fitlmematrix
Fit linear mixed-effects model
Description
creates
a linear mixed-effects model of the responses lme
= fitlmematrix(X
,y
,Z
,[])y
using
the fixed-effects design matrix X
and random-effects
design matrix or matrices in Z
.
[]
implies that there is one group. That
is, the grouping variable G
is ones(n,1)
,
where n is the number of observations. Using fitlmematrix(X,Y,Z,[])
without
a specified covariance pattern most likely results in a nonidentifiable
model. This syntax is recommended only if you build the grouping information
into the random effects design Z
and specify a
covariance pattern for the random effects using the 'CovariancePattern'
name-value
pair argument.
also
creates a linear mixed-effects model with additional options specified
by one or more lme
= fitlmematrix(___,Name,Value
)Name,Value
pair arguments, using
any of the previous input arguments.
For example, you can specify the names of the response, predictor, and grouping variables. You can also specify the covariance pattern, fitting method, or the optimization algorithm.
Examples
No Grouping Variable Specified
Load the sample data.
load carsmall
Fit a linear mixed-effects model, where miles per gallon (MPG) is the response, weight is the predictor variable, and the intercept varies by model year. First, define the design matrices. Then, fit the model using the specified design matrices.
y = MPG; X = [ones(size(Weight)), Weight]; Z = ones(size(y)); lme = fitlmematrix(X,y,Z,Model_Year)
lme = Linear mixed-effects model fit by ML Model information: Number of observations 94 Fixed effects coefficients 2 Random effects coefficients 3 Covariance parameters 2 Formula: y ~ x1 + x2 + (z11 | g1) Model fit statistics: AIC BIC LogLikelihood Deviance 486.09 496.26 -239.04 478.09 Fixed effects coefficients (95% CIs): Name Estimate SE tStat DF pValue Lower Upper {'x1'} 43.575 2.3038 18.915 92 1.8371e-33 39 48.151 {'x2'} -0.0067097 0.0004242 -15.817 92 5.5373e-28 -0.0075522 -0.0058672 Random effects covariance parameters (95% CIs): Group: g1 (3 Levels) Name1 Name2 Type Estimate Lower Upper {'z11'} {'z11'} {'std'} 3.301 1.4448 7.5421 Group: Error Name Estimate Lower Upper {'Res Std'} 2.8997 2.5075 3.3532
Now, fit the same model by building the grouping into the Z
matrix.
Z = double([Model_Year==70, Model_Year==76, Model_Year==82]); lme = fitlmematrix(X,y,Z,[],'Covariancepattern','Isotropic')
lme = Linear mixed-effects model fit by ML Model information: Number of observations 94 Fixed effects coefficients 2 Random effects coefficients 3 Covariance parameters 2 Formula: y ~ x1 + x2 + (z11 + z12 + z13 | g1) Model fit statistics: AIC BIC LogLikelihood Deviance 486.09 496.26 -239.04 478.09 Fixed effects coefficients (95% CIs): Name Estimate SE tStat DF pValue Lower Upper {'x1'} 43.575 2.3038 18.915 92 1.8371e-33 39 48.151 {'x2'} -0.0067097 0.0004242 -15.817 92 5.5373e-28 -0.0075522 -0.0058672 Random effects covariance parameters (95% CIs): Group: g1 (1 Levels) Name1 Name2 Type Estimate Lower Upper {'z11'} {'z11'} {'std'} 3.301 1.4448 7.5421 Group: Error Name Estimate Lower Upper {'Res Std'} 2.8997 2.5075 3.3532
Longitudinal Study with a Covariate
Load the sample data.
load('weight.mat');
weight
contains data from a longitudinal study, where 20 subjects are randomly assigned 4 exercise programs (A, B, C, D) and their weight loss is recorded over six 2-week time periods. This is simulated data.
