Linear Mixed Models in Stata
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1 Linear Mixed Models in Stata Roberto G. Gutierrez Director of Statistics StataCorp LP Fourth German Stata Users Group Meeting R. Gutierrez (StataCorp) Linear Mixed Models in Stata March 31, / 30
2 Outline 1 The Linear Mixed Model 2 One-Level Models 3 Two-Level Models 4 Factor Notation 5 A Glimpse at the Future R. Gutierrez (StataCorp) Linear Mixed Models in Stata March 31, / 30
3 Outline 1 The Linear Mixed Model 2 One-Level Models 3 Two-Level Models 4 Factor Notation 5 A Glimpse at the Future R. Gutierrez (StataCorp) Linear Mixed Models in Stata March 31, / 30
4 Outline 1 The Linear Mixed Model 2 One-Level Models 3 Two-Level Models 4 Factor Notation 5 A Glimpse at the Future R. Gutierrez (StataCorp) Linear Mixed Models in Stata March 31, / 30
5 Outline 1 The Linear Mixed Model 2 One-Level Models 3 Two-Level Models 4 Factor Notation 5 A Glimpse at the Future R. Gutierrez (StataCorp) Linear Mixed Models in Stata March 31, / 30
6 Outline 1 The Linear Mixed Model 2 One-Level Models 3 Two-Level Models 4 Factor Notation 5 A Glimpse at the Future R. Gutierrez (StataCorp) Linear Mixed Models in Stata March 31, / 30
7 Definition y = Xβ + Zu + ɛ where y is the n 1 vector of responses X is the n p fixed-effects design matrix β are the fixed effects Z is the n q random-effects design matrix u are the random effects ɛ is the n 1 vector of errors such that [ ] ( [ u G 0 N 0, ɛ 0 σɛ 2 I n ]) R. Gutierrez (StataCorp) Linear Mixed Models in Stata March 31, / 30
8 Variance components Random effects are not directly estimated, but instead characterized by the elements of G, known as variance components You can, however predict random effects As such, you fit a mixed model by estimating β, σ 2 ɛ, and the variance components in G R. Gutierrez (StataCorp) Linear Mixed Models in Stata March 31, / 30
9 Variance components Random effects are not directly estimated, but instead characterized by the elements of G, known as variance components You can, however predict random effects As such, you fit a mixed model by estimating β, σ 2 ɛ, and the variance components in G R. Gutierrez (StataCorp) Linear Mixed Models in Stata March 31, / 30
10 Variance components Random effects are not directly estimated, but instead characterized by the elements of G, known as variance components You can, however predict random effects As such, you fit a mixed model by estimating β, σ 2 ɛ, and the variance components in G R. Gutierrez (StataCorp) Linear Mixed Models in Stata March 31, / 30
11 Panel representation Classical representation has roots in the design literature, but can make model specification hard When the data can be thought of as M independent panels, it is more convenient to express the mixed model as (for i = 1,..., M) y i = X i β + Z i u i + ɛ i where u i N(0, S), for q q variance S, and Z Z 2 0 u 1 Z = ; u =. ; u Z M M G = I M S R. Gutierrez (StataCorp) Linear Mixed Models in Stata March 31, / 30
12 Panel representation Classical representation has roots in the design literature, but can make model specification hard When the data can be thought of as M independent panels, it is more convenient to express the mixed model as (for i = 1,..., M) y i = X i β + Z i u i + ɛ i where u i N(0, S), for q q variance S, and Z Z 2 0 u 1 Z = ; u =. ; u Z M M G = I M S R. Gutierrez (StataCorp) Linear Mixed Models in Stata March 31, / 30
13 Using the panel representation For example, take a random intercept model. In the classical framework, the random intercepts are random coefficients on indicator variables identifying each panel It is better to just think at the panel level and consider M realizations of a random intercept This generalizes to more than one level of nested panels Issue of terminology for multi-level models R. Gutierrez (StataCorp) Linear Mixed Models in Stata March 31, / 30
14 Using the panel representation For example, take a random intercept model. In the classical framework, the random intercepts are random coefficients on indicator variables identifying each panel It is better to just think at the panel level and consider M realizations of a random intercept This generalizes to more than one level of nested panels Issue of terminology for multi-level models R. Gutierrez (StataCorp) Linear Mixed Models in Stata March 31, / 30
