fitlm

Fit linear regression model

Syntax

mdl = fitlm(tbl)

mdl = fitlm(X,y)

mdl = fitlm(___,modelspec)

mdl = fitlm(___,Name,Value)

Description

mdl = fitlm(tbl) returns a linear regression model fit to variables in the table or dataset array tbl. By default, fitlm takes the last variable as the response variable.

mdl = fitlm(X,y) returns a linear regression model of the responses y, fit to the data matrix X.

mdl = fitlm(___,modelspec) defines the model specification using any of the input argument combinations in the previous syntaxes.

mdl = fitlm(___,Name,Value) specifies additional options using one or more name-value pair arguments. For example, you can specify which variables are categorical, perform robust regression, or use observation weights.

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                        

Fit Linear Regression Using Data in Table

Load the sample data.

load carsmall

Store the variables in a table.

tbl = table(Weight,Acceleration,MPG,'VariableNames',{'Weight','Acceleration','MPG'});

Display the first five rows of the table.

tbl(1:5,:)

ans=5×3 table
    Weight    Acceleration    MPG
    ______    ____________    ___

     3504           12        18 
     3693         11.5        15 
     3436           11        18 
     3433           12        16 
     3449         10.5        17 

Fit a linear regression model for miles per gallon (MPG). Specify the model formula by using Wilkinson notation.

lm = fitlm(tbl,'MPG~Weight+Acceleration')

lm = 
Linear regression model:
    MPG ~ 1 + Weight + Acceleration

Estimated Coefficients:
                     Estimate         SE         tStat       pValue  
                    __________    __________    _______    __________

    (Intercept)         45.155        3.4659     13.028    1.6266e-22
    Weight          -0.0082475    0.00059836    -13.783    5.3165e-24
    Acceleration       0.19694       0.14743     1.3359       0.18493


Number of observations: 94, Error degrees of freedom: 91
Root Mean Squared Error: 4.12
R-squared: 0.743,  Adjusted R-Squared: 0.738
F-statistic vs. constant model: 132, p-value = 1.38e-27

The model 'MPG~Weight+Acceleration' in this example is equivalent to set the model specification as 'linear'. For example,

lm2 = fitlm(tbl,'linear');

If you use a character vector for model specification and you do not specify the response variable, then fitlm accepts the last variable in tbl as the response variable and the other variables as the predictor variables.

Fit Linear Regression Using Specified Model Formula

Fit a linear regression model using a model formula specified by Wilkinson notation.

Load the sample data.

load carsmall

Store the variables in a table.

tbl = table(Weight,Acceleration,Model_Year,MPG,'VariableNames',{'Weight','Acceleration','Model_Year','MPG'});

Fit a linear regression model for miles per gallon (MPG) with weight and acceleration as the predictor variables.

lm = fitlm(tbl,'MPG~Weight+Acceleration')

lm = 
Linear regression model:
    MPG ~ 1 + Weight + Acceleration

Estimated Coefficients:
                     Estimate         SE         tStat       pValue  
                    __________    __________    _______    __________

    (Intercept)         45.155        3.4659     13.028    1.6266e-22
    Weight          -0.0082475    0.00059836    -13.783    5.3165e-24
    Acceleration       0.19694       0.14743     1.3359       0.18493


Number of observations: 94, Error degrees of freedom: 91
Root Mean Squared Error: 4.12
R-squared: 0.743,  Adjusted R-Squared: 0.738
F-statistic vs. constant model: 132, p-value = 1.38e-27

The p-value of 0.18493 indicates that Acceleration does not have a significant impact on MPG.

Remove Acceleration from the model, and try improving the model by adding the predictor variable Model_Year. First define Model_Year as a categorical variable.

tbl.Model_Year = categorical(tbl.Model_Year);
lm = fitlm(tbl,'MPG~Weight+Model_Year')

lm = 
Linear regression model:
    MPG ~ 1 + Weight + Model_Year

Estimated Coefficients:
                      Estimate         SE         tStat       pValue  
                     __________    __________    _______    __________

    (Intercept)           40.11        1.5418     26.016    1.2024e-43
    Weight           -0.0066475    0.00042802    -15.531    3.3639e-27
    Model_Year_76        1.9291       0.74761     2.5804      0.011488
    Model_Year_82        7.9093       0.84975     9.3078    7.8681e-15


Number of observations: 94, Error degrees of freedom: 90
Root Mean Squared Error: 2.92
R-squared: 0.873,  Adjusted R-Squared: 0.868
F-statistic vs. constant model: 206, p-value = 3.83e-40

Specifying modelspec using Wilkinson notation enables you to update the model without having to change the design matrix. fitlm uses only the variables that are specified in the formula. It also creates the necessary two dummy indicator variables for the categorical variable Model_Year.

