LinearModel

Linear regression model

Description

LinearModel is a fitted linear regression model object. A regression model describes the relationship between a response and predictors. The linearity in a linear regression model refers to the linearity of the predictor coefficients.

Use the properties of a LinearModel object to investigate a fitted linear regression model. The object properties include information about coefficient estimates, summary statistics, fitting method, and input data. Use the object functions to predict responses and to modify, evaluate, and visualize the linear regression model.

Creation

Create a LinearModel object by using fitlm or stepwiselm.

fitlm fits a linear regression model to data using a fixed model specification. Use addTerms, removeTerms, or step to add or remove terms from the model. Alternatively, use stepwiselm to fit a model using stepwise linear regression.

Properties

Coefficient Estimates

CoefficientCovariance — Covariance matrix of coefficient estimates

numeric matrix

This property is read-only.

Covariance matrix of coefficient estimates, specified as a p-by-p matrix of numeric values. p is the number of coefficients in the fitted model.

For details, see Coefficient Standard Errors and Confidence Intervals.

Data Types: single | double

CoefficientNames — Coefficient names

cell array of character vectors

This property is read-only.

Coefficient names, specified as a cell array of character vectors, each containing the name of the corresponding term.

Data Types: cell

Coefficients — Coefficient values

table

This property is read-only.

Coefficient values, specified as a table. Coefficients contains one row for each coefficient and these columns:

• Estimate — Estimated coefficient value

• SE — Standard error of the estimate

• tStat — t-statistic for a test that the coefficient is zero

• pValue — p-value for the t-statistic

Use anova (only for a linear regression model) or coefTest to perform other tests on the coefficients. Use coefCI to find the confidence intervals of the coefficient estimates.

To obtain any of these columns as a vector, index into the property using dot notation. For example, obtain the estimated coefficient vector in the model mdl:

beta = mdl.Coefficients.Estimate

Data Types: table

NumCoefficients — Number of model coefficients

positive integer

This property is read-only.

Number of model coefficients, specified as a positive integer. NumCoefficients includes coefficients that are set to zero when the model terms are rank deficient.

Data Types: double

NumEstimatedCoefficients — Number of estimated coefficients

positive integer

This property is read-only.

Number of estimated coefficients in the model, specified as a positive integer. NumEstimatedCoefficients does not include coefficients that are set to zero when the model terms are rank deficient. NumEstimatedCoefficients is the degrees of freedom for regression.

Data Types: double

Summary Statistics

DFE — Degrees of freedom for error

positive integer

This property is read-only.

Degrees of freedom for the error (residuals), equal to the number of observations minus the number of estimated coefficients, specified as a positive integer.

Data Types: double

Diagnostics — Observation diagnostics

table

This property is read-only.

Observation diagnostics, specified as a table that contains one row for each observation and the columns described in this table.

Column	Meaning	Description
Leverage	Diagonal elements of HatMatrix	Leverage for each observation indicates to what extent the fit is determined by the observed predictor values. A value close to 1 indicates that the fit is largely determined by that observation, with little contribution from the other observations. A value close to 0 indicates that the fit is largely determined by the other observations. For a model with P coefficients and N observations, the average value of Leverage is P/N. A Leverage value greater than 2*P/N indicates high leverage.
CooksDistance	Cook's distance	CooksDistance is a measure of scaled change in fitted values. An observation with CooksDistance greater than three times the mean Cook's distance can be an outlier.
Dffits	Delete-1 scaled differences in fitted values	Dffits is the scaled change in the fitted values for each observation that results from excluding that observation from the fit. Values greater than 2*sqrt(P/N) in absolute value can be considered influential.
S2_i	Delete-1 variance	S2_i is a set of residual variance estimates obtained by deleting each observation in turn. These estimates can be compared with the mean squared error (MSE) value, stored in the MSE property.
CovRatio	Delete-1 ratio of determinant of covariance	CovRatio is the ratio of the determinant of the coefficient covariance matrix, with each observation deleted in turn, to the determinant of the covariance matrix for the full model. Values greater than 1 + 3*P/N or less than 1 – 3*P/N indicate influential points.
Dfbetas	Delete-1 scaled differences in coefficient estimates	Dfbetas is an N-by-P matrix of the scaled change in the coefficient estimates that results from excluding each observation in turn. Values greater than 3/sqrt(N) in absolute value indicate that the observation has a significant influence on the corresponding coefficient.
HatMatrix	Projection matrix to compute fitted from observed responses	HatMatrix is an N-by-N matrix such that Fitted = HatMatrix*Y, where Y is the response vector and Fitted is the vector of fitted response values.