15 Using the panel representation For example, take a random intercept model. In the classical framework, the random intercepts are random coefficients on indicator variables identifying each panel It is better to just think at the panel level and consider M realizations of a random intercept This generalizes to more than one level of nested panels Issue of terminology for multi-level models R. Gutierrez (StataCorp) Linear Mixed Models in Stata March 31, / 30
16 Using the panel representation For example, take a random intercept model. In the classical framework, the random intercepts are random coefficients on indicator variables identifying each panel It is better to just think at the panel level and consider M realizations of a random intercept This generalizes to more than one level of nested panels Issue of terminology for multi-level models R. Gutierrez (StataCorp) Linear Mixed Models in Stata March 31, / 30
17 One-level models Example Consider the Junior School Project data which compares math scores of various schools in the third and fifth years Data on n = 887 pupils in M = 48 schools Let s fit the model math5 ij = β 0 + β 1 math3 ij + u i + ɛ ij for i = 1,..., 48 schools and j = 1,..., n i pupils. u i is a random effect (intercept) at the school level R. Gutierrez (StataCorp) Linear Mixed Models in Stata March 31, / 30
18 One-level models Example Consider the Junior School Project data which compares math scores of various schools in the third and fifth years Data on n = 887 pupils in M = 48 schools Let s fit the model math5 ij = β 0 + β 1 math3 ij + u i + ɛ ij for i = 1,..., 48 schools and j = 1,..., n i pupils. u i is a random effect (intercept) at the school level R. Gutierrez (StataCorp) Linear Mixed Models in Stata March 31, / 30
19 . xtmixed math5 math3 school: Performing EM optimization: Performing gradient-based optimization: Iteration 0: log restricted-likelihood = Iteration 1: log restricted-likelihood = Computing standard errors: Mixed-effects REML regression Number of obs = 887 Group variable: school Number of groups = 48 Obs per group: min = 5 avg = 18.5 max = 62 Wald chi2(1) = Log restricted-likelihood = Prob > chi2 = math5 Coef. Std. Err. z P> z [95% Conf. Interval] math _cons Random-effects Parameters Estimate Std. Err. [95% Conf. Interval] school: Identity sd(_cons) sd(residual) LR test vs. linear regression: chibar2(01) = Prob >= chibar2 = R. Gutierrez (StataCorp) Linear Mixed Models in Stata March 31, / 30
20 Adding a random slope For the most part, the previous is what you would get using xtreg Consider instead the model math5 ij = β 0 + β 1 math3 ij + u 0i + u 1i math3 ij + ɛ ij In essence, each school has its own random regression line such that the intercept is N(β 0, σ 2 0 ) and the slope on math3 is N(β 1, σ 2 1 ) R. Gutierrez (StataCorp) Linear Mixed Models in Stata March 31, / 30
21 Adding a random slope For the most part, the previous is what you would get using xtreg Consider instead the model math5 ij = β 0 + β 1 math3 ij + u 0i + u 1i math3 ij + ɛ ij In essence, each school has its own random regression line such that the intercept is N(β 0, σ 2 0 ) and the slope on math3 is N(β 1, σ 2 1 ) R. Gutierrez (StataCorp) Linear Mixed Models in Stata March 31, / 30
22 Adding a random slope For the most part, the previous is what you would get using xtreg Consider instead the model math5 ij = β 0 + β 1 math3 ij + u 0i + u 1i math3 ij + ɛ ij In essence, each school has its own random regression line such that the intercept is N(β 0, σ 2 0 ) and the slope on math3 is N(β 1, σ 2 1 ) R. Gutierrez (StataCorp) Linear Mixed Models in Stata March 31, / 30
23 . xtmixed math5 math3 school: math3 (output omitted ) Mixed-effects REML regression Number of obs = 887 Group variable: school Number of groups = 48 Obs per group: min = 5 avg = 18.5 max = 62 Wald chi2(1) = Log restricted-likelihood = Prob > chi2 = math5 Coef. Std. Err. z P> z [95% Conf. Interval] math _cons Random-effects Parameters Estimate Std. Err. [95% Conf. Interval] school: Independent sd(math3) sd(_cons) sd(residual) LR test vs. linear regression: chi2(2) = Prob > chi2 = Note: LR test is conservative and provided only for reference R. Gutierrez (StataCorp) Linear Mixed Models in Stata March 31, / 30
24 Predict Random effects are not estimated, but the can be predicted (BLUPs). predict r1 r0, reffects. describe r* storage display value variable name type format label variable label r1 float %9.0g BLUP r.e. for school: math3 r0 float %9.0g BLUP r.e. for school: _cons. gen b0 = _b[_cons] + r0. gen b1 = _b[math3] + r1. bysort school: gen tolist = _n==1 R. Gutierrez (StataCorp) Linear Mixed Models in Stata March 31, / 30