Fit Linear Regression Using Terms Matrix

Fit a linear regression model using a terms matrix.

Terms Matrix for Table Input

If the model variables are in a table, then a column of 0s in a terms matrix represents the position of the response variable.

Load the hospital data set.

load hospital

Store the variables in a table.

t = table(hospital.Sex,hospital.BloodPressure(:,1),hospital.Age,hospital.Smoker, ...
    'VariableNames',{'Sex','BloodPressure','Age','Smoker'});

Represent the linear model 'BloodPressure ~ 1 + Sex + Age + Smoker' using a terms matrix. The response variable is in the second column of the table, so the second column of the terms matrix must be a column of 0s for the response variable.

T = [0 0 0 0;1 0 0 0;0 0 1 0;0 0 0 1]

T = 4×4

     0     0     0     0
     1     0     0     0
     0     0     1     0
     0     0     0     1

Fit a linear model.

mdl1 = fitlm(t,T)

mdl1 = 
Linear regression model:
    BloodPressure ~ 1 + Sex + Age + Smoker

Estimated Coefficients:
                   Estimate       SE        tStat        pValue  
                   ________    ________    ________    __________

    (Intercept)      116.14      2.6107      44.485    7.1287e-66
    Sex_Male       0.050106     0.98364    0.050939       0.95948
    Age            0.085276    0.066945      1.2738        0.2058
    Smoker_1           9.87      1.0346      9.5395    1.4516e-15


Number of observations: 100, Error degrees of freedom: 96
Root Mean Squared Error: 4.78
R-squared: 0.507,  Adjusted R-Squared: 0.492
F-statistic vs. constant model: 33, p-value = 9.91e-15

Terms Matrix for Matrix Input

If the predictor and response variables are in a matrix and column vector, then you must include 0 for the response variable at the end of each row in a terms matrix.

Load the carsmall data set and define the matrix of predictors.

load carsmall
X = [Acceleration,Weight];

Specify the model 'MPG ~ Acceleration + Weight + Acceleration:Weight + Weight^2' using a terms matrix. This model includes the main effect and two-way interaction terms for the variables Acceleration and Weight, and a second-order term for the variable Weight.

T = [0 0 0;1 0 0;0 1 0;1 1 0;0 2 0]

T = 5×3

     0     0     0
     1     0     0
     0     1     0
     1     1     0
     0     2     0

Fit a linear model.

mdl2 = fitlm(X,MPG,T)

mdl2 = 
Linear regression model:
    y ~ 1 + x1*x2 + x2^2

Estimated Coefficients:
                    Estimate          SE         tStat       pValue  
                   ___________    __________    _______    __________

    (Intercept)         48.906        12.589     3.8847    0.00019665
    x1                 0.54418       0.57125    0.95261       0.34337
    x2               -0.012781     0.0060312    -2.1192      0.036857
    x1:x2          -0.00010892    0.00017925    -0.6076         0.545
    x2^2            9.7518e-07    7.5389e-07     1.2935       0.19917


Number of observations: 94, Error degrees of freedom: 89
Root Mean Squared Error: 4.1
R-squared: 0.751,  Adjusted R-Squared: 0.739
F-statistic vs. constant model: 67, p-value = 4.99e-26

Only the intercept and x2 term, which corresponds to the Weight variable, are significant at the 5% significance level.

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.

Specify Response and Predictor Variables for Linear Model

Fit a linear regression model to sample data. Specify the response and predictor variables, and include only pairwise interaction terms in the model.

