# residuals

Residuals of fitted linear mixed-effects model

## Description

returns
the residuals from the linear mixed-effects model `R`

= residuals(`lme`

,`Name,Value`

)`lme`

with
additional options specified by one or more `Name,Value`

pair
arguments.

For example, you can specify Pearson or standardized residuals, or residuals with contributions from only fixed effects.

## Input Arguments

`lme`

— Linear mixed-effects model

`LinearMixedModel`

object

Linear mixed-effects model, specified as a `LinearMixedModel`

object constructed using `fitlme`

or `fitlmematrix`

.

### Name-Value Arguments

Specify optional pairs of arguments as
`Name1=Value1,...,NameN=ValueN`

, where `Name`

is
the argument name and `Value`

is the corresponding value.
Name-value arguments must appear after other arguments, but the order of the
pairs does not matter.

*
Before R2021a, use commas to separate each name and value, and enclose*
`Name`

*in quotes.*

`Conditional`

— Indicator for conditional residuals

`True`

(default) | `False`

Indicator for conditional residuals, specified as the comma-separated
pair consisting of `'Conditional'`

and one of the
following.

`True` | Contribution from both fixed effects and random effects (conditional) |

`False` | Contribution from only fixed effects (marginal) |

**Example: **`'Conditional,'False'`

`ResidualType`

— Residual type

`'Raw'`

(default) | `'Pearson'`

| `'Standardized'`

Residual type, specified by the comma-separated pair consisting
of `ResidualType`

and one of the following.

Residual Type | Conditional | Marginal |
---|---|---|

`'Raw'` |
$${r}_{i}^{C}={\left[y-X\widehat{\beta}-Z\widehat{b}\right]}_{i}$$ |
$${r}_{i}^{M}={\left[y-X\widehat{\beta}\right]}_{i}$$ |

`'Pearson'` |
$$p{r}_{i}^{C}=\frac{{r}_{i}^{C}}{{\sqrt{\left[{\widehat{Var}}_{y,b}\left(y-X\beta -Zb\right)\right]}}_{ii}}$$ |
$$p{r}_{i}^{M}=\frac{{r}_{i}^{M}}{\sqrt{{\left[{\widehat{Var}}_{y}\left(y-X\beta \right)\right]}_{ii}}}$$ |

`'Standardized'` |
$$s{t}_{i}^{C}=\frac{{r}_{i}^{C}}{\sqrt{{\left[{\widehat{Var}}_{y}\left({r}^{C}\right)\right]}_{ii}}}$$ |
$$s{t}_{i}^{M}=\frac{{r}_{i}^{M}}{\sqrt{{\left[{\widehat{Var}}_{y}\left({r}^{M}\right)\right]}_{ii}}}$$ |

For more information on the conditional and marginal residuals
and residual variances, see `Definitions`

at the
end of this page.

**Example: **`'ResidualType','Standardized'`

## Output Arguments

`R`

— Residuals

*n*-by-1 vector

Residuals of the fitted linear mixed-effects model `lme`

returned
as an *n*-by-1 vector, where *n* is
the number of observations.

## Examples

### Plot Residuals vs. Fitted Values

Load the sample data.

`load('weight.mat');`

`weight`

contains data from a longitudinal study, where 20 subjects are randomly assigned to 4 exercise programs, and their weight loss is recorded over six 2-week time periods. This is simulated data.

Store the data in a table. Define `Subject`

and `Program`

as categorical variables.

tbl = table(InitialWeight,Program,Subject,Week,y); tbl.Subject = nominal(tbl.Subject); tbl.Program = nominal(tbl.Program);

Fit a linear mixed-effects model where the initial weight, type of program, week, and the interaction between the week and type of program are the fixed effects. The intercept and week vary by subject.

`lme = fitlme(tbl,'y ~ InitialWeight + Program*Week + (Week|Subject)');`

Compute the fitted values and raw residuals.

