An introduction to simple linear regression
Regression models describe the relationship between variables by fitting a line to the observed data. Linear regression models use a straight line, while logistic and nonlinear regression models use a curved line. Regression allows you to estimate how a dependent variable changes as the independent variable(s) change.
Simple linear regression is used to estimate the relationship between two quantitative variables. You can use simple linear regression when you want to know:
 How strong the relationship is between two variables (e.g. the relationship between rainfall and soil erosion).
 The value of the dependent variable at a certain value of the independent variable (e.g. the amount of soil erosion at a certain level of rainfall).
If you have more than one independent variable, use multiple linear regression instead.
Assumptions of simple linear regression
Simple linear regression is a parametric test, meaning that it makes certain assumptions about the data. These assumptions are:
 Homogeneity of variance (homoscedasticity): the size of the error in our prediction doesn’t change significantly across the values of the independent variable.
 Independence of observations: the observations in the dataset were collected using statistically valid sampling methods, and there are no hidden relationships among observations.
 Normality: The data follows a normal distribution.
Linear regression makes one additional assumption:
 The relationship between the independent and dependent variable is linear: the line of best fit through the data points is a straight line (rather than a curve or some sort of grouping factor).
If your data do not meet the assumptions of homoscedasticity or normality, you may be able to use a nonparametric test instead, such as the Spearman rank test.
If your data violate the assumption of independence of observations (e.g. if observations are repeated over time), you may be able to perform a linear mixedeffects model that accounts for the additional structure in the data.
How to perform a simple linear regression
Simple linear regression formula
The formula for a simple linear regression is:
 y is the predicted value of the dependent variable (y) for any given value of the independent variable (x).
 B_{0} is the intercept, the predicted value of y when the x is 0.
 B_{1} is the regression coefficient – how much we expect y to change as x increases.
 x is the independent variable ( the variable we expect is influencing y).
 e is the error of the estimate, or how much variation there is in our estimate of the regression coefficient.
Linear regression finds the line of best fit line through your data by searching for the regression coefficient (B_{1}) that minimizes the total error (e) of the model.
While you can perform a linear regression by hand, this is a tedious process, so most people use statistical programs to help them quickly analyze the data.
Simple linear regression in R
R is a free, powerful, and widelyused statistical program. Download the dataset to try it yourself using our income and happiness example.
Dataset for simple linear regression (.csv)
Load the income.data dataset into your R environment, and then run the following command to generate a linear model describing the relationship between income and happiness:
This code takes the data you have collected data = income.data
and calculates the effect that the independent variable income
has on the dependent variable happiness
using the equation for the linear model: lm()
.
To learn more, follow our full stepbystep guide to linear regression in R.
Interpreting the results
To view the results of the model, you can use the summary()
function in R:
This function takes the most important parameters from the linear model and puts them into a table, which looks like this:
This output table first repeats the formula that was used to generate the results (‘Call’), then summarizes the model residuals (‘Residuals’), which give an idea of how well the model fits the real data.
Next is the ‘Coefficients’ table. The first row gives the estimates of the yintercept, and the second row gives the regression coefficient of the model.
Row 1 of the table is labeled (Intercept)
. This is the yintercept of the regression equation, with a value of 0.20. You can plug this into your regression equation if you want to predict happiness values across the range of income that you have observed:
The next row in the ‘Coefficients’ table is income. This is the row that describes the estimated effect of income on reported happiness:
The Estimate
column is the estimated effect, also called the regression coefficient or r^{2} value. The number in the table (0.713) tells us that for every one unit increase in income (where one unit of income = $10,000) there is a corresponding 0.71unit increase in reported happiness (where happiness is a scale of 1 to 10).
The Std. Error
column displays the standard error of the estimate. This number shows how much variation there is in our estimate of the relationship between income and happiness.
The t value
column displays the test statistic. Unless you specify otherwise, the test statistic used in linear regression is the tvalue from a twosided ttest. The larger the test statistic, the less likely it is that our results occurred by chance.
The Pr(> t )
column shows the pvalue. This number tells us how likely we are to see the estimated effect of income on happiness if the null hypothesis of no effect were true.
Because the pvalue is so low (p < 0.001), we can reject the null hypothesis and conclude that income has a statistically significant effect on happiness.
The last three lines of the model summary are statistics about the model as a whole. The most important thing to notice here is the pvalue of the model. Here it is significant (p < 0.001), which means that this model is a good fit for the observed data.
Presenting the results
When reporting your results, include the estimated effect (i.e. the regression coefficient), standard error of the estimate, and the pvalue. You should also interpret your numbers to make it clear to your readers what your regression coefficient means:
It can also be helpful to include a graph with your results. For a simple linear regression, you can simply plot the observations on the x and y axis and then include the regression line and regression function:
Can you predict values outside the range of your data?
No! We often say that regression models can be used to predict the value of the dependent variable at certain values of the independent variable. However, this is only true for the range of values where we have actually measured the response.
We can use our income and happiness regression analysis as an example. Between $15,000 and $75,000, we found an r^{2} of 0.73 ± 0.0193. But what if we did a second survey of people making between $75,000 and $150,000?
The r^{2} for the relationship between income and happiness is now 0.21, or a 0.21unit increase in reported happiness for every $10,000 increase in income. While the relationship is still statistically significant (p<0.001), the slope is much smaller than before.
What if we hadn’t measured this group, and instead extrapolated the line from the 15–75k incomes to the 70–150k incomes?
You can see that if we simply extrapolated from the 15–75k income data, we would overestimate the happiness of people in the 75–150k income range.
If we instead fit a curve to the data, it seems to fit the actual pattern much better.
It looks as though happiness actually levels off at higher incomes, so we can’t use the same regression line we calculated from our lowerincome data to predict happiness at higher levels of income.
Even when you see a strong pattern in your data, you can’t know for certain whether that pattern continues beyond the range of values you have actually measured. Therefore, it’s important to avoid extrapolating beyond what the data actually tell you.
Frequently asked questions about simple linear regression
 What is a regression model?

A regression model is a statistical model that estimates the relationship between one dependent variable and one or more independent variables using a line (or a plane in the case of two or more independent variables).
A regression model can be used when the dependent variable is quantitative, except in the case of logistic regression, where the dependent variable is binary.
 What is simple linear regression?

Simple linear regression is a regression model that estimates the relationship between one independent variable and one dependent variable using a straight line. Both variables should be quantitative.
For example, the relationship between temperature and the expansion of mercury in a thermometer can be modeled using a straight line: as temperature increases, the mercury expands. This linear relationship is so certain that we can use mercury thermometers to measure temperature.
 How is the error calculated in a linear regression model?

Linear regression most often uses meansquare error (MSE) to calculate the error of the model. MSE is calculated by:
 measuring the distance of the observed yvalues from the predicted yvalues at each value of x;
 squaring each of these distances;
 calculating the mean of each of the squared distances.
Linear regression fits a line to the data by finding the regression coefficient that results in the smallest MSE.