1. Visualizing model fit
Several plots can quantify the performance of a model. We'll look at these plots and their interpretation first, then the code to draw them.
2. Residual properties of a good fit
If a linear regression model is a good fit, then the residuals are approximately normally distributed, with mean zero.
3. Bream and perch again
Earlier, we ran models on the bream and perch datasets. From looking at the scatter plots with linear trend lines, it appeared that the bream model was a good fit, but the perch model wasn't because the observed masses increased faster than linearly with the lengths.
4. Residuals vs. fitted
The first diagnostic plot is of residuals versus fitted values. The blue line is a LOWESS trend line, which is a smooth curve following the data. These aren't good for making predictions but are useful for visualizing trends.
If residuals met the assumption that they are normally distributed with mean zero, then the trend line should closely follow the y equals zero line on the plot. For the bream dataset, this is true.
By contrast, the perch model doesn't meet the assumption. The residuals are above zero when the fitted value is small or big and below zero in the middle.
5. Q-Q plot
The second diagnostic plot is called a Q-Q plot. It shows whether or not the residuals follow a normal distribution.
On the x-axis, the points are quantiles from the normal distribution. On the y-axis, you get the sample quantiles, which are the quantiles derived from your dataset.
It sounds technical, but interpreting this plot is straightforward. If the points track along the straight line, they are normally distributed. If not, they aren't.
Here, most of the bream points follow the line closely. Two points at each extreme don't follow the line. These correspond to the rows of the bream dataset with the highest residuals.
The perch dataset doesn't track the line as closely. In particular, you can see on the right-hand side of the plot that the residuals are larger than expected. That means the model is a particularly poor fit for the longer lengths of perch.
6. Scale-location plot
The third plot shows the square root of the standardized residuals versus the fitted values. It's often called a scale-location plot, because that's easier to say.
Where the first plot showed whether or not the residuals go positive or negative as the fitted values change, this plot shows whether the size of the residuals gets bigger or smaller.
The residuals for the bream dataset get a little bigger as the fitted values increase, but it's not a huge change.
Again, the plot of the perch model has a trend line that goes up and down all over the place, indicating a poor fit.
7. residplot()
To create the residuals vs. fitted plot, you can use the residplot function from seaborn. It takes the usual x, y, and data arguments, in addition to the lowess argument. This will add a smooth curve following the data, visualizing the trend of your residuals.
You'll also need to specify the x and y labels manually.
8. qqplot()
To draw a Q-Q plot, you can use the qqplot function from the statsmodels package. You set the residuals of the model as your data argument and the fit argument to True. This will compare the data quantiles to a normal distribution. The last argument is optional, but when set to "45", set as a string, it will draw a 45-degree line on your plot, making it easier to interpret the pattern.
9. Scale-location plot
The last plot, scale-location, requires a bit more preprocessing. You first need to extract the normalized residuals from the model, which you can get by using the get_influence method, then accessing the resid_studentized_internal attribute. Don't worry about this too much now, we'll come back to that in the following lesson.
You then take the absolute values and take the square root of these normalized residuals to standardize them. Next, you can call sns dot regplot, passing in mdl_bream dot fittedvalues for x, and the standardized residuals for y. Again, you can also include a lowess argument to make interpretation easier. Lastly, you specify the axes manually.
10. Let's practice!
Time to interpret and create diagnostic plots yourself.