Create linear regression model using stepwise regression
returns
a linear model for the variables in the table or dataset array mdl
= stepwiselm(tbl
,modelspec
)tbl
using
stepwise regression to add or remove predictors. stepwiselm
uses
forward and backward stepwise regression to determine a final model.
At each step, the function searches for terms to add to or remove
from the model based on the value of the 'Criterion'
argument. modelspec
is
the starting model for the stepwise procedure.
creates
a linear model for any of the inputs in the previous syntaxes, with
additional options specified by one or more mdl
= stepwiselm(___,Name,Value
)Name,Value
pair
arguments.
For example, you can specify the categorical variables, the
smallest or largest set of terms to use in the model, the maximum
number of steps to take, or the criterion stepwiselm
uses
to add or remove terms.
Load the sample data.
load hald
hald
contains hardening data for 13 different
concrete compositions. heat
is the heat of hardening
after 180 days. ingredients
is the percentage of
each different ingredient in the cement sample.
Fit a linear model to the data. Set the criterion value to enter the model as 0.06.
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 the constant model. stepwiselm
performs
forward selection and x4
, x1
,
and x2
, respectively, as the corresponding pvalues
are less than the PEnter
value of 0.06. stepwiselm
later
uses backward elimination and eliminates x4
from
the model. Because, given that x2
is in the model,
the pvalue of x4
is higher
than the default value of PRemove
, 0.1.
Perform stepwise regression with variables in a dataset array. Specify the starting model using formula, and identify the response and predictor variables with optional arguments.
Load the sample data.
load hospital
The hospital dataset array includes the gender, age, weight, and smoking status of patients.
Fit a linear model with a starting model of a constant
term and Smoker
as the predictor variable. Specify
the response variable, Weight
, and categorical
predictor variables, Sex
, Age
,
and Smoker
.
mdl = stepwiselm(hospital,'Weight~1+Smoker',... 'ResponseVar','Weight','PredictorVars',{'Sex','Age','Smoker'},... 'CategoricalVar',{'Sex','Smoker'})
1. Adding Sex, FStat = 770.0158, pValue = 6.262758e48 2. Removing Smoker, FStat = 0.21224, pValue = 0.64605 mdl = Linear regression model: Weight ~ 1 + Sex Estimated Coefficients: Estimate SE tStat pValue (Intercept) 130.47 1.1995 108.77 5.2762e104 Sex_Male 50.06 1.7496 28.612 2.2464e49 Number of observations: 100, Error degrees of freedom: 98 Root Mean Squared Error: 8.73 Rsquared: 0.893, Adjusted RSquared 0.892 Fstatistic vs. constant model: 819, pvalue = 2.25e49
At each step, stepwiselm
searches for terms
to add and remove. At first step, stepwise algorithm adds Sex
to
the model with a pvalue of 6.26e48. Then, removes
Smoker from the model, since given Sex
in the model,
the variable Smoker
becomes redundant. stepwiselm
only
includes Sex
in the final linear model. The weight
of the patients do not seem to differ significantly according to age
or the status of smoking.
tbl
— Input datatable  dataset arrayInput 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
namevalue
pair argument. To use a subset of the columns as predictors, use the PredictorVars
namevalue
pair argument.
Data Types: single
 double
 logical
X
— Predictor variablesmatrixPredictor variables, specified as an nbyp 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 variablevectorResponse variable, specified as an nby1
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
— Starting modelstring specifying the model  tby(p+1) terms matrix  string of the form 'Y ~ terms'
Starting model for the stepwise regression, specified as one of the following:
String specifying the type of starting 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. 
'poly  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. 
If you want to specify the smallest or largest set of terms
in the model, use the Lower
and Upper
namevalue
pair arguments.
tby(p+1) matrix, namely a terms matrix, specifying terms to include in model, where t is the number of terms and p is the number of predictor variables, and plus one is for the response variable.
String representing a formula in the form
'
,Y
~ terms
'
where
the terms
are in Wilkinson Notation.
Specify optional commaseparated 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
.
