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Fit linear mixed-effects model

`lme = fitlme(tbl,formula)`

`lme = fitlme(tbl,formula,Name,Value)`

returns
a linear mixed-effects model with additional options specified by
one or more `lme`

= fitlme(`tbl`

,`formula`

,`Name,Value`

)`Name,Value`

pair arguments.

For example, you can specify the covariance pattern of the random-effects terms, the method to use in estimating the parameters, or options for the optimization algorithm.

Load the sample data.

`load imports-85`

Store the variables in a table.

tbl = table(X(:,12),X(:,14),X(:,24),'VariableNames',{'Horsepower','CityMPG','EngineType'});

Display the first five rows of the table.

tbl(1:5,:)

`ans=`*5×3 table*
Horsepower CityMPG EngineType
__________ _______ __________
111 21 13
111 21 13
154 19 37
102 24 35
115 18 35

Fit a linear mixed-effects model for miles per gallon in the city, with fixed effects for horsepower, and uncorrelated random effect for intercept and horsepower grouped by the engine type.

`lme = fitlme(tbl,'CityMPG~Horsepower+(1|EngineType)+(Horsepower-1|EngineType)');`

In this model, `CityMPG`

is the response variable, horsepower is the predictor variable, and engine type is the grouping variable. The fixed-effects portion of the model corresponds to `1 + Horsepower`

, because the intercept is included by default.

Since the random-effect terms for intercept and horsepower are uncorrelated, these terms are specified separately. Because the second random-effect term is only for horsepower, you must include a `–1`

to eliminate the intercept from the second random-effect term.

Display the model.

lme

lme = Linear mixed-effects model fit by ML Model information: Number of observations 203 Fixed effects coefficients 2 Random effects coefficients 14 Covariance parameters 3 Formula: CityMPG ~ 1 + Horsepower + (1 | EngineType) + (Horsepower | EngineType) Model fit statistics: AIC BIC LogLikelihood Deviance 1099.5 1116 -544.73 1089.5 Fixed effects coefficients (95% CIs): Name Estimate SE tStat DF pValue '(Intercept)' 37.276 2.8556 13.054 201 1.3147e-28 'Horsepower' -0.12631 0.02284 -5.53 201 9.8848e-08 Lower Upper 31.645 42.906 -0.17134 -0.081269 Random effects covariance parameters (95% CIs): Group: EngineType (7 Levels) Name1 Name2 Type Estimate Lower '(Intercept)' '(Intercept)' 'std' 5.7338 2.3773 Upper 13.829 Group: EngineType (7 Levels) Name1 Name2 Type Estimate Lower 'Horsepower' 'Horsepower' 'std' 0.050357 0.02307 Upper 0.10992 Group: Error Name Estimate Lower Upper 'Res Std' 3.226 2.9078 3.5789

Note that the random-effects covariance parameters for intercept and horsepower are separate in the display.

Now, fit a linear mixed-effects model for miles per gallon in the city, with the same fixed-effects term and potentially correlated random effect for intercept and horsepower grouped by the engine type.

`lme2 = fitlme(tbl,'CityMPG~Horsepower+(Horsepower|EngineType)');`

Because the random-effect term includes the intercept by default, you do not have to add `1`

, the random effect term is equivalent to `(1 + Horsepower|EngineType)`

.

Display the model.

lme2

lme2 = Linear mixed-effects model fit by ML Model information: Number of observations 203 Fixed effects coefficients 2 Random effects coefficients 14 Covariance parameters 4 Formula: CityMPG ~ 1 + Horsepower + (1 + Horsepower | EngineType) Model fit statistics: AIC BIC LogLikelihood Deviance 1089 1108.9 -538.52 1077 Fixed effects coefficients (95% CIs): Name Estimate SE tStat DF pValue '(Intercept)' 33.824 4.0181 8.4178 201 7.1678e-15 'Horsepower' -0.1087 0.032912 -3.3029 201 0.0011328 Lower Upper 25.901 41.747 -0.1736 -0.043806 Random effects covariance parameters (95% CIs): Group: EngineType (7 Levels) Name1 Name2 Type Estimate Lower '(Intercept)' '(Intercept)' 'std' 9.4952 4.7022 'Horsepower' '(Intercept)' 'corr' -0.96843 -0.99568 'Horsepower' 'Horsepower' 'std' 0.078874 0.039917 Upper 19.174 -0.78738 0.15585 Group: Error Name Estimate Lower Upper 'Res Std' 3.1845 2.8774 3.5243

Note that the random effects covariance parameters for intercept and horsepower are together in the display, and it includes the correlation (`'corr'`

) between the intercept and horsepower.

Load the sample data.

