# fitlme

Fit linear mixed-effects model

## Syntax

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

## Description

example

````lme = fitlme(tbl,formula)` returns a linear mixed-effects model, specified by `formula`, fitted to the variables in the table or dataset array `tbl`.```

example

````lme = fitlme(tbl,formula,Name,Value)` returns a linear mixed-effects model with additional options specified by one or more `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.```

## Examples

collapse all

`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 {'(Intercept)'} {'(Intercept)'} {'std'} 5.7338 Lower Upper 2.3773 13.829 Group: EngineType (7 Levels) Name1 Name2 Type Estimate {'Horsepower'} {'Horsepower'} {'std'} 0.050357 Lower Upper 0.02307 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 {'(Intercept)'} {'(Intercept)'} {'std' } 9.4952 {'Horsepower' } {'(Intercept)'} {'corr'} -0.96843 {'Horsepower' } {'Horsepower' } {'std' } 0.078874 Lower Upper 4.7022 19.174 -0.99568 -0.78738 0.039917 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 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

`${y}_{im}={\beta }_{0}+{\beta }_{1}{WtdILI}_{im}+{b}_{0m}+{\epsilon }_{im},\phantom{\rule{1em}{0ex}}i=1,2,...,468,\phantom{\rule{1em}{0ex}}m=1,2,...,52,$`

where ${y}_{im}$ is the observation $i$ for level $m$ of grouping variable `Date`, ${b}_{0m}$ is the random effect for level $m$ of the grouping variable `Date`, and ${\epsilon }_{im}$ is the observation error for observation $i$. The random effect has the prior distribution,

`${b}_{0m}\sim N\left(0,{\sigma }_{b}^{2}\right),$`

and the error term has the distribution,

`${\epsilon }_{im}\sim N\left(0,{\sigma }^{2}\right).$`

`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 {'(Intercept)'} {'(Intercept)'} {'std'} 0.17146 Lower Upper 0.13227 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 ${\sigma }_{b}$ 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

`$\begin{array}{l}{\underset{}{\overset{ˆ}{y}}}_{28}={\underset{}{\overset{ˆ}{\beta }}}_{0}+{\underset{}{\overset{ˆ}{\beta }}}_{1}{WtdILI}_{28}+{\underset{}{\overset{ˆ}{b}}}_{10/30/2005}\\ \phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}=0.1639+0.7236*\left(1.343\right)+0.3318\\ \phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}=1.46749,\end{array}$`

where $\underset{}{\overset{ˆ}{b}}$ 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('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, $Shift_Evening$ and $Shift_Morning$, where