Load sample data.

load hospital

Fit a linear model with interaction terms to the data. Specify weight as the response variable, and sex, age, and smoking status as the predictor variables. Also, specify that sex and smoking status are categorical variables.

mdl = fitlm(hospital,'interactions','ResponseVar','Weight',...
    'PredictorVars',{'Sex','Age','Smoker'},...
    'CategoricalVar',{'Sex','Smoker'})

mdl = 
Linear regression model:
    Weight ~ 1 + Sex*Age + Sex*Smoker + Age*Smoker

Estimated Coefficients:
                         Estimate      SE        tStat        pValue  
                         ________    _______    ________    __________

    (Intercept)             118.7     7.0718      16.785     6.821e-30
    Sex_Male               68.336     9.7153      7.0339    3.3386e-10
    Age                   0.31068    0.18531      1.6765      0.096991
    Smoker_1               3.0425     10.446     0.29127       0.77149
    Sex_Male:Age         -0.49094    0.24764     -1.9825      0.050377
    Sex_Male:Smoker_1      0.9509     3.8031     0.25003       0.80312
    Age:Smoker_1         -0.07288    0.26275    -0.27737       0.78211


Number of observations: 100, Error degrees of freedom: 93
Root Mean Squared Error: 8.75
R-squared: 0.898,  Adjusted R-Squared: 0.892
F-statistic vs. constant model: 137, p-value = 6.91e-44

The weight of the patients do not seem to differ significantly according to age, or the status of smoking, or interaction of these factors with patient sex at the 5% significance level.

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.

Compute Mean Absolute Error Using Cross-Validation

Compute the mean absolute error of a regression model by using 10-fold cross-validation.

Load the carsmall data set. Specify the Acceleration and Displacement variables as predictors and the Weight variable as the response.

load carsmall
X1 = Acceleration;
X2 = Displacement;
y = Weight;

Create the custom function regf (shown at the end of this example). This function fits a regression model to training data and then computes predicted car weights on a test set. The function compares the predicted car weight values to the true values, and then computes the mean absolute error (MAE) and the MAE adjusted to the range of the test set car weights.

Note: If you use the live script file for this example, the regf function is already included at the end of the file. Otherwise, you need to create this function at the end of your .m file or add it as a file on the MATLAB® path.

By default, crossval performs 10-fold cross-validation. For each of the 10 training and test set partitions of the data in X1, X2, and y, compute the MAE and adjusted MAE values using the regf function. Find the mean MAE and mean adjusted MAE.

rng('default') % For reproducibility
values = crossval(@regf,X1,X2,y)

values = 10×2

  319.2261    0.1132
  342.3722    0.1240
  214.3735    0.0902
  174.7247    0.1128
  189.4835    0.0832
  249.4359    0.1003
  194.4210    0.0845
  348.7437    0.1700
  283.1761    0.1187
  210.7444    0.1325

mean(values)

ans = 1×2

  252.6701    0.1129

This code creates the function regf.

function errors = regf(X1train,X2train,ytrain,X1test,X2test,ytest)
tbltrain = table(X1train,X2train,ytrain, ...
    'VariableNames',{'Acceleration','Displacement','Weight'});
tbltest = table(X1test,X2test,ytest, ...
    'VariableNames',{'Acceleration','Displacement','Weight'});
mdl = fitlm(tbltrain,'Weight ~ Acceleration + Displacement');
yfit = predict(mdl,tbltest);
MAE = mean(abs(yfit-tbltest.Weight));
adjMAE = MAE/range(tbltest.Weight);
errors = [MAE adjMAE];
end

Input Arguments

tbl — Input data

table | dataset array

Input data including predictor and response variables, specified as a table or dataset array. The predictor variables can be numeric, logical, categorical, character, or string. The response variable must be numeric or logical.

• By default, fitlm takes the last variable as the response variable and the others as the predictor variables.

• To set a different column as the response variable, use the ResponseVar name-value pair argument.

• To use a subset of the columns as predictors, use the PredictorVars name-value pair argument.

• To define a model specification, set the modelspec argument using a formula or terms matrix. The formula or terms matrix specifies which columns to use as the predictor or response variables.

The variable names in a table do not have to be valid MATLAB^® identifiers. However, if the names are not valid, you cannot use a formula when you fit or adjust a model; for example:

• You cannot specify modelspec using a formula.

