Be the only one, not the best one

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

mdl = fitlm(tbl,modelspec) returns a linear model of the type you specify in modelspec fit to variables in the table or dataset array tbl.

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

mdl = fitlm(X,y,modelspec) returns a linear model of the type you specify in modelspec for the responses y, fit to the data matrix X.

mdl = fitlm(___,Name,Value) returns a linear model with additional options specified by 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 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 = 

    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).

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

This syntax uses Wilkinson notation to specify the modelspec.

The model 'MPG~Weight+Acceleration' in this example is equivalent to fitting the model using the string 'linear' as modelspec. For example,

lm2 = fitlm(tbl,'linear');

When you use a string as modelspec and do not specify the response variable, fitlm by default accepts the last variable in tbl as the response variable and the other variables as the predictor variables. If there are any categorical variables and you use 'linear' as the modelspec, then you must explicitly specify those variables as categorical variables using the CategoricalVars name-value pair argument.

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 -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 nominal 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.

Linear Regression with Categorical Predictor

Fit a model of a table that contains a categorical predictor.

Load the carsmall data.

load carsmall

Construct a table containing continuous predictor variable Weight, nominal predictor variable Year, and response variable MPG.

tbl = table(MPG,Weight);
tbl.Year = nominal(Model_Year);

Create a fitted model of MPG as a function of Year, Weight, and Weight^2. (You don't have to include Weight explicitly in your formula because it is a lower-order term of Weight^2) and is included automatically.

mdl = fitlm(tbl,'MPG ~ Year + Weight^2')

mdl = 


Linear regression model:
    MPG ~ 1 + Weight + Year + Weight^2

Estimated Coefficients:
                    Estimate         SE         tStat       pValue  
                   __________    __________    _______    __________

    (Intercept)        54.206        4.7117     11.505    2.6648e-19
    Weight          -0.016404     0.0031249    -5.2493    1.0283e-06
    Year_76            2.0887       0.71491     2.9215     0.0044137
    Year_82            8.1864       0.81531     10.041    2.6364e-16
    Weight^2       1.5573e-06    4.9454e-07      3.149     0.0022303


Number of observations: 94, Error degrees of freedom: 89
Root Mean Squared Error: 2.78
R-squared: 0.885,  Adjusted R-Squared 0.88
F-statistic vs. constant model: 172, p-value = 5.52e-41

fitlm creates two dummy (indicator) variables for the nominal variate, Year. The dummy variable Year_76 takes the value 1 if model year is 1976 and takes the value 0 if it is not. The dummy variable Year_82 takes the value 1 if model year is 1982 and takes the value 0 if it is not. And the year 1970 is the reference year. The corresponding model is

$\hat MPG = 54.206 - 0.0164(Weight) + 2.0887(Year\_76) + 8.1864(Year\_82) + 1.557e-06(Weigh{t^2})$

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 a Robust Linear Regression Model

Fit a linear regression model using a robust fitting method.

Load the sample data.

load hald

The hald data measures the effect of cement composition on its hardening heat. The matrix ingredients contains the percent composition of four chemicals present in the cement. The array heat contains the heat of hardening after 180 days for each cement sample.

Fit a robust linear model to the data.

mdl = fitlm(ingredients,heat,'linear','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

Related Examples

Input Arguments

`tbl` — Input datatable | dataset array

Input data, specified as a table or dataset array. When modelspec is a formula, it specifies the variables to be used as the predictors and response. Otherwise, if you do not specify the predictor and response variables, the last variable is the response variable and the others are the predictor variables by default.

Predictor variables can be numeric, or any grouping variable type, such as logical or categorical (see Grouping Variables). The response must be numeric or logical.

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.

Data Types: single | double | logical

`X` — Predictor variablesmatrix

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 | logical

`y` — Response variablevector

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

`modelspec` — Model specification`'linear'` (default) | string naming the model | t-by-(p + 1) terms matrix | string of the form `'Y ~ terms'`

Model specification, specified as one of the following.

A string naming the model.