`load flu`

The `flu`

dataset array has a `Date`

variable, and 10 variables containing estimated influenza rates (in 9 different regions, estimated from Google® searches, plus a nationwide estimate from the Centers for Disease Control and Prevention, CDC).

To fit a linear-mixed effects model, your data must be in a properly formatted dataset array. To fit a linear mixed-effects model with the influenza rates as the responses, combine the nine columns corresponding to the regions into an array. The new dataset array, `flu2`

, must have the new response variable `FluRate`

, the nominal variable `Region`

that shows which region each estimate is from, the nationwide estimate `WtdILI`

, and the grouping variable `Date`

.

flu2 = stack(flu,2:10,'NewDataVarName','FluRate',... 'IndVarName','Region'); flu2.Date = nominal(flu2.Date);

Display the first six rows of `flu2`

.

flu2(1:6,:)

ans = Date WtdILI Region FluRate 10/9/2005 1.182 NE 0.97 10/9/2005 1.182 MidAtl 1.025 10/9/2005 1.182 ENCentral 1.232 10/9/2005 1.182 WNCentral 1.286 10/9/2005 1.182 SAtl 1.082 10/9/2005 1.182 ESCentral 1.457

Fit a linear mixed-effects model with a fixed-effects term for the nationwide estimate, `WtdILI`

, and a random intercept that varies by `Date`

. The model corresponds to

where is the observation for level of grouping variable `Date`

, is the random effect for level of the grouping variable `Date`

, and is the observation error for observation . The random effect has the prior distribution,

and the error term has the distribution,

`lme = fitlme(flu2,'FluRate ~ 1 + WtdILI + (1|Date)')`

lme = Linear mixed-effects model fit by ML Model information: Number of observations 468 Fixed effects coefficients 2 Random effects coefficients 52 Covariance parameters 2 Formula: FluRate ~ 1 + WtdILI + (1 | Date) Model fit statistics: AIC BIC LogLikelihood Deviance 286.24 302.83 -139.12 278.24 Fixed effects coefficients (95% CIs): Name Estimate SE tStat DF pValue '(Intercept)' 0.16385 0.057525 2.8484 466 0.0045885 'WtdILI' 0.7236 0.032219 22.459 466 3.0502e-76 Lower Upper 0.050813 0.27689 0.66028 0.78691 Random effects covariance parameters (95% CIs): Group: Date (52 Levels) Name1 Name2 Type Estimate Lower '(Intercept)' '(Intercept)' 'std' 0.17146 0.13227 Upper 0.22226 Group: Error Name Estimate Lower Upper 'Res Std' 0.30201 0.28217 0.32324

Estimated covariance parameters are displayed in the section titled "Random effects covariance parameters". The estimated value of is 0.17146 and its 95% confidence interval is [0.13227, 0.22226]. Since this interval does not include 0, the random-effects term is significant. You can formally test the significance of any random-effects term using a likelihood ratio test via the `compare`

method.

The estimated response at an observation is the sum of the fixed effects and the random-effect value at the grouping variable level corresponding to that observation. For example, the estimated flu rate for observation 28 is

where is the estimated best linear unbiased predictor (BLUP) of the random effects for the intercept. You can compute this value as follows.

beta = fixedEffects(lme); [~,~,STATS] = randomEffects(lme); % Compute the random-effects statistics (STATS) STATS.Level = nominal(STATS.Level); y_hat = beta(1) + beta(2)*flu2.WtdILI(28) + STATS.Estimate(STATS.Level=='10/30/2005')

y_hat = 1.4674

You can display the fitted value using the `fitted`

method.

F = fitted(lme); F(28)

ans = 1.4674

Load the sample data.

load(fullfile(matlabroot,'examples','stats','shift.mat'))

The data shows the absolute deviations from the target quality characteristic measured from the products each of five operators manufacture during three shifts: morning, evening, and night. This is a randomized block design, where the operators are the blocks. The experiment is designed to study the impact of the time of shift on the performance. The performance measure is the absolute deviations of the quality characteristics from the target value. This is simulated data.

Fit a linear mixed-effects model with a random intercept grouped by operator to assess if performance significantly differs according to the time of the shift. Use the restricted maximum likelihood method and `'effects'`

contrasts.

