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## Examining Residuals for Model Verification

You can examine the `stats` structure, which is returned by both `nlmefit` and `nlmefitsa`, to determine the quality of your model. The `stats` structure contains fields with conditional weighted residuals (`cwres` field) and individual weighted residuals (`iwres` field). Since the model assumes that residuals are normally distributed, you can examine the residuals to see how well this assumption holds.

This example generates synthetic data using normal distributions. It shows how the fit statistics look:

• Good when testing against the same type of model as generates the data

• Poor when tested against incorrect data models

1. Initialize a 2-D model with 100 individuals:

```nGroups = 100; % 100 Individuals nlmefun = @(PHI,t)(PHI(:,1)*5 + PHI(:,2)^2.*t); % Regression fcn REParamsSelect = [1 2]; % Both Parameters have random effect errorParam = .03; beta0 = [ 1.5 5]; % Parameter means psi = [ 0.35 0; ... % Covariance Matrix 0 0.51 ]; time =[0.25;0.5;0.75;1;1.25;2;3;4;5;6]; nParameters = 2; rng(0,'twister') % for reproducibility```
2. Generate the data for fitting with a proportional error model:

```b_i = mvnrnd(zeros(1, numel(REParamsSelect)), psi, nGroups); individualParameters = zeros(nGroups,nParameters); individualParameters(:, REParamsSelect) = ... bsxfun(@plus,beta0(REParamsSelect), b_i); groups = repmat(1:nGroups,numel(time),1); groups = vertcat(groups(:)); y = zeros(numel(time)*nGroups,1); x = zeros(numel(time)*nGroups,1); for i = 1:nGroups idx = groups == i; f = nlmefun(individualParameters(i,:), time); % Make a proportional error model for y: y(idx) = f + errorParam*f.*randn(numel(f),1); x(idx) = time; end P = [ 1 0 ; 0 1 ];```
3. Fit the data using the same regression function and error model as the model generator:

```[~,~,stats] = nlmefit(x,y,groups, ... [],nlmefun,[1 1],'REParamsSelect',REParamsSelect,... 'ErrorModel','Proportional','CovPattern',P);```
4. Create a plotting routine by copying the following function definition, and creating a file `plotResiduals.m` on your MATLAB® path:

```function plotResiduals(stats) pwres = stats.pwres; iwres = stats.iwres; cwres = stats.cwres; figure subplot(2,3,1); normplot(pwres); title('PWRES') subplot(2,3,4); createhistplot(pwres); subplot(2,3,2); normplot(cwres); title('CWRES') subplot(2,3,5); createhistplot(cwres); subplot(2,3,3); normplot(iwres); title('IWRES') subplot(2,3,6); createhistplot(iwres); title('IWRES') function createhistplot(pwres) h = histogram(pwres); % x is the probability/height for each bin x = h.Values/sum(h.Values*h.BinWidth) % n is the center of each bin n = h.BinEdges + (0.5*h.BinWidth) n(end) = []; bar(n,x); ylim([0 max(x)*1.05]); hold on; x2 = -4:0.1:4; f2 = normpdf(x2,0,1); plot(x2,f2,'r'); end end```
5. Plot the residuals using the `plotResiduals` function:

`plotResiduals(stats);`

The upper probability plots look straight, meaning the residuals are normally distributed. The bottom histogram plots match the superimposed normal density plot. So you can conclude that the error model matches the data.

6. For comparison, fit the data using a constant error model, instead of the proportional model that created the data:

```[~,~,stats] = nlmefit(x,y,groups, ... [],nlmefun,[0 0],'REParamsSelect',REParamsSelect,... 'ErrorModel','Constant','CovPattern',P); plotResiduals(stats);```

The upper probability plots are not straight, indicating the residuals are not normally distributed. The bottom histogram plots are fairly close to the superimposed normal density plots.

7. For another comparison, fit the data to a different structural model than the one that created the data:

```nlmefun2 = @(PHI,t)(PHI(:,1)*5 + PHI(:,2).*t.^4); [~,~,stats] = nlmefit(x,y,groups, ... [],nlmefun2,[0 0],'REParamsSelect',REParamsSelect,... 'ErrorModel','constant', 'CovPattern',P); plotResiduals(stats);```

The upper probability plots are not straight. Also, the histogram plots are quite skewed compared to the superimposed normal density plots. These residuals are not normally distributed, and do not match the model.

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