25 Predict Random effects are not estimated, but the can be predicted (BLUPs). predict r1 r0, reffects. describe r* storage display value variable name type format label variable label r1 float %9.0g BLUP r.e. for school: math3 r0 float %9.0g BLUP r.e. for school: _cons. gen b0 = _b[_cons] + r0. gen b1 = _b[math3] + r1. bysort school: gen tolist = _n==1 R. Gutierrez (StataCorp) Linear Mixed Models in Stata March 31, / 30
26 Predict. list school b0 b1 if school<=10 & tolist school b0 b R. Gutierrez (StataCorp) Linear Mixed Models in Stata March 31, / 30
27 Option fitted We could use these intercepts and slopes to plot the estimated lines for each school. Equivalently, we could just plot the fitted values. predict math5hat, fitted. sort school math3. twoway connected math5hat math3 if school<=10, connect(l) R. Gutierrez (StataCorp) Linear Mixed Models in Stata March 31, / 30
28 Option fitted We could use these intercepts and slopes to plot the estimated lines for each school. Equivalently, we could just plot the fitted values. predict math5hat, fitted. sort school math3. twoway connected math5hat math3 if school<=10, connect(l) R. Gutierrez (StataCorp) Linear Mixed Models in Stata March 31, / 30
29 Fitted values: xb + Zu math3 R. Gutierrez (StataCorp) Linear Mixed Models in Stata March 31, / 30
30 Covariance structures In our previous model, it was assumed that u 0i and u 1i are independent. That is, [ ] σ 2 S = σ1 2 But, what if we also wanted to estimate a covariance? R. Gutierrez (StataCorp) Linear Mixed Models in Stata March 31, / 30
31 Covariance structures In our previous model, it was assumed that u 0i and u 1i are independent. That is, [ ] σ 2 S = σ1 2 But, what if we also wanted to estimate a covariance? R. Gutierrez (StataCorp) Linear Mixed Models in Stata March 31, / 30
32 . xtmixed math5 math3 school: math3, cov(unstructured) var mle Mixed-effects ML regression Number of obs = 887 Group variable: school Number of groups = 48 Obs per group: min = 5 avg = 18.5 max = 62 Wald chi2(1) = Log likelihood = Prob > chi2 = math5 Coef. Std. Err. z P> z [95% Conf. Interval] math _cons Random-effects Parameters Estimate Std. Err. [95% Conf. Interval] school: Unstructured var(math3) var(_cons) cov(math3,_cons) var(residual) LR test vs. linear regression: chi2(3) = Prob > chi2 = Note: LR test is conservative and provided only for reference R. Gutierrez (StataCorp) Linear Mixed Models in Stata March 31, / 30
33 Notes We also added options variance and mle to fully reproduce the results found in the gllamm manual Again, we can compare this model with previous using lrtest Available covariance structures are Independent (default), Identity, Exchangeable, and Unstructured R. Gutierrez (StataCorp) Linear Mixed Models in Stata March 31, / 30
34 Notes We also added options variance and mle to fully reproduce the results found in the gllamm manual Again, we can compare this model with previous using lrtest Available covariance structures are Independent (default), Identity, Exchangeable, and Unstructured R. Gutierrez (StataCorp) Linear Mixed Models in Stata March 31, / 30
35 Notes We also added options variance and mle to fully reproduce the results found in the gllamm manual Again, we can compare this model with previous using lrtest Available covariance structures are Independent (default), Identity, Exchangeable, and Unstructured R. Gutierrez (StataCorp) Linear Mixed Models in Stata March 31, / 30
36 ML vs. REML ML is based on standard normal theory With REML, the likelihood is that of a set of linear constrasts of y that do not depend on the fixed effects REML variance components are less biased in small samples, since they incorporate degrees of freedom used to estimated fixed effects REML estimates are unbiased in balanced data LR tests are always valid with ML, not so with REML Very much a matter of personal taste R. Gutierrez (StataCorp) Linear Mixed Models in Stata March 31, / 30
37 ML vs. REML ML is based on standard normal theory With REML, the likelihood is that of a set of linear constrasts of y that do not depend on the fixed effects REML variance components are less biased in small samples, since they incorporate degrees of freedom used to estimated fixed effects REML estimates are unbiased in balanced data LR tests are always valid with ML, not so with REML Very much a matter of personal taste R. Gutierrez (StataCorp) Linear Mixed Models in Stata March 31, / 30
