# LinearMixedModel class

Linear mixed-effects model class

## Description

A `LinearMixedModel` object represents a model of a response variable with fixed and random effects. It comprises data, a model description, fitted coefficients, covariance parameters, design matrices, residuals, residual plots, and other diagnostic information for a linear mixed-effects model. You can predict model responses with the `predict` function and generate random data at new design points using the `random` function.

## Construction

You can fit a linear mixed-effects model using `fitlme(tbl,formula)` if your data is in a table or dataset array. Alternatively, 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)`.

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### `tbl` — Input datatable | `dataset` array

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: `single` | `double` | `char` | `cell`

### `formula` — Formula for model specificationstring of the form ```'y ~ fixed + (random1|grouping1) + ... + (randomR|groupingR)'```

Formula for model specification, specified as a string of the form `'y ~ fixed + (random1|grouping1) + ... + (randomR|groupingR)'`. For a full description, see Formula.

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

### `X` — Fixed-effects design matrixn-by-p matrix

Fixed-effects design matrix, specified as an n-by-p matrix, where n is the number of observations, and p is the number of fixed-effects predictor variables. Each row of `X` corresponds to one observation, and each column of `X` corresponds to one variable.

Data Types: `single` | `double`

### `y` — Response valuesn-by-1 vector

Response values, specified as an n-by-1 vector, where n is the number of observations.

Data Types: `single` | `double`

### `Z` — Random-effects designn-by-q matrix | cell array of R n-by-q(r) matrices, r = 1, 2, ..., R

Random-effects design, specified as either of the following.

• If there is one random-effects term in the model, then `Z` must be an n-by-q matrix, where n is the number of observations and q is the number of variables in the random-effects term.

• If there are R random-effects terms, then `Z` must be a cell array of length R. Each cell of `Z` contains an n-by-q(r) design matrix `Z{r}`, r = 1, 2, ..., R, corresponding to each random-effects term. Here, q(r) is the number of random effects term in the rth random effects design matrix, `Z{r}`.

Data Types: `single` | `double` | `cell`

### `G` — Grouping variable or variablesn-by-1 vector | cell array of R n-by-1 vectors

Grouping variable or variables, specified as either of the following.

• If there is one random-effects term, then `G` must be an n-by-1 vector corresponding to a single grouping variable with M levels or groups.

`G` can be a categorical vector, numeric vector, character array, or cell array of strings.

• If there are multiple random-effects terms, then `G` must be a cell array of length R. Each cell of `G` contains a grouping variable `G{r}`, r = 1, 2, ..., R, with M(r) levels.

`G{r}` can be a categorical vector, numeric vector, character array, or cell array of strings.

Data Types: `single` | `double` | `char` | `cell`

## Properties

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### `Coefficients` — Fixed-effects coefficient estimatesdataset array

Fixed-effects coefficient estimates and related statistics, stored as a dataset array containing the following fields.

 `Name` Name of the term. `Estimate` Estimated value of the coefficient. `SE` Standard error of the coefficient. `tStat` t-statistics for testing the null hypothesis that the coefficient is equal to zero. `DF` Degrees of freedom for the t-test. Method to compute `DF` is specified by the `'DFMethod'` name-value pair argument. `Coefficients` always uses the `'Residual'` method for `'DFMethod'`. `pValue` p-value for the t-test. `Lower` Lower limit of the confidence interval for coefficient. `Coefficients` always uses the 95% confidence level, i.e.`'alpha'` is 0.05. `Upper` Upper limit of confidence interval for coefficient. `Coefficients` always uses the 95% confidence level, i.e.`'alpha'` is 0.05.

You can change `'DFMethod'` and `'alpha'` while computing confidence intervals for or testing hypotheses involving fixed- and random-effects, using the `coefCI` and `coefTest` methods.

### `CoefficientCovariance` — Covariance of the estimated fixed-effects coefficientsp-by-p matrix

Covariance of the estimated fixed-effects coefficients of the linear mixed-effects model, stored as a p-by-p matrix, where p is the number of fixed-effects coefficients.

You can display the covariance parameters associated with the random effects using the `covarianceParameters` method.

Data Types: `double`

### `CoefficientNames` — Names of the fixed-effects coefficients1-by-p cell array of strings

Names of the fixed-effects coefficients of a linear mixed-effects model, stored as a 1-by-p cell array of strings.

Data Types: `cell`

### `DFE` — Residual degrees of freedompositive integer value

Residual degrees of freedom, stored as a positive integer value. DFE = np, where n is the number of observations, and p is the number of fixed-effects coefficients.