Diagnostics contains information that is helpful in finding outliers and influential observations. Delete-1 diagnostics capture the changes that result from excluding each observation in turn from the fit. For more details, see Hat Matrix and Leverage, Cook’s Distance, and Delete-1 Statistics.

Use plotDiagnostics to plot observation diagnostics.

Rows not used in the fit because of missing values (in ObservationInfo.Missing) or excluded values (in ObservationInfo.Excluded) contain NaN values in the CooksDistance, Dffits, S2_i, and CovRatio columns and zeros in the Leverage, Dfbetas, and HatMatrix columns.

To obtain any of these columns as an array, index into the property using dot notation. For example, obtain the delete-1 variance vector in the model mdl:

S2i = mdl.Diagnostics.S2_i;

Data Types: table

Fitted — Fitted response values based on input data

numeric vector

This property is read-only.

Fitted (predicted) response values based on input data, specified as an n-by-1 numeric vector. n is the number of observations in the input data. Use predict to compute predictions for other predictor values, or to compute confidence bounds on Fitted.

Data Types: single | double

LogLikelihood — Loglikelihood

numeric value

This property is read-only.

Loglikelihood of response values, specified as a numeric value, based on the assumption that each response value follows a normal distribution. The mean of the normal distribution is the fitted (predicted) response value, and the variance is the MSE.

Data Types: single | double

ModelCriterion — Criterion for model comparison

structure

This property is read-only.

Criterion for model comparison, specified as a structure with these fields:

• AIC — Akaike information criterion. AIC = –2*logL + 2*m, where logL is the loglikelihood and m is the number of estimated parameters.

• AICc — Akaike information criterion corrected for the sample size. AICc = AIC + (2*m*(m + 1))/(n – m – 1), where n is the number of observations.

• BIC — Bayesian information criterion. BIC = –2*logL + m*log(n).

• CAIC — Consistent Akaike information criterion. CAIC = –2*logL + m*(log(n) + 1).

Information criteria are model selection tools that you can use to compare multiple models fit to the same data. These criteria are likelihood-based measures of model fit that include a penalty for complexity (specifically, the number of parameters). Different information criteria are distinguished by the form of the penalty.

When you compare multiple models, the model with the lowest information criterion value is the best-fitting model. The best-fitting model can vary depending on the criterion used for model comparison.

To obtain any of the criterion values as a scalar, index into the property using dot notation. For example, obtain the AIC value aic in the model mdl:

aic = mdl.ModelCriterion.AIC

Data Types: struct

MSE — Mean squared error

numeric value

This property is read-only.

Mean squared error (residuals), specified as a numeric value.

MSE = SSE / DFE,

where MSE is the mean squared error, SSE is the sum of squared errors, and DFE is the degrees of freedom.

Data Types: single | double

Residuals — Residuals for fitted model

table

This property is read-only.

Residuals for the fitted model, specified as a table that contains one row for each observation and the columns described in this table.

Column	Description
Raw	Observed minus fitted values
Pearson	Raw residuals divided by the root mean squared error (RMSE)
Standardized	Raw residuals divided by their estimated standard deviation
Studentized	Raw residual divided by an independent estimate of the residual standard deviation. The residual for observation i is divided by an estimate of the error standard deviation based on all observations except observation i.

Use plotResiduals to create a plot of the residuals. For details, see Residuals.

Rows not used in the fit because of missing values (in ObservationInfo.Missing) or excluded values (in ObservationInfo.Excluded) contain NaN values.

To obtain any of these columns as a vector, index into the property using dot notation. For example, obtain the raw residual vector r in the model mdl:

r = mdl.Residuals.Raw

Data Types: table

RMSE — Root mean squared error

numeric value

This property is read-only.

