Documentation

plot

Scatter plot or added variable plot of linear regression model

Description

example

plot(mdl) creates a plot of the linear regression model mdl. The plot type depends on the number of predictor variables.

• If mdl includes multiple predictor variables, plot creates an Added Variable Plot for the whole model except the constant (intercept) term, equivalent to plotAdded(mdl).

• If mdl includes a single predictor variable, plot creates a scatter plot of the data along with a fitted curve and confidence bounds.

• If mdl does not include a predictor, plot creates a histogram of the residuals, equivalent to plotResiduals(mdl).

h = plot(mdl) returns graphics objects for the lines or patch in the plot. Use h to modify the properties of a specific line or patch after you create the plot. For a list of properties, see Line Properties and Patch Properties.

Examples

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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.

Year = categorical(Model_Year);
tbl = table(MPG,Weight,Year);
mdl = fitlm(tbl,'MPG ~ Year + Weight^2');

Create an added variable plot of the model.

plot(mdl) 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 plotAdded function. 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.

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 plot function.

plot(mdl) The plot illustrates that the model is significant because a horizontal line does not fit between the confidence bounds, which is consistent with the pValue result.

Create the same plot by using the plotAdded function. 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 plot function.

Input Arguments

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Linear regression model, specified as a LinearModel object created using fitlm or stepwiselm.

Output Arguments

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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.

If mdl includes a one or more predictors, then h(1), h(2), h(3), and h(4) correspond to adjusted data points, the fitted line, and the lower and upper bounds of the fitted line, respectively.

If mdl does not include a predictor, then h corresponds to the histogram of residuals.

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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.

$\begin{array}{c}{\stackrel{˜}{y}}_{i}=\overline{y}+{r}_{yi},\\ {\stackrel{˜}{x}}_{1i}={\overline{x}}_{1}+{r}_{xi},\end{array}$

where ${\overline{x}}_{1}$ and $\overline{y}$ represent the average of x1 and y, respectively.

plotAdded plots a scatter plot of (${\stackrel{˜}{x}}_{1i}$, ${\stackrel{˜}{y}}_{i}$), a fitted line for $\stackrel{˜}{y}$ as a function of ${\stackrel{˜}{x}}_{1}$ (that is, ${\beta }_{1}{\stackrel{˜}{x}}_{1}$), 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 predictors.

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 s.

plot creates an added variable plot for the model as a whole (except a constant term ) if the model includes multiple terms.

Tips

• 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.

Alternative Functionality

• A LinearModel object provides multiple plotting functions.

• When creating a model, use plotAdded to understand the effect of adding or removing a predictor variable.

• When verifying a model, use plotDiagnostics to find questionable data and to understand the effect of each observation. Also, use plotResiduals to analyze the residuals of the model.

• After fitting a model, use plotAdjustedResponse, plotPartialDependence, and 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.

• The plot function creates an added variable plot for the model as a whole (except a constant term) if the model includes multiple terms. Use plotAdded to select particular predictors for an added variable plot.