This corresponds to the `'Residual'` method of calculating degrees of freedom in the `fixedEffects` and `randomEffects` methods.

Data Types: `double`

### `FitMethod` — Method used to fit the linear mixed-effects model`ML` | `REML`

Method used to fit the linear mixed-effects model, stored as either of the following strings.

• `ML`, if the fitting method is maximum likelihood

• `REML`, if the fitting method is restricted maximum likelihood

Data Types: `char`

### `Formula` — Specification of the fixed- and random-effects terms, and grouping variablesobject

Specification of the fixed-effects terms, random-effects terms, and grouping variables that define the linear mixed-effects model, stored as an object.

For more information on how to specify the model to fit using a formula, see Formula.

### `LogLikelihood` — Maximized log or restricted log likelihoodscalar value

Maximized log likelihood or maximized restricted log likelihood of the fitted linear mixed-effects model depending on the fitting method you choose, stored as a scalar value.

Data Types: `double`

### `ModelCriterion` — Model criteriondataset array

Model criterion to compare fitted linear mixed-effects models, stored as a dataset array with the following columns.

 `AIC` Akaike Information Criterion `BIC` Bayesian Information Criterion `Loglikelihood` Log likelihood value of the model `Deviance` –2 times the log likelihood of the model

If n is the number of observations used in fitting the model, and p is the number of fixed-effects coefficients, then for calculating AIC and BIC,

• The total number of parameters is nc + p + 1, where nc is the total number of parameters in the random-effects covariance excluding the residual variance

• The effective number of observations is

• n, when the fitting method is maximum likelihood (ML)

• np, when the fitting method is restricted maximum likelihood (REML)

### `MSE` — ML or REML estimatepositive scalar value

ML or REML estimate, based on the fitting method used for estimating σ2, stored as a positive scalar value. σ2 is the residual variance or variance of the observation error term of the linear mixed-effects model.

Data Types: `double`

### `NumCoefficients` — Number of fixed-effects coefficientspositive integer value

Number of fixed-effects coefficients in the fitted linear mixed-effects model, stored as a positive integer value.

Data Types: `double`

### `NumEstimatedCoefficients` — Number of estimated fixed-effects coefficientspositive integer value

Number of estimated fixed-effects coefficients in the fitted linear mixed-effects model, stored as a positive integer value.

Data Types: `double`

### `NumObservations` — Number of observationspositive integer value

Number of observations used in the fit, stored as a positive integer value. This is the number of rows in the table or dataset array, or the design matrices minus the excluded rows or rows with `NaN` values.

Data Types: `double`

### `NumPredictors` — Number of predictorspositive integer value

Number of variables used as predictors in the linear mixed-effects model, stored as a positive integer value.

Data Types: `double`

### `NumVariables` — Total number of variablespositive integer value

Total number of variables including the response and predictors, stored as a positive integer value.

• If the sample data is in a table or dataset array `tbl`, `NumVariables` is the total number of variables in `tbl` including the response variable.

• If the fit is based on matrix input, `NumVariables` is the total number of columns in the predictor matrix or matrices, and response vector.

`NumVariables` includes variables, if there are any, that are not used as predictors or as the response.

Data Types: `double`

### `ObservationInfo` — Information about the observationstable

Information about the observations used in the fit, stored as a table.

`ObservationInfo` has one row for each observation and the following four columns.

 `Weights` The value of the weighted variable for that observation. Default value is 1. `Excluded` `true`, if the observation was excluded from the fit using the `'Exclude'` name-value pair argument, `false`, otherwise. 1 stands for `true` and 0 stands for `false`. `Missing` `true`, if the observation was excluded from the fit because any response or predictor value is missing, `false`, otherwise. Missing values include `NaN` for numeric variables, empty cells for cell arrays, blank rows for character arrays, and the `` value for categorical arrays. `Subset` `true`, if the observation was used in the fit, `false`, if it was not used because it is missing or excluded.

Data Types: `table`

### `ObservationNames` — Names of observationscell array of strings

Names of observations used in the fit, stored as a cell array of strings.

• If the data is in a table or dataset array, `tbl`, containing observation names, `ObservationNames` has those names.

• If the data is provided in matrices, or a table or dataset array without observation names, then `ObservationNames` is an empty cell array.

Data Types: `cell`

### `PredictorNames` — Names of predictorscell array of strings

Names of the variables that you use as predictors in the fit, stored as a cell array of strings that has the same length as `NumPredictors`.

Data Types: `cell`

### `ResponseName` — Names of response variablecharacter string

Name of the variable used as the response variable in the fit, stored as a character string.