'Criterion','aic','Upper',interactions,'Verbose',1
instructs stepwiselm
to
use the Akaike information criterion, display the action it takes
at each step, and include at most the interaction terms in the model.'CategoricalVars'
— Categorical variablescell array of strings  logical or numeric index vectorCategorical variables in the fit, specified as the commaseparated
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 namevalue 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
'Criterion'
— Criterion to add or remove terms'sse'
(default)  'aic'
 'bic'
 'rsquared'
 'adjrsquared'
Criterion to add or remove terms, specified as the commaseparated
pair consisting of 'Criterion'
and one of the following:
'sse'
— Default for stepwiselm
. pvalue
for an Ftest of the change in the sum of squared
error by adding or removing the term.
'aic'
— Change in the value
of Akaike information criterion (AIC).
'bic'
— Change in the value
of Bayesian information criterion (BIC).
'rsquared'
— Increase in
the value of R^{2}.
'adjrsquared'
— Increase
in the value of adjusted R^{2}.
Example: 'Criterion','bic'
'Exclude'
— Observations to excludelogical or numeric index vectorObservations to exclude from the fit, specified as the commaseparated
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 termtrue
(default)  false
Indicator the for constant term (intercept) in the fit, specified
as the commaseparated 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
'Lower'
— Model specification describing terms that cannot be removed
from model'constant'
(default)Model specification describing terms that cannot be removed
from the model, specified as the commaseparated pair consisting of 'Lower'
and
one of the string options for modelspec
naming
the model.
Example: 'Lower','linear'
'NSteps'
— Number of steps to take1
(default)  positive integerNumber of steps to take, specified as the commaseparated pair
consisting of 'NSteps'
and a positive integer.
Data Types: single
 double
'PEnter'
— Improvement measure for adding termscalar valueImprovement measure for adding a term, specified as the commaseparated
pair consisting of 'PEnter'
and a scalar value.
The default values are below.
Criterion  Default value  Decision 

'Deviance'  0.05  If the pvalue of F or
chisquared statistic is smaller than PEnter , add
the term to the model. 
'SSE'  0.05  If the SSE of the model is smaller than PEnter ,
add the term to the model. 
'AIC'  0  If the change in the AIC of the model is smaller than PEnter ,
add the term to the model. 
'BIC'  0  If the change in the BIC of the model is smaller than PEnter ,
add the term to the model. 
'Rsquared'  0.1  If the increase in the Rsquared of the model is larger than PEnter ,
add the term to the model. 
'AdjRsquared'  0  If the increase in the adjusted Rsquared of the model is larger
than PEnter , add the term to the model. 
For more information on the criteria, see Criterion
namevalue
pair argument.
Example: 'PEnter',0.075
'PredictorVars'
— Predictor variablescell array of strings  logical or numeric index vectorPredictor variables to use in the fit, specified as the commaseparated
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'
namevalue
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
'PRemove'
— Improvement measure for removing termscalar valueImprovement measure for removing a term, specified as the commaseparated
pair consisting of 'PRemove'
and a scalar value.
Criterion  Default value  Decision 

'Deviance'  0.10  If the pvalue of F or
chisquared statistic is larger than PRemove , remove
the term from the model. 
'SSE'  0.10  If the pvalue of the F statistic is larger
than PRemove , remove the term from the model. 
'AIC'  0.01  If the change in the AIC of the model is larger than PRemove ,
remove the term from the model. 
'BIC'  0.01  If the change in the BIC of the model is larger than PRemove ,
remove the term from the model. 
'Rsquared'  0.05  If the increase in the Rsquared value of the model is smaller
than PRemove , remove the term from the model. 
'AdjRsquared'  0.05  If the increase in the adjusted Rsquared value of the model
is smaller than PRemove , remove the term from the
model. 
At each step, stepwise algorithm also checks whether any term is redundant (linearly dependent) with other terms in the current model. When any term is linearly dependent with other terms in the current model, it is removed, regardless of the criterion value.
For more information on the criteria, see Criterion
namevalue
pair argument.
Example: 'PRemove',0.05
'ResponseVar'
— Response variablelast column in tbl
(default)  string for variable name  logical or numeric index vectorResponse variable to use in the fit, specified as the commaseparated
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
'Upper'
— Model specification describing largest set of terms in fit'interaction'
(default)  stringModel specification describing the largest set of terms in the
fit, specified as the commaseparated pair consisting of 'Upper'
and
one of the string options for modelspec
naming
the model.
Example: 'Upper','quadratic'
'VarNames'
— Names of variables in fit{'x1','x2',...,'xn','y'}
(default)  cell array of stringsNames of variables in fit, specified as the commaseparated
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
'Verbose'
— Control for display of information1
(default)  0
 2
Control for display of information, specified as the commaseparated
pair consisting of 'Verbose'
and one of the following:
0
— Suppress all display.