38 ML vs. REML ML is based on standard normal theory With REML, the likelihood is that of a set of linear constrasts of y that do not depend on the fixed effects REML variance components are less biased in small samples, since they incorporate degrees of freedom used to estimated fixed effects REML estimates are unbiased in balanced data LR tests are always valid with ML, not so with REML Very much a matter of personal taste R. Gutierrez (StataCorp) Linear Mixed Models in Stata March 31, / 30
39 ML vs. REML ML is based on standard normal theory With REML, the likelihood is that of a set of linear constrasts of y that do not depend on the fixed effects REML variance components are less biased in small samples, since they incorporate degrees of freedom used to estimated fixed effects REML estimates are unbiased in balanced data LR tests are always valid with ML, not so with REML Very much a matter of personal taste R. Gutierrez (StataCorp) Linear Mixed Models in Stata March 31, / 30
40 ML vs. REML ML is based on standard normal theory With REML, the likelihood is that of a set of linear constrasts of y that do not depend on the fixed effects REML variance components are less biased in small samples, since they incorporate degrees of freedom used to estimated fixed effects REML estimates are unbiased in balanced data LR tests are always valid with ML, not so with REML Very much a matter of personal taste R. Gutierrez (StataCorp) Linear Mixed Models in Stata March 31, / 30
41 ML vs. REML ML is based on standard normal theory With REML, the likelihood is that of a set of linear constrasts of y that do not depend on the fixed effects REML variance components are less biased in small samples, since they incorporate degrees of freedom used to estimated fixed effects REML estimates are unbiased in balanced data LR tests are always valid with ML, not so with REML Very much a matter of personal taste R. Gutierrez (StataCorp) Linear Mixed Models in Stata March 31, / 30
42 Two-level models Example Baltagi et al. (2001) estimate a Cobb-Douglas production function examining the productivity of public capital in each state s private output. For y equal to the log of the gross state product measured each year from , the model is y ij = X ij β + u i + v j(i) + ɛ ij for j = 1,..., M i states nested within i = 1,..., 9 regions. X consists of various economic factors treated as fixed effects. R. Gutierrez (StataCorp) Linear Mixed Models in Stata March 31, / 30
43 Estimated fixed effects. xtmixed gsp private emp hwy water other unemp region: state: Mixed-effects REML regression Number of obs = 816 No. of Observations per Group Group Variable Groups Minimum Average Maximum region state Wald chi2(6) = Log restricted-likelihood = Prob > chi2 = gsp Coef. Std. Err. z P> z [95% Conf. Interval] private emp hwy water other unemp _cons R. Gutierrez (StataCorp) Linear Mixed Models in Stata March 31, / 30
44 Estimated variance components Random-effects Parameters Estimate Std. Err. [95% Conf. Interval] region: Identity state: Identity sd(_cons) sd(_cons) sd(residual) LR test vs. linear regression: chi2(2) = Prob > chi2 = R. Gutierrez (StataCorp) Linear Mixed Models in Stata March 31, / 30
45 Constraints on variance components We begin by adding some random coefficients at the region level. xtmixed gsp private emp hwy water other unemp region: hwy unemp state:, > nolog nogroup nofetable Mixed-effects REML regression Number of obs = 816 Wald chi2(6) = Log restricted-likelihood = Prob > chi2 = Random-effects Parameters Estimate Std. Err. [95% Conf. Interval] region: Independent sd(hwy) sd(unemp) sd(_cons) state: Identity sd(_cons) sd(residual) LR test vs. linear regression: chi2(4) = Prob > chi2 = R. Gutierrez (StataCorp) Linear Mixed Models in Stata March 31, / 30
46 Constraints on variance components We can constrain the variance components on hwy and unemp to be equal with. xtmixed gsp private emp hwy water other unemp region: hwy unemp, cov(ident > ity) region: state:, nolog nogroup nofetable Mixed-effects REML regression Number of obs = 816 Wald chi2(6) = Log restricted-likelihood = Prob > chi2 = Random-effects Parameters Estimate Std. Err. [95% Conf. Interval] region: Identity sd(hwy unemp) region: Identity state: Identity sd(_cons) sd(_cons) sd(residual) LR test vs. linear regression: chi2(3) = Prob > chi2 = R. Gutierrez (StataCorp) Linear Mixed Models in Stata March 31, / 30
