MATLAB: Where is the coefficient for the reference condition when using ‘fitlm’ to perform ANOVA with intercept omitted

anovafitlminterceptMATLABmissing group

I want to compare the means for a variable among five different groups. To do so, I use fitlm to perform an ANOVA. The following code provides an example that closely resembles my own data:

% simulate data
x = repmat([1:5]',[10,1]);
y = (x-4).^2 + randn([size(x),1]);
grp = categorical(x);
% fit model
fitlm(grp,y)

This yields an intercept, which represents the mean for grp 1, and coefficients for the other groups that represent the difference of the mean for those groups relative to the intercept.

I am unsure (and would also like to know) how to then calculate the standard error (SE) of the group means from the combination of the SE for the intercept and SE for the coefficients. I therefore tried omitting the intercept, expecting coefficients for each group

fitlm(grp,y,'Intercept',false)

Here there is indeed no intercept, but also no coefficient for group 1. Where did it go?

Aside, as a workaround, I tried dummy coding the group variable

grp_dummy = dummyvar(grp);
fitlm(grp_dummy,y,'Intercept',false)

This gives me the expected result, but the SE correspond to the SE of the intercept in the first model. How then do the SE of the coefficients of the first model fit in with this latter model?

Best Answer

Why not just use this?

for iGrp=1:5
    stderr = std(y(x==iGrp)) / sqrt( sum(x==iGrp) )
end

Related Solutions

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: ANOVA Cofficients: anovan stats.coeffs

More information about the n-way anova model:

http://www.mathworks.com/help/stats/anova.html#bqttd3j-1

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