1
— Display the action taken
at each step.
2
— Also display the actions
evaluated at each step.
Example: 'Verbose',2
'Weights'
— Observation weightsones(n,1)
(default)  nby1 vector of nonnegative scalar valuesObservation weights, specified as the commaseparated pair consisting
of 'Weights'
and an nby1 vector
of nonnegative scalar values, where n is the number
of observations.
Data Types: single
 double
mdl
— Linear modelLinearModel
objectLinear model representing a leastsquares fit of the response
to the data, returned as a LinearModel
object.
For the properties and methods of the linear model object, mdl
,
see the LinearModel
class
page.
You can construct a model using fitlm
,
and then manually adjust the model using step
, addTerms
, or removeTerms
.
A terms matrix 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 plus one is for the response variable.
The value of T(i,j)
is the exponent of variable j
in
term i
. Suppose there are three predictor variables A
, B
,
and C
:
[0 0 0 0] % Constant term or intercept [0 1 0 0] % B; equivalently, A^0 * B^1 * C^0 [1 0 1 0] % A*C [2 0 0 0] % A^2 [0 1 2 0] % B*(C^2)
0
at the
end of each term represents the response variable. In general,If you have the variables in a table or dataset array,
then 0
must represent the response variable depending
on the position of the response variable. The following example illustrates
this.
Load the sample data and define the dataset array.
load hospital ds = dataset(hospital.Sex,hospital.BloodPressure(:,1),hospital.Age,... hospital.Smoker,'VarNames',{'Sex','BloodPressure','Age','Smoker'});
Represent the linear model 'BloodPressure ~ 1 + Sex
+ Age + Smoker'
in a terms matrix. The response variable
is in the second column of the dataset array, so there must be a column
of 0s for the response variable in the second column of the terms
matrix.
T = [0 0 0 0;1 0 0 0;0 0 1 0;0 0 0 1]
T = 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 1
Redefine the dataset array.
ds = dataset(hospital.BloodPressure(:,1),hospital.Sex,hospital.Age,... hospital.Smoker,'VarNames',{'BloodPressure','Sex','Age','Smoker'});
Now, the response variable is the first term in the dataset
array. Specify the same linear model, 'BloodPressure ~ 1
+ Sex + Age + Smoker'
, using a terms matrix.
T = [0 0 0 0;0 1 0 0;0 0 1 0;0 0 0 1]
T = 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1
If you have the predictor and response variables in
a matrix and column vector, then you must include 0
for
the response variable at the end of each term. The following example
illustrates this.
Load the sample data and define the matrix of predictors.
load carsmall
X = [Acceleration,Weight];
Specify the model 'MPG ~ Acceleration + Weight + Acceleration:Weight
+ Weight^2'
using a term matrix and fit the model to the
data. This model includes the main effect and twoway interaction
terms for the variables, Acceleration
and Weight
,
and a secondorder term for the variable, Weight
.
T = [0 0 0;1 0 0;0 1 0;1 1 0;0 2 0]
T = 0 0 0 1 0 0 0 1 0 1 1 0 0 2 0
Fit a linear model.
mdl = fitlm(X,MPG,T)
mdl = 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 9.7518e07 7.5389e07 1.2935 0.19917 Number of observations: 94, Error degrees of freedom: 89 Root Mean Squared Error: 4.1 Rsquared: 0.751, Adjusted RSquared 0.739 Fstatistic vs. constant model: 67, pvalue = 4.99e26
Only the intercept and x2
term, which correspond
to the Weight
variable, are significant at the
5% significance level.
Now, perform a stepwise regression with a constant model as the starting model and a linear model with interactions as the upper model.
T = [0 0 0;1 0 0;0 1 0;1 1 0];
mdl = stepwiselm(X,MPG,[0 0 0],'upper',T)
1. Adding x2, FStat = 259.3087, pValue = 1.643351e28 mdl = Linear regression model: y ~ 1 + x2 Estimated Coefficients: Estimate SE tStat pValue (Intercept) 49.238 1.6411 30.002 2.7015e49 x2 0.0086119 0.0005348 16.103 1.6434e28 Number of observations: 94, Error degrees of freedom: 92 Root Mean Squared Error: 4.13 Rsquared: 0.738, Adjusted RSquared 0.735 Fstatistic vs. constant model: 259, pvalue = 1.64e28
The results of the stepwise regression are consistent with the
results of fitlm
in the previous step.