• You cannot use a formula to specify the terms to add or remove when you use the addTerms function or the removeTerms function, respectively.

• You cannot use a formula to specify the lower and upper bounds of the model when you use the step or stepwiselm function with the name-value pair arguments 'Lower' and 'Upper', respectively.

You can verify the variable names in tbl by using the isvarname function. If the variable names are not valid, then you can convert them by using the matlab.lang.makeValidName function.

X — Predictor variables

matrix

Predictor variables, specified as an n-by-p matrix, where n is the number of observations and p is the number of predictor variables. Each column of X represents one variable, and each row represents one observation.

By default, there is a constant term in the model, unless you explicitly remove it, so do not include a column of 1s in X.

Data Types: single | double

y — Response variable

vector

Response variable, specified as an n-by-1 vector, where n is the number of observations. Each entry in y is the response for the corresponding row of X.

Data Types: single | double | logical

modelspec — Model specification

'linear' (default) | character vector or string scalar naming the model | t-by-(p + 1) terms matrix | character vector or string scalar formula in the form 'y ~ terms'

Model specification, specified as one of these values.

• A character vector or string scalar naming the model.

  Value	Model Type
  'constant'	Model contains only a constant (intercept) term.
  'linear'	Model contains an intercept and linear term for each predictor.
  'interactions'	Model contains an intercept, linear term for each predictor, and all products of pairs of distinct predictors (no squared terms).
  'purequadratic'	Model contains an intercept term and linear and squared terms for each predictor.
  'quadratic'	Model contains an intercept term, linear and squared terms for each predictor, and all products of pairs of distinct predictors.
  'polyijk'	Model is a polynomial with all terms up to degree i in the first predictor, degree j in the second predictor, and so on. Specify the maximum degree for each predictor by using numerals 0 though 9. The model contains interaction terms, but the degree of each interaction term does not exceed the maximum value of the specified degrees. For example, 'poly13' has an intercept and x1, x2, x22, x23, x1*x2, and x1*x22 terms, where x1 and x2 are the first and second predictors, respectively.

• A t-by-(p + 1) matrix, or a Terms Matrix, specifying terms in the model, where t is the number of terms and p is the number of predictor variables, and +1 accounts for the response variable. A terms matrix is convenient when the number of predictors is large and you want to generate the terms programmatically.

• A character vector or string scalar Formula in the form

  'y ~ terms',

  where the terms are in Wilkinson Notation. The variable names in the formula must be variable names in tbl or variable names specified by Varnames. Also, the variable names must be valid MATLAB identifiers.

  The software determines the order of terms in a fitted model by using the order of terms in tbl or X. Therefore, the order of terms in the model can be different from the order of terms in the specified formula.

Example: 'quadratic'

Example: 'y ~ x1 + x2^2 + x1:x2'

Data Types: single | double | char | string

Name-Value Pair Arguments

Specify optional comma-separated pairs of Name,Value arguments. Name is the argument name and Value is the corresponding value. Name must appear inside quotes. You can specify several name and value pair arguments in any order as Name1,Value1,...,NameN,ValueN.

Example: 'Intercept',false,'PredictorVars',[1,3],'ResponseVar',5,'RobustOpts','logistic' specifies a robust regression model with no constant term, where the algorithm uses the logistic weighting function with the default tuning constant, first and third variables are the predictor variables, and fifth variable is the response variable.

'CategoricalVars' — Categorical variable list

string array | cell array of character vectors | logical or numeric index vector

Categorical variable list, specified as the comma-separated pair consisting of 'CategoricalVars' and either a string array or cell array of character vectors containing categorical variable names in the table or dataset array tbl, or a logical or numeric index vector indicating which columns are categorical.

• If data is in a table or dataset array tbl, then, by default, fitlm treats all categorical values, logical values, character arrays, string arrays, and cell arrays of character vectors as categorical variables.

• If data is in matrix X, then the default value of 'CategoricalVars' is an empty matrix []. That is, no variable is categorical unless you specify it as categorical.

For example, you can specify the second and third variables out of six as categorical using either of the following:

Example: 'CategoricalVars',[2,3]

Example: 'CategoricalVars',logical([0 1 1 0 0 0])

Data Types: single | double | logical | string | cell

'Exclude' — Observations to exclude

logical or numeric index vector

Observations to exclude from the fit, specified as the comma-separated pair consisting of 'Exclude' and a logical or numeric index vector indicating which observations to exclude from the fit.

