After fitting a model, examine the result and make adjustments.
Model Display
A linear regression model shows several diagnostics when you enter its name or enter disp(mdl)
. This display gives some of the basic information to check whether the fitted model represents the data adequately.
For example, fit a linear model to data constructed with two out of five predictors not present and with no intercept term:
X = randn(100,5); y = X*[1;0;3;0;-1] + randn(100,1); mdl = fitlm(X,y)
mdl = Linear regression model: y ~ 1 + x1 + x2 + x3 + x4 + x5 Estimated Coefficients: Estimate SE tStat pValue _________ ________ ________ __________ (Intercept) 0.038164 0.099458 0.38372 0.70205 x1 0.92794 0.087307 10.628 8.5494e-18 x2 -0.075593 0.10044 -0.75264 0.45355 x3 2.8965 0.099879 29 1.1117e-48 x4 0.045311 0.10832 0.41831 0.67667 x5 -0.99708 0.11799 -8.4504 3.593e-13 Number of observations: 100, Error degrees of freedom: 94 Root Mean Squared Error: 0.972 R-squared: 0.93, Adjusted R-Squared: 0.926 F-statistic vs. constant model: 248, p-value = 1.5e-52
Notice that:
The display contains the estimated values of each coefficient in the Estimate
column. These values are reasonably near the true values [0;1;0;3;0;-1]
.
There is a standard error column for the coefficient estimates.
The reported pValue
(which are derived from the t statistics (tStat
) under the assumption of normal errors) for predictors 1, 3, and 5 are extremely small. These are the three predictors that were used to create the response data y
.
The pValue
for (Intercept)
, x2
and x4
are much larger than 0.01. These three predictors were not used to create the response data y
.
The display contains R2, adjusted R2, and F statistics.
ANOVA
To examine the quality of the fitted model, consult an ANOVA table. For example, use anova
on a linear model with five predictors:
tbl = anova(mdl)
tbl=6×5 table
SumSq DF MeanSq F pValue
_______ __ _______ _______ __________
x1 106.62 1 106.62 112.96 8.5494e-18
x2 0.53464 1 0.53464 0.56646 0.45355
x3 793.74 1 793.74 840.98 1.1117e-48
x4 0.16515 1 0.16515 0.17498 0.67667
x5 67.398 1 67.398 71.41 3.593e-13
Error 88.719 94 0.94382
This table gives somewhat different results than the model display. The table clearly shows that the effects of x2
and x4
are not significant. Depending on your goals, consider removing x2
and x4
from the model.
Diagnostic Plots
Diagnostic plots help you identify outliers, and see other problems in your model or fit. For example, load the carsmall
data, and make a model of MPG
as a function of Cylinders
(categorical) and Weight
:
load carsmall tbl = table(Weight,MPG,Cylinders); tbl.Cylinders = categorical(tbl.Cylinders); mdl = fitlm(tbl,'MPG ~ Cylinders*Weight + Weight^2');
Make a leverage plot of the data and model.
plotDiagnostics(mdl)
There are a few points with high leverage. But this plot does not reveal whether the high-leverage points are outliers.
Look for points with large Cook’s distance.
plotDiagnostics(mdl,'cookd')
There is one point with large Cook’s distance. Identify it and remove it from the model. You can use the Data Cursor to click the outlier and identify it, or identify it programmatically:
[~,larg] = max(mdl.Diagnostics.CooksDistance); mdl2 = fitlm(tbl,'MPG ~ Cylinders*Weight + Weight^2','Exclude',larg);
Residuals — Model Quality for Training Data
There are several residual plots to help you discover errors, outliers, or correlations in the model or data. The simplest residual plots are the default histogram plot, which shows the range of the residuals and their frequencies, and the probability plot, which shows how the distribution of the residuals compares to a normal distribution with matched variance.
Examine the residuals:
plotResiduals(mdl)
The observations above 12 are potential outliers.
plotResiduals(mdl,'probability')
The two potential outliers appear on this plot as well. Otherwise, the probability plot seems reasonably straight, meaning a reasonable fit to normally distributed residuals.
You can identify the two outliers and remove them from the data:
outl = find(mdl.Residuals.Raw > 12)
outl = 2×1
90
97
To remove the outliers, use the Exclude
name-value pair:
mdl3 = fitlm(tbl,'MPG ~ Cylinders*Weight + Weight^2','Exclude',outl);
Examine a residuals plot of mdl2:
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