47 Factor Notation Sometimes random effects are crossed rather than nested Example Consider a dataset consisting of weight measurements on 48 pigs at each of 9 weeks. We wish to fit the following model weight ij = β 0 + β 1 week ij + u i + v j + ɛ ij for i = 1,..., 48 pigs and j = 1,..., 9 weeks Note that the week random effects v j are not nested within pigs, they are assumed to be the same for each pig R. Gutierrez (StataCorp) Linear Mixed Models in Stata March 31, / 30
48 Factor Notation Sometimes random effects are crossed rather than nested Example Consider a dataset consisting of weight measurements on 48 pigs at each of 9 weeks. We wish to fit the following model weight ij = β 0 + β 1 week ij + u i + v j + ɛ ij for i = 1,..., 48 pigs and j = 1,..., 9 weeks Note that the week random effects v j are not nested within pigs, they are assumed to be the same for each pig R. Gutierrez (StataCorp) Linear Mixed Models in Stata March 31, / 30
49 Factor Notation Sometimes random effects are crossed rather than nested Example Consider a dataset consisting of weight measurements on 48 pigs at each of 9 weeks. We wish to fit the following model weight ij = β 0 + β 1 week ij + u i + v j + ɛ ij for i = 1,..., 48 pigs and j = 1,..., 9 weeks Note that the week random effects v j are not nested within pigs, they are assumed to be the same for each pig R. Gutierrez (StataCorp) Linear Mixed Models in Stata March 31, / 30
50 Fitting the model One approach to fitting this model is to consider the data as a whole and treat the random effects as random coefficients on lots of indicator variables, that is u 1. [ ] u = u 48 σ 2 v 1 N(0, G); G = u I σvi 2 9. v 9 We could generate these indicator variables, but luckily xtmixed has factor notation to avoid this R. Gutierrez (StataCorp) Linear Mixed Models in Stata March 31, / 30
51 Fitting the model One approach to fitting this model is to consider the data as a whole and treat the random effects as random coefficients on lots of indicator variables, that is u 1. [ ] u = u 48 σ 2 v 1 N(0, G); G = u I σvi 2 9. v 9 We could generate these indicator variables, but luckily xtmixed has factor notation to avoid this R. Gutierrez (StataCorp) Linear Mixed Models in Stata March 31, / 30
52 . xtmixed weight week _all: R.id _all: R.week Mixed-effects REML regression Number of obs = 432 Group variable: _all Number of groups = 1 Obs per group: min = 432 avg = max = 432 Wald chi2(1) = Log restricted-likelihood = Prob > chi2 = weight Coef. Std. Err. z P> z [95% Conf. Interval] week _cons Random-effects Parameters Estimate Std. Err. [95% Conf. Interval] _all: Identity _all: Identity sd(r.id) sd(r.week) sd(residual) LR test vs. linear regression: chi2(2) = Prob > chi2 = Note: LR test is conservative and provided only for reference R. Gutierrez (StataCorp) Linear Mixed Models in Stata March 31, / 30
53 Some notes all tells xtmixed to treat the whole data as one big panel R.varname is the random-effects analog of xi. It creates an (overparameterized) set of indicator variables, but unlike xi, does this behind the scenes When you use R.varname, covariance structure reverts to Identity There are alternate ways to fit this model with lower dimension The trick is to realize that all effects are nested within the data as a whole R. Gutierrez (StataCorp) Linear Mixed Models in Stata March 31, / 30
54 Some notes all tells xtmixed to treat the whole data as one big panel R.varname is the random-effects analog of xi. It creates an (overparameterized) set of indicator variables, but unlike xi, does this behind the scenes When you use R.varname, covariance structure reverts to Identity There are alternate ways to fit this model with lower dimension The trick is to realize that all effects are nested within the data as a whole R. Gutierrez (StataCorp) Linear Mixed Models in Stata March 31, / 30
55 Some notes all tells xtmixed to treat the whole data as one big panel R.varname is the random-effects analog of xi. It creates an (overparameterized) set of indicator variables, but unlike xi, does this behind the scenes When you use R.varname, covariance structure reverts to Identity There are alternate ways to fit this model with lower dimension The trick is to realize that all effects are nested within the data as a whole R. Gutierrez (StataCorp) Linear Mixed Models in Stata March 31, / 30
56 Some notes all tells xtmixed to treat the whole data as one big panel R.varname is the random-effects analog of xi. It creates an (overparameterized) set of indicator variables, but unlike xi, does this behind the scenes When you use R.varname, covariance structure reverts to Identity There are alternate ways to fit this model with lower dimension The trick is to realize that all effects are nested within the data as a whole R. Gutierrez (StataCorp) Linear Mixed Models in Stata March 31, / 30
