Linear regression model
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.
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.
CoefficientCovariance
— Covariance matrix of coefficient estimatesThis property is readonly.
Covariance matrix of coefficient estimates, specified as a pbyp 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 namesThis property is readonly.
Coefficient names, specified as a cell array of character vectors, each containing the name of the corresponding term.
Data Types: cell
Coefficients
— Coefficient valuesThis property is readonly.
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
— tstatistic for a test that the
coefficient is zero
pValue
— pvalue for the
tstatistic
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 coefficientsThis property is readonly.
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 coefficientsThis property is readonly.
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
DFE
— Degrees of freedom for errorThis property is readonly.
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 diagnosticsThis property is readonly.
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  Delete1 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  Delete1 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  Delete1 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  Delete1 scaled differences in coefficient estimates  Dfbetas is an
N byP 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 byN 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. Delete1 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 Delete1 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 delete1 variance vector in the model
mdl
:
S2i = mdl.Diagnostics.S2_i;
Data Types: table
Fitted
— Fitted response values based on input dataThis property is readonly.
Fitted (predicted) response values based on input data, specified as an
nby1 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
— LoglikelihoodThis property is readonly.
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 comparisonThis property is readonly.
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 likelihoodbased 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 bestfitting model. The bestfitting 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 errorThis property is readonly.
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 modelThis property is readonly.
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 errorThis property is readonly.
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
— Rsquared value for modelThis property is readonly.
Rsquared value for the model, specified as a structure with two fields:
Ordinary
— Ordinary (unadjusted) Rsquared
Adjusted
— Rsquared adjusted for the number of
coefficients
The Rsquared value is the proportion of the total sum of squares explained by the
model. The ordinary Rsquared 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 (RSquared).
To obtain either of these values as a scalar, index into the property using dot
notation. For example, obtain the adjusted Rsquared value in the model
mdl
:
r2 = mdl.Rsquared.Adjusted
Data Types: struct
SSE
— Sum of squared errorsThis property is readonly.
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 squaresThis property is readonly.
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 squaresThis property is readonly.
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
Robust
— Robust fit informationThis property is readonly.
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 informationThis property is readonly.
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:

TermName 

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  Fstatistic that leads to the step 
PValue  pvalue of the Fstatistic 
The structure is empty unless you fit the model using stepwise regression.
Data Types: struct
Formula
— Model informationLinearFormula
objectThis property is readonly.
Model information, specified as a LinearFormula
object.
Display the formula of the fitted model mdl
using dot
notation:
mdl.Formula
NumObservations
— Number of observationsThis property is readonly.
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'
namevalue pair
argument) or rows with missing values.
Data Types: double
NumPredictors
— Number of predictor variablesThis property is readonly.
Number of predictor variables used to fit the model, specified as a positive integer.
Data Types: double
NumVariables
— Number of variablesThis property is readonly.
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 informationThis property is readonly.
Observation information, specified as an nby4 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' namevalue 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 namesThis property is readonly.
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 modelThis property is readonly.
Names of predictors used to fit the model, specified as a cell array of character vectors.
Data Types: cell
ResponseName
— Response variable nameThis property is readonly.
Response variable name, specified as a character vector.
Data Types: char
VariableInfo
— Information about variablesThis property is readonly.
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

