Pearson and Spearman Rank Correlation Coefficient — Explained

Relationship between random variables.

Correlation Coefficient is a statistical measure to find the relationship between two random variables. Correlation between two random variables can be used to compare the relationship between the two. By observing the correlation coefficient, the strength of the relationship can be measured.

The value of the correlation coefficient ranges from -1 to +1.

• The value close to +1 denotes a high linear relationship, and with an increase of one random variable, the second random variable also increases.
• The value close to -1 denotes a high linear relationship, and with an increase of one random variable, the second random variable decreases.
• The value close or equal to 0, denotes no relationship between the two random variables.

Some prerequisites — Covariance:

Covariance is a measure used to determine how much two random variables differ by its respective mean. It is affected by a change in scale. The value of the covariance coefficient lies between -∞ and +∞.

Notation,
X, Y: Two random variables
X_bar: mean of random variable X
Y_bar: mean of random variable Y
n: length of random variable X, Y

Here covariance of height vs weight >0 which is 114.24, which means with an increase in height, weight increases.

Hence covariance compares two variables in terms of the deviations from their mean value.

There is one limitation of covariance that its value ranges between -∞ and +∞, hence

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Effective Way To Replace Correlation With Predictive Power Score(PPS)

The strength of a linear relationship between two quantitative variables can be measured using Correlation. It is a statistical method that is very easy in order to calculate and to interpret. It is generally represented by ‘r’ known as the coefficient of correlation.

This is the reason why it is highly misused by professionals because correlation cannot be termed for causation. It is not necessary that if two variables have a correlation then one is dependent on the other and similarly if there is no correlation between two variables it is possible that they might have some relation. This is where PPS(Predictive Power Score) comes into the role.

Predictive Power Score works similar to the coefficient of correlation but has some additional functionalities like:

• It works on both Linear and Non-Linear Relationships
• Can be applied to both Numeric and Categorical columns
• It finds more patterns in the data.

In this article, we will explore how we can use the Predictive Power Score to replace correlation.

Implementation:

PPS is an open-source python library so we will install it like any other python library using pip install ppscore.

1. Importing required libraries

We will import ppscore along with pandas to load a dataset that we will work on.

import ppscore as pps

import pandas as pd

1. Loading the Dataset

We will be using different datasets to explore different functionalities of PPS. We will first import an advertising dataset of an MNC which contains the target variable as ‘Sales’ and features like ‘TV’, ‘Radio’, etc.

df = pd.read_csv(‘advertising.csv’)

df.head()

1. Finding Relation using PPScore

We will use some basic functions defined in ppscore.

1. Finding the Relationship score

PP Score lies between 0(No Predictive Power) to 1(perfect predictive power), in this step we will find PPScore/Relationship between the target variable and the featured variable in the given dataset.

pps.score(df, "Sales", "TV")

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Pearson and Spearman Rank Correlation Coefficient — Explained

Relationship between random variables.

Correlation Coefficient is a statistical measure to find the relationship between two random variables. Correlation between two random variables can be used to compare the relationship between the two. By observing the correlation coefficient, the strength of the relationship can be measured.

The value of the correlation coefficient ranges from -1 to +1.

• The value close to +1 denotes a high linear relationship, and with an increase of one random variable, the second random variable also increases.
• The value close to -1 denotes a high linear relationship, and with an increase of one random variable, the second random variable decreases.
• The value close or equal to 0, denotes no relationship between the two random variables.

Some prerequisites — Covariance:

Covariance is a measure used to determine how much two random variables differ by its respective mean. It is affected by a change in scale. The value of the covariance coefficient lies between -∞ and +∞.

Notation,
X, Y: Two random variables
X_bar: mean of random variable X
Y_bar: mean of random variable Y
n: length of random variable X, Y

Here covariance of height vs weight >0 which is 114.24, which means with an increase in height, weight increases.

Hence covariance compares two variables in terms of the deviations from their mean value.