Define Subject
and Program
as categorical variables. Create the design matrices for a linear mixed-effects model, with the initial weight, type of program, week, and the interaction between the week and type of program as the fixed effects. The intercept and coefficient of week vary by subject.
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:
Subject = nominal(Subject); Program = nominal(Program); D = dummyvar(Program); % Create dummy variables for Program X = [ones(120,1), InitialWeight, D(:,2:4), Week,... D(:,2).*Week, D(:,3).*Week, D(:,4).*Week]; Z = [ones(120,1), Week]; G = Subject;
Since the model has an intercept, you only need the dummy variables for programs B, C, and D. This is also known as the 'reference'
method of coding dummy variables.
Fit the model using fitlmematrix
with the defined design matrices and grouping variables.
lme = fitlmematrix(X,y,Z,G,'FixedEffectPredictors',... {'Intercept','InitWeight','PrgB','PrgC','PrgD','Week','Week_PrgB','Week_PrgC','Week_PrgD'},... 'RandomEffectPredictors',{{'Intercept','Week'}},'RandomEffectGroups',{'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 ~ Intercept + InitWeight + PrgB + PrgC + PrgD + Week + Week_PrgB + Week_PrgC + Week_PrgD + (Intercept + 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 pValue Lower Upper {'Intercept' } 0.66105 0.25892 2.5531 111 0.012034 0.14798 1.1741 {'InitWeight'} 0.0031879 0.0013814 2.3078 111 0.022863 0.00045067 0.0059252 {'PrgB' } 0.36079 0.13139 2.746 111 0.0070394 0.10044 0.62113 {'PrgC' } -0.033263 0.13117 -0.25358 111 0.80029 -0.29319 0.22666 {'PrgD' } 0.11317 0.13132 0.86175 111 0.39068 -0.14706 0.3734 {'Week' } 0.1732 0.067454 2.5677 111 0.011567 0.039536 0.30686 {'Week_PrgB' } 0.038771 0.095394 0.40644 111 0.68521 -0.15026 0.2278 {'Week_PrgC' } 0.030543 0.095394 0.32018 111 0.74944 -0.15849 0.21957 {'Week_PrgD' } 0.033114 0.095394 0.34713 111 0.72915 -0.15592 0.22214 Random effects covariance parameters (95% CIs): Group: Subject (20 Levels) Name1 Name2 Type Estimate Lower Upper {'Intercept'} {'Intercept'} {'std' } 0.18407 0.12281 0.27587 {'Week' } {'Intercept'} {'corr'} 0.66841 0.21077 0.88573 {'Week' } {'Week' } {'std' } 0.15033 0.11004 0.20537 Group: Error Name Estimate Lower Upper {'Res Std'} 0.10261 0.087882 0.11981
Examine the fixed effects coefficients table. The row labeled 'InitWeight'
has a -value of 0.0228, and the row labeled 'Week'
has a -value of 0.0115. These -values indicate significant effects of the initial weights of the subjects and the time factor 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 zero, thus they seem significant. You can also test the significance of the random-effects using the compare
method.
Random Intercept Model
Load the sample data.
load flu
The flu
dataset array has a Date
variable, and 10 variables for 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, where the influenza rates are the responses, combine the nine columns corresponding to the regions into an array that has a single response variable, FluRate
, and a nominal variable, Region
, the nationwide estimate WtdILI
, 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);
Define the design matrices for a random-intercept linear mixed-effects model, where the intercept varies by Date
. The corresponding model is
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,
y = flu2.FluRate; X = [ones(468,1) flu2.WtdILI]; Z = [ones(468,1)]; G = flu2.Date;
Fit the linear mixed-effects model.
