# simulateLM

simulateLM simulates linear regression data with prespecified values of statistical indexes.

## Syntax

• y=simulateLM(n)example
• y=simulateLM(n,Name,Value)example
• [y, X]=simulateLM(___)example

## Description

simulateLM simulates linear regression data. It is possible to specify:

1) the requested value of R2;

2) the values of the beta coefficients;

3) the correlation (covariance) matrix among the explanatory variables.

4) the value of the intercept term.

5) the distribution to use to generate the Xs;

6) the distribution to use to generate the ys.

 y =simulateLM(n) Use all defaul options.

 y =simulateLM(n, Name, Value) Simulate with prefixed value of R2.

 [y, X] =simulateLM(___) Use prefixed correlation matrix for cov(X).

## Examples

expand all

### Use all defaul options.

Simulate 100 observations y and X (uncorrelated with y) using standard normal distribution.

[y,X]=simulateLM(100,'plots',true); ### Simulate with prefixed value of R2.

Set value of R2;

R2=0.82;
n=10000;
[y,X]=simulateLM(n,'R2',R2);
out=fitlm(X,y);

### Use prefixed correlation matrix for cov(X).

Set value of R2;

R2=0.26;
n=10000;
A = gallery('moler',5,0.2);
[y,X]=simulateLM(n,'R2',R2,'SigmaX',A);
out=fitlm(X,y)
out =

Linear regression model:
y ~ 1 + x1 + x2 + x3 + x4 + x5

Estimated Coefficients:
Estimate        SE        tStat       pValue
_________    ________    _______    __________

(Intercept)    -0.076653    0.053898    -1.4222       0.15501
x1                1.0515    0.056414     18.638    3.0647e-76
x2               0.92447    0.056001     16.508    2.0145e-60
x3                1.0394    0.055297     18.797      1.72e-77
x4               0.92012    0.055248     16.654    1.8903e-61
x5                 1.029    0.054653     18.828    9.8022e-78

Number of observations: 10000, Error degrees of freedom: 9994
Root Mean Squared Error: 5.39
F-statistic vs. constant model: 679, p-value = 0


## Related Examples

expand all

### Use prefixed values of R2, beta and intercept.

Set value of R2.

R2=0.92;
beta=[3; 4; 5; 2; 7];
intercept=43;
n=100000;
[y,X]=simulateLM(n,'R2',R2,'beta',beta);
out=fitlm(X,y);

### Sim study.

Compare the distribution of values of R2 with data generated from Normal with those generated from Student T with 5 degrees of freedom.

% Set value of R2.
R2=0.92;
beta=[3; 4; 5; 2; 7; 2; 3];
nsimul=1000;
R2all=zeros(nsimul,2);
n=100;
df=5;
for j=1:nsimul
% Data generated from Normal
[y,X]=simulateLM(n,'R2',R2,'beta',beta);
out=fitlm(X,y);
R2all(j,1)=out.Rsquared.Ordinary;
% Data generated from T(5)
[y,X]=simulateLM(n,'R2',R2,'beta',beta,'distriby','T','distribypars',df);
out=fitlm(X,y);
R2all(j,2)=out.Rsquared.Ordinary;
end
boxplot(R2all,'Labels',{'Normal', 'T(5)'});

## Input Arguments

### n — sample size. Scalar.

n is a positive integer which defines the length of the simulated data. For example if n=100, y will be 100x1 and X will be 100xp.

Data Types: single| double

### 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:  'R2',0.90 , 'beta',[3 5 8] , 'Sigma', gallery('lehmer',5) , 'distribX', 'Beta' , 'distribXpars', '[0.2 0.6]' , 'distriby', 'Lognormal' , 'distribypars', '[2 10]' , 'distribypars', '[2 10]' , 'intercept', '10' , 'plots',false 

### R2 —Squared multiple correlation coefficient (R2).scalar.

The requested value of R2. A number in the interval [0 1] which specifies the requested value of R2.

The default is to simulate regression data with R2=0;

Example:  'R2',0.90 

Data Types: double

### beta —the values of the beta coefficients.vector.

Vector which contains the values of the regression coefficients. The default is a vector of ones.

Example:  'beta',[3 5 8] 

Data Types: double

### SigmaX —the correlation matrix.matrix.

Positive definite matrix which contains the correlation matrix among regressors. The default is the identity matrix.

Example:  'Sigma', gallery('lehmer',5) 

Data Types: double

### distribX —distribution to use to simulate the regressors.character.

Character which specifies the distribution to use to simulate the values of the explanatory variables.

For the list of valid names see MATLAB function random.

Default is to use the Standard normal distribution.

Example:  'distribX', 'Beta' 

Data Types: double

### distribXpars —parameters of the distribution to use in distribX.vector.

Scalar value or array of scalar values containing the distribution parameters specified in distribX.

Example:  'distribXpars', '[0.2 0.6]' 

Data Types: double

### distriby —distribution to use to simulate the response.character.

Character which specifies the distribution to use to simulate the values of the explanatory variables. The default is to use the Standard normal distribution.

Example:  'distriby', 'Lognormal' 

Data Types: double

### distribypars —parameters of the distribution to use in distriby.vector.

Scalar value or array of scalar values containing the distribution parameters specified in distriby. For examples if distriby is 'Lognormal' and 'distribypars' is [2 10], the errors are generated according to a Log Normal distribution with parameters mu and sigma respectively equal to 2 and 10.

Example:  'distribypars', '[2 10]' 

Data Types: double

### nexpl —number of explanatory varibles.if vector beta is supplied nexpl is equal to length(beta).

Similarly if sigmaX is supplied nexpl is set equal to size(sigmaX,1).

Note that both nexpl is supplied together with beta and SigmaX it is check that nexpl =length(beta) = size(SigmaX,1). If options beta and sigmaX are empty nexpl is set equal to 3.

Example:  'distribypars', '[2 10]' 

Data Types: double

### intercept —value of the intercept to use.scalar.

The default value for intercept is 0.

Example:  'intercept', '10' 

Data Types: double

### plots —Plot on the screen.boolean.

If plots = true, the yXplot which shows the response against all the explanatory variables s shown on the screen. The default value for plots is false, that is no plot is shown on the screen.

Example:  'plots',false 

Data Types: single | double

## Output Arguments

### y —simulated response.  Vector

Column vector of length n containing the response.

### X —simulated regressors.  Matrix

Matrix of size n-times-nexpl containing the values of the regressors.