`'effects'`

contrasts mean that the coefficients sum to 0, and `fitlme`

creates a matrix called a *fixed effects design matrix* to describe the effect of shift. This matrix has two columns, and , where

The model corresponds to

where represents the observations, and represents the operators, = 1, 2, ..., 15, and = 1, 2, ..., 5. The random effects and the observation error have the following distributions:

and

lme = fitlme(shift,'QCDev ~ Shift + (1|Operator)',... 'FitMethod','REML','DummyVarCoding','effects')

lme = Linear mixed-effects model fit by REML Model information: Number of observations 15 Fixed effects coefficients 3 Random effects coefficients 5 Covariance parameters 2 Formula: QCDev ~ 1 + Shift + (1 | Operator) Model fit statistics: AIC BIC LogLikelihood Deviance 58.913 61.337 -24.456 48.913 Fixed effects coefficients (95% CIs): Name Estimate SE tStat DF pValue '(Intercept)' 3.6525 0.94109 3.8812 12 0.0021832 'Shift_Evening' -0.53293 0.31206 -1.7078 12 0.11339 'Shift_Morning' -0.91973 0.31206 -2.9473 12 0.012206 Lower Upper 1.6021 5.703 -1.2129 0.14699 -1.5997 -0.23981 Random effects covariance parameters (95% CIs): Group: Operator (5 Levels) Name1 Name2 Type Estimate Lower '(Intercept)' '(Intercept)' 'std' 2.0457 0.98207 Upper 4.2612 Group: Error Name Estimate Lower Upper 'Res Std' 0.85462 0.52357 1.395

Compute the best linear unbiased predictor (BLUP) estimates of random effects.

B = randomEffects(lme)

`B = `*5×1*
0.5775
1.1757
-2.1715
2.3655
-1.9472

The estimated absolute deviation from the target quality characteristics for the third operator working the evening shift is

You can also display this value as follows.

F = fitted(lme); F(shift.Shift=='Evening' & shift.Operator=='3')

ans = 0.9481

Similarly, you can calculate the estimated absolute deviation from the target quality characteristics for the third operator working the morning shift as

You can also display this value as follows.

F(shift.Shift=='Morning' & shift.Operator=='3')

ans = 0.5613

The operator tends to make a smaller magnitude of error during the morning shift.

Load the sample data.

load(fullfile(matlabroot,'examples','stats','fertilizer.mat'))

The dataset array includes data from a split-plot experiment, where soil is divided into three blocks based on the soil type: sandy, silty, and loamy. Each block is divided into five plots, where five types of tomato plants (cherry, heirloom, grape, vine, and plum) are randomly assigned to these plots. The tomato plants in the plots are then divided into subplots, where each subplot is treated by one of four fertilizers. This is simulated data.

Store the data in a dataset array called `ds`

, and define `Tomato`

, `Soil`

, and `Fertilizer`

as categorical variables.

ds = fertilizer; ds.Tomato = nominal(ds.Tomato); ds.Soil = nominal(ds.Soil); ds.Fertilizer = nominal(ds.Fertilizer);

Fit a linear mixed-effects model, where `Fertilizer`

and `Tomato`

are the fixed-effects variables, and the mean yield varies by the block (soil type) and the plots within blocks (tomato types within soil types) independently.

This model corresponds to

where = 1, 2, ..., 60, index corresponds to the fertilizer types, corresponds to the tomato types, and = 1, 2, 3 corresponds to the blocks (soil). represents the th soil type, and represents the th tomato type nested in the th soil type. is the dummy variable representing level of the fertilizer. Similarly, is the dummy variable representing level of the tomato type.

The random effects and observation error have these prior distributions: ~N(0, ), ~N(0, ), and ~ N(0, ).