Root mean squared error (residuals), specified as a numeric value.

RMSE = sqrt(MSE),

where RMSE is the root mean squared error and MSE is the mean squared error.

Data Types: single | double

Rsquared — R-squared value for model

structure

This property is read-only.

R-squared value for the model, specified as a structure with two fields:

• Ordinary — Ordinary (unadjusted) R-squared

• Adjusted — R-squared adjusted for the number of coefficients

The R-squared value is the proportion of the total sum of squares explained by the model. The ordinary R-squared value relates to the SSR and SST properties:

Rsquared = SSR/SST,

where SST is the total sum of squares, and SSR is the regression sum of squares.

For details, see Coefficient of Determination (R-Squared).

To obtain either of these values as a scalar, index into the property using dot notation. For example, obtain the adjusted R-squared value in the model mdl:

r2 = mdl.Rsquared.Adjusted

Data Types: struct

SSE — Sum of squared errors

numeric value

This property is read-only.

Sum of squared errors (residuals), specified as a numeric value.

The Pythagorean theorem implies

SST = SSE + SSR,

where SST is the total sum of squares, SSE is the sum of squared errors, and SSR is the regression sum of squares.

Data Types: single | double

SSR — Regression sum of squares

numeric value

This property is read-only.

Regression sum of squares, specified as a numeric value. The regression sum of squares is equal to the sum of squared deviations of the fitted values from their mean.

The Pythagorean theorem implies

SST = SSE + SSR,

where SST is the total sum of squares, SSE is the sum of squared errors, and SSR is the regression sum of squares.

Data Types: single | double

SST — Total sum of squares

numeric value

This property is read-only.

Total sum of squares, specified as a numeric value. The total sum of squares is equal to the sum of squared deviations of the response vector y from the mean(y).

The Pythagorean theorem implies

SST = SSE + SSR,

where SST is the total sum of squares, SSE is the sum of squared errors, and SSR is the regression sum of squares.

Data Types: single | double

Fitting Method

Robust — Robust fit information

structure

This property is read-only.

Robust fit information, specified as a structure with the fields described in this table.

Field	Description
WgtFun	Robust weighting function, such as 'bisquare' (see 'RobustOpts')
Tune	Tuning constant. This field is empty ([]) if WgtFun is 'ols' or if WgtFun is a function handle for a custom weight function with the default tuning constant 1.
Weights	Vector of weights used in the final iteration of robust fit. This field is empty for a CompactLinearModel object.

This structure is empty unless you fit the model using robust regression.

Data Types: struct

Steps — Stepwise fitting information

structure

This property is read-only.

Stepwise fitting information, specified as a structure with the fields described in this table.

Field	Description
Start	Formula representing the starting model
Lower	Formula representing the lower bound model. The terms in Lower must remain in the model.
Upper	Formula representing the upper bound model. The model cannot contain more terms than Upper.
Criterion	Criterion used for the stepwise algorithm, such as 'sse'
PEnter	Threshold for Criterion to add a term
PRemove	Threshold for Criterion to remove a term
History	Table representing the steps taken in the fit

The History table contains one row for each step, including the initial fit, and the columns described in this table.

Column	Description
Action	Action taken during the step: 'Start' — First step 'Add' — A term is added 'Remove' — A term is removed
TermName	If Action is 'Start', TermName specifies the starting model specification. If Action is 'Add' or 'Remove', TermName specifies the term added or removed in the step.
Terms	Model specification in a Terms Matrix
DF	Regression degrees of freedom after the step
delDF	Change in regression degrees of freedom from the previous step (negative for steps that remove a term)
Deviance	Deviance (residual sum of squares) at the step (only for a generalized linear regression model)
FStat	F-statistic that leads to the step
PValue	p-value of the F-statistic

The structure is empty unless you fit the model using stepwise regression.

Data Types: struct

Input Data

Formula — Model information

LinearFormula object

This property is read-only.

Model information, specified as a LinearFormula object.

Display the formula of the fitted model mdl using dot notation:

mdl.Formula

NumObservations — Number of observations

positive integer

This property is read-only.