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 variablesThis property is readonly.
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'
namevalue 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 dataThis property is readonly.
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
CompactLinearModel
compact  Compact linear regression 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 
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  DurbinWatson test with linear regression model object 
partialDependence  Compute partial dependence 
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  Gather properties of linear or generalized linear regression model 
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 4.8957e21 x1 0.0065416 0.0011274 5.8023 9.8742e08 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 Rsquared: 0.752, Adjusted RSquared: 0.744 Fstatistic vs. constant model: 90, pvalue = 7.38e27
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 $\mathit{y}={\beta}_{0}+{\beta}_{1}{\mathit{X}}_{1}+{\beta}_{2}{\mathit{X}}_{2}+{\beta}_{3}{\mathit{X}}_{3}+\u03f5$.
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 4.8957e21
x1 0.0065416 0.0011274 5.8023 9.8742e08
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
— tstatistic 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 tstatistic for the intercept is 47.977/3.8785 = 12.37.
pValue
— pvalue for the tstatistic of the hypothesis test that the corresponding coefficient is equal to zero or not. For example, the pvalue of the tstatistic 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.
Rsquared
and Adjusted Rsquared
— Coefficient of determination and adjusted coefficient of determination, respectively. For example, the Rsquared
value suggests that the model explains approximately 75% of the variability in the response variable MPG
.
Fstatistic vs. constant model
— Test statistic for the Ftest on the regression model, which tests whether the model fits significantly better than a degenerate model consisting of only a constant term.
pvalue
— pvalue for the Ftest on the model. For example, the model is significant with a pvalue of 7.3816e27.
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 7.3816e27
Residual 1488.8 89 16.728
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'
namevalue 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 3.2371e30 Model_Year_76 3.8839 1.4059 2.7625 0.0069402 Model_Year_82 14.02 1.4369 9.7571 8.2164e16 Number of observations: 94, Error degrees of freedom: 91 Root Mean Squared Error: 5.56 Rsquared: 0.531, Adjusted RSquared: 0.521 Fstatistic vs. constant model: 51.6, pvalue = 1.07e15
The model formula in the display, MPG ~ 1 + Model_Year
, corresponds to
$\mathrm{MPG}={\beta}_{0}+{\beta}_{1}{{\rm I}}_{\mathrm{Year}=76}+{\beta}_{2}{{\rm I}}_{\mathrm{Year}=82}+\u03f5$,
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:
$\mathit{y}={\beta}_{0}{{\rm I}}_{{\mathit{x}}_{1}=70}+\left({\beta}_{0}+{\beta}_{1}\right){{\rm I}}_{{\mathit{x}}_{1}=76}+\left({{\beta}_{0}+\beta}_{2}\right){{\rm I}}_{{\mathit{x}}_{2}=82}+\u03f5$.
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 3.2371e30 Model_Year_76 21.574 0.95387 22.617 4.0156e39 Model_Year_82 31.71 0.99896 31.743 5.2234e51 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 4.0156e39 Model_Year_70 3.8839 1.4059 2.7625 0.0069402 Model_Year_82 10.136 1.3812 7.3385 8.7634e11 Number of observations: 94, Error degrees of freedom: 91 Root Mean Squared Error: 5.56 Rsquared: 0.531, Adjusted RSquared: 0.521 Fstatistic vs. constant model: 51.6, pvalue = 1.07e15
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 pvalue of each term to test whether or not the corresponding coefficient is equal to zero. Each pvalue 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 1.0694e15
Error 2815.2 91 30.936
The component ANOVA table includes the pvalue of the Model_Year
variable, which is smaller than the pvalues of the indicator variables.
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 Rsquared: 0.979, Adjusted RSquared: 0.969 Fstatistic vs. constant model: 94.6, pvalue = 9.03e07
For more details, see the topic Reduce Outlier Effects Using Robust Regression, which compares the results of a robust fit to a standard leastsquares fit.
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 = 1.105281e06 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 5.4566e10 x1 1.4683 0.1213 12.105 2.6922e07 x2 0.66225 0.045855 14.442 5.029e08 Number of observations: 13, Error degrees of freedom: 10 Root Mean Squared Error: 2.41 Rsquared: 0.979, Adjusted RSquared: 0.974 Fstatistic vs. constant model: 230, pvalue = 4.41e09
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 pvalues 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 pvalue of x4
is greater than the default value of PRemove
, 0.1.
A terms matrix T
is a
tby(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.
For reduced computation time on highdimensional 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 highdimensional 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.
Usage notes and limitations:
When you fit a model by using fitlm
or stepwiselm
, you cannot supply training
data in a table that contains a logical vector, character array, categorical array,
string array, or cell array of character vectors. Also, you cannot use the 'CategoricalVars'
namevalue pair argument. Code generation does not
support categorical predictors. To include
categorical predictors in a model, preprocess the categorical predictors by
using dummyvar
before fitting the
model.
For more information, see Introduction to Code Generation.
Usage notes and limitations:
The following object functions fully support GPU arrays:
The following object functions support model objects fitted with GPU array input arguments:
For more information, see Run MATLAB Functions on a GPU (Parallel Computing Toolbox).
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