F = fitted(lme); R = residuals(lme);

Plot the residuals versus the fitted values.

plot(F,R,'bx') xlabel('Fitted Values') ylabel('Residuals')

Now, plot the residuals versus the fitted values, grouped by program.

figure(); gscatter(F,R,Program)

The residuals seem to behave similarly across levels of the program as expected.

### Compute Conditional and Marginal Pearson Residuals

Load the sample data.

`load carbig`

Store the variables for miles per gallon (MPG), acceleration, horsepower, cylinders, and model year in a table.

tbl = table(MPG,Acceleration,Horsepower,Cylinders,Model_Year);

Fit a linear mixed-effects model for miles per gallon (MPG), with fixed effects for acceleration, horsepower, and the cylinders, and potentially correlated random effects for intercept and acceleration grouped by model year.

`lme = fitlme(tbl,'MPG ~ Acceleration + Horsepower + Cylinders + (Acceleration|Model_Year)');`

Compute the conditional Pearson residuals and display the first five residuals.

PR = residuals(lme,'ResidualType','Pearson'); PR(1:5)

`ans = `*5×1*
-0.0533
0.0652
0.3655
-0.0106
-0.3340

Compute the marginal Pearson residuals and display the first five residuals.

PRM = residuals(lme,'ResidualType','Pearson','Conditional',false); PRM(1:5)

`ans = `*5×1*
-0.1250
0.0130
0.3242
-0.0861
-0.3006

### Examine Residuals

Load the sample data.

`load carbig`

Store the variables for miles per gallon (MPG), acceleration, horsepower, cylinders, and model year in a table.

tbl = table(MPG,Acceleration,Horsepower,Cylinders,Model_Year);

Fit a linear mixed-effects model for miles per gallon (MPG), with fixed effects for acceleration, horsepower, and the cylinders, and potentially correlated random effects for intercept and acceleration grouped by model year.

`lme = fitlme(tbl,'MPG ~ Acceleration + Horsepower + Cylinders + (Acceleration|Model_Year)');`

Draw a histogram of the raw residuals with a normal fit.

r = residuals(lme); histfit(r)

Normal distribution seems to be a good fit for the residuals.

Compute the conditional Pearson and standardized residuals and create box plots of all three types of residuals.

pr = residuals(lme,'ResidualType','Pearson'); st = residuals(lme,'ResidualType','Standardized'); X = [r pr st]; boxplot(X)

Red plus signs show the observations with residuals above or below $$q3+1.5(q3-q1)$$ and $$q1-1.5(q3-q1)$$, where $$q1$$ and $$q3$$ are the 25th and 75th percentiles, respectively.

Find the observations with residuals that are 2.5 standard deviations above and below the mean.

find(r > mean(r,'omitnan') + 2.5*std(r,'omitnan'))

`ans = `*7×1*
62
252
255
330
337
341
396

find(r < mean(r,'omitnan') - 2.5*std(r,'omitnan'))

`ans = `*3×1*
119
324
375

## More About

### Conditional and Marginal Residuals

Conditional residuals include contributions from both fixed and random effects, whereas marginal residuals include contribution from only fixed effects.

Suppose the linear mixed-effects model `lme`

has
an *n*-by-*p* fixed-effects design
matrix *X* and an *n*-by-*q* random-effects
design matrix *Z*. Also, suppose the *p*-by-1
estimated fixed-effects vector is $$\widehat{\beta}$$, and the *q*-by-1
estimated best linear unbiased predictor (BLUP) vector of random effects
is $$\widehat{b}$$. The fitted conditional response
is

$${\widehat{y}}_{Cond}=X\widehat{\beta}+Z\widehat{b},$$

and the fitted marginal response is

$${\widehat{y}}_{Mar}=X\widehat{\beta},$$

`residuals`

can return three types of residuals:
raw, Pearson, and standardized. For any type, you can compute the
conditional or the marginal residuals. For example, the conditional
raw residual is

$${r}_{Cond}=y-X\widehat{\beta}-Z\widehat{b},$$

and the marginal raw residual is

$${r}_{Mar}=y-X\widehat{\beta}.$$

For more information on other types of residuals, see the `ResidualType`

name-value
pair argument.

## See Also

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