For example, you can exclude observations 2 and 3 out of 6 using either of the following examples.

Example: 'Exclude',[2,3]

Example: 'Exclude',logical([0 1 1 0 0 0])

Data Types: single | double | logical

'Intercept' — Indicator for constant term

true (default) | false

Indicator for the constant term (intercept) in the fit, specified as the comma-separated pair consisting of 'Intercept' and either true to include or false to remove the constant term from the model.

Use 'Intercept' only when specifying the model using a character vector or string scalar, not a formula or matrix.

Example: 'Intercept',false

'PredictorVars' — Predictor variables

string array | cell array of character vectors | logical or numeric index vector

Predictor variables to use in the fit, specified as the comma-separated pair consisting of 'PredictorVars' and either a string array or cell array of character vectors of the variable names in the table or dataset array tbl, or a logical or numeric index vector indicating which columns are predictor variables.

The string values or character vectors should be among the names in tbl, or the names you specify using the 'VarNames' name-value pair argument.

The default is all variables in X, or all variables in tbl except for ResponseVar.

For example, you can specify the second and third variables as the predictor variables using either of the following examples.

Example: 'PredictorVars',[2,3]

Example: 'PredictorVars',logical([0 1 1 0 0 0])

Data Types: single | double | logical | string | cell

'ResponseVar' — Response variable

last column in tbl (default) | character vector or string scalar containing variable name | logical or numeric index vector

Response variable to use in the fit, specified as the comma-separated pair consisting of 'ResponseVar' and either a character vector or string scalar containing the variable name in the table or dataset array tbl, or a logical or numeric index vector indicating which column is the response variable. You typically need to use 'ResponseVar' when fitting a table or dataset array tbl.

For example, you can specify the fourth variable, say yield, as the response out of six variables, in one of the following ways.

Example: 'ResponseVar','yield'

Example: 'ResponseVar',[4]

Example: 'ResponseVar',logical([0 0 0 1 0 0])

Data Types: single | double | logical | char | string

'RobustOpts' — Indicator of robust fitting type

'off' (default) | 'on' | character vector | string scalar | structure

Indicator of the robust fitting type to use, specified as the comma-separated pair consisting of 'RobustOpts' and one of these values.

• 'off' — No robust fitting. fitlm uses ordinary least squares.

• 'on' — Robust fitting using the 'bisquare' weight function with the default tuning constant.

• Character vector or string scalar — Name of a robust fitting weight function from the following table. fitlm uses the corresponding default tuning constant specified in the table.

• Structure with the two fields RobustWgtFun and Tune.

  • The RobustWgtFun field contains the name of a robust fitting weight function from the following table or a function handle of a custom weight function.

  • The Tune field contains a tuning constant. If you do not set the Tune field, fitlm uses the corresponding default tuning constant.

  Weight Function	Description	Default Tuning Constant
  'andrews'	w = (abs(r)<pi) .* sin(r) ./ r	1.339
  'bisquare'	w = (abs(r)<1) .* (1 - r.^2).^2 (also called biweight)	4.685
  'cauchy'	w = 1 ./ (1 + r.^2)	2.385
  'fair'	w = 1 ./ (1 + abs(r))	1.400
  'huber'	w = 1 ./ max(1, abs(r))	1.345
  'logistic'	w = tanh(r) ./ r	1.205
  'ols'	Ordinary least squares (no weighting function)	None
  'talwar'	w = 1 * (abs(r)<1)	2.795
  'welsch'	w = exp(-(r.^2))	2.985
  function handle	Custom weight function that accepts a vector r of scaled residuals, and returns a vector of weights the same size as r	1

  • The default tuning constants of built-in weight functions give coefficient estimates that are approximately 95% as statistically efficient as the ordinary least-squares estimates, provided the response has a normal distribution with no outliers. Decreasing the tuning constant increases the downweight assigned to large residuals; increasing the tuning constant decreases the downweight assigned to large residuals.