57 Some notes all tells xtmixed to treat the whole data as one big panel R.varname is the random-effects analog of xi. It creates an (overparameterized) set of indicator variables, but unlike xi, does this behind the scenes When you use R.varname, covariance structure reverts to Identity There are alternate ways to fit this model with lower dimension The trick is to realize that all effects are nested within the data as a whole R. Gutierrez (StataCorp) Linear Mixed Models in Stata March 31, / 30
58 . xtmixed weight week _all: R.id week: Mixed-effects REML regression Number of obs = 432 No. of Observations per Group Group Variable Groups Minimum Average Maximum _all week Wald chi2(1) = Log restricted-likelihood = Prob > chi2 = weight Coef. Std. Err. z P> z [95% Conf. Interval] week _cons Random-effects Parameters Estimate Std. Err. [95% Conf. Interval] _all: Identity week: Identity sd(r.id) sd(_cons) sd(residual) LR test vs. linear regression: chi2(2) = Prob > chi2 = Note: LR test is conservative and provided only for reference R. Gutierrez (StataCorp) Linear Mixed Models in Stata March 31, / 30
59 . xtmixed weight week _all: R.week id: Mixed-effects REML regression Number of obs = 432 No. of Observations per Group Group Variable Groups Minimum Average Maximum _all id Wald chi2(1) = Log restricted-likelihood = Prob > chi2 = weight Coef. Std. Err. z P> z [95% Conf. Interval] week _cons Random-effects Parameters Estimate Std. Err. [95% Conf. Interval] _all: Identity id: Identity sd(r.week) sd(_cons) sd(residual) LR test vs. linear regression: chi2(2) = Prob > chi2 = Note: LR test is conservative and provided only for reference R. Gutierrez (StataCorp) Linear Mixed Models in Stata March 31, / 30
60 A glimpse at the future You can welcome Stata to the game. We hope you like the syntax and output. Some things to look for in future versions Correlated errors and heteroskedasticity Exploiting matrix sparsity/very large problems Factor variables Degrees of freedom calculations Generalized linear mixed models. Adding family() and link() options to what we have here R. Gutierrez (StataCorp) Linear Mixed Models in Stata March 31, / 30
61 A glimpse at the future You can welcome Stata to the game. We hope you like the syntax and output. Some things to look for in future versions Correlated errors and heteroskedasticity Exploiting matrix sparsity/very large problems Factor variables Degrees of freedom calculations Generalized linear mixed models. Adding family() and link() options to what we have here R. Gutierrez (StataCorp) Linear Mixed Models in Stata March 31, / 30
62 A glimpse at the future You can welcome Stata to the game. We hope you like the syntax and output. Some things to look for in future versions Correlated errors and heteroskedasticity Exploiting matrix sparsity/very large problems Factor variables Degrees of freedom calculations Generalized linear mixed models. Adding family() and link() options to what we have here R. Gutierrez (StataCorp) Linear Mixed Models in Stata March 31, / 30
63 A glimpse at the future You can welcome Stata to the game. We hope you like the syntax and output. Some things to look for in future versions Correlated errors and heteroskedasticity Exploiting matrix sparsity/very large problems Factor variables Degrees of freedom calculations Generalized linear mixed models. Adding family() and link() options to what we have here R. Gutierrez (StataCorp) Linear Mixed Models in Stata March 31, / 30
64 A glimpse at the future You can welcome Stata to the game. We hope you like the syntax and output. Some things to look for in future versions Correlated errors and heteroskedasticity Exploiting matrix sparsity/very large problems Factor variables Degrees of freedom calculations Generalized linear mixed models. Adding family() and link() options to what we have here R. Gutierrez (StataCorp) Linear Mixed Models in Stata March 31, / 30
65 A glimpse at the future You can welcome Stata to the game. We hope you like the syntax and output. Some things to look for in future versions Correlated errors and heteroskedasticity Exploiting matrix sparsity/very large problems Factor variables Degrees of freedom calculations Generalized linear mixed models. Adding family() and link() options to what we have here R. Gutierrez (StataCorp) Linear Mixed Models in Stata March 31, / 30
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