import pandas as pd
import numpy as np
import matplotlib.pyplot as plt
import seaborn as sns


## Correlation

• Correlation coefficient
• Quantifies the linear relationship between two variables
• Number between -1 and 1
• Magnitude corresponds to strength of relationship
• Sign (+ or -) corresponds to direction of relationship
• Pearson product-moment correlation($r$)

• Most Common
• $\bar{x}$ = mean of $x$
• $\sigma_x$ = standard deviation of $x$

$$r = \sum_{i=1}^{n} \frac{(x_i - \bar{x})(y_i - \bar{y})}{\sigma_x \times \sigma_y}$$

• Variation

• Kendall's Tau
• Spearman's rho

### Relationships between variables

In this chapter, you'll be working with a dataset world_happiness containing results from the 2019 World Happiness Report. The report scores various countries based on how happy people in that country are. It also ranks each country on various societal aspects such as social support, freedom, corruption, and others. The dataset also includes the GDP per capita and life expectancy for each country.

In this exercise, you'll examine the relationship between a country's life expectancy (life_exp) and happiness score (happiness_score) both visually and quantitatively.

world_happiness = pd.read_csv('./dataset/world_happiness.csv', index_col=0)

country social_support freedom corruption generosity gdp_per_cap life_exp happiness_score
1 Finland 2.0 5.0 4.0 47.0 42400 81.8 155
2 Denmark 4.0 6.0 3.0 22.0 48300 81.0 154
3 Norway 3.0 3.0 8.0 11.0 66300 82.6 153
4 Iceland 1.0 7.0 45.0 3.0 47900 83.0 152
5 Netherlands 15.0 19.0 12.0 7.0 50500 81.8 151
sns.scatterplot(x='life_exp', y='happiness_score', data=world_happiness);

sns.lmplot(x='life_exp', y='happiness_score', data=world_happiness, ci=None);

cor = world_happiness['life_exp'].corr(world_happiness['happiness_score'])
print(cor)

0.7802249053272061


## Correlation caveats

• Correlation only accounts for linear relationships
• Transformation
• Certain statistical methods rely on variables having a linear relationship
• Correlation coefficient
• Linear regression
• Correlation does not imply causation
• $x$ is correlated with $y$ does not mean $x$ causes $y$

### What can't correlation measure?

While the correlation coefficient is a convenient way to quantify the strength of a relationship between two variables, it's far from perfect. In this exercise, you'll explore one of the caveats of the correlation coefficient by examining the relationship between a country's GDP per capita (gdp_per_cap) and happiness score.

sns.scatterplot(x='gdp_per_cap', y='life_exp', data=world_happiness);

# Correlation between gdp_per_cap and life_exp
cor = world_happiness['gdp_per_cap'].corr(world_happiness['life_exp'])
print(cor)

0.7019547642148014


### Transforming variables

When variables have skewed distributions, they often require a transformation in order to form a linear relationship with another variable so that correlation can be computed. In this exercise, you'll perform a transformation yourself.

sns.scatterplot(x='gdp_per_cap', y='happiness_score', data=world_happiness);

# Calculate correlation
cor = world_happiness['gdp_per_cap'].corr(world_happiness['happiness_score'])
print(cor)

0.7279733012222975

world_happiness['log_gdp_per_cap'] = np.log(world_happiness['gdp_per_cap'])

# Scatterplot of log_gdp_per_cap and happiness_score
sns.scatterplot(x='log_gdp_per_cap', y='happiness_score', data=world_happiness);

# Calculate correlation
cor = world_happiness['log_gdp_per_cap'].corr(world_happiness['happiness_score'])
print(cor)

0.8043146004918288


### Does sugar improve happiness?

A new column has been added to world_happiness called grams_sugar_per_day, which contains the average amount of sugar eaten per person per day in each country. In this exercise, you'll examine the effect of a country's average sugar consumption on its happiness score.

world_happiness = pd.read_csv('./dataset/world_happiness_add_sugar.csv', index_col=0)
world_happiness

country social_support freedom corruption generosity gdp_per_cap life_exp happiness_score grams_sugar_per_day
Unnamed: 0
1 Finland 2 5 4.0 47 42400 81.8 155 86.8
2 Denmark 4 6 3.0 22 48300 81.0 154 152.0
3 Norway 3 3 8.0 11 66300 82.6 153 120.0
4 Iceland 1 7 45.0 3 47900 83.0 152 132.0
5 Netherlands 15 19 12.0 7 50500 81.8 151 122.0
... ... ... ... ... ... ... ... ... ...
129 Yemen 100 147 83.0 155 2340 68.1 5 77.9
130 Rwanda 144 21 2.0 90 2110 69.1 4 14.1
131 Tanzania 131 78 34.0 49 2980 67.7 3 28.0
132 Afghanistan 151 155 136.0 137 1760 64.1 2 24.5
133 Central African Republic 155 133 122.0 113 794 52.9 1 22.4

133 rows × 9 columns

sns.scatterplot(x='grams_sugar_per_day', y='happiness_score', data=world_happiness);

# Correlation between grams_sugar_per_day and happiness_score
cor = world_happiness['grams_sugar_per_day'].corr(world_happiness['happiness_score'])
print(cor)

0.6939100021829635


## Design of experiments

• Vocabulary
• Experiment aims to answer: What is the effect of the treatment on the response?
• Treatment: explanatory / independent variable
• Response: response / dependent variable
• E.g.: What is the effect of an advertisement on the number of products purchased?
• Response: number of products purchased
• Controlled experiments
• Participants are assigned by researchers to either treatment group or control group
• Control group does not
• Group should be comparable so that causation can be inferred
• If groups are not comparable, this could lead to confounding (bias)
• Gold standard of experiment
• Randomized controlled trial
• Participants are assigned to treatment/control randomly, not based on any other characteristics
• Choosing randomly helps ensure that groups are comparable
• Placebo
• Resembles treatement, but has no effect
• Participants will not know which group they're in
• Double-blind trial
• Person administering the treatment/running the study doesn't know whether the treatment is real or a placebo
• Prevents bias in the response and/or analysis of results
• Fewopportunities for bias = more reliable conclusion about causation
• Observational studies
• Participants are not assigned randomly to groups
• Participants assign themselves, usually based on pre-existing characteristics
• Many research questions are not conductive to a controlled experiment
• Cannot force someone to smoke or have a disease
• Establish association, not causation
• Effects can be confounded by factors that got certain people into the control or treatment group
• There are ways to control for confounders to get more reliable conclusions about association
• Longitudinal vs. cross-sectional studies
• Longitudinal study
• Participants are followed over a period of time to examine effect of treatment on response
• Effect of age on height is not confounded by generation
• More expensive, results take longer
• Cross-sectional study
• Data on participants is collected from a single snapshot in time
• Effect of age on height is confounded by generation
• Cheaper, fater, more convenient