Q-Q plots are graphical tools that help you assess the validity of some assumptions in regression models, such as normality, linearity, and homoscedasticity. In this article, you will learn how to create and interpret Q-Q plots using a simple example.
A Q-Q plot, short for quantile-quantile plot, is a scatterplot that compares the quantiles of two distributions. One distribution is usually the observed data, and the other is a theoretical or reference distribution, such as the normal distribution. The idea is to see how well the data fit the expected distribution by checking if the points lie on or near a straight line.
The idea of Q-Q (quantile-quantile plot) is to compare distributions of a given set versus an ideal true gaussian distribution with same mean and deviation. (quantile scale, 2D) It allows you to inspect how much your distribution fits to a Gaussian, and where it deviates more.
To create a Q-Q plot, you need to sort the data from smallest to largest and assign them ranks. Then, you need to calculate the expected quantiles of the reference distribution for each rank. For example, if you use the normal distribution, you can use the inverse cumulative distribution function (CDF) to find the expected quantiles. Finally, you need to plot the observed data on the y-axis and the expected quantiles on the x-axis.
To interpret a Q-Q plot, you need to look at the shape and pattern of the points. If the points lie on or close to a 45-degree line, it means that the data follow the reference distribution closely. If the points deviate from the line, it means that there are some differences between the data and the reference distribution. For example, if the points are curved, it means that the data are skewed or have heavy tails. If the points are scattered or have gaps, it means that the data have outliers or are multimodal.
A Q-Q plot can be used in regression models to check some of the assumptions that are required for valid inference. For example, you can use a Q-Q plot to check if the residuals of the model are normally distributed, which is an assumption for many parametric tests and confidence intervals. You can also use a Q-Q plot to check if the residuals have a constant variance, which is an assumption for the homoscedasticity of the model. To do this, you need to create a Q-Q plot for the residuals of the model and compare them with the normal distribution.
Q-Q plots are useful tools, but they also have some limitations that you should be aware of. For example, Q-Q plots can be sensitive to the sample size and the choice of the reference distribution. A small sample size may not reveal the true shape of the data distribution, and a different reference distribution may give a different impression of the fit. Moreover, Q-Q plots can be subjective and hard to quantify. Different people may have different opinions on how close or far the points are from the line, and there is no clear rule on how to measure the deviation.
Q-Q plots are not the only way to check the assumptions of regression models. There are other graphical and numerical methods that can complement or supplement Q-Q plots. For example, you can use histograms, boxplots, or density plots to visualize the distribution of the data or the residuals. You can also use tests, such as the Kolmogorov-Smirnov test, the Shapiro-Wilk test, or the Breusch-Pagan test, to test the hypotheses of normality or homoscedasticity. However, these methods also have their own advantages and disadvantages, so you should use them with caution and in combination with Q-Q plots.