Data Types: `char`

### `Rsquared` — Proportion of variability in the response explained by the fitted modelstructure

Proportion of variability in the response explained by the fitted model, stored as a structure. It is the multiple correlation coefficient or R-squared. `Rsquared` has two fields.

 `Ordinary` R-squared value, stored as a scalar value in a structure. ```Rsquared.Ordinary = 1 – SSE./SST``` `Adjusted` R-squared value adjusted for the number of fixed-effects coefficients, stored as a scalar value in a structure. ```Rsquared.Adjusted = 1 – (SSE./SST)*(DFT./DFE)```, where ```DFE = n – p```, `DFT = n – 1`, and `n` is the total number of observations, `p` is the number of fixed-effects coefficients.

Data Types: `struct`

### `SSE` — Error sum of squarespositive scalar value

Error sum of squares, that is, sum of the squared conditional residuals, stored as a positive scalar value.

`SSE = sum((y – F).^2)`, where `y` is the response vector, and `F` is the fitted conditional response of the linear mixed-effects model. The conditional model has contributions from both fixed and random effects.

Data Types: `double`

### `SSR` — Regression sum of squarespositive scalar value

Regression sum of squares, that is, the sum of squares explained by the linear mixed-effects regression, stored as a positive scalar value. It is the sum of squared deviations of the conditional fitted values from their mean.

`SSR = sum((F – mean(F)).^2)`, where `F` is the fitted conditional response of the linear mixed-effects model. The conditional model has contributions from both fixed and random effects.

Data Types: `double`

### `SST` — Total sum of squarespositive scalar value

Total sum of squares, that is, the sum of the squared deviations of the observed response values from their mean, stored as a positive scalar value.

`SST = sum((y – mean(y)).^2) = SSR + SSE`, where `y` is the response vector.

Data Types: `double`

### `Variables` — Variablestable

Variables, stored as a table.

• If the fit is based on a table or dataset array `tbl`, then `Variables` is identical to `tbl`.

• If the fit is based on matrix input, then `Variables` is a table containing all the variables in the predictor matrix or matrices, and response variable.

Data Types: `table`

### `VariableInfo` — Information about the variablestable

Information about the variables used in the fit, stored as a table.

`VariableInfo` has one row for each variable and contains the following four columns.

 `Class` Class of the variable (`'double'`, `'cell'`, `'nominal'`, and so on). `Range` Value range of the variable. For a numerical variable, it is a two-element vector of the form `[min,max]`.For a cell or categorical variable, it is a cell or categorical array containing all unique values of the variable. `InModel` `true`, if the variable is a predictor in the fitted model.`false`, if the variable is not in the fitted model. `IsCategorical` `true`, if the variable has a type that is treated as a categorical predictor, such as cell, logical, or categorical, or if it is specified as categorical by the `'Categorical'` name-value pair argument of the `fit` method.`false`, if it is a continuous predictor.

Data Types: `table`

### `VariableNames` — Names of the variablescell array of strings

Names of the variables used in the fit, stored as a cell array of strings.

• If sample data is in a table or dataset array `tbl`, `VariableNames` contains the names of the variables in `tbl`.

• If sample data is in matrix format, then `VariableInfo` includes variable names you supply while fitting the model. If you do not supply the variable names, then `VariableInfo` contains the default names.

Data Types: `cell`

## Methods

 anova Analysis of variance for linear mixed-effects model coefCI Confidence intervals for coefficients of linear mixed-effects model coefTest Hypothesis test on fixed and random effects of linear mixed-effects model compare Compare linear mixed-effects models covarianceParameters Extract covariance parameters of linear mixed-effects model designMatrix Fixed- and random-effects design matrices disp Display linear mixed-effects model fit Fit linear mixed-effects model using tables fitmatrix Fit linear mixed-effects model using design matrices fitted Fitted responses from a linear mixed-effects model fixedEffects Estimates of fixed effects and related statistics plotResiduals Plot residuals of linear mixed-effects model predict Predict response of linear mixed-effects model random Generate random responses from fitted linear mixed-effects model randomEffects Estimates of random effects and related statistics residuals Residuals of fitted linear mixed-effects model response Response vector of the linear mixed-effects model

## Definitions

### Formula

In general, a formula for model specification is a string 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, or cell arrays of strings.

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.

## Copy Semantics

Value. To learn how value classes affect copy operations, see Copying Objects in the MATLAB® documentation.

