Correlation and Experimental Design¶

In this chapter, you'll learn how to quantify the strength of a linear relationship between two variables, and explore how confounding variables can affect the relationship between two other variables. You'll also see how a study’s design can influence its results, change how the data should be analyzed, and potentially affect the reliability of your conclusions. This is the Summary of lecture "Introduction to Statistics in Python", via datacamp.

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author: Chanseok Kang
categories: [Python, Datacamp, Statistics]
image: images/lmplot_wh.png

In [1]:

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.

In [2]:

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

Out[2]:

	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

In [3]:

# Create a scatterplot of happiness_score vs. life_exp and show
sns.scatterplot(x='life_exp', y='happiness_score', data=world_happiness);

In [5]:

# Create scatterplot of happiness_score vs. life_exp with trendline
sns.lmplot(x='life_exp', y='happiness_score', data=world_happiness, ci=None);

In [6]:

# Correlation between life_exp and happiness_score
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.

In [7]:

# Scatterplot of gdp_per_cap and life_exp
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.

In [8]:

# Scatterplot of happiness_score vs. gdp_per_cap
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

In [9]:

# Create log_gdp_per_cap column
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.

In [10]:

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

Out[10]:

	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

In [11]:

# Scatterplot of grams_sugar_per_day and happiness_score
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?
  - Treatment: advertisement
  - Response: number of products purchased
Controlled experiments
- Participants are assigned by researchers to either treatment group or control group
  - Treatment group sees advertisement
  - 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