WK2 Lecture 1 - Scatterplots

Components

A scatterplot is a graphical representation that displays the relationship between two numeric continuous variables.

Components include:

Data points - each point is a unit
X axis - independent (explanatory) variable
Y axis - dependent (response) variable
Axes labels and title

Describing Relationships

The relationship between two numeric variables visualised in a scatterplot is described based on the direction, shape, and strength.

Scatterplot Features.pngcenter
Investigating relationships (aka associations) requires statistical inference to understand if there's evidence of a relationship or not.

Correlation

A scatterplot is the visual component in identifying correlations and relationships between two numeric variables. The correlation coefficient $(r)$ is the numerical value of the strength of the linear relationship.

The equation for determining correlation coefficient is as follows:

r = \frac{n \sum X Y - (\sum X) (\sum Y)}{\sqrt{[n \sum X^{2} - (\sum X)^{2}] [n \sum Y^{2} - (\sum Y)^{2}]}}

However this is often not an efficient way of determining correlation, it can be calculated in r as follows

cor (DATASET$X-AXIS ~ DATASET$Y-AXIS)

or by looking for multiple R squared on a summary of the data, note that this gives strength but not direction.

Properties of correlation coefficients are:

$r$ lies between -1 and 1
$r = 1$ if the relationship between x and y is a straight line as both increase
$r = - 1$ if the relationship is a straight line as x increases and y decreases
$r = 0$ if x and y are not linearly related (no correlation, or quadratic relationship)
$r$ indicates the slope and strength of the linear relationship

When evaluating a scatterplot by sight alone, it is sometimes difficult to determine the strength of the relationship. One can describe it as "appears to be a strong/moderate/weak relationship". When describing the strength based on the correlation coefficient the strength of the values are subjective as a general rule:

Absolute value of R	strength of the relationship
0 - 0.2	Very weak
0.2 - 0.4	Weak
0.4 - 0.6	Moderate
0.6 - 0.8	Strong
0.8 - 1.0	Very strong

Correlation and Causation

It's important to note that correlation is not causation.
Meaning that just because there is a correlation between two variables, does not mean that one is causing change in the other.
A third or more variable/s may be influencing both variables.
Unseen factors causing influence on both correlated variables creating misleading impressions of causation
E.g. ice cream sales and drowning deaths rising are correlated, but this does not address the third variable; rising weather temperatures
Establishing causation needs more evidence; controlled experiments are necessary to confirm if a cause and effect relationship exists
Advanced analyses can be used in some circumstances
Relying solely on correlations can lead to incorrect conclusions and inappropriate actions.

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