Number of observations the fitting function used in fitting, specified as a positive integer. NumObservations is the number of observations supplied in the original table, dataset, or matrix, minus any excluded rows (set with the 'Exclude' name-value pair argument) or rows with missing values.

Data Types: double

NumPredictors — Number of predictor variables

positive integer

This property is read-only.

Number of predictor variables used to fit the model, specified as a positive integer.

Data Types: double

NumVariables — Number of variables

positive integer

This property is read-only.

Number of variables in the input data, specified as a positive integer. NumVariables is the number of variables in the original table or dataset, or the total number of columns in the predictor matrix and response vector.

NumVariables also includes any variables that are not used to fit the model as predictors or as the response.

Data Types: double

ObservationInfo — Observation information

table

This property is read-only.

Observation information, specified as an n-by-4 table, where n is equal to the number of rows of input data. ObservationInfo contains the columns described in this table.

Column	Description
Weights	Observation weights, specified as a numeric value. The default value is 1.
Excluded	Indicator of excluded observations, specified as a logical value. The value is true if you exclude the observation from the fit by using the 'Exclude' name-value pair argument.
Missing	Indicator of missing observations, specified as a logical value. The value is true if the observation is missing.
Subset	Indicator of whether or not the fitting function uses the observation, specified as a logical value. The value is true if the observation is not excluded or missing, meaning the fitting function uses the observation.

To obtain any of these columns as a vector, index into the property using dot notation. For example, obtain the weight vector w of the model mdl:

w = mdl.ObservationInfo.Weights

Data Types: table

ObservationNames — Observation names

cell array of character vectors

This property is read-only.

Observation names, specified as a cell array of character vectors containing the names of the observations used in the fit.

• If the fit is based on a table or dataset containing observation names, ObservationNames uses those names.

• Otherwise, ObservationNames is an empty cell array.

Data Types: cell

PredictorNames — Names of predictors used to fit model

cell array of character vectors

This property is read-only.

Names of predictors used to fit the model, specified as a cell array of character vectors.

Data Types: cell

ResponseName — Response variable name

character vector

This property is read-only.

Response variable name, specified as a character vector.

Data Types: char

VariableInfo — Information about variables

table

This property is read-only.

Information about variables contained in Variables, specified as a table with one row for each variable and the columns described in this table.

Column	Description
Class	Variable class, specified as a cell array of character vectors, such as 'double' and 'categorical'
Range	Variable range, specified as a cell array of vectors Continuous variable — Two-element vector [min,max], the minimum and maximum values Categorical variable — Vector of distinct variable values
InModel	Indicator of which variables are in the fitted model, specified as a logical vector. The value is true if the model includes the variable.
IsCategorical	Indicator of categorical variables, specified as a logical vector. The value is true if the variable is categorical.

VariableInfo also includes any variables that are not used to fit the model as predictors or as the response.

Data Types: table

VariableNames — Names of variables

cell array of character vectors

This property is read-only.

Names of variables, specified as a cell array of character vectors.

• If the fit is based on a table or dataset, this property provides the names of the variables in the table or dataset.

• If the fit is based on a predictor matrix and response vector, VariableNames contains the values specified by the 'VarNames' name-value pair argument of the fitting method. The default value of 'VarNames' is {'x1','x2',...,'xn','y'}.

VariableNames also includes any variables that are not used to fit the model as predictors or as the response.

Data Types: cell

Variables — Input data

table

This property is read-only.

Input data, specified as a table. Variables contains both predictor and response values. If the fit is based on a table or dataset array, Variables contains all the data from the table or dataset array. Otherwise, Variables is a table created from the input data matrix X and the response vector y.

Variables also includes any variables that are not used to fit the model as predictors or as the response.

