How to find residual variance from fitlm

Hi, I was wondering how you read the fitlm output after using data to get the residual variance. I thought that it might be the R-squared output or the Adjusted R-squared but apparently that is not the case. here is an example with the output

%% data

x=[353.65 389.76 385.83 419.45 427.53 354.28 570.05 375.29 530.73 444.68 386.08 490.73 595.65 353.09 387.71 565.55 548.73 362.69 443.09 421.94 510.90 414.92 471.25 530.13 566.98 514.40 526.64 429.74 428.03 438.46 550.17 505.43 429.94 594.41 387.85 439.18 353.81 412.47 380.51 581.10 445.24 528.21 534.49 551.51 373.46 448.30 564.95 373.99 595.85 509.58 589.94 388.65 508.03 375.07 596.75 563.80 502.94 373.95 545.65 443.27];

y=[0.32432 0.05176 0.16317 0.45606 0.37908 0.40913 0.52598 0.24144 0.31293 0.32629 0.39778 0.21505 0.31383 0.45964 0.34407 0.55298 0.40881 0.24382 0.30078 0.26296 0.70163 0.17748 0.38921 0.41283 0.47194 0.64746 0.64146 0.25919 0.41283 0.15618 0.29238 0.61447 0.31118 0.53470 0.06382 0.26518 0.36523 0.31896 0.28402 0.68385 0.69655 0.45083 0.41407 0.45839 0.15570 0.59098 0.48110 0.08150 0.51529 0.29448 0.34008 0.16886 0.31172 0.41063 0.41504 0.54690 0.44654 0.02075 0.68367 0.35737];

%% fit lenear models

xy_lm=fitlm(x,y)

output:

xy_lm =

Linear regression model:

y ~ 1 + x1

Estimated Coefficients:

Estimate SE tStat pValue

_________ __________ _______ __________

(Intercept) -0.15828 0.10942 -1.4465 0.15341

x1 0.0011405 0.00023052 4.9473 6.7976e-06

Number of observations: 60, Error degrees of freedom: 58

Root Mean Squared Error: 0.141

R-squared: 0.297, Adjusted R-Squared: 0.285

F-statistic vs. constant model: 24.5, p-value = 6.8e-06

Answer 1

[Torsten]

%% data
x = [353.65 389.76 385.83 419.45 427.53 354.28 570.05 375.29 530.73 444.68 386.08 490.73 595.65 353.09 387.71 565.55 548.73 362.69 443.09 421.94 510.90 414.92 471.25 530.13 566.98 514.40 526.64 429.74 428.03 438.46 550.17 505.43 429.94 594.41 387.85 439.18 353.81 412.47 380.51 581.10 445.24 528.21 534.49 551.51 373.46 448.30 564.95 373.99 595.85 509.58 589.94 388.65 508.03 375.07 596.75 563.80 502.94 373.95 545.65 443.27];
y = [0.32432 0.05176 0.16317 0.45606 0.37908 0.40913 0.52598 0.24144 0.31293 0.32629 0.39778 0.21505 0.31383 0.45964 0.34407 0.55298 0.40881 0.24382 0.30078 0.26296 0.70163 0.17748 0.38921 0.41283 0.47194 0.64746 0.64146 0.25919 0.41283 0.15618 0.29238 0.61447 0.31118 0.53470 0.06382 0.26518 0.36523 0.31896 0.28402 0.68385 0.69655 0.45083 0.41407 0.45839 0.15570 0.59098 0.48110 0.08150 0.51529 0.29448 0.34008 0.16886 0.31172 0.41063 0.41504 0.54690 0.44654 0.02075 0.68367 0.35737];
[x,ix] = sort(x)

x = 1×60

353.0900 353.6500 353.8100 354.2800 362.6900 373.4600 373.9500 373.9900 375.0700 375.2900 380.5100 385.8300 386.0800 387.7100 387.8500 388.6500 389.7600 412.4700 414.9200 419.4500 421.9400 427.5300 428.0300 429.7400 429.9400 438.4600 439.1800 443.0900 443.2700 444.6800

ix = 1×60

14 1 37 6 18 45 58 48 54 8 39 3 11 15 35 52 2 38 22 4 20 5 29 28 33 30 36 19 60 10

y = y(ix)

y = 1×60

0.4596 0.3243 0.3652 0.4091 0.2438 0.1557 0.0208 0.0815 0.4106 0.2414 0.2840 0.1632 0.3978 0.3441 0.0638 0.1689 0.0518 0.3190 0.1775 0.4561 0.2630 0.3791 0.4128 0.2592 0.3112 0.1562 0.2652 0.3008 0.3574 0.3263

plot(x,y)

A = [ones(numel(x),1),x.'];
b = y.';
c = A\b;

ans = 60

format long
%Residual variance for linear regression model
var(y-(c(1)+c(2)*x))*(numel(x)-1)/(numel(x)-2)

ans = 
   0.019752198890146

%% fit linear model
xy_lm=fitlm(x,y)

xy_lm = 
Linear regression model:
    y ~ 1 + x1

Estimated Coefficients:
                        Estimate                   SE                   tStat                 pValue       
                   ___________________    ____________________    _________________    ____________________

    (Intercept)      -0.15828398677887        0.10942366710812    -1.44652423887853       0.153413107689524
    x1             0.00114045590675123    0.000230521588155199     4.94728461606562    6.79762178610595e-06


Number of observations: 60, Error degrees of freedom: 58
Root Mean Squared Error: 0.141
R-squared: 0.297,  Adjusted R-Squared: 0.285
F-statistic vs. constant model: 24.5, p-value = 6.8e-06

%Residual variance = MeanSq of Residual
anova(xy_lm,'summary')

ans = 3×5 table
                      SumSq          DF          MeanSq                 F                  pValue       
                _________________    __    __________________    ________________    ___________________

    Total        1.62907495001833    59    0.0276114398308192                                           
    Model       0.483447414389884     1     0.483447414389884    24.4756250723596    6.7976217861058e-06
    Residual     1.14562753562845    58    0.0197521988901457                                           

[Dylan]

Thanks Torsten, but what does that all mean?

[Torsten]

You wanted the residual variance - I gave you two ways to calculate it:

A = [ones(numel(x),1),x.'];
b = y.';
c = A\b;
format long
%Residual variance for linear regression model
var(y-(c(1)+c(2)*x))*(numel(x)-1)/(numel(x)-2)

or with fitlm

%% fit linear model
xy_lm=fitlm(x,y);
%Residual variance = MeanSq of Residual
anova(xy_lm,'summary')

[Dylan]

Ohhh thankyou! I didnt realise that the annova table gives you residual