  • The value r in the weight functions is

    r = resid/(tune*s*sqrt(1–h)),

    where resid is the vector of residuals from the previous iteration, tune is the tuning constant, h is the vector of leverage values from a least-squares fit, and s is an estimate of the standard deviation of the error term given by

    s = MAD/0.6745.

    MAD is the median absolute deviation of the residuals from their median. The constant 0.6745 makes the estimate unbiased for the normal distribution. If X has p columns, the software excludes the smallest p absolute deviations when computing the median.

For robust fitting, fitlm uses M-estimation to formulate estimating equations and solves them using the method of Iteratively Reweighted Least Squares (IRLS).

Example: 'RobustOpts','andrews'

'VarNames' — Names of variables

{'x1','x2',...,'xn','y'} (default) | string array | cell array of character vectors

Names of variables, specified as the comma-separated pair consisting of 'VarNames' and a string array or cell array of character vectors including the names for the columns of X first, and the name for the response variable y last.

'VarNames' is not applicable to variables in a table or dataset array, because those variables already have names.

The variable names do not have to be valid MATLAB identifiers. However, if the names are not valid, you cannot use a formula when you fit or adjust a model; for example:

• You cannot use a formula to specify the terms to add or remove when you use the addTerms function or the removeTerms function, respectively.

• You cannot use a formula to specify the lower and upper bounds of the model when you use the step or stepwiselm function with the name-value pair arguments 'Lower' and 'Upper', respectively.

Before specifying 'VarNames',varNames, you can verify the variable names in varNames by using the isvarname function. If the variable names are not valid, then you can convert them by using the matlab.lang.makeValidName function.

Example: 'VarNames',{'Horsepower','Acceleration','Model_Year','MPG'}

Data Types: string | cell

'Weights' — Observation weights

ones(n,1) (default) | n-by-1 vector of nonnegative scalar values

Observation weights, specified as the comma-separated pair consisting of 'Weights' and an n-by-1 vector of nonnegative scalar values, where n is the number of observations.

Data Types: single | double

Output Arguments

mdl — Linear model

LinearModel object

Linear model representing a least-squares fit of the response to the data, returned as a LinearModel object.

If the value of the 'RobustOpts' name-value pair is not [] or 'ols', the model is not a least-squares fit, but uses the robust fitting function.

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.

Formula

A formula for model specification is a character vector or string scalar of the form 'y ~ terms'.

• y is the response name.

• terms represents the predictor terms in a model using Wilkinson notation.

To represent predictor and response variables, use the variable names of the table input tbl or the variable names specified by using VarNames. The default value of VarNames is {'x1','x2',...,'xn','y'}.

For example:

• 'y ~ x1 + x2 + x3' specifies a three-variable linear model with intercept.

• 'y ~ x1 + x2 + x3 – 1' specifies a three-variable linear model without intercept. Note that formulas include a constant (intercept) term by default. To exclude a constant term from the model, you must include –1 in the formula.

A formula includes a constant term unless you explicitly remove the term using –1.

Wilkinson Notation

Wilkinson notation describes the terms present in a model. The notation relates to the terms present in a model, not to the multipliers (coefficients) of those terms.

Wilkinson notation uses these symbols:

• + means include the next variable.

• – means do not include the next variable.

• : defines an interaction, which is a product of terms.

• * defines an interaction and all lower-order terms.

• ^ raises the predictor to a power, exactly as in * repeated, so ^ includes lower-order terms as well.

• () groups terms.

This table shows typical examples of Wilkinson notation.

Wilkinson Notation	Terms in Standard Notation
1	Constant (intercept) term
x1^k, where k is a positive integer	x1, x12, ..., x1k
x1 + x2	x1, x2
x1*x2	x1, x2, x1*x2
x1:x2	x1*x2 only
–x2	Do not include x2
x1*x2 + x3	x1, x2, x3, x1*x2
x1 + x2 + x3 + x1:x2	x1, x2, x3, x1*x2
x1*x2*x3 – x1:x2:x3	x1, x2, x3, x1*x2, x1*x3, x2*x3
x1*(x2 + x3)	x1, x2, x3, x1*x2, x1*x3

For more details, see Wilkinson Notation.