A formula for model specification is a string
of the form '
Y
~ terms
'
where
Y
is the response name.
terms
contains
Variable names
+
means include the next variable

means do not include the next
variable
:
defines an interaction, a product
of terms
*
defines an interaction and all lowerorder terms
^
raises the predictor to a power,
exactly as in *
repeated, so ^
includes
lower order terms as well
()
groups terms
Note:
Formulas include a constant (intercept) term by default. To
exclude a constant term from the model, include 
For example,
'Y ~ A + B + C'
means a threevariable
linear model with intercept.'Y ~ A + B +
C  1'
is a threevariable linear model without intercept.'Y ~ A + B + C + B^2'
is a threevariable
model with intercept and a B^2
term.'Y
~ A + B^2 + C'
is the same as the previous example because B^2
includes
a B
term.'Y ~ A + B +
C + A:B'
includes an A*B
term.'Y
~ A*B + C'
is the same as the previous example because A*B
= A + B + A:B
.'Y ~ A*B*C  A:B:C'
has
all interactions among A
, B
,
and C
, except the threeway interaction.'Y
~ A*(B + C + D)'
has all linear terms, plus products of A
with
each of the other variables.
Wilkinson notation describes the factors present in models. The notation relates to factors present in models, not to the multipliers (coefficients) of those factors.
Wilkinson Notation  Factors in Standard Notation 

1  Constant (intercept) term 
A^k , where k is a positive
integer  A , A^{2} ,
..., A^{k} 
A + B  A , B 
A*B  A , B , A*B 
A:B  A*B only 
B  Do not include B 
A*B + C  A , B , C , A*B 
A + B + C + A:B  A , B , C , A*B 
A*B*C  A:B:C  A , B , C , A*B , A*C , B*C 
A*(B + C)  A , B , C , A*B , A*C 
Statistics and Machine Learning Toolbox™ notation always includes a constant term
unless you explicitly remove the term using 1
.
You cannot use robust regression with stepwise regression.
Check your data for outliers before using stepwiselm
.
For other methods such as anova
,
or properties of the LinearModel
object, see LinearModel
.
Stepwise regression is a systematic method
for adding and removing terms from a linear or generalized linear
model based on their statistical significance in explaining the response
variable. The method begins with an initial model, specified using modelspec
,
and then compares the explanatory power of incrementally larger and
smaller models.
MATLAB^{®} uses forward and backward stepwise regression to
determine a final model. At each step, the method searches for terms
to add to or remove from the model based on the value of the 'Criterion'
argument.
The default value of 'Criterion'
is 'sse'
,
and in this case, stepwiselm
uses the pvalue
of an Fstatistic to test models with and without
a potential term at each step. If a term is not currently in the model,
the null hypothesis is that the term would have a zero coefficient
if added to the model. If there is sufficient evidence to reject the
null hypothesis, the term is added to the model. Conversely, if a
term is currently in the model, the null hypothesis is that the term
has a zero coefficient. If there is insufficient evidence to reject
the null hypothesis, the term is removed from the model.
Here is how stepwise proceeds when 'Criterion'
is 'sse'
:
Fit the initial model.
If any terms not in the model have pvalues less than an entrance tolerance (that is, if it is unlikely that they would have zero coefficient if added to the model), add the one with the smallest pvalue and repeat this step; otherwise, go to step 3.
If any terms in the model have pvalues greater than an exit tolerance (that is, the hypothesis of a zero coefficient can be rejected), remove the one with the largest pvalue and go to step 2; otherwise, end.
The default for stepwiseglm
is 'Deviance'
and
it follows a similar procedure for adding or removing terms.
There are several other criteria available, which you can specify
using the 'Criterion'
argument. You can use the
change in the value of the Akaike information criterion, Bayesian
information criterion, Rsquared, adjusted Rsquared as a criterion
to add or remove terms.
Depending on the terms included in the initial model and the order in which terms are moved in and out, the method might build different models from the same set of potential terms. The method terminates when no single step improves the model. There is no guarantee, however, that a different initial model or a different sequence of steps will not lead to a better fit. In this sense, stepwise models are locally optimal, but might not be globally optimal.
fitlm
 LinearModel
 step