## Examples

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### Random Intercept Model with Categorical Predictor

`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 Center 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 and region as the predictor variable, combine the nine columns corresponding to the regions into a tall array. The new dataset array, `flu2`, must have the response variable, `FluRate`, the nominal variable, `Region`, that shows which region each estimate is from, and the grouping variable `Date`.

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

Fit a linear mixed-effects model with fixed effects for region and a random intercept that varies by `Date`.

Because region is a nominal variable, `fitlme` takes the first region, `NE`, as the reference and creates eight dummy variables representing the other eight regions. For example, I[MidAtl] is the dummy variable representing the region `MidAtl`. For details, see Dummy Indicator Variables.

The corresponding model is

$\begin{array}{l}{y}_{im}={\beta }_{0}+{\beta }_{1}I{\left[MidAtl\right]}_{i}+{\beta }_{2}I{\left[ENCentral\right]}_{i}+{\beta }_{3}I{\left[WNCentral\right]}_{i}+{\beta }_{4}I{\left[SAtl\right]}_{i}\\ \text{ }\text{ }+{\beta }_{5}I{\left[ESCentral\right]}_{i}+{\beta }_{6}I{\left[WSCentral\right]}_{i}+{\beta }_{7}I{\left[Mtn\right]}_{i}+{\beta }_{8}I{\left[Pac\right]}_{i}+{b}_{0m}+{\epsilon }_{im},\text{ }m=1,2,...,52,\end{array}$

where yim is the observation i for level m of grouping variable `Date`, βj, j = 0, 1, ..., 8, are the fixed-effects coefficients, b0m is the random effect for level m of the grouping variable `Date`, and εim is the observation error for observation i. The random effect has the prior distribution, b ~ N(0,σ2b) and the error term has the distribution, ε ~ N(0,σ2).

`lme = fitlme(flu2,'FluRate ~ 1 + Region + (1|Date)')`
```Linear mixed-effects model fit by ML Model information: Number of observations 468 Fixed effects coefficients 9 Random effects coefficients 52 Covariance parameters 2 Formula: FluRate ~ 1 + Region + (1|Date) Model fit statistics: AIC BIC LogLikelihood Deviance 318.71 364.35 -148.36 296.71 Fixed effects coefficients (95% CIs): Name Estimate SE tStat DF pValue Lower Upper '(Intercept)' 1.2233 0.096678 12.654 459 1.085e-31 1.0334 1.4133 'Region_MidAtl' 0.010192 0.052221 0.19518 459 0.84534 -0.092429 0.11281 'Region_ENCentral' 0.051923 0.052221 0.9943 459 0.3206 -0.050698 0.15454 'Region_WNCentral' 0.23687 0.052221 4.5359 459 7.3324e-06 0.13424 0.33949 'Region_SAtl' 0.075481 0.052221 1.4454 459 0.14902 -0.02714 0.1781 'Region_ESCentral' 0.33917 0.052221 6.495 459 2.1623e-10 0.23655 0.44179 'Region_WSCentral' 0.069 0.052221 1.3213 459 0.18705 -0.033621 0.17162 'Region_Mtn' 0.046673 0.052221 0.89377 459 0.37191 -0.055948 0.14929 'Region_Pac' -0.16013 0.052221 -3.0665 459 0.0022936 -0.26276 -0.057514 Random effects covariance parameters (95% CIs): Group: Date (52 Levels) Name1 Name2 Type Estimate Lower Upper '(Intercept)' '(Intercept)' 'std' 0.6443 0.5297 0.78368 Group: Error Name Estimate Lower Upper 'Res Std' 0.26627 0.24878 0.285```

The p-values 7.3324e-06 and 2.1623e-10 respectively show that the fixed effects of the flu rates in regions `WNCentral` and `ESCentral` are significantly different relative to the flu rates in region `NE`.

The confidence limits for the standard deviation of the random-effects term, σ2b, do not include 0 (0.5297, 0.78368), which indicates that the random-effects term is significant. You can also test the significance of the random-effects terms using the `compare` method.

The estimated value of 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 best linear unbiased predictor (BLUP) of the flu rate for region `WNCentral` in week 10/9/2005 is

$\begin{array}{l}{\stackrel{^}{y}}_{WNCentral,10/9/2005}={\stackrel{^}{\beta }}_{0}+{\stackrel{^}{\beta }}_{3}I\left[WNCentral\right]+{\stackrel{^}{b}}_{10/9/2005}\\ \text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }=1.2233+0.23687-0.1718\\ \text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }=1.28837.\end{array}$

This is the fitted conditional response, since it includes contribution to the estimate from both the fixed and random effects. 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(4) + STATS.Estimate(STATS.Level=='10/9/2005')```
```y_hat = 1.2884```