`lme = fitlme(ds,'Yield ~ Fertilizer * Tomato + (1|Soil) + (1|Soil:Tomato)')`

lme = Linear mixed-effects model fit by ML Model information: Number of observations 60 Fixed effects coefficients 20 Random effects coefficients 18 Covariance parameters 3 Formula: Yield ~ 1 + Tomato*Fertilizer + (1 | Soil) + (1 | Soil:Tomato) Model fit statistics: AIC BIC LogLikelihood Deviance 522.57 570.74 -238.29 476.57 Fixed effects coefficients (95% CIs): Name Estimate SE tStat DF '(Intercept)' 77 8.5836 8.9706 40 'Tomato_Grape' -16 11.966 -1.3371 40 'Tomato_Heirloom' -6.6667 11.966 -0.55714 40 'Tomato_Plum' 32.333 11.966 2.7022 40 'Tomato_Vine' -13 11.966 -1.0864 40 'Fertilizer_2' 34.667 8.572 4.0442 40 'Fertilizer_3' 33.667 8.572 3.9275 40 'Fertilizer_4' 47.667 8.572 5.5607 40 'Tomato_Grape:Fertilizer_2' -2.6667 12.123 -0.21997 40 'Tomato_Heirloom:Fertilizer_2' -8 12.123 -0.65992 40 'Tomato_Plum:Fertilizer_2' -15 12.123 -1.2374 40 'Tomato_Vine:Fertilizer_2' -16 12.123 -1.3198 40 'Tomato_Grape:Fertilizer_3' 16.667 12.123 1.3748 40 'Tomato_Heirloom:Fertilizer_3' 3.3333 12.123 0.27497 40 'Tomato_Plum:Fertilizer_3' 3.6667 12.123 0.30246 40 'Tomato_Vine:Fertilizer_3' 3 12.123 0.24747 40 'Tomato_Grape:Fertilizer_4' 13.333 12.123 1.0999 40 'Tomato_Heirloom:Fertilizer_4' -19 12.123 -1.5673 40 'Tomato_Plum:Fertilizer_4' -2.6667 12.123 -0.21997 40 'Tomato_Vine:Fertilizer_4' 8.6667 12.123 0.71492 40 pValue Lower Upper 4.0206e-11 59.652 94.348 0.18873 -40.184 8.1837 0.58053 -30.85 17.517 0.010059 8.1496 56.517 0.28379 -37.184 11.184 0.00023272 17.342 51.991 0.00033057 16.342 50.991 1.9567e-06 30.342 64.991 0.82701 -27.167 21.834 0.51309 -32.501 16.501 0.22317 -39.501 9.5007 0.19439 -40.501 8.5007 0.17683 -7.8341 41.167 0.78476 -21.167 27.834 0.76387 -20.834 28.167 0.80581 -21.501 27.501 0.27796 -11.167 37.834 0.12492 -43.501 5.5007 0.82701 -27.167 21.834 0.47881 -15.834 33.167 Random effects covariance parameters (95% CIs): Group: Soil (3 Levels) Name1 Name2 Type Estimate Lower '(Intercept)' '(Intercept)' 'std' 2.5028 0.027711 Upper 226.05 Group: Soil:Tomato (15 Levels) Name1 Name2 Type Estimate Lower '(Intercept)' '(Intercept)' 'std' 10.225 6.1497 Upper 17.001 Group: Error Name Estimate Lower Upper 'Res Std' 10.499 8.5389 12.908

The -values corresponding to the last 12 rows in the fixed-effects coefficients display (0.82701 to 0.47881) indicate that interaction coefficients between the tomato and fertilizer types are not significant. To test for the overall interaction between tomato and fertilizer, use the `anova`

method after refitting the model using `'effects'`

contrasts.

The confidence interval for the standard deviations of the random-effects terms ( ), where the intercept is grouped by soil, is very large. This term does not appear significant.

Refit the model after removing the interaction term `Tomato:Fertilizer`

and the random-effects term `(1 | Soil)`

.

`lme = fitlme(ds,'Yield ~ Fertilizer + Tomato + (1|Soil:Tomato)')`

lme = Linear mixed-effects model fit by ML Model information: Number of observations 60 Fixed effects coefficients 8 Random effects coefficients 15 Covariance parameters 2 Formula: Yield ~ 1 + Tomato + Fertilizer + (1 | Soil:Tomato) Model fit statistics: AIC BIC LogLikelihood Deviance 511.06 532 -245.53 491.06 Fixed effects coefficients (95% CIs): Name Estimate SE tStat DF pValue '(Intercept)' 77.733 7.3293 10.606 52 1.3108e-14 'Tomato_Grape' -9.1667 9.6045 -0.95441 52 0.34429 'Tomato_Heirloom' -12.583 9.6045 -1.3102 52 0.1959 'Tomato_Plum' 28.833 9.6045 3.0021 52 0.0041138 'Tomato_Vine' -14.083 9.6045 -1.4663 52 0.14858 'Fertilizer_2' 26.333 4.5004 5.8514 52 3.3024e-07 'Fertilizer_3' 39 4.5004 8.6659 52 1.1459e-11 'Fertilizer_4' 47.733 4.5004 10.607 52 1.308e-14 Lower Upper 63.026 92.441 -28.439 10.106 -31.856 6.6895 9.5605 48.106 -33.356 5.1895 17.303 35.364 29.969 48.031 38.703 56.764 Random effects covariance parameters (95% CIs): Group: Soil:Tomato (15 Levels) Name1 Name2 Type Estimate Lower '(Intercept)' '(Intercept)' 'std' 10.02 6.0812 Upper 16.509 Group: Error Name Estimate Lower Upper 'Res Std' 12.325 10.024 15.153

You can compare the two models using the `compare`

method with the simulated likelihood ratio test since both a fixed-effect and a random-effect term are tested.

Load the sample data.

load(fullfile(matlabroot,'examples','stats','weight.mat'))

`weight`

contains data from a longitudinal study, where 20 subjects are randomly assigned to 4 exercise programs (A, B, C, D), 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.