lme = fitlmematrix(X,y,Z,G,'FixedEffectPredictors',{'Intercept','NationalRate'},... 'RandomEffectPredictors',{{'Intercept'}},'RandomEffectGroups',{'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: y ~ Intercept + NationalRate + (Intercept | 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 Lower Upper {'Intercept' } 0.16385 0.057525 2.8484 466 0.0045885 0.050813 0.27689 {'NationalRate'} 0.7236 0.032219 22.459 466 3.0502e-76 0.66028 0.78691 Random effects covariance parameters (95% CIs): Group: Date (52 Levels) Name1 Name2 Type Estimate Lower Upper {'Intercept'} {'Intercept'} {'std'} 0.17146 0.13227 0.22226 Group: Error Name Estimate Lower Upper {'Res Std'} 0.30201 0.28217 0.32324
The confidence limits of the standard deviation of the random-effects term , do not include zero (0.13227, 0.22226), which indicates that the random-effects term is significant. You can also test the significance of the random-effects using compare
method.
The estimated value of an observation is the sum of the fixed-effects values and value of the random effect at the grouping variable level corresponding to that observation. For example, the estimated flu rate for observation 28
where is the 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 simply display the fitted value using the fitted(lme)
method.
F = fitted(lme); F(28)
ans = 1.4674
Randomized Block Design
Load the sample data.
load('shift.mat');
The data shows the deviations from the target quality characteristic measured from the products that 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 deviations of the quality characteristics from the target value. This is simulated data.
Define the design matrices for a linear mixed-effects model with a random intercept grouped by operator, and shift as the fixed effects. Use the 'effects'
contrasts. 'effects'
contrasts mean that the coefficients sum to 0. You need to create two contrast coded variables in the fixed-effects design matrix, X1
and X2
, 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
S = shift.Shift; X1 = (S=='Morning') - (S=='Night'); X2 = (S=='Evening') - (S=='Night'); X = [ones(15,1), X1, X2]; y = shift.QCDev; Z = ones(15,1); G = shift.Operator;
Fit a linear mixed-effects model using the specified design matrices and restricted maximum likelihood method.
lme = fitlmematrix(X,y,Z,G,'FitMethod','REML','FixedEffectPredictors',.... {'Intercept','S_Morning','S_Evening'},'RandomEffectPredictors',{{'Intercept'}},... 'RandomEffectGroups',{'Operator'},'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: y ~ Intercept + S_Morning + S_Evening + (Intercept | 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 Lower Upper {'Intercept'} 3.6525 0.94109 3.8812 12 0.0021832 1.6021 5.703 {'S_Morning'} -0.91973 0.31206 -2.9473 12 0.012206 -1.5997 -0.23981 {'S_Evening'} -0.53293 0.31206 -1.7078 12 0.11339 -1.2129 0.14699 Random effects covariance parameters (95% CIs): Group: Operator (5 Levels) Name1 Name2 Type Estimate Lower Upper {'Intercept'} {'Intercept'} {'std'} 2.0457 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 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
Correlated and Uncorrelated Random-Effects Terms
Load the sample data.
load carbig
Fit a linear mixed-effects model for miles per gallon (MPG), with fixed effects for acceleration and horsepower, and uncorrelated random effect for intercept and acceleration grouped by the model year. This model corresponds to
with the random-effects terms having the following prior distributions:
where represents the model year.
First, prepare the design matrices for fitting the linear mixed-effects model.
X = [ones(406,1) Acceleration Horsepower]; Z = {ones(406,1),Acceleration}; G = {Model_Year,Model_Year}; Model_Year = nominal(Model_Year);
Now, fit the model using fitlmematrix
with the defined design matrices and grouping variables.
