Correlation and Regression

Correlation and Scatter Graphs

When we have bivariate data, one variable will be the independent (or explanatory) variable, and another will be the dependent (or response) variable. The independent variable is the one that you can control, and goes on the $x$ axis. The dependent variable is the one that is being affected, and it goes on the $y$ axis.

There won’t always be a clear independent and dependent variable, and in that case it is not as important which way round they go on a scatter graph.

Correlation comes in three flavours: positive, negative and no correlation. In positive correlation, as one variable goes up so does the other. In negative correlation, as one variable goes up the other goes down. In no correlation, there is no clear link between the variables.

Regression

The regression line of $mathbf{y}$ on $mathbf{x}$ is the line of best fit. It is always written in the same form:

$y=a+bx$

$a$ is the $y$ intercept

$b$ is the gradient

The regression line can be used to predict values of the dependent variable. This comes in two flavours:

• Interpolation – If the value of $x$ being used in the prediction falls inside the range of the values of $x$ in the data. The predicted value should be reliable.
• Extrapolation – If the value of $x$ being used in the prediction falls outside of the range of the values of $x$ in the data. The predicted value might be unreliable.

Regression with Coded Data

Regression can also be done on coded data. All we do is we substitute the coding into our regression line, then rearrange to get it back in straight line form.

Example: $y=4+10x$, with coding $s=2y$, $t=x-3$ becomes:

$begin{aligned}dfrac{s}{2}&=4+10(t+3)[1.2em]&=4+10t+30[1.2em]&=34+10tend{aligned}$

$s=68+20t$

Also, we can form regression lines from non-linear data in some cases.

Example: $y=ax^{n}$ becomes $log(y)=log(a)+nlog(x)$ which is regression in $log(y)$ and $log(x)$.