`fitlme`

uses program A as a reference and creates the necessary dummy variables [.]. Since the model already has an intercept, `fitlme`

only creates dummy variables for programs B, C, and D. This is also known as the `'reference'`

method of coding dummy variables.

This model corresponds to

where = 1, 2, ..., 120, and = 1, 2, ..., 20. are the fixed-effects coefficients, = 0, 1, ..., 8, and and are random effects. stands for initial weight and is a dummy variable representing a type of program. For example, is the dummy variable representing program type B. The random effects and observation error have the following prior distributions:

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

lme = Linear mixed-effects model fit by ML Model information: Number of observations 120 Fixed effects coefficients 9 Random effects coefficients 40 Covariance parameters 4 Formula: y ~ 1 + InitialWeight + Program*Week + (1 + Week | Subject) Model fit statistics: AIC BIC LogLikelihood Deviance -22.981 13.257 24.49 -48.981 Fixed effects coefficients (95% CIs): Name Estimate SE tStat DF '(Intercept)' 0.66105 0.25892 2.5531 111 'InitialWeight' 0.0031879 0.0013814 2.3078 111 'Program_B' 0.36079 0.13139 2.746 111 'Program_C' -0.033263 0.13117 -0.25358 111 'Program_D' 0.11317 0.13132 0.86175 111 'Week' 0.1732 0.067454 2.5677 111 'Program_B:Week' 0.038771 0.095394 0.40644 111 'Program_C:Week' 0.030543 0.095394 0.32018 111 'Program_D:Week' 0.033114 0.095394 0.34713 111 pValue Lower Upper 0.012034 0.14798 1.1741 0.022863 0.00045067 0.0059252 0.0070394 0.10044 0.62113 0.80029 -0.29319 0.22666 0.39068 -0.14706 0.3734 0.011567 0.039536 0.30686 0.68521 -0.15026 0.2278 0.74944 -0.15849 0.21957 0.72915 -0.15592 0.22214 Random effects covariance parameters (95% CIs): Group: Subject (20 Levels) Name1 Name2 Type Estimate Lower '(Intercept)' '(Intercept)' 'std' 0.18407 0.12281 'Week' '(Intercept)' 'corr' 0.66841 0.21076 'Week' 'Week' 'std' 0.15033 0.11004 Upper 0.27587 0.88573 0.20537 Group: Error Name Estimate Lower Upper 'Res Std' 0.10261 0.087882 0.11981

The -values 0.022863 and 0.011567 indicate significant effects of subject initial weights and time in the amount of weight lost. The weight loss of subjects who are in program B is significantly different relative to the weight loss of subjects who are in program A. The lower and upper limits of the covariance parameters for the random effects do not include 0, thus they are significant. You can also test the significance of the random effects using the `compare`

method.

`tbl`

— Input datatable |

`dataset`

arrayInput data, which includes the response variable, predictor
variables, and grouping variables, specified as a table or `dataset`

array. The predictor
variables can be continuous or grouping variables (see Grouping Variables). You
must specify the model for the variables using `formula`

.

**Data Types: **`table`

`formula`

— Formula for model specificationcharacter vector or string scalar of the form

```
'y ~ fixed +
(random1|grouping1) + ... + (randomR|groupingR)'
```

Formula for model specification, specified as a character vector or string scalar of the form
```
'y ~ fixed + (random1|grouping1) + ... +
(randomR|groupingR)'
```

. The formula is case sensitive. For a
full description, see Formula.

**Example: **`'y ~ treatment + (1|block)'`

Specify optional
comma-separated pairs of `Name,Value`

arguments. `Name`

is
the argument name and `Value`

is the corresponding value.
`Name`

must appear inside quotes. You can specify several name and value
pair arguments in any order as `Name1,Value1,...,NameN,ValueN`

.

`'CovariancePattern','Diagonal','Optimizer','fminunc','OptimizerOptions',opt`

specifies
a model, where the random-effects terms have a diagonal covariance
matrix structure, and `fitlme`

uses the `fminunc`

optimization
algorithm with the custom optimization parameters defined in variable `opt`

.`'CovariancePattern'`

— Pattern of covariance matrix`'FullCholesky'`

(default) | character vector | string scalar | square symmetric logical matrix | string array | cell array of character vectors or logical matricesPattern of the covariance matrix of the random effects, specified as the comma-separated pair
consisting of `'CovariancePattern'`

and a character vector, a string
scalar, a square symmetric logical matrix, a string array, or a cell array of character
vectors or logical matrices.

If there are *R* random-effects terms, then the value of
`'CovariancePattern'`

must be a string array or cell array of
length *R*, where each element *r* of the array
specifies the pattern of the covariance matrix of the random-effects vector associated
with the *r*th random-effects term. The options for each element
follow.