lme = fitlmematrix(X,MPG,Z,G,'FixedEffectPredictors',.... {'Intercept','Acceleration','Horsepower'},'RandomEffectPredictors',... {{'Intercept'},{'Acceleration'}},'RandomEffectGroups',{'Model_Year','Model_Year'})
lme = Linear mixed-effects model fit by ML Model information: Number of observations 392 Fixed effects coefficients 3 Random effects coefficients 26 Covariance parameters 3 Formula: y ~ Intercept + Acceleration + Horsepower + (Intercept | Model_Year) + (Acceleration | Model_Year) Model fit statistics: AIC BIC LogLikelihood Deviance 2194.5 2218.3 -1091.3 2182.5 Fixed effects coefficients (95% CIs): Name Estimate SE tStat DF pValue Lower Upper {'Intercept' } 49.839 2.0518 24.291 389 5.6168e-80 45.806 53.873 {'Acceleration'} -0.58565 0.10846 -5.3995 389 1.1652e-07 -0.7989 -0.3724 {'Horsepower' } -0.16534 0.0071227 -23.213 389 1.9755e-75 -0.17934 -0.15133 Random effects covariance parameters (95% CIs): Group: Model_Year (13 Levels) Name1 Name2 Type Estimate Lower Upper {'Intercept'} {'Intercept'} {'std'} 8.0929e-07 NaN NaN Group: Model_Year (13 Levels) Name1 Name2 Type Estimate Lower Upper {'Acceleration'} {'Acceleration'} {'std'} 0.18783 0.17509 0.2015 Group: Error Name Estimate Lower Upper {'Res Std'} 3.7258 3.4731 3.9969
Note that the random effects covariance parameters for intercept and acceleration are separate in the display. The standard deviation of the random effect for the intercept does not seem significant.
Refit the model with potentially correlated random effects for intercept and acceleration. In this case, the random-effects terms has this prior distribution
where represents the model year.
First, prepare the random-effects design matrix and grouping variable.
Z = [ones(406,1) Acceleration]; G = Model_Year; lme = fitlmematrix(X,MPG,Z,G,'FixedEffectPredictors',.... {'Intercept','Acceleration','Horsepower'},'RandomEffectPredictors',... {{'Intercept','Acceleration'}},'RandomEffectGroups',{'Model_Year'})
lme = Linear mixed-effects model fit by ML 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 2193.5 2221.3 -1089.7 2179.5 Fixed effects coefficients (95% CIs): Name Estimate SE tStat DF pValue Lower Upper {'Intercept' } 50.133 2.2652 22.132 389 7.7727e-71 45.679 54.586 {'Acceleration'} -0.58327 0.13394 -4.3545 389 1.7075e-05 -0.84661 -0.31992 {'Horsepower' } -0.16954 0.0072609 -23.35 389 5.188e-76 -0.18382 -0.15527 Random effects covariance parameters (95% CIs): Group: Model_Year (13 Levels) Name1 Name2 Type Estimate Lower Upper {'Intercept' } {'Intercept' } {'std' } 3.3475 1.2862 8.7119 {'Acceleration'} {'Intercept' } {'corr'} -0.87971 -0.98501 -0.29675 {'Acceleration'} {'Acceleration'} {'std' } 0.33789 0.1825 0.62558 Group: Error Name Estimate Lower Upper {'Res Std'} 3.6874 3.4298 3.9644
Note that the random effects covariance parameters for intercept and acceleration are together in the display, with an addition of the correlation between the intercept and acceleration. The confidence intervals for the standard deviations and the correlation between the random effects for intercept and acceleration do not include 0s, hence they seem significant. You can compare these two models using the compare
method.
Specify the Covariance Pattern
Load the sample data.
load('weight.mat');
weight
contains data from a longitudinal study, where 20 subjects are randomly assigned 4 exercise programs, and their weight loss is recorded over six 2-week time periods. This is simulated data.
Define Subject
and Program
as categorical variables.
Subject = nominal(Subject); Program = nominal(Program);
Create the design matrices for a linear mixed-effects model, with the initial weight, type of program, and week as the fixed effects.
