MATLAB: What is the regress function doing

regressregression

I don't understand what the regress function is returning. My understanding that it should be the gradient of the line of best fit. For example:

x = [1:5]'
y = [2 4 6 8 10]'
b = regress(y, x)

returns b = 2 as expected, whereas:

x = [1:5]'
y = [5 4 3 2 1]'
b = regress(y, x)

returns b = 0.6364 as opposed to -1.

What is the regress function calculating in this case? Where does this value for b come from?

Best Answer

The reason the first data set returned the slope you expected is that the intercept was zero, and your design forced a zero intercept. The reason the second data set is not returning -1 for the slope is because you are forcing it to have an intercept at zero.

If you include an intercept term, you get the expected -1 slope with a y-intercept of +6:

x = [1:5]'
y = [5 4 3 2 1]'
b = regress(y, [x ones(size(x))])
b =
           -1
            6

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MATLAB: How does MATLAB calculate standard error in ‘fitlm’

The following code can be used to calculate the standard error on mock data both in the case that the intercept is set to 'false' and when an intercept is present:

%%Sample Data Definition
X = [1 2 3 4 5 6 7 8 9 10];
Y = [1 1.5 2.1 4 5 6.3 7.3 7.8 9 9.9];
%%Calculation of Standard Error Without Intercept
n = length(X);                                       %  Number of observations

Slope = sum(X.*Y)/sum(X.^2);                         %  Calculates slope
yfit=X*Slope;                                        %  Fitted response values based on the slope

r = Y - yfit;                                        %  r is the residuals, which is the observed minus fitted values

SSE = sum(r.^2);                                     %  SSE is the sum of squared errors

MSE=SSE/(n-1);                                       %  Mean Squared Error 

Code_NoIntercept_SE=sqrt(MSE/sum(X.^2))              %  Standard error of the slope
model1 = fitlm(X,Y,'Intercept',false);               %  Calls 'fitlm'

MATLAB_NoIntercept_SE=model1.Coefficients.SE         %  Queries the standard error of MATLAB's model

%%Clear Variables then Redefine Data
clear all
X = [1 2 3 4 5 6 7 8 9 10];
Y = [1 1.5 2.1 4 5 6.3 7.3 7.8 9 9.9];
%%Calculation of Standard Error With Intercept
n = length(X);                                      %  Number of observations
XBar=mean(X);                                       %  Calculates mean of X
YBar=mean(Y);                                       %  Calculates mean of Y
Sxx=sum((X-XBar).^2);  
Sxy=sum((X-XBar).*(Y-YBar));
Slope = Sxy/Sxx;                                    %  Calculates Slope
Intercept= YBar-Slope*XBar;                         %  Calculates Intercept
yfit=Intercept + X*Slope;                           %  Fitted response values based on the slope
r = Y - yfit;                                       %  r is the residuals, which is the observed minus fitted values
SSE = sum(r.^2);                                    %  SSE is the sum of squared errors
MSE=SSE/(n-2);                                      %  Mean Squared Error 
Code_Intercept_SE=[                                 %  Standard Error of the regression coefficients
    sqrt(MSE*sum(X.^2)/(n*Sxx));                    %  Standard Error of the intercept coefficient
    sqrt(MSE/Sxx)]                                  %  Standard Error of the slope coefficient
model2 = fitlm(X,Y,'Intercept',true);               %  Calls 'fitlm'
MATLAB_Intercept_SE=model2.Coefficients.SE          %  Queries the standard error of MATLAB's model

MATLAB: How to force the intercept of a regression line to zero

There are 2 main ways you can do this:

dlm = fitlm(X,y,'Intercept',false);

or using Wilkinson notation:

dlm = fitlm(X,y,'y~x1-1');

I would highly suggest learning the Wilkinson notation, as this allows you to fit models and specify the form of the equation you would like to fit.

Best Answer

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MATLAB: How does MATLAB calculate standard error in ‘fitlm’

MATLAB: How to force the intercept of a regression line to zero

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