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Create linear regression model using stepwise regression
mdl = stepwiselm(tbl,modelspec) returns a linear model for the variables in the table or dataset array 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.
mdl = stepwiselm(X,y,modelspec) creates a linear model of the responses y to the predictor variables in the data matrix X, using stepwise regression to add or remove predictors. modelspec is the starting model for the stepwise procedure.
mdl = stepwiselm(___,Name,Value) creates a linear model for any of the inputs in the previous syntaxes, with additional options specified by one or more 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 = 1.105281e-06 3. Adding x2, FStat = 5.0259, pValue = 0.051687 4. Removing x4, FStat = 1.8633, pValue = 0.2054 mdl = Linear regression model: y ~ 1 + x1 + x2 Estimated Coefficients: Estimate SE tStat pValue (Intercept) 52.577 2.2862 22.998 5.4566e-10 x1 1.4683 0.1213 12.105 2.6922e-07 x2 0.66225 0.045855 14.442 5.029e-08 Number of observations: 13, Error degrees of freedom: 10 Root Mean Squared Error: 2.41 R-squared: 0.979, Adjusted R-Squared 0.974 F-statistic vs. constant model: 230, p-value = 4.41e-09
By default, the starting model is the constant model. stepwiselm performs forward selection and x4, x1, and x2, respectively, as the corresponding p-values 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 p-value 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 = 6.262758e-48 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 5.2762e-104 Sex_Male 50.06 1.7496 28.612 2.2464e-49 Number of observations: 100, Error degrees of freedom: 98 Root Mean Squared Error: 8.73 R-squared: 0.893, Adjusted R-Squared 0.892 F-statistic vs. constant model: 819, p-value = 2.25e-49
At each step, stepwiselm searches for terms to add and remove. At first step, stepwise algorithm adds Sex to the model with a p-value of 6.26e-48. 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.
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
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
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
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. |
'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. |
If you want to specify the smallest or largest set of terms in the model, use the Lower and Upper name-value pair arguments.
t-by-(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 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: '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.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
Criterion to add or remove terms, specified as the comma-separated pair consisting of 'Criterion' and one of the following:
'Deviance' — Default for stepwiseglm. p-value for F or chi-squared test of the change in the deviance by adding or removing the term. F-test is for testing a single model. Chi-squared test is for comparing two different models. This option is not valid for stepwiselm.
'sse' — Default for stepwiselm. p-value for an F-test 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'
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
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
Model specification describing terms that cannot be removed from the model, specified as the comma-separated pair consisting of 'Lower' and one of the string options for modelspec naming the model.
Example: 'Lower','linear'
Number of steps to take, specified as the comma-separated pair consisting of 'NSteps' and a positive integer.
Data Types: single | double
Improvement measure for adding a term, specified as the comma-separated pair consisting of 'PEnter' and a scalar value. The default values are below.
Criterion | Default value | Decision |
---|---|---|
'Deviance' | 0.05 | If the p-value of F or chi-squared 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 R-squared of the model is larger than PEnter, add the term to the model. |
'AdjRsquared' | 0 | If the increase in the adjusted R-squared of the model is larger than PEnter, add the term to the model. |
For more information on the criteria, see Criterion name-value pair argument.
Example: 'PEnter',0.075
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
Improvement measure for removing a term, specified as the comma-separated pair consisting of 'PRemove' and a scalar value.
Criterion | Default value | Decision |
---|---|---|
'Deviance' | 0.10 | If the p-value of F or chi-squared statistic is larger than PRemove, remove the term from the model. |
'SSE' | 0.10 | If the p-value 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 R-squared value of the model is smaller than PRemove, remove the term from the model. |
'AdjRsquared' | -0.05 | If the increase in the adjusted R-squared 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 name-value pair argument.
Example: 'PRemove',0.05
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
Model specification describing the largest set of terms in the fit, specified as the comma-separated pair consisting of 'Upper' and one of the string options for modelspec naming the model.
Example: 'Upper','quadratic'
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
Control for display of information, specified as the comma-separated 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
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
Linear model representing a least-squares 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 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 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)
The 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 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 = 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 9.7518e-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 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 = 1.643351e-28 mdl = Linear regression model: y ~ 1 + x2 Estimated Coefficients: Estimate SE tStat pValue (Intercept) 49.238 1.6411 30.002 2.7015e-49 x2 -0.0086119 0.0005348 -16.103 1.6434e-28 Number of observations: 94, Error degrees of freedom: 92 Root Mean Squared Error: 4.13 R-squared: 0.738, Adjusted R-Squared 0.735 F-statistic vs. constant model: 259, p-value = 1.64e-28
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 lower-order 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 -1 in the formula. |
For example,
'Y ~ A + B + C' means a three-variable
linear model with intercept.
'Y ~ A + B +
C - 1' is a three-variable linear model without intercept.
'Y ~ A + B + C + B^2' is a three-variable
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 three-way 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 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 p-value of an F-statistic 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 p-values 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 p-value and repeat this step; otherwise, go to step 3.
If any terms in the model have p-values greater than an exit tolerance (that is, the hypothesis of a zero coefficient can be rejected), remove the one with the largest p-value 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, R-squared, adjusted R-squared 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