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Simple linear regression is a statistical method used to analyze the relationship between two variables, typically a dependent variable and an independent variable. It involves fitting a straight line to a set of data points in order to determine the strength and direction of the relationship between the two variables. This method is often used to make predictions and understand the impact of the independent variable on the dependent variable. It is a commonly used technique in various fields such as economics, social sciences, and business. Simple linear regression is a fundamental tool in statistical analysis and can provide valuable insights into data patterns and relationships.

## What is Simple Linear Regression?

**Simple Linear Regression** is a statistical test used to predict a single variable using one other variable. It also is used to determine the numerical relationship between two variables. The variable you want to predict should be continuous and your data should meet the other assumptions listed below.

## Assumptions for Simple Linear Regression

Every statistical method has assumptions. Assumptions mean that your data must satisfy certain properties in order for statistical method results to be accurate.

The assumptions for Simple Linear Regression include:

- Linearity
- No Outliers
- Similar Spread across Range
- Independence
- Normality of Residuals

Let’s dive in to each one of these separately.

**Linearity**

The variables that you care about must be related linearly. This means that if you plot the variables, you will be able to draw a straight line that fits the shape of the data.

**No Outliers**

The variables that you care about must not contain outliers. Linear Regression is sensitive to outliers, or data points that have unusually large or small values. You can tell if your variables have outliers by plotting them and observing if any points are far from all other points.

**Similar Spread across Range**

In statistics this is called homoscedasticity, which describes when variables have a similar spread across their ranges.

**Independence**

Each of your observations (data points) should be independent. This means that each value of your variables doesn’t “depend” on any of the others. For example, this assumption is usually violated when there are multiple data points over time from the same unit of observation (e.g. subject/participant/customer/store), because the data points from the same unit of observation are likely to be related or affect one another.

*If your data have repeated measures over time from the same units of observation, you should use a Mixed Effects Model.*

**Normality of Residuals**

The word “residuals” refers to the values resulting from subtracting the expected (or predicted) dependent variables from the actual values. The distribution of these values should match a normal (or bell curve) distribution shape.

Meeting this assumption assures that the results of the regression are equally applicable across the full spread of the data and that there is no systematic bias in the prediction.

## When to use Simple Linear Regression?

You should use Simple Linear Regression in the following scenario:

- You want to use one variable in a
**prediction**of another, or you want to quantify the numerical relationship between two variables - The variable you want to predict (your dependent variable) is
**continuous** - You have
**one independent variable**, or one variable that you are using as a predictor

Let’s clarify these to help you know when to use Simple Linear Regression

**Prediction**

You are looking for a statistical test to predict one variable using another. This is a prediction question. Other types of analyses include examining the strength of the relationship between two variables (correlation) or examining differences between groups (difference).

**Continuous Dependent Variable**

The variable you want to predict must be continuous. Continuous means that your variable of interest can basically take on any value, such as heart rate, height, weight, number of ice cream bars you can eat in 1 minute, etc.

Types of data that are NOT continuous include ordered data (such as finishing place in a race, best business rankings, etc.), categorical data (gender, eye color, race, etc.), or binary data (purchased the product or not, has the disease or not, etc.).

*If your dependent variable is binary, you should use Simple Logistic Regression, and if your dependent variable is categorical, then you should use Multinomial Logistic Regression or Linear Discriminant Analysis.*

**One Independent Variable**

Simple Linear Regression is used when there is one predictor variable measured at a single point in time.

*If you have more than one independent variable, you should use another variant of linear regression called Multiple Linear Regression instead, and if you have one independent variable but it is measured for the same group at multiple points in time, then you should use a Mixed Effects Model.*

## Simple Linear Regression Example

**Dependent Variable**: Revenue

**Independent Variable**: Dollars spent on advertising by city

The null hypothesis, which is statistical lingo for what would happen if the treatment does nothing, is that there is no relationship between spend on advertising and revenue within a city. Our test will assess the likelihood of this hypothesis being true.

We gather our data and after assuring that the assumptions of linear regression are met, we perform the analysis.

When we run this analysis, we get beta coefficients and p-values for each term in the model. For any linear regression model, you will have one beta coefficient that equals the intercept of your linear regression line (often labelled with a 0 as β_{0}). This is simply where the regression line crosses the y-axis. In the case of simple linear regression, there is only one other beta coefficient (β_{1}), which represents the relationship between the independent and dependent variables.

This second beta coefficient is the slope of the regression line and is the key to understanding the numerical relationship between your variables. Essentially, for each unit (value of 1) increase in your independent variable, your dependent variable is expected to change by the value of β_{1}.

The p-value associated with β_{1} is the chance of seeing our results assuming there is actually no relationship between dollars spent on advertising and revenue. A p-value less than or equal to 0.05 means that our result is statistically significant and we can trust that the difference is not due to chance alone.

In addition, this analysis will result in an R-Squared (R^{2}) value. This value can range from 0-1 and represents how well your linear regression line fits your data points. The higher the R^{2}, the better your model fits your data.