Data Types: table

Object Functions

Create CompactLinearModel

compact

Compact linear regression model

Add or Remove Terms from Linear Model

addTerms	Add terms to linear regression model
removeTerms	Remove terms from linear regression model
step	Improve linear regression model by adding or removing terms

Predict Responses

feval	Predict responses of linear regression model using one input for each predictor
predict	Predict responses of linear regression model
random	Simulate responses with random noise for linear regression model

Evaluate Linear Model

anova	Analysis of variance for linear regression model
coefCI	Confidence intervals of coefficient estimates of linear regression model
coefTest	Linear hypothesis test on linear regression model coefficients
dwtest	Durbin-Watson test with linear regression model object
partialDependence	Compute partial dependence

Visualize Linear Model and Summary Statistics

plot	Scatter plot or added variable plot of linear regression model
plotAdded	Added variable plot of linear regression model
plotAdjustedResponse	Adjusted response plot of linear regression model
plotDiagnostics	Plot observation diagnostics of linear regression model
plotEffects	Plot main effects of predictors in linear regression model
plotInteraction	Plot interaction effects of two predictors in linear regression model
plotPartialDependence	Create partial dependence plot (PDP) and individual conditional expectation (ICE) plots
plotResiduals	Plot residuals of linear regression model
plotSlice	Plot of slices through fitted linear regression surface

Gather Properties of Linear Model

gather

Gather properties of Statistics and Machine Learning Toolbox object from GPU

Examples

Fit Linear Regression Using Data in Matrix

Fit a linear regression model using a matrix input data set.

Load the carsmall data set, a matrix input data set.

load carsmall
X = [Weight,Horsepower,Acceleration];

Fit a linear regression model by using fitlm.

mdl = fitlm(X,MPG)

mdl = 
Linear regression model:
    y ~ 1 + x1 + x2 + x3

Estimated Coefficients:
                    Estimate        SE          tStat        pValue  
                   __________    _________    _________    __________

    (Intercept)        47.977       3.8785        12.37    4.8957e-21
    x1             -0.0065416    0.0011274      -5.8023    9.8742e-08
    x2              -0.042943     0.024313      -1.7663       0.08078
    x3              -0.011583      0.19333    -0.059913       0.95236


Number of observations: 93, Error degrees of freedom: 89
Root Mean Squared Error: 4.09
R-squared: 0.752,  Adjusted R-Squared: 0.744
F-statistic vs. constant model: 90, p-value = 7.38e-27

The model display includes the model formula, estimated coefficients, and model summary statistics.

The model formula in the display, y ~ 1 + x1 + x2 + x3, corresponds to $\mathit{y}={\beta }_0+{\beta }_1{\mathit{X}}_1+{\beta }_2{\mathit{X}}_2+{\beta }_3{\mathit{X}}_3+ϵ$.

The model display also shows the estimated coefficient information, which is stored in the Coefficients property. Display the Coefficients property.

mdl.Coefficients

ans=4×4 table
                    Estimate        SE          tStat        pValue  
                   __________    _________    _________    __________

    (Intercept)        47.977       3.8785        12.37    4.8957e-21
    x1             -0.0065416    0.0011274      -5.8023    9.8742e-08
    x2              -0.042943     0.024313      -1.7663       0.08078
    x3              -0.011583      0.19333    -0.059913       0.95236

The Coefficient property includes these columns:

• Estimate — Coefficient estimates for each corresponding term in the model. For example, the estimate for the constant term (intercept) is 47.977.

• SE — Standard error of the coefficients.

• tStat — t-statistic for each coefficient to test the null hypothesis that the corresponding coefficient is zero against the alternative that it is different from zero, given the other predictors in the model. Note that tStat = Estimate/SE. For example, the t-statistic for the intercept is 47.977/3.8785 = 12.37.

• pValue — p-value for the t-statistic of the hypothesis test that the corresponding coefficient is equal to zero or not. For example, the p-value of the t-statistic for x2 is greater than 0.05, so this term is not significant at the 5% significance level given the other terms in the model.

The summary statistics of the model are:

• Number of observations — Number of rows without any NaN values. For example, Number of observations is 93 because the MPG data vector has six NaN values and the Horsepower data vector has one NaN value for a different observation, where the number of rows in X and MPG is 100.

• Error degrees of freedom — n – p, where n is the number of observations, and p is the number of coefficients in the model, including the intercept. For example, the model has four predictors, so the Error degrees of freedom is 93 – 4 = 89.

• Root mean squared error — Square root of the mean squared error, which estimates the standard deviation of the error distribution.

