How do I extend the confidence boundaries of fitlm? Also help with general confidence boundaries of Least Squares?

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Hugh Wheaton 2020-6-16

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回答： John D'Errico 2020-6-16

I am not as experienced with stats as I'd like to be, so unfortunately I don't know too much about fitting confidence bounds, how they are calculated and whether I'm after an observational/ functional or simultaneous/ non-simultaneous or not; I'd like to, but for now I really just need to get these confidence bounds on a plot I need for uni.

I can't figure out how to calculate them in time, nor can I seem to find any resources about the boundaries of a linear regression method in matlab, so I just want to ask here: How do I extend the bounds plotted using the fitlm() function?

I have a model, 'mdl3', which is superimposed on the data itself using "hold on" followed by "plot(mdl3)". The entire point of this regression model is specifically to measure the x-intercept, so although it is useful having the bounds there to graphically demonstrate the fit, it would be ideal if I could extrapolate them all the way to the edge(s) of the axis limit(s). I understand, from my recent scowering, that the default boundaries are likely "simultaneous"; meaning, from my vague understanding, this is not intended for extrapolation: In this case, would it be possible to just superimpose another, non-simultanous, confidence boundary?

P.S.: As you may have read, I also need results for the uncertainty of the x-intercept, and this is also an aspect I am rather light on (I had to drop out of first year stats due to illness and haven't managed to catch up yet). I am using the taylor series approximation, using the "*.CoefficientCovariance(1,2)" as the

value; if there is a better method/ inbuilt matlab function, I'd greatly appreciate that as well. Cheers!

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Answer 1

John D'Errico 2020-6-16

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You needed to scower more.

Plotting the result of a call to fitlm then uses predict to compute confidence intervals when plotted.
Reading the help for predict, by default the confidence intervals are NON-simultaneous.
If you look at the code for LinearModel/plot (do NOT edit the code, use type instead) you will see the call to predict did NOT specify a change from the default.

Oh. "Simultaneous" does not indicate whether extrapolation is intended. Extrapolation of confidence intervals is probably a really bad idea anyway you would do it.

>> N = 25;
>> x = rand(N,1);
>> y = randn(size(x)) + x;
>> mdl = fitlm(x,y)
mdl = 
Linear regression model:
    y ~ 1 + x1
Estimated Coefficients:
                       Estimate                SE                  tStat                pValue       
                   _________________    _________________    _________________    ___________________
    (Intercept)    -0.51960353138985    0.351018534063917    -1.48027377749585      0.152369194934559
    x1              1.76582214020889    0.620664611608085     2.84505046233232    0.00916753871041665
Number of observations: 25, Error degrees of freedom: 23
Root Mean Squared Error: 0.915
R-squared: 0.26,  Adjusted R-Squared: 0.228
F-statistic vs. constant model: 8.09, p-value = 0.00917
>> xci = linspace(-.5,1.5)';
>> [ypred,yci] = predict(mdl,xci,'simul',true);
>> plot(mdl)
>> hold on
>> plot(xci,yci,'g-')