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This example shows how to fit a linear regression model. A typical workflow involves the following: import data, fit a regression, test its quality, modify it to improve the quality, and share it.

`hospital.xls`

is an Excel® spreadsheet containing patient names, sex, age, weight, blood pressure, and dates of treatment in an experimental protocol. First read the data into a table.

patients = readtable('hospital.xls','ReadRowNames',true);

Examine the five rows of data.

patients(1:5,:)

`ans=`*5×11 table*
name sex age wgt smoke sys dia trial1 trial2 trial3 trial4
____________ _____ ___ ___ _____ ___ ___ ______ ______ ______ ______
YPL-320 {'SMITH' } {'m'} 38 176 1 124 93 18 -99 -99 -99
GLI-532 {'JOHNSON' } {'m'} 43 163 0 109 77 11 13 22 -99
PNI-258 {'WILLIAMS'} {'f'} 38 131 0 125 83 -99 -99 -99 -99
MIJ-579 {'JONES' } {'f'} 40 133 0 117 75 6 12 -99 -99
XLK-030 {'BROWN' } {'f'} 49 119 0 122 80 14 23 -99 -99

The `sex`

and `smoke`

fields seem to have two choices each. So change these fields to categorical.

patients.smoke = categorical(patients.smoke,0:1,{'No','Yes'}); patients.sex = categorical(patients.sex);

Your goal is to model the systolic pressure as a function of a patient's age, weight, sex, and smoking status. Create a linear formula for `'sys'`

as a function of `'age'`

, `'wgt'`

, `'sex'`

, and `'smoke'`

.

```
modelspec = 'sys ~ age + wgt + sex + smoke';
mdl = fitlm(patients,modelspec)
```

mdl = Linear regression model: sys ~ 1 + sex + age + wgt + smoke Estimated Coefficients: Estimate SE tStat pValue _________ ________ ________ __________ (Intercept) 118.28 7.6291 15.504 9.1557e-28 sex_m 0.88162 2.9473 0.29913 0.76549 age 0.08602 0.06731 1.278 0.20438 wgt -0.016685 0.055714 -0.29947 0.76524 smoke_Yes 9.884 1.0406 9.498 1.9546e-15 Number of observations: 100, Error degrees of freedom: 95 Root Mean Squared Error: 4.81 R-squared: 0.508, Adjusted R-Squared: 0.487 F-statistic vs. constant model: 24.5, p-value = 5.99e-14

The sex, age, and weight predictors have rather high $$p$$-values, indicating that some of these predictors might be unnecessary.

See if there are outliers in the data that should be excluded from the fit. Plot the residuals.

plotResiduals(mdl)

There is one possible outlier, with a value greater than 12. This is probably not truly an outlier. For demonstration, here is how to find and remove it.

Find the outlier.

outlier = mdl.Residuals.Raw > 12; find(outlier)

ans = 84

Remove the outlier.

mdl = fitlm(patients,modelspec,... 'Exclude',84); mdl.ObservationInfo(84,:)

`ans=`*1×4 table*
Weights Excluded Missing Subset
_______ ________ _______ ______
WXM-486 1 true false false

Observation 84 is no longer in the model.

Try to obtain a simpler model, one with fewer predictors but the same predictive accuracy. `step`

looks for a better model by adding or removing one term at a time. Allow `step`

take up to 10 steps.

`mdl1 = step(mdl,'NSteps',10)`

1. Removing wgt, FStat = 4.6001e-05, pValue = 0.9946 2. Removing sex, FStat = 0.063241, pValue = 0.80199

mdl1 = Linear regression model: sys ~ 1 + age + smoke Estimated Coefficients: Estimate SE tStat pValue ________ ________ ______ __________ (Intercept) 115.11 2.5364 45.383 1.1407e-66 age 0.10782 0.064844 1.6628 0.09962 smoke_Yes 10.054 0.97696 10.291 3.5276e-17 Number of observations: 99, Error degrees of freedom: 96 Root Mean Squared Error: 4.61 R-squared: 0.536, Adjusted R-Squared: 0.526 F-statistic vs. constant model: 55.4, p-value = 1.02e-16

`step`

took two steps. This means it could not improve the model further by adding or subtracting a single term.

Plot the effectiveness of the simpler model on the training data.

plotResiduals(mdl1)

The residuals look about as small as those of the original model.

Suppose you have four new people, aged 25, 30, 40, and 65, and the first and third smoke. Predict their systolic pressure using `mdl1`

.

ages = [25;30;40;65]; smoker = {'Yes';'No';'Yes';'No'}; systolicnew = feval(mdl1,ages,smoker)

`systolicnew = `*4×1*
127.8561
118.3412
129.4734
122.1149

To make predictions, you need only the variables that `mdl1`

uses.

You might want others to be able to use your model for prediction. Access the terms in the linear model.

coefnames = mdl1.CoefficientNames

`coefnames = `*1x3 cell*
{'(Intercept)'} {'age'} {'smoke_Yes'}

View the model formula.

mdl1.Formula

ans = sys ~ 1 + age + smoke

Access the coefficients of the terms.

coefvals = mdl1.Coefficients(:,1).Estimate

`coefvals = `*3×1*
115.1066
0.1078
10.0540

The model is `sys = 115.1066 + 0.1078*age + 10.0540*smoke`

, where `smoke`

is `1`

for a smoker, and `0`

otherwise.

`fitlm`

| `LinearModel`

| `feval`

| `step`

| `plotResiduals`