# Test a Basic Linear Regression Model¶

```
Author: Samuel M.H. <samuel.mh@gmail.com>
Date: 28-02-2016
```

## Instructions

This week's assignment asks you to test a basic linear regression model for the association between your primary explanatory variable and a response variable, and to create a blog entry describing your results.

### Data preparation for this assignment:

If your explanatory variable is categorical with more than two categories, you will need to collapse it down to two categories, or subset your data to select observations from 2 categories (next week you'll learn how to analyze categorical explanatory variable with more than 2 categories).

If your response variable is categorical, you will need to identify a quantitative variable in the data set that you can use as a response variable for this assignment. Variables with response scales with 4-5 values that represent a change in magnitude (for example, "strongly disagree to strongly agree", "never to often") can be considered quantitative for the assignment.

### The assignment:

If you have a categorical explanatory variable, make sure one of your categories is coded "0" and generate a frequency table for this variable to check your coding. If you have a quantitative explanatory variable, center it so that the mean = 0 (or really close to 0) by subtracting the mean, and then calculate the mean to check your centering.

Test a linear regression model and summarize the results in a couple of sentences. Make sure to include statistical results (regression coefficients and p-values) in your summary.

## What to submit

Create a blog entry where you 1) post your program and output, and 2) post a frequency table for your (recoded) categorical explanatory variable or report the mean for your centered explanatory variable. 3) Write a few sentences describing the results of your linear regression analysis.

## Intro

In this exercise I will test the association between the age at which a person gets married for the first time and the alcohol they consumed during last year.

## Dataset

- National Epidemiological Survey on Alcohol and Related Conditions (NESARC)
- CSV file
- File description

### Variables

**Explanatory**: S1Q4A, age at first marriage.- Quantitative: 14-94 years, 31794 samples
- Unknown values (99) and never married people are discarded.

**Response**: ETOTLCA2, average daily volume of ethanol consumed in past year, from all types of alcoholic beverages combined.- Quantitative: 0.0003-219.955 oz. ethanol/day
- Unkwnown values are discarded observations. Could be mapped to the mean, but don't add much.

```
%matplotlib inline
import numpy as np
import pandas as pd
import statsmodels.formula.api as smf
import seaborn as sns
import matplotlib.pyplot as plt
#import scipy
```

## Data

```
# Load data
data = pd.read_csv('../datasets/NESARC/nesarc_pds.csv', usecols=['S1Q4A','ETOTLCA2'])
# Custom dataframe
df = pd.DataFrame()
df['AGE'] = data['S1Q4A'].replace(' ',np.NaN).replace('99',np.NaN).astype(float)
df['ALCOHOL'] = data['ETOTLCA2'].replace(' ',np.NaN).astype(float)
df = df.dropna()
# Center the explanatory variable
print('Original')
print(df.describe())
df['AGE'] = df['AGE']- df['AGE'].mean()
print('\n\nCentered AGE')
print(df.describe())
```

We see the final population contains 19277 samples and the AGE variable has been centered, zero mean.

### Outliers

In order to improve the generalization an obtain better results, I will perform an analysis of the outliers.

These are the boxplots:

```
# Boxplots
fig, (ax1,ax2)= plt.subplots(1,2)
bp_age = sns.boxplot(x='AGE',data=df, whis=1.5, orient='v', ax=ax1)
bp_alcohol = sns.boxplot(x='ALCOHOL',data=df, whis=1.5, orient='v', ax=(ax2))
plt.subplots_adjust(bottom=0.1, right=1.8, top=1.5)
fig.show()
```

It is possible to see that there are several outliers in the AGE variable as well as in the ALCOHOL one. The procedure will be calculating the interquartile range and discarding the observations greater than 1.5 these values (the blue points).

Values $v$ that satisfy any of these properties are discarded: $$ v_{max} > Q_3 + 1.5 (Q_3-Q_1)\\ v_{min} < Q_1 - 1.5 (Q_3-Q_1) $$

```
#Calculate minimum and maximum values for outliers
def outlier_limits(data, whis=1.5):
q3 = np.percentile(data, 75)
q1 = np.percentile(data, 25)
iqr = q3-q1
return(
q1-(whis*iqr),
q3+(whis*iqr),
)
```

```
# Outlier limits
print('Allowed ranges')
print('{0} < AGE < {1}'.format(*outlier_limits(df['AGE'])))
print('{0} < ALCOHOL < {1}'.format(*outlier_limits(df['ALCOHOL'])))
```

```
age_lower,age_upper= outlier_limits(df['AGE'])
alc_lower,alc_upper= outlier_limits(df['ALCOHOL'])
df = df[
(df['AGE']>age_lower) &
(df['AGE']<age_upper) &
(df['ALCOHOL']>alc_lower) &
(df['ALCOHOL']<alc_upper)
]
df['AGE'] = df['AGE']- df['AGE'].mean()
df.describe()
```

It is possible to see that all the values are in range, outliers have been removed and the population is formed by 16451 samples.

## Regression model

```
model = smf.ols(formula='ALCOHOL~AGE',data=df).fit()
print model.summary()
```

## Summary

The p-value (nearly zero) is lower than 0.05. Therefore, the null hypothesis can be rejected as there is statistical evidence of the association between the variables.

The R-squared value is nearly 0, so, almost no variance can be explained by the AGE and the predictive equation $$ \hat{ALCOHOL}=AGE*0.0051+0.1898 $$ is useless. There is no correlation between variables.

This is a scatter plot with the fitted line.

```
sns.lmplot(x='AGE', y='ALCOHOL', data=df)
```

It is possible to see the plot does not comply with the homoscedasticity assumption for the linear regression model.

The linear model cannot decribe the relationship between the age at which one gets married and the alcohol they drank during last year, but, the relationship has statistical significance (it did not happen by chance).