Tips

• To access the model properties of the LinearModel object mdl, you can use dot notation. For example, mdl.Residuals returns a table of the raw, Pearson, Studentized, and standardized residual values for the model.

• After training a model, you can generate C/C++ code that predicts responses for new data. Generating C/C++ code requires MATLAB Coder™. For details, see Introduction to Code Generation.

Algorithms

• The main fitting algorithm is QR decomposition. For robust fitting, fitlm uses M-estimation to formulate estimating equations and solves them using the method of Iteratively Reweighted Least Squares (IRLS).

• fitlm treats a categorical predictor as follows:

  • A model with a categorical predictor that has L levels (categories) includes L – 1 indicator variables. The model uses the first category as a reference level, so it does not include the indicator variable for the reference level. If the data type of the categorical predictor is categorical, then you can check the order of categories by using categories and reorder the categories by using reordercats to customize the reference level. For more details about creating indicator variables, see Automatic Creation of Dummy Variables.

  • fitlm treats the group of L – 1 indicator variables as a single variable. If you want to treat the indicator variables as distinct predictor variables, create indicator variables manually by using dummyvar. Then use the indicator variables, except the one corresponding to the reference level of the categorical variable, when you fit a model. For the categorical predictor X, if you specify all columns of dummyvar(X) and an intercept term as predictors, then the design matrix becomes rank deficient.

  • Interaction terms between a continuous predictor and a categorical predictor with L levels consist of the element-wise product of the L – 1 indicator variables with the continuous predictor.

  • Interaction terms between two categorical predictors with L and M levels consist of the (L – 1)*(M – 1) indicator variables to include all possible combinations of the two categorical predictor levels.

  • You cannot specify higher-order terms for a categorical predictor because the square of an indicator is equal to itself.

• fitlm considers NaN, '' (empty character vector), "" (empty string), <missing>, and <undefined> values in tbl, X, and Y to be missing values. fitlm does not use observations with missing values in the fit. The ObservationInfo property of a fitted model indicates whether or not fitlm uses each observation in the fit.

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.

References

[1] DuMouchel, W. H., and F. L. O'Brien. “Integrating a Robust Option into a Multiple Regression Computing Environment.” Computer Science and Statistics: Proceedings of the 21st Symposium on the Interface. Alexandria, VA: American Statistical Association, 1989.

[2] Holland, P. W., and R. E. Welsch. “Robust Regression Using Iteratively Reweighted Least-Squares.” Communications in Statistics: Theory and Methods, A6, 1977, pp. 813–827.

[3] Huber, P. J. Robust Statistics. Hoboken, NJ: John Wiley & Sons, Inc., 1981.

[4] Street, J. O., R. J. Carroll, and D. Ruppert. “A Note on Computing Robust Regression Estimates via Iteratively Reweighted Least Squares.” The American Statistician. Vol. 42, 1988, pp. 152–154.

Extended Capabilities

Tall Arrays

Calculate with arrays that have more rows than fit in memory.

This function supports tall arrays for out-of-memory data with some limitations.

• If any input argument to fitlm is a tall array, then all of the other inputs must be tall arrays as well. This includes nonempty variables supplied with the 'Weights' and 'Exclude' name-value pairs.

• The 'RobustOpts' name-value pair is not supported with tall arrays.

• For tall data, fitlm returns a CompactLinearModel object that contains most of the same properties as a LinearModel object. The main difference is that the compact object is sensitive to memory requirements. The compact object does not include properties that include the data, or that include an array of the same size as the data. The compact object does not contain these LinearModel properties:

  • Diagnostics

  • Fitted

  • ObservationInfo

  • ObservationNames

  • Residuals

  • Steps

  • Variables

  You can compute the residuals directly from the compact object returned by LM = fitlm(X,Y) using

  RES = Y - predict(LM,X);
  S = LM.RMSE;
  histogram(RES,linspace(-3*S,3*S,51))
  

• If the CompactLinearModel object is missing lower order terms that include categorical factors:

  • The plotEffects and plotInteraction methods are not supported.

  • The anova method with the 'components' option is not supported.

For more information, see Tall Arrays for Out-of-Memory Data.

GPU Arrays

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

This function fully supports GPU arrays. For more information, see Run MATLAB Functions on a GPU (Parallel Computing Toolbox).