D = dummyvar(Program); X = [ones(120,1), InitialWeight, D(:,2:4), Week]; Z = [ones(120,1) Week]; G = Subject;
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:
Fit the model using fitlmematrix
with the defined design matrices and grouping variables. Assume the repeated observations collected on a subject have common variance along diagonals.
lme = fitlmematrix(X,y,Z,G,'FixedEffectPredictors',... {'Intercept','InitWeight','PrgB','PrgC','PrgD','Week'},... 'RandomEffectPredictors',{{'Intercept','Week'}},... 'RandomEffectGroups',{'Subject'},'CovariancePattern','Isotropic')
lme = Linear mixed-effects model fit by ML Model information: Number of observations 120 Fixed effects coefficients 6 Random effects coefficients 40 Covariance parameters 2 Formula: y ~ Intercept + InitWeight + PrgB + PrgC + PrgD + Week + (Intercept + Week | Subject) Model fit statistics: AIC BIC LogLikelihood Deviance -24.783 -2.483 20.391 -40.783 Fixed effects coefficients (95% CIs): Name Estimate SE tStat DF pValue Lower Upper {'Intercept' } 0.4208 0.28169 1.4938 114 0.13799 -0.13723 0.97883 {'InitWeight'} 0.0045552 0.0015338 2.9699 114 0.0036324 0.0015168 0.0075935 {'PrgB' } 0.36993 0.12119 3.0525 114 0.0028242 0.12986 0.61 {'PrgC' } -0.034009 0.1209 -0.28129 114 0.77899 -0.27351 0.2055 {'PrgD' } 0.121 0.12111 0.99911 114 0.31986 -0.11891 0.36091 {'Week' } 0.19881 0.037134 5.3538 114 4.5191e-07 0.12525 0.27237 Random effects covariance parameters (95% CIs): Group: Subject (20 Levels) Name1 Name2 Type Estimate Lower Upper {'Intercept'} {'Intercept'} {'std'} 0.16561 0.12896 0.21269 Group: Error Name Estimate Lower Upper {'Res Std'} 0.10272 0.088014 0.11987
Input Arguments
X
— Fixed-effects design matrix
n-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 values
n-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 design
n-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 ofZ
contains an n-by-q(r) design matrixZ{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 variables
n-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, logical vector, numeric vector, character array, string array, or cell array of character vectors.If there are multiple random-effects terms, then
G
must be a cell array of length R. Each cell ofG
contains a grouping variableG{r}
, r = 1, 2, ..., R, with M(r) levels.G{r}
can be a categorical vector, logical vector, numeric vector, character array, string array, or cell array of character vectors.
Data Types: categorical
| logical
| single
| double
| char
| string
| cell
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','DummyVarCoding','full','Optimizer','fminunc'
specifies
a random-effects covariance pattern with zero off-diagonal elements,
creates a dummy variable for each level of a categorical variable,
and uses the fminunc
optimization algorithm.
FixedEffectPredictors
— Names of columns in fixed-effects design matrix
{'x1','x2',...,'xP'}
(default) | string array or cell array of length p
Names of columns in the fixed-effects design matrix X
, specified as the
comma-separated pair consisting of 'FixedEffectPredictors'
and a
string array or cell array of length p.
For example, if you have a constant term and two predictors,
say TimeSpent
and Gender
, where Female
is
the reference level for Gender
, as the fixed effects,
then you can specify the names of your fixed effects in the following
way. Gender_Male
represents the dummy variable
you must create for category Male
. You can choose
different names for these variables.
Example: 'FixedEffectPredictors',{'Intercept','TimeSpent','Gender_Male'}
,
Data Types: string
| cell
RandomEffectPredictors
— Names of columns in random-effects design matrix or cell array
string array or cell array of length q | cell array of length R with elements of length
q(r), r = 1, 2, ...,
R
Names of columns in the random-effects design matrix or cell
array Z
, specified as the comma-separated pair
consisting of 'RandomEffectPredictors'
and either
of the following:
A string array or cell array of length q when
Z
is an n-by-q design matrix. In this case, the default is{'z1','z2',...,'zQ'}
.A cell array of length R, when
Z
is a cell array of length R with each elementZ{r}
of length q(r), r = 1, 2, ..., R. In this case, the default is{'z11','z12',...,'z1Q(1)'},...,{'zr1','zr2',...,'zrQ(r)'}
.
