Scatter plot or added variable plot of linear regression model
plot( creates a plot of the linear
mdl. The plot type depends on the number of
mdl includes multiple predictor variables,
plot creates an Added Variable Plot for the whole model except the
constant (intercept) term, equivalent to
mdl includes a single predictor variable,
plot creates a scatter plot of the data along
with a fitted curve and confidence bounds.
mdl does not include a predictor,
plot creates a histogram of the residuals,
Create a linear regression model of car mileage as a function of weight and model year. Then create an added variable plot to see the significance of the model.
Create a linear regression model of mileage from the
carsmall data set.
load carsmall Year = categorical(Model_Year); tbl = table(MPG,Weight,Year); mdl = fitlm(tbl,'MPG ~ Year + Weight^2');
Create an added variable plot of the model.
The plot illustrates that the model is significant because a horizontal line does not fit between the confidence bounds.
Create the same plot by using the
Create a scatter plot of data along with a fitted curve and confidence bounds for a simple linear regression model. A simple linear regression model includes only one predictor variable.
Create a simple linear regression model of mileage from the
carsmall data set.
load carsmall tbl = table(MPG,Weight); mdl = fitlm(tbl,'MPG ~ Weight')
mdl = Linear regression model: MPG ~ 1 + Weight Estimated Coefficients: Estimate SE tStat pValue __________ _________ _______ __________ (Intercept) 49.238 1.6411 30.002 2.7015e-49 Weight -0.0086119 0.0005348 -16.103 1.6434e-28 Number of observations: 94, Error degrees of freedom: 92 Root Mean Squared Error: 4.13 R-squared: 0.738, Adjusted R-Squared: 0.735 F-statistic vs. constant model: 259, p-value = 1.64e-28
pValue of the
Weight variable is very small, which means that the variable is statistically significant in the model. Visualize this result by creating a scatter plot of the data, along with a fitted curve and its 95% confidence bounds, using the
The plot illustrates that the model is significant because a horizontal line does not fit between the confidence bounds, which is consistent with the
Create the same plot by using the
When a model includes only one term in addition to the constant term, an adjusted value is equivalent to its original value. Therefore, this added variable plot is the same as the scatter plot created by the
h— Graphics objects
Graphics objects corresponding to the lines or patch in the plot, returned as a graphics array. Use dot notation to query and set properties of graphics objects. For details, see Line Properties and Patch Properties.
mdl includes a one or more predictors, then
h(4) correspond to
adjusted data points, the fitted line, and the lower and upper bounds of the
fitted line, respectively.
mdl does not include a predictor, then
h corresponds to the histogram of residuals.
An added variable plot, also known as a partial regression leverage plot, illustrates the incremental effect on the response of specified terms caused by removing the effects of all other terms.
An added variable plot created by
plotAdded with a single selected term
corresponding to a single predictor variable includes these plots:
Scatter plot of adjusted response values against adjusted predictor variable values
Fitted line for adjusted response values as a function of adjusted predictor variable values
95% confidence bounds of the fitted line
The adjusted values are equal to the average of the variable plus the residuals of the variable fit to all predictors except the selected predictor. For example, consider an added variable plot for the first predictor variable x1. Fit the response variable y and the selected predictor variable x1 to all predictors except x1 as follows:
yi = gy(x2i, x3i, …, xpi) + ryi,
x1i = gx(x2i, x3i, …, xpi) + rxi,
where gy and gx are the fit of y and x1, respectively, against all predictors except the selected predictor (x1). ry and rx are the corresponding residual vectors. The subscript i represents the observation number. The adjusted value is the sum of the average value and the residual for each observation.
where and represent the average of x1 and y, respectively.
plotAdded plots a scatter plot of (, ), a fitted line for as a function of (that is, ), and the 95% confidence bounds of the fitted line. The coefficient
β1 is the same as the coefficient estimate of
x1 in the full model, which includes all
ryi represents the part of the response values unexplained by the predictors (except x1), and rxi represents the part of the x1 values unexplained by the other predictors. Therefore, the fitted line represents how the new information introduced by adding x1 can explain the unexplained part of the response values. If the slope of the fitted line is close to zero and the confidence bounds can include a horizontal line, then the plot indicates that the new information from x1 does not explain the unexplained part of the response values well. That is, x1 is not significant in the model fit.
plotAdded also supports an extension of the added variable plot so that
you can select multiple terms instead of a single term. Therefore, you can also specify a
categorical predictor, all terms that involve a specific predictor, or the model as a whole
(except a constant (intercept) term). Consider a set of predictors X with
a coefficient vector β, where
βi is the coefficient estimate of
xi in the full model if you specify the
ith coefficient for an added variable plot; otherwise,
βi is zero. Define a unit direction vector
u as u =
β/s where s = norm(β). Then, Xβ =
(Xu)s. Treat Xu as a single predictor with
a coefficient s, and create an added variable plot for
Xu in the same way as creating the plot for a
single term. The coefficient of the fitted line in the added variable plot corresponds to
plot creates an added variable plot for the model as a whole
(except a constant term ) if the model includes multiple terms.
The data cursor displays the values of the selected plot point in a data tip (small text box located next to the data point). The data tip includes the x-axis and y-axis values for the selected point, along with the observation name or number.
LinearModel object provides multiple plotting functions.
When creating a model, use
plotAdded to understand the effect of adding or removing a predictor
After fitting a model, use
plotEffects to understand the effect of a particular predictor. Use
plotInteraction to understand the
interaction effect between two predictors. Also, use
plotSlice to plot slices through the prediction surface.
plot function creates an added variable plot for the
model as a whole (except a constant term) if the model includes multiple terms.
plotAdded to select particular
predictors for an added variable plot.