• R-squared and Adjusted R-squared — Coefficient of determination and adjusted coefficient of determination, respectively. For example, the R-squared value suggests that the model explains approximately 75% of the variability in the response variable MPG.

• F-statistic vs. constant model — Test statistic for the F-test on the regression model, which tests whether the model fits significantly better than a degenerate model consisting of only a constant term.

• p-value — p-value for the F-test on the model. For example, the model is significant with a p-value of 7.3816e-27.

You can find these statistics in the model properties (NumObservations, DFE, RMSE, and Rsquared) and by using the anova function.

anova(mdl,'summary')

ans=3×5 table
                SumSq     DF    MeanSq      F         pValue  
                ______    __    ______    ______    __________

    Total       6004.8    92    65.269                        
    Model         4516     3    1505.3    89.987    7.3816e-27
    Residual    1488.8    89    16.728                        

Use plot to create an added variable plot (partial regression leverage plot) for the whole model except the constant (intercept) term.

plot(mdl)

Linear Regression with Categorical Predictor

Fit a linear regression model that contains a categorical predictor. Reorder the categories of the categorical predictor to control the reference level in the model. Then, use anova to test the significance of the categorical variable.

Model with Categorical Predictor

Load the carsmall data set and create a linear regression model of MPG as a function of Model_Year. To treat the numeric vector Model_Year as a categorical variable, identify the predictor using the 'CategoricalVars' name-value pair argument.

load carsmall
mdl = fitlm(Model_Year,MPG,'CategoricalVars',1,'VarNames',{'Model_Year','MPG'})

mdl = 
Linear regression model:
    MPG ~ 1 + Model_Year

Estimated Coefficients:
                     Estimate      SE      tStat       pValue  
                     ________    ______    ______    __________

    (Intercept)        17.69     1.0328    17.127    3.2371e-30
    Model_Year_76     3.8839     1.4059    2.7625     0.0069402
    Model_Year_82      14.02     1.4369    9.7571    8.2164e-16


Number of observations: 94, Error degrees of freedom: 91
Root Mean Squared Error: 5.56
R-squared: 0.531,  Adjusted R-Squared: 0.521
F-statistic vs. constant model: 51.6, p-value = 1.07e-15

The model formula in the display, MPG ~ 1 + Model_Year, corresponds to

$\mathrm{MPG}={\beta }_0+{\beta }_1{{\rm I}}_{\mathrm{Year}=76}+{\beta }_2{{\rm I}}_{\mathrm{Year}=82}+ϵ$,

where ${{\rm I}}_{\mathrm{Year}=76}$ and ${{\rm I}}_{\mathrm{Year}=82}$ are indicator variables whose value is one if the value of Model_Year is 76 and 82, respectively. The Model_Year variable includes three distinct values, which you can check by using the unique function.

unique(Model_Year)

ans = 3×1

    70
    76
    82

fitlm chooses the smallest value in Model_Year as a reference level ('70') and creates two indicator variables ${{\rm I}}_{\mathrm{Year}=76}$ and ${{\rm I}}_{\mathrm{Year}=82}$. The model includes only two indicator variables because the design matrix becomes rank deficient if the model includes three indicator variables (one for each level) and an intercept term.

Model with Full Indicator Variables

You can interpret the model formula of mdl as a model that has three indicator variables without an intercept term:

$\mathit{y}={\beta }_0{{\rm I}}_{{\mathit{x}}_1=70}+\left({\beta }_0+{\beta }_1\right){{\rm I}}_{{\mathit{x}}_1=76}+\left({{\beta }_0+\beta }_2\right){{\rm I}}_{{\mathit{x}}_2=82}+ϵ$.