For example, suppose you have correlated random effects for
intercept and a variable named Acceleration
. Then,
you can specify the random-effects predictor names as follows.
Example: 'RandomEffectPredictors',{'Intercept','Acceleration'}
If you have two random effects terms, one for the intercept
and the variable Acceleration
grouped by variable g1
,
and the second for the intercept, grouped by the variable g2
,
then you specify the random-effects predictor names as follows.
Example: 'RandomEffectPredictors',{{'Intercept','Acceleration'},{'Intercept'}}
Data Types: string
| cell
ResponseVarName
— Name of response variable
'y'
(default) | character vector | string scalar
Name of response variable, specified as the comma-separated pair consisting of
'ResponseVarName'
and a character vector or string scalar.
For example, if your response variable name is score
,
then you can specify it as follows.
Example: 'ResponseVarName','score'
Data Types: char
| string
RandomEffectGroups
— Names of random effects grouping variables
'g'
or
{'g1','g2',...,'gR'}
(default) | character vector | string scalar | string array | cell array of character vectors
Names of random effects grouping variables, specified as the
comma-separated pair 'RandomEffectGroups'
and either
of the following:
Character vector or string scalar — If there is only one random-effects term, that is, if
G
is a vector, then the value of'RandomEffectGroups'
is the name for the grouping variableG
. The default is'g'
.String array or cell array of character vectors — If there are multiple random-effects terms, that is, if
G
is a cell array of length R, then the value of'RandomEffectGroups'
is a string array or cell array of length R, where each element is the name for the grouping variableG{r}
. The default is{'g1','g2',...,'gR'}
.
For example, if you have two random-effects terms, z1
and z2
,
grouped by the grouping variables sex
and subject
,
then you can specify the names of your grouping variables as follows.
Example: 'RandomEffectGroups',{'sex','subject'}
Data Types: char
| string
| cell
CovariancePattern
— Pattern of covariance matrix
'FullCholesky'
(default) | character vector | string scalar | square symmetric logical matrix | string array | cell array of character vectors or logical matrices
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. |
'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 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 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
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 weights
vector 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 exclude
use all rows without NaNs
(default) | vector of integer or logical values
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
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
variables in this table.
Value | Description |
---|---|
'reference' (default) | fitlmematrix 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' | fitlmematrix 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' | fitlmematrix 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'
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 algorithm
structure 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 useoptimoptions('fminunc')
to change the options of the optimization algorithm. Seeoptimoptions
for the options'fminunc'
uses. If'Optimizer'
is'fminunc'
and you do not supply'OptimizerOptions'
, then the default forLinearMixedModel
is the default options created byoptimoptions('fminunc')
with'Algorithm'
set to'quasi-newton'
.If
'Optimizer'
is'quasinewton'
, then usestatset('LinearMixedModel')
to change the optimization parameters. If you don’t change the optimization parameters, thenLinearMixedModel
uses the default options created bystatset('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 value
Relative 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 value
Absolute tolerance on the step size, specified as a positive scalar value.
MaxIter
— Maximum number of iterations allowed
10000
(default) | positive scalar value
Maximum 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
Output Arguments
lme
— Linear mixed-effects model
LinearMixedModel
object
Linear mixed-effects model, returned as a LinearMixedModel
object.
More About
Cholesky Parameterization
One of the assumptions of linear mixed-effects models is that the random effects have the following prior distribution.
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
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
then
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,
Alternative Functionality
You can also fit a linear mixed-effects model using fitlme(tbl,formula)
,
where tbl
is a table or dataset array containing
the response y
, the predictor variables X
,
and the grouping variables, and formula
is of
the form 'y ~ fixed + (random1|g1)
+ ... + (randomR|gR)'
.
Version History
Introduced in R2013b
See Also
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