You can simply display the fitted value using the `fitted` method.

```F = fitted(lme); F(flu2.Date == '10/9/2005' & flu2.Region == 'WNCentral')```
```ans = 1.2884```

Compute the fitted marginal response for region `WNCentral` in week 10/9/2005.

```F = fitted(lme,'Conditional',false); F(flu2.Date == '10/9/2005' & flu2.Region == 'WNCentral')```
```ans = 1.4602```

### Linear Mixed-Effects Model with a Random Slope

`load carbig`

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

$MP{G}_{im}={\beta }_{0}+{\beta }_{1}Ac{c}_{i}+{\beta }_{2}HP+{b}_{0m}+{b}_{1}{}_{m}Ac{c}_{im}+{\epsilon }_{im},\text{ }m=1,2,3,$

with the random-effects terms having the following prior distribution

${b}_{m}=\left(\begin{array}{l}{b}_{0m}\\ {b}_{1m}\end{array}\right)~N\left(0,\left(\begin{array}{cc}{\sigma }_{0}^{2}& {\sigma }_{0,1}\\ {\sigma }_{0,1}& {\sigma }_{1}^{2}\end{array}\right)\right),$

where D(θ) is the covariance matrix.

First, prepare the design matrices for fitting the linear mixed-effects model.

```X = [ones(406,1) Acceleration Horsepower]; Z = [ones(406,1) Acceleration]; Model_Year = nominal(Model_Year); G = Model_Year;```

Now, fit the model using `fitlmematrix` with the defined design matrices and grouping variables. Use the `'fminunc'` optimization algorithm.

```lme = fitlmematrix(X,MPG,Z,G,'FixedEffectPredictors',.... {'Intercept','Acceleration','Horsepower'},'RandomEffectPredictors',... {{'Intercept','Acceleration'}},'RandomEffectGroups',{'Model_Year'},... 'FitMethod','REML')```
```lme = Linear mixed-effects model fit by REML Model information: Number of observations 392 Fixed effects coefficients 3 Random effects coefficients 26 Covariance parameters 4 Formula: y ~ Intercept + Acceleration + Horsepower + (Intercept + Acceleration | Model_Year) Model fit statistics: AIC BIC LogLikelihood Deviance 2202.9 2230.7 -1094.5 2188.9 Fixed effects coefficients (95% CIs): Name Estimate SE tStat DF pValue Lower Upper 'Intercept' 50.064 2.3176 21.602 389 1.4185e-68 45.507 54.62 'Acceleration' -0.57897 0.13843 -4.1825 389 3.5654e-05 -0.85112 -0.30681 'Horsepower' -0.16958 0.0073242 -23.153 389 3.5289e-75 -0.18398 -0.15518 Random effects covariance parameters (95% CIs): Group: Model_Year (13 Levels) Name1 Name2 Type Estimate Lower Upper 'Intercept' 'Intercept' 'std' 3.72 1.5215 9.0954 'Acceleration' 'Intercept' 'corr' -0.8769 -0.98275 -0.33845 'Acceleration' 'Acceleration' 'std' 0.3593 0.19418 0.66483 Group: Error Name Estimate Lower Upper 'Res Std' 3.6913 3.4331 3.9688```

The fixed effects coefficients display includes the estimate, standard errors (`SE`), and the 95% confidence interval limits (`Lower` and `Upper`). The p-values for (`pValue`) indicate that all three fixed-effects coefficients are significant.

The confidence intervals for the standard deviations and the correlation between the random effects for intercept and acceleration do not include zeros, hence they seem significant. Use the `compare` method to test for the random effects.

Display the covariance matrix of the estimated fixed-effects coefficients.

`lme.CoefficientCovariance`
```ans = 5.3711 -0.2809 -0.0126 -0.2809 0.0192 0.0005 -0.0126 0.0005 0.0001```

The diagonal elements show the variances of the fixed-effects coefficient estimates. For example, the variance of the estimate of the intercept is 5.3711. Note that the standard errors of the estimates are the square roots of the variances. For example, the standard error of the intercept is 2.3176, which is `sqrt(5.3711)`.

The off-diagonal elements show the correlation between the fixed-effects coefficient estimates. For example, the correlation between the intercept and acceleration is –0.2809 and the correlation between acceleration and horsepower is 0.0005.

Display the coefficient of determination for the model.

`lme.Rsquared`
```ans = Ordinary: 0.7826 Adjusted: 0.7815```

The adjusted value is the R-squared value adjusted for the number of predictors in the model.