There is one limitation of covariance that its value ranges between -∞ and +∞, hence

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Pearson correlation coefficient

In my previous blog, we learnt about Covariance to measure relationship between two random variables.

Covariance

In this blog, we’ll try to understand how to measure relationships between random variables.

medium.com

As Covariance has limitation to quantify the relationship, there is another concept called Pearson correlation coefficient (PCC) that overcome this limitation. It’s often represented with the Greek alphabet ρ. So the Pearson correlation coefficient between two random variables x and y is nothing but the covariance( X, Y) divided by the standard deviation of x and the standard deviation of y. Here is the mathematical formula for ρ.

Now, you might ask, why are we defining a new metric? Because covariance doesn’t take variability in account, and here we use the standard deviation of x and y in denominator.

What exactly standard deviation of x is? It is nothing but square root of variance of x, and variance is all about variability.

When you measure covariance, you’re not measuring the variability within x&y. But just a small modification on covariance i.e (dividing your covariance by a standard deviation of x and standard deviation of y) will give you variability and interpretability.

As we saw in last blog on Covariance i.e, as x increases, if y also increase, then covariance is going to be positive. But how much positive? It could be very, very positive or very negative, right? Similarly, I know that as x increases, y decreases, my covariance is going to be negative. Right? But I don’t know how much negative…

So PCC is a very nice idea to quantify the relationship, Below graph gives a better understanding on PCC.

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Clearly explained: Pearson V/S Spearman Correlation Coefficient

I recently came across a scenario where I educated myself about the difference between the Pearson and Spearman correlation coefficient. I felt that is one piece of information that a lot of people in the data science fraternity on the medium can make use of. I’ll explain thoroughly the difference between the two and the exact scenarios where the use of each one is suitable. Read on!Contents of this post:

1. Definition of CorrelationComparative analysis between Pearson and Spearman correlation coefficients

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Definition of Correlation

Correlation is the degree to which two variables are linearly related. This is an important step in bi-variate data analysis. In the broadest sense correlation is actually any statistical relationship, whether causal or not, between two random variables in bivariate data.

An important rule to remember is that Correlation doesn’t imply causation

Let’s understand through two examples as to what it actually implies.

1. The consumption of ice-cream increases during the summer months. There is a strong correlation between the sales of ice-cream units. In this particular example, we see there is a causal relationship also as the extreme summers do push the sale of ice-creams up.Ice-creams sales also have a strong correlation with shark attacks. Now as we can see very clearly here, the shark attacks are most definitely not caused due to ice-creams. So, there is no causation here.

Hence, we can understand that the correlation doesn’t ALWAYS imply causation!

What is the Correlation Coefficient?

The correlation coefficient is a statistical measure of the strength of the relationship between the relative movements of two variables. The values range between -1.0 and 1.0. A correlation of -1.0 shows a perfect negative correlation, while a correlation of 1.0 shows a perfect positive correlation. A correlation of 0.0 shows no linear relationship between the movement of the two variables.

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2 Important Correlation Coefficients — Pearson & Spearman

1. Pearson Correlation Coefficient

**Wikipedia Definition: **In statistics, the Pearson correlation coefficient also referred to as Pearson’s _r _or the bivariate correlation is a statistic that measures the linear correlation between two variables X and Y. It has a value between +1 and −1. A value of +1 is a total positive linear correlation, 0 is no linear correlation, and −1 is a total negative linear correlation.

_Important Inference to keep in mind: _The Pearson correlation can evaluate ONLY a linear relationship between two continuous variables (A relationship is linear only when a change in one variable is associated with a proportional change in the other variable)Example use case:_ We can use the Pearson correlation to evaluate whether an increase in age leads to an increase in blood pressure._

Below is an example of how the Pearson correlation coefficient ® varies with the **strength and the direction of the relationship **between the two variables. Note that when no linear relationship could be established (refer to graphs in the third column), the Pearson coefficient yields a value of zero.

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