I will explain when a quadratic regression is a suitable model. You will learn how to perform a quadratic regression and how to interpret the results. Quadratic regression is an extension to the linear regression you learned in the series of videos on regression. There, we studied two variables that we could model using a straight line. Quadratic regression makes it possible to study a curved line. Let's take a look at an example. You are working at a factory that produces decaf coffee. The percentage of caffeine is an important measure, as it has to be below a certain level to be able to sell your coffee as decaf coffee. This will be our CTQ. The coffee is being decaffeinated in an extraction process. Let's study whether the number of times that a batch of coffee has been put into the extraction process has an influence on your caffeine percentage. Both these variables are numerical, so we have to perform a regression analysis. We have performed a linear regression analysis, and we see in the results that they are significant. So there is statistical proof of a relationship between extractions and caffeine percentage. And as you can expect, more extractions mean less caffeine. Also the number of extractions is a strong predictor of the percentage of caffeine, as the R-squared is 76.9%. However, we also see that the line does not fit a data point really well. If you look closely, you see that the shape of the line between the data points could maybe be curved instead of straight. And this is confirmed by our residual analysis. We can see in the probability plot that the distribution of the residuals is not normally distributed and in the time graph, that we have some outliers. This shows us that the regular linear regression that we did is not a perfect fit for this data. The alternative I am going to show you is the quadratic regression model. This model is one possible extension to the simple linear regression model we already know. If you perform a quadratic regression analysis, it means that you add an extra term to your formula. This changes the shape of the fitted line, it makes it curved. Let's perform such an analysis in Minitab. So pause your movie, load your data before you continue. This is what your data looks like with caffeine percentage in one column, and the number of extractions in the other column. Let's perform a quadratic regression analysis. Of course, you can find these under the Stat menu, as it's a statistical analysis. You go to the Regression menu > Fitted Line Plot. What is your y? That's caffeine percentage. And, what's your influence factor x? That's extractions. Now, you can fill in the type of regression model that we are interested in, and that's of course, Quadratic. To be a bit quicker, let's go immediately to Graphs and ask for a four in one plot, so that we have this ready for our residual analysis later on. OK, OK and this is your output. You get your four in one here and you have your fitted line plot, of course, as well, with the curved quadratic line. In your session window, you will see that you also have some output. Let's take a look at the output. Our fitted line plot, that Minitab computed, looks like this. As you can see, we now added the extra quadratic term to the graph. And the curved line fits the data points better than the straight line did. Physically speaking, this also makes sense because we are looking at extraction of caffeine. And of course, the first few extractions, this will go very fast. However, as caffeine percentage gets lower, it will be more tough to extract it. Hence, a curved relationship. If we look at the session window, we see that the p value indicates that the quadratic term in the model is significant. The R squared increased to 92.2%. Remember that it was 76.9% in the linear model, so our quadratic model fits our data better than the linear model did. Let's take a look if the assumptions on the line regression are now met. This is our four in one plot for the residual analysis. It looks the residuals are now normally distributed. And we see that the residuals show no outliers, patterns or other irregularities. So the assumptions are met and the results from the quadratic regression analysis are valid. Let's summarize. If the linear regression analysis does not meet the assumption, you will have to apply an alternative analysis technique. One of these alternatives is the quadratic regression analysis. This technique fits a curved line to your data instead of a straight line. After you've performed the regression analysis, you always have to perform the residual analysis again to see if the assumptions are met.