Alternatively, you can create a model that has three indicator variables without an intercept term by manually creating indicator variables and specifying the model formula.

temp_Year = dummyvar(categorical(Model_Year));
Model_Year_70 = temp_Year(:,1);
Model_Year_76 = temp_Year(:,2);
Model_Year_82 = temp_Year(:,3);
tbl = table(Model_Year_70,Model_Year_76,Model_Year_82,MPG);
mdl = fitlm(tbl,'MPG ~ Model_Year_70 + Model_Year_76 + Model_Year_82 - 1')

mdl = 
Linear regression model:
    MPG ~ Model_Year_70 + Model_Year_76 + Model_Year_82

Estimated Coefficients:
                     Estimate      SE       tStat       pValue  
                     ________    _______    ______    __________

    Model_Year_70      17.69      1.0328    17.127    3.2371e-30
    Model_Year_76     21.574     0.95387    22.617    4.0156e-39
    Model_Year_82      31.71     0.99896    31.743    5.2234e-51


Number of observations: 94, Error degrees of freedom: 91
Root Mean Squared Error: 5.56

Choose Reference Level in Model

You can choose a reference level by modifying the order of categories in a categorical variable. First, create a categorical variable Year.

Year = categorical(Model_Year);

Check the order of categories by using the categories function.

categories(Year)

ans = 3x1 cell
    {'70'}
    {'76'}
    {'82'}

If you use Year as a predictor variable, then fitlm chooses the first category '70' as a reference level. Reorder Year by using the reordercats function.

Year_reordered = reordercats(Year,{'76','70','82'});
categories(Year_reordered)

ans = 3x1 cell
    {'76'}
    {'70'}
    {'82'}

The first category of Year_reordered is '76'. Create a linear regression model of MPG as a function of Year_reordered.

mdl2 = fitlm(Year_reordered,MPG,'VarNames',{'Model_Year','MPG'})

mdl2 = 
Linear regression model:
    MPG ~ 1 + Model_Year

Estimated Coefficients:
                     Estimate      SE        tStat       pValue  
                     ________    _______    _______    __________

    (Intercept)       21.574     0.95387     22.617    4.0156e-39
    Model_Year_70    -3.8839      1.4059    -2.7625     0.0069402
    Model_Year_82     10.136      1.3812     7.3385    8.7634e-11


Number of observations: 94, Error degrees of freedom: 91
Root Mean Squared Error: 5.56
R-squared: 0.531,  Adjusted R-Squared: 0.521
F-statistic vs. constant model: 51.6, p-value = 1.07e-15

mdl2 uses '76' as a reference level and includes two indicator variables ${{\rm I}}_{\mathrm{Year}=70}$ and ${{\rm I}}_{\mathrm{Year}=82}$.

Evaluate Categorical Predictor

The model display of mdl2 includes a p-value of each term to test whether or not the corresponding coefficient is equal to zero. Each p-value examines each indicator variable. To examine the categorical variable Model_Year as a group of indicator variables, use anova. Use the 'components'(default) option to return a component ANOVA table that includes ANOVA statistics for each variable in the model except the constant term.

anova(mdl2,'components')

ans=2×5 table
                  SumSq     DF    MeanSq      F        pValue  
                  ______    __    ______    _____    __________

    Model_Year    3190.1     2    1595.1    51.56    1.0694e-15
    Error         2815.2    91    30.936                       

The component ANOVA table includes the p-value of the Model_Year variable, which is smaller than the p-values of the indicator variables.

Fit Robust Linear Regression Model

Load the hald data set, which measures the effect of cement composition on its hardening heat.

load hald

This data set includes the variables ingredients and heat. The matrix ingredients contains the percent composition of four chemicals present in the cement. The vector heat contains the values for the heat hardening after 180 days for each cement sample.

Fit a robust linear regression model to the data.

mdl = fitlm(ingredients,heat,'RobustOpts','on')

mdl = 
Linear regression model (robust fit):
    y ~ 1 + x1 + x2 + x3 + x4

Estimated Coefficients:
                   Estimate      SE        tStat       pValue 
                   ________    _______    ________    ________

    (Intercept)       60.09     75.818     0.79256      0.4509
    x1               1.5753    0.80585      1.9548    0.086346
    x2               0.5322    0.78315     0.67957     0.51596
    x3              0.13346     0.8166     0.16343     0.87424
    x4             -0.12052     0.7672    -0.15709     0.87906


Number of observations: 13, Error degrees of freedom: 8
Root Mean Squared Error: 2.65
R-squared: 0.979,  Adjusted R-Squared: 0.969
F-statistic vs. constant model: 94.6, p-value = 9.03e-07

For more details, see the topic Reduce Outlier Effects Using Robust Regression, which compares the results of a robust fit to a standard least-squares fit.