The model corresponds to

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

`${b}_{0m}\sim N\left(0,{\sigma }_{b}^{2}\right)$`

and

`${\epsilon }_{im}\sim N\left(0,{\sigma }^{2}\right).$`

```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 {'(Intercept)'} {'(Intercept)'} {'std'} 2.0457 Lower Upper 0.98207 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

`$\begin{array}{l}{\underset{}{\overset{ˆ}{y}}}_{\text{Evening},\text{Operator}3}={\underset{}{\overset{ˆ}{\beta }}}_{0}+{\underset{}{\overset{ˆ}{\beta }}}_{1}\text{Shift}\text{_}\text{Evening}+{\underset{}{\overset{ˆ}{b}}}_{03}\\ \phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{0.2777777777777778em}{0ex}}=3.6525-0.53293-2.1715\\ \phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{0.2777777777777778em}{0ex}}=0.94807.\end{array}$`

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

`$\begin{array}{l}{\underset{}{\overset{ˆ}{y}}}_{\text{Morning},\text{Operator}3}={\underset{}{\overset{ˆ}{\beta }}}_{0}+{\underset{}{\overset{ˆ}{\beta }}}_{2}\text{Shift}\text{_}\text{Morning}+{\underset{}{\overset{ˆ}{b}}}_{03}\\ \phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{0.2777777777777778em}{0ex}}=3.6525-0.91973-2.1715\\ \phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{0.2777777777777778em}{0ex}}=0.56127.\end{array}$`

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('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

`$\begin{array}{l}{y}_{imjk}={\beta }_{0}+\sum _{m=2}^{4}{\beta }_{1m}I{\left[F\right]}_{im}+\sum _{j=2}^{5}{\beta }_{2j}I{\left[T\right]}_{ij}+\sum _{j=2}^{5}\sum _{m=2}^{4}{\beta }_{3mj}I{\left[F\right]}_{im}I{\left[T\right]}_{ij}\\ \phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}+{b}_{0k}{S}_{k}+{b}_{0jk}\left(S*T{\right)}_{jk}+{\epsilon }_{imjk},\end{array}$`

where $i$ = 1, 2, ..., 60, index $m$ corresponds to the fertilizer types, $j$ corresponds to the tomato types, and $k$ = 1, 2, 3 corresponds to the blocks (soil). ${S}_{k}$ represents the $k$ th soil type, and $\left(S*T{\right)}_{jk}$ represents the $j$ th tomato type nested in the $k$ th soil type. $I\left[F{\right]}_{im}$ is the dummy variable representing level $m$ of the fertilizer. Similarly, $I\left[T{\right]}_{ij}$ is the dummy variable representing level $j$ of the tomato type.

The random effects and observation error have these prior distributions: ${b}_{0k}$~N(0, ${\sigma }_{S}^{2}$ ), ${b}_{0jk}$~N(0, ${\sigma }_{S*T}^{2}$ ), and ${ϵ}_{imjk}$ ~ N(0, ${\sigma }^{2}$ ).

`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 {'(Intercept)'} {'(Intercept)'} {'std'} 2.5028 Lower Upper 0.027711 226.05 Group: Soil:Tomato (15 Levels) Name1 Name2 Type Estimate {'(Intercept)'} {'(Intercept)'} {'std'} 10.225 Lower Upper 6.1497 17.001 Group: Error Name Estimate Lower Upper {'Res Std'} 10.499 8.5389 12.908 ```

The $p$-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 ( ${\sigma }_{S}^{2}$ ), 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 {'(Intercept)' } 77.733 7.3293 10.606 52 {'Tomato_Grape' } -9.1667 9.6045 -0.95441 52 {'Tomato_Heirloom'} -12.583 9.6045 -1.3102 52 {'Tomato_Plum' } 28.833 9.6045 3.0021 52 {'Tomato_Vine' } -14.083 9.6045 -1.4663 52 {'Fertilizer_2' } 26.333 4.5004 5.8514 52 {'Fertilizer_3' } 39 4.5004 8.6659 52 {'Fertilizer_4' } 47.733 4.5004 10.607 52 pValue Lower Upper 1.3108e-14 63.026 92.441 0.34429 -28.439 10.106 0.1959 -31.856 6.6895 0.0041138 9.5605 48.106 0.14858 -33.356 5.1895 3.3024e-07 17.303 35.364 1.1459e-11 29.969 48.031 1.308e-14 38.703 56.764 Random effects covariance parameters (95% CIs): Group: Soil:Tomato (15 Levels) Name1 Name2 Type Estimate {'(Intercept)'} {'(Intercept)'} {'std'} 10.02 Lower Upper 6.0812 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('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 $I$[.]. 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

`$\begin{array}{l}{y}_{im}={\beta }_{0}+{\beta }_{1}I{W}_{i}+{\beta }_{2}Wee{k}_{i}+{\beta }_{3}I{\left[PB\right]}_{i}+{\beta }_{4}I{\left[PC\right]}_{i}+{\beta }_{5}I{\left[PD\right]}_{i}\\ \phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}+{\beta }_{6}\left(Wee{k}_{i}*I{\left[PB\right]}_{i}\right)+{\beta }_{7}\left(Wee{k}_{i}*I{\left[PC\right]}_{i}\right)+{\beta }_{8}\left(Wee{k}_{i}*I{\left[PD\right]}_{i}\right)\\ \phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}+b{}_{0m}+\phantom{\rule{0.16666666666666666em}{0ex}}{b}_{1m}Wee{k}_{im}+{\epsilon }_{im},\end{array}$`

where $i$ = 1, 2, ..., 120, and $m$ = 1, 2, ..., 20. ${\beta }_{j}$ are the fixed-effects coefficients, $j$ = 0, 1, ..., 8, and ${b}_{0m}$ and ${b}_{1m}$ are random effects. $IW$ stands for initial weight and $I\left[\cdot \right]$ is a dummy variable representing a type of program. For example, $I\left[PB{\right]}_{i}$ is the dummy variable representing program type B. The random effects and observation error have the following prior distributions:

`${b}_{0m}\sim N\left(0,{\sigma }_{0}^{2}\right)$`

`${b}_{1m}\sim N\left(0,{\sigma }_{1}^{2}\right)$`

`${\epsilon }_{im}\sim N\left(0,{\sigma }^{2}\right).$`

`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 {'(Intercept)'} {'(Intercept)'} {'std' } 0.18407 {'Week' } {'(Intercept)'} {'corr'} 0.66841 {'Week' } {'Week' } {'std' } 0.15033 Lower Upper 0.12281 0.27587 0.21076 0.88573 0.11004 0.20537 Group: Error Name Estimate Lower Upper {'Res Std'} 0.10261 0.087882 0.11981 ```

The $p$-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.

## Input Arguments

collapse all

Input 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 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)'`

### 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.

Example: `'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`.

Pattern 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 rth 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 σ2b is the common variance of the random-effects terms. `'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 σ2b1 is the common variance of the random-effects terms and σb1,b2 is the common covariance between any two random-effects term . `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`

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'`

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`

Indices 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`

Coding to use for dummy variables created from the categorical variables, specified as the comma-separated pair consisting of `'DummyVarCoding'` and one of the variables in this table.

ValueDescription
`'reference'` (default)`fitlme` creates dummy variables with a reference group. This scheme treats the first category as a reference group and creates one less dummy variables than the number of categories. You can check the category order of a categorical variable by using the `categories` function, and change the order by using the `reordercats` function.
`'effects'``fitlme` creates dummy variables using effects coding. This scheme uses –1 to represent the last category. This scheme creates one less dummy variables than the number of categories.
`'full'``fitlme` creates full dummy variables. This scheme creates one dummy variable for each category.

For more details about creating dummy variables, see Automatic Creation of Dummy Variables.

Example: `'DummyVarCoding','effects'`

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'`

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')`.

Relative tolerance on the gradient of the objective function, specified as a positive scalar value.

Absolute tolerance on the step size, specified as a positive scalar value.

Maximum number of iterations allowed, specified as a positive scalar value.

Level of display, specified as one of `'off'`, `'iter'`, or `'final'`.

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

ValueDescription
`'default'`An internally defined default value
`'random'`A random initial value

Example: `'StartMethod','random'`

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`

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`

## Output Arguments

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Linear mixed-effects model, returned as a `LinearMixedModel` object.

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### Formula

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, `Xj`, which can be continuous or grouping variables

• Grouping variables, `g1`, `g2`, ..., `gR`,

where the grouping variables in `Xj` and `gr` can be categorical, logical, character arrays, string arrays, or cell arrays of character vectors.

Then, in a formula of the form, ```'y ~ fixed + (random1|g1) + ... + (randomR|gR)'```, the term `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 NotationFactors in Standard Notation
`1`Constant (intercept) term
`X^k`, where `k` is a positive integer`X`, `X2`, ..., `Xk`
`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:

`'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.

### Cholesky Parameterization

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 σ2 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

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

### Log-Cholesky Parameterization

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 Lii ≥ 0,

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

## Alternatives

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)`.

## References

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

## Version History

Introduced in R2013b