReg, and we will specify the quantile we want to estimate using the q argument.

For example, to estimate the median (50th percentile), we would set q=0.5.

```
quantiles = [0.1, 0.25, 0.5, 0.75, 0.9]
x = df['hours_studied']
y = df['exam_score']
models = []
for q in quantiles:
model = sm.quantreg('exam_score ~ hours_studied', df)
res = model.fit(q=q)
models.append(res)
```

The above code fits a separate quantile regression model for each specified quantile (10th, 25th, 50th, 75th, and 90th percentiles) and saves the model results in a list called `models`.

## Predicting the Expected 90th Percentile of Exam Scores

We can use the fitted quantile regression model to predict the expected 90th percentile of exam scores for a given number of hours studied. Suppose we want to predict the expected 90th percentile of exam scores for a student who studied 7 hours:

```
x_pred = pd.DataFrame({'hours_studied': [7]})
y_pred = []
for model in models:
y_pred.append(model.predict(x_pred))
y_pred = np.array(y_pred)
y_pred.mean()
```

The above code creates a DataFrame with the predictor variable (hours studied) that we want to predict for (7 hours), and then uses the `predict` function to estimate the exam score for each quantile regression model.

We then take the average of the predicted values across all quantiles to estimate the expected 90th percentile of exam scores for a student who studied 7 hours.

## Upper and Lower Confidence Limits for the Intercept and Predictor Variable

We can calculate upper and lower confidence limits for the intercept and predictor variable using the fitted quantile regression model. Suppose we want to calculate the 95% confidence limits for the intercept and predictor variable:

```
conf_ints = []
for model in models:
conf_ints.append(model.conf_int())
conf_ints = np.array(conf_ints)
conf_ints.mean(axis=0)
```

The above code calculates the 95% confidence limits for the intercept and predictor variable for each quantile regression model using the `conf_int` function.

We then take the average of the confidence intervals across all quantiles to estimate the overall confidence limits.

## Scatterplot with the Fitted Quantile Regression Equation

We can also create a scatterplot with the fitted quantile regression equation to visualize the relationship between hours studied and exam scores. Suppose we want to create a scatterplot with the fitted quantile regression equation for the 50th percentile:

```
x_range = np.arange(df['hours_studied'].min(), df['hours_studied'].max()+1)
y_pred = models[2].predict(pd.DataFrame({'hours_studied': x_range}))
plt.scatter(x, y, alpha=0.5)
plt.plot(x_range, y_pred, color='red')
plt.xlabel('Hours Studied')
plt.ylabel('Exam Score')
plt.show()
```

The above code creates a range of predictor variable values (hours studied) using `np.arange`, predicts the expected values of the dependent variable (exam scores) for each value of the predictor variable using the `predict` function for the 50th percentile model, and then creates a scatterplot of the data points with the fitted quantile regression equation overlaid in red.

## Conclusion

In this section, we learned how to perform quantile regression using Python and the statsmodels package. We fit a quantile regression model to estimate the relationship between hours studied and exam scores, predicted the expected 90th percentile of exam scores, calculated upper and lower confidence limits for the intercept and predictor variable, and created a scatterplot with the fitted quantile regression equation.

By performing quantile regression, we gain a deeper understanding of the relationship between the predictor variable (hours studied) and the dependent variable (exam score) at different points of the distribution. This can be useful for making more accurate predictions and identifying potential areas for improvement.

Quantile regression is a statistical technique that allows us to estimate the relationship between a dependent variable and one or more independent variables for different quantiles of the distribution. Unlike traditional linear regression, which assumes a linear relationship between the variables, quantile regression is useful when the relationship is non-linear or we want to understand how it varies across the distribution.

By using Python and packages like NumPy, Pandas, Statsmodels, and Matplotlib, we can perform quantile regression, predict a specific quantile of the dependent variable, calculate confidence intervals, and visualize the fitted regression equation. Quantile regression is helpful in making more accurate predictions and identifying potential areas for improvement.