Fit Linear Model Using Stepwise Regression

Load the hald data set, which measures the effect of cement composition on its hardening heat.

load hald

This data set includes the variables ingredients and heat. The matrix ingredients contains the percent composition of four chemicals present in the cement. The vector heat contains the values for the heat hardening after 180 days for each cement sample.

Fit a stepwise linear regression model to the data. Specify 0.06 as the threshold for the criterion to add a term to the model.

 mdl = stepwiselm(ingredients,heat,'PEnter',0.06)

1. Adding x4, FStat = 22.7985, pValue = 0.000576232
2. Adding x1, FStat = 108.2239, pValue = 1.105281e-06
3. Adding x2, FStat = 5.0259, pValue = 0.051687
4. Removing x4, FStat = 1.8633, pValue = 0.2054

mdl = 
Linear regression model:
    y ~ 1 + x1 + x2

Estimated Coefficients:
                   Estimate       SE       tStat       pValue  
                   ________    ________    ______    __________

    (Intercept)     52.577       2.2862    22.998    5.4566e-10
    x1              1.4683       0.1213    12.105    2.6922e-07
    x2             0.66225     0.045855    14.442     5.029e-08


Number of observations: 13, Error degrees of freedom: 10
Root Mean Squared Error: 2.41
R-squared: 0.979,  Adjusted R-Squared: 0.974
F-statistic vs. constant model: 230, p-value = 4.41e-09

By default, the starting model is a constant model. stepwiselm performs forward selection and adds the x4, x1, and x2 terms (in that order), because the corresponding p-values are less than the PEnter value of 0.06. stepwiselm then uses backward elimination and removes x4 from the model because, once x2 is in the model, the p-value of x4 is greater than the default value of PRemove, 0.1.

More About

Terms Matrix

A terms matrix T is a t-by-(p + 1) matrix specifying terms in a model, where t is the number of terms, p is the number of predictor variables, and +1 accounts for the response variable. The value of T(i,j) is the exponent of variable j in term i.

For example, suppose that an input includes three predictor variables x1, x2, and x3 and the response variable y in the order x1, x2, x3, and y. Each row of T represents one term:

• [0 0 0 0] — Constant term or intercept

• [0 1 0 0] — x2; equivalently, x1^0 * x2^1 * x3^0

• [1 0 1 0] — x1*x3

• [2 0 0 0] — x1^2

• [0 1 2 0] — x2*(x3^2)

The 0 at the end of each term represents the response variable. In general, a column vector of zeros in a terms matrix represents the position of the response variable. If you have the predictor and response variables in a matrix and column vector, then you must include 0 for the response variable in the last column of each row.

Alternative Functionality

• For reduced computation time on high-dimensional data sets, fit a linear regression model using the fitrlinear function.

• To regularize a regression, use fitrlinear, lasso, ridge, or plsregress.

  • fitrlinear regularizes a regression for high-dimensional data sets using lasso or ridge regression.

  • lasso removes redundant predictors in linear regression using lasso or elastic net.

  • ridge regularizes a regression with correlated terms using ridge regression.

  • plsregress regularizes a regression with correlated terms using partial least squares.

Extended Capabilities

C/C++ Code Generation

Generate C and C++ code using MATLAB® Coder™.

Usage notes and limitations:

• The predict and random functions support code generation.

For more information, see Introduction to Code Generation.

GPU Arrays

Accelerate code by running on a graphics processing unit (GPU) using Parallel Computing Toolbox™.

Usage notes and limitations:

• The following object functions fully support GPU arrays:

  • feval

  • predict

  • random

  • partialDependence

  • plotPartialDependence

• The following object functions support model objects fitted with GPU array input arguments:

  • compact

  • addTerms

  • removeTerms

  • step

  • anova

  • coefCI

  • coefTest

  • dwtest

  • plot

  • plotAdded

  • plotAdjustedResponse

  • plotDiagnostics

  • plotEffects

  • plotInteraction

  • plotResiduals

  • plotSlice

  • gather

For more information, see Run MATLAB Functions on a GPU (Parallel Computing Toolbox).