`'FullCholesky'` | Default. Full covariance matrix using the Cholesky parameterization. `fitlme` estimates
all elements of the covariance matrix. |

`'Full'` | Full covariance matrix, using the log-Cholesky parameterization. `fitlme` estimates
all elements of the covariance matrix. |

`'Diagonal'` |
Diagonal covariance matrix. That is, off-diagonal elements of the covariance matrix are constrained to be 0. $$\left(\begin{array}{ccc}{\sigma}_{b1}^{2}& 0& 0\\ 0& {\sigma}_{b2}^{2}& 0\\ 0& 0& {\sigma}_{b3}^{2}\end{array}\right)$$ |

`'Isotropic'` |
Diagonal covariance matrix with equal variances. That is, off-diagonal elements of the covariance matrix are constrained to be 0, and the diagonal elements are constrained to be equal. For example, if there are three random-effects terms with an isotropic covariance structure, this covariance matrix looks like $$\left(\begin{array}{ccc}{\sigma}_{b}^{2}& 0& 0\\ 0& {\sigma}_{b}^{2}& 0\\ 0& 0& {\sigma}_{b}^{2}\end{array}\right)$$ where
σ |

`'CompSymm'` |
Compound symmetry structure. That is, common variance along diagonals and equal correlation between all random effects. For example, if there are three random-effects terms with a covariance matrix having a compound symmetry structure, this covariance matrix looks like $$\left(\begin{array}{ccc}{\sigma}_{b1}^{2}& {\sigma}_{b1,b2}& {\sigma}_{b1,b2}\\ {\sigma}_{b1,b2}& {\sigma}_{b1}^{2}& {\sigma}_{b1,b2}\\ {\sigma}_{b1,b2}& {\sigma}_{b1,b2}& {\sigma}_{b1}^{2}\end{array}\right)$$ where
σ |