String	Model Type
`'constant'`	Model contains only a constant (intercept) term.
`'linear'`	Model contains an intercept and linear terms for each predictor.
`'interactions'`	Model contains an intercept, linear terms, and all products of pairs of distinct predictors (no squared terms).
`'purequadratic'`	Model contains an intercept, linear terms, and squared terms.
`'quadratic'`	Model contains an intercept, linear terms, interactions, and squared terms.
`'polyijk'`	Model is a polynomial with all terms up to degree `i` in the first predictor, degree `j` in the second predictor, etc. Use numerals `0` through `9`. For example, `'poly2111'` has a constant plus all linear and product terms, and also contains terms with predictor 1 squared.

t-by-(p + 1) matrix, namely terms matrix, specifying terms to include in the model, where t is the number of terms and p is the number of predictor variables, and plus 1 is for the response variable.
A string representing a formula in the form

'Y ~ terms',

where the terms are in Wilkinson Notation.

Example: 'quadratic'

Example: 'y ~ X1 + X2^2 + X1:X2'

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 single 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 variablescell array of strings | logical or numeric index vector

Categorical variables in the fit, specified as the comma-separated pair consisting of 'CategoricalVars' and either a cell array of strings of the names of the categorical variables 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 the default is to treat all categorical or logical variables, character arrays, or cell arrays of strings as categorical variables.
If data is in matrix X, then the default value of this name-value pair argument is an empty matrix []. That is, no variable is categorical unless you specify it.

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

Example: 'CategoricalVars',[2,3]

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

Data Types: single | double | logical

`'Exclude'` — Observations to excludelogical 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 the for 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 string, not a formula or matrix.

Example: 'Intercept',false

`'PredictorVars'` — Predictor variablescell array of strings | logical or numeric index vector

Predictor variables to use in the fit, specified as the comma-separated pair consisting of 'PredictorVars' and either a cell array of strings 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 strings 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 | cell

`'ResponseVar'` — Response variablelast column in `tbl` (default) | string for 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 string of 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

`'RobustOpts'` — Indicator of robust fitting type`'off'` (default) | `'on'` | string | structure with string or function handle

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

'off' — No robust fitting. fitlm uses ordinary least squares.
'on' — Robust fitting. When you use robust fitting, 'bisquare' weight function is the default.
String — Name of the robust fitting weight function from the following table. fitlm uses the corresponding default tuning constant in the table.

Structure with the string RobustWgtFun containing the name of the robust fitting weight function from the following table and optional scalar Tune fields — fitlm uses the RobustWgtFun weight function and Tune tuning constant from the structure. You can choose the name of the robust fitting weight function from this table. If you do not supply a Tune field, the fitting function uses the corresponding default tuning constant.

Weight Function	Equation	Default Tuning Constant
`'andrews'`	`w = (abs(r)<pi) .* sin(r) ./ r`	1.339
`'bisquare'` (default)	`w = (abs(r)<1) .* (1 - r.^2).^2`	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

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, 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 there are p columns in X, the smallest p absolute deviations are excluded when computing the median.

Default tuning constants 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.

Structure with the function handle RobustWgtFun and optional scalar Tune fields — You can specify a custom weight function. fitlm uses the RobustWgtFun weight function and Tune tuning constant from the structure. Specify RobustWgtFun as a function handle that accepts a vector of residuals, and returns a vector of weights the same size. The fitting function scales the residuals, dividing by the tuning constant (default 1) and by an estimate of the error standard deviation before it calls the weight function.

Example: 'RobustOpts','andrews'

`'VarNames'` — Names of variables in fit`{'x1','x2',...,'xn','y'}` (default) | cell array of strings

Names of variables in fit, specified as the comma-separated pair consisting of 'VarNames' and a cell array of strings 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.

For example, if in your data, horsepower, acceleration, and model year of the cars are the predictor variables, and miles per gallon (MPG) is the response variable, then you can name the variables as follows.

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

Data Types: 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.

For properties and methods of the linear model object, mdl, see the LinearModel class page.

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