`PAT` | Square symmetric logical matrix. If `'CovariancePattern'` is
defined by the matrix `PAT` , and if ```
PAT(a,b)
= false
``` , then the `(a,b)` element of the
corresponding covariance matrix is constrained to be 0. |

**Example: **`'CovariancePattern','Diagonal'`

**Example: **`'CovariancePattern',{'Full','Diagonal'}`

**Data Types: **`char`

| `string`

| `logical`

| `cell`

`'FitMethod'`

— Method for estimating parameters`'ML'`

(default) | `'REML'`

Method for estimating parameters of the linear mixed-effects
model, specified as the comma-separated pair consisting of `'FitMethod'`

and
either of the following.

`'ML'` | Default. Maximum likelihood estimation |

`'REML'` | Restricted maximum likelihood estimation |

**Example: **`'FitMethod','REML'`

`'Weights'`

— Observation weightsvector of scalar values

Observation weights, specified as the comma-separated pair consisting
of `'Weights'`

and a vector of length *n*,
where *n* is the number of observations.

**Data Types: **`single`

| `double`

`'Exclude'`

— Indices for rows to excludeuse all rows without

`NaNs`

(default) | vector of integer or logical valuesIndices for rows to exclude from the linear mixed-effects model
in the data, specified as the comma-separated pair consisting of `'Exclude'`

and
a vector of integer or logical values.

For example, you can exclude the 13th and 67th rows from the fit as follows.

**Example: **`'Exclude',[13,67]`

**Data Types: **`single`

| `double`

| `logical`

`'DummyVarCoding'`

— Coding to use for dummy variables`'reference'`

(default) | `'effects'`

| `'full'`

Coding to use for dummy variables created from the categorical
variables, specified as the comma-separated pair consisting of `'DummyVarCoding'`

and
one of the following.

Value | Description |
---|---|

`'reference'` | Default. Coefficient for first category set to 0. |

`'effects'` | Coefficients sum to 0. |

`'full'` | One dummy variable for each category. |

**Example: **`'DummyVarCoding','effects'`

`'Optimizer'`

— Optimization algorithm`'quasinewton'`

(default) | `'fminunc'`

Optimization algorithm, specified as the comma-separated pair
consisting of `'Optimizer'`

and either of the following.

`'quasinewton'` | Default. Uses a trust region based quasi-Newton optimizer.
Change the options of the algorithm using `statset('LinearMixedModel')` .
If you don’t specify the options, then `LinearMixedModel` uses
the default options of `statset('LinearMixedModel')` . |

`'fminunc'` | You must have Optimization
Toolbox™ to specify this option.
Change the options of the algorithm using `optimoptions('fminunc')` .
If you don’t specify the options, then `LinearMixedModel` uses
the default options of `optimoptions('fminunc')` with `'Algorithm'` set
to `'quasi-newton'` . |

**Example: **`'Optimizer','fminunc'`

`'OptimizerOptions'`

— Options for optimization algorithmstructure returned by

`statset`

| object returned by `optimoptions`

Options for the optimization algorithm, specified as the comma-separated
pair consisting of `'OptimizerOptions'`

and a structure
returned by `statset('LinearMixedModel')`

or an object
returned by `optimoptions('fminunc')`

.

If

`'Optimizer'`

is`'fminunc'`

, then use`optimoptions('fminunc')`

to change the options of the optimization algorithm. See`optimoptions`

for the options`'fminunc'`

uses. If`'Optimizer'`

is`'fminunc'`

and you do not supply`'OptimizerOptions'`

, then the default for`LinearMixedModel`

is the default options created by`optimoptions('fminunc')`

with`'Algorithm'`

set to`'quasi-newton'`

.If

`'Optimizer'`

is`'quasinewton'`

, then use`statset('LinearMixedModel')`

to change the optimization parameters. If you don’t change the optimization parameters, then`LinearMixedModel`

uses the default options created by`statset('LinearMixedModel')`

:

The `'quasinewton'`

optimizer uses the following
fields in the structure created by `statset('LinearMixedModel')`

.

`TolFun`

— Relative tolerance on gradient of objective function`1e-6`

(default) | positive scalar valueRelative tolerance on the gradient of the objective function, specified as a positive scalar value.

`TolX`

— Absolute tolerance on step size`1e-12`

(default) | positive scalar valueAbsolute tolerance on the step size, specified as a positive scalar value.

`MaxIter`

— Maximum number of iterations allowed`10000`

(default) | positive scalar valueMaximum number of iterations allowed, specified as a positive scalar value.

`Display`

— Level of display`'off'`

(default) | `'iter'`

| `'final'`

Level of display, specified as one of `'off'`

, `'iter'`

,
or `'final'`

.

`'StartMethod'`

— Method to start iterative optimization`'default'`

(default) | `'random'`

Method to start iterative optimization, specified as the comma-separated
pair consisting of `'StartMethod'`

and either of
the following.

Value | Description |
---|---|

`'default'` | An internally defined default value |

`'random'` | A random initial value |

**Example: **`'StartMethod','random'`

`'Verbose'`

— Indicator to display optimization process on screen`false`

(default) | `true`

Indicator to display the optimization process on screen, specified
as the comma-separated pair consisting of `'Verbose'`

and
either `false`

or `true`

. Default
is `false`

.

The setting for `'Verbose'`

overrides the field `'Display'`

in `'OptimizerOptions'`

.

**Example: **`'Verbose',true`

`'CheckHessian'`

— Indicator to check positive definiteness of Hessian`false`

(default) | `true`

Indicator to check the positive definiteness of the Hessian
of the objective function with respect to unconstrained parameters
at convergence, specified as the comma-separated pair consisting of `'CheckHessian'`

and
either `false`

or `true`

. Default
is `false`

.

Specify `'CheckHessian'`

as `true`

to
verify optimality of the solution or to determine if the model is
overparameterized in the number of covariance parameters.

**Example: **`'CheckHessian',true`

`lme`

— Linear mixed-effects model`LinearMixedModel`

objectLinear mixed-effects model, returned as a `LinearMixedModel`

object.

In general, a formula for model specification is a character vector or string
scalar of the form `'y ~ terms'`

. For the linear mixed-effects models, this
formula is in the form ```
'y ~ fixed + (random1|grouping1) + ... +
(randomR|groupingR)'
```

, where `fixed`

and
`random`

contain the fixed-effects and the random-effects terms.

Suppose a table `tbl`

contains the following:

A response variable,

`y`

Predictor variables,

`X`

, which can be continuous or grouping variables_{j}Grouping variables,

`g`

,_{1}`g`

, ...,_{2}`g`

,_{R}

where the grouping variables in
`X`

and
_{j}`g`

can be
categorical, logical, character arrays, string arrays, or cell arrays of character
vectors._{r}

Then, in a formula of the form, `'y ~ fixed + (random`

,
the term _{1}|g_{1})
+ ... + (random_{R}|g_{R})'`fixed`

corresponds to a specification of
the fixed-effects design matrix `X`

, `random`

_{1} is
a specification of the random-effects design matrix `Z`

_{1} corresponding
to grouping variable `g`

_{1},
and similarly `random`

_{R} is
a specification of the random-effects design matrix `Z`

_{R} corresponding
to grouping variable `g`

_{R}.
You can express the `fixed`

and `random`

terms
using Wilkinson notation.

Wilkinson notation describes the factors present in models. The notation relates to factors present in models, not to the multipliers (coefficients) of those factors.

Wilkinson Notation | Factors in Standard Notation |
---|---|

`1` | Constant (intercept) term |

`X^k` , where `k` is a positive
integer | `X` , `X` ,
..., `X` |

`X1 + X2` | `X1` , `X2` |

`X1*X2` | `X1` , `X2` , ```
X1.*X2
(elementwise multiplication of X1 and X2)
``` |

`X1:X2` | `X1.*X2` only |

`- X2` | Do not include `X2` |

`X1*X2 + X3` | `X1` , `X2` , `X3` , `X1*X2` |

`X1 + X2 + X3 + X1:X2` | `X1` , `X2` , `X3` , `X1*X2` |

`X1*X2*X3 - X1:X2:X3` | `X1` , `X2` , `X3` , `X1*X2` , `X1*X3` , `X2*X3` |

`X1*(X2 + X3)` | `X1` , `X2` , `X3` , `X1*X2` , `X1*X3` |

Statistics and Machine
Learning Toolbox™ notation always includes a constant term
unless you explicitly remove the term using `-1`

.
Here are some examples for linear mixed-effects model specification.

**Examples:**

Formula | Description |
---|---|

`'y ~ X1 + X2'` | Fixed effects for the intercept, `X1` and `X2` .
This is equivalent to `'y ~ 1 + X1 + X2'` . |

`'y ~ -1 + X1 + X2'` | No intercept and fixed effects for `X1` and `X2` .
The implicit intercept term is suppressed by including `-1` . |

`'y ~ 1 + (1 | g1)'` | Fixed effects for the intercept plus random effect for the
intercept for each level of the grouping variable `g1` . |

`'y ~ X1 + (1 | g1)'` | Random intercept model with a fixed slope. |

`'y ~ X1 + (X1 | g1)'` | Random intercept and slope, with possible correlation between
them. This is equivalent to `'y ~ 1 + X1 + (1 + X1|g1)'` . |

`'y ~ X1 + (1 | g1) + (-1 + X1 | g1)' ` | Independent random effects terms for intercept and slope. |

`'y ~ 1 + (1 | g1) + (1 | g2) + (1 | g1:g2)'` | Random intercept model with independent main effects for `g1` and `g2` ,
plus an independent interaction effect. |

One of the assumptions of linear mixed-effects models is that the random effects have the following prior distribution.

$$b~N\left(0,{\sigma}^{2}D\left(\theta \right)\right),$$

where *D* is a *q*-by-*q* symmetric
and positive semidefinite matrix, parameterized by a variance component
vector *θ*, *q* is the number
of variables in the random-effects term, and *σ*^{2} is
the observation error variance. Since the covariance matrix of the
random effects, *D*, is symmetric, it has *q*(*q*+1)/2
free parameters. Suppose *L* is the lower triangular
Cholesky factor of *D*(*θ*)
such that

$$D\left(\theta \right)=L\left(\theta \right)L{\left(\theta \right)}^{T},$$

then the *q**(*q*+1)/2-by-1
unconstrained parameter vector *θ* is formed
from elements in the lower triangular part of *L*.

For example, if

$$L=\left[\begin{array}{ccc}{L}_{11}& 0& 0\\ {L}_{21}& {L}_{22}& 0\\ {L}_{31}& {L}_{32}& {L}_{33}\end{array}\right],$$

then

$$\theta =\left[\begin{array}{c}{L}_{11}\\ {L}_{21}\\ {L}_{31}\\ {L}_{22}\\ {L}_{32}\\ {L}_{33}\end{array}\right].$$

When the diagonal elements of *L* in
Cholesky parameterization are constrained to be positive, then the
solution for *L* is unique. Log-Cholesky parameterization
is the same as Cholesky parameterization except that the logarithm
of the diagonal elements of *L* are used to guarantee
unique parameterization.

For example, for the 3-by-3 example in Cholesky parameterization,
enforcing *L*_{ii} ≥
0,

$$\theta =\left[\begin{array}{c}\mathrm{log}\left({L}_{11}\right)\\ {L}_{21}\\ {L}_{31}\\ \mathrm{log}\left({L}_{22}\right)\\ {L}_{32}\\ \mathrm{log}\left({L}_{33}\right)\end{array}\right].$$

If your model is not easily described using a formula, you can
create matrices to define the fixed and random effects, and fit the
model using `fitlmematrix(X,y,Z,G)`

.

[1] Pinherio, J. C., and D. M. Bates. “Unconstrained
Parametrizations for Variance-Covariance Matrices”. *Statistics
and Computing*, Vol. 6, 1996, pp. 289–296.

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