Generalised linear latent and mixed models -------------------------------------------
^gllamm^ depvar [varlist] [^if^ exp] [^in^ range] ^,^ ^i(^varlist^)^ [ ^nocons^tant ^o^ffset^(^varname^)^ ^nr^f^(^#^,^...^,^#^)^ ^e^qs^(^eqnames^)^ ^frload^(^#^,^...^,^#^)^ ^ip(^string^)^ ^ni^p^(^#^,^...^,^#^)^ ^pe^qs^(^eqname^)^ ^bmat^rix^(^matrix^)^ ^ge^qs^(^eqnames^)^ ^nocor^rel ^c^onstraints^(^clist^)^ ^we^ight^(^varname^)^ ^pwe^ight^(^varname^ > )^ ^f^amily^(^family^)^ ^fv(^varname^)^ ^de^nom^(^varname^)^ ^s(^eqname^)^ ^l^ink^(^link^)^ ^lv(^varname^)^ ^expa^nded^(^varname varname string^)^ ^b^asecategory^(^#^)^ ^comp^osite^(^varnames^)^ ^th^resh^(^eqnames^)^ ^eth^resh^(^eqnames > ^)^ ^fr^om^(^matrix^)^ ^copy^ ^skip^ ^long^ ^lf0(^#^ ^#^)^ ^ga^teaux^(^#^ ^#^ ^#^)^ ^se^arch^(^#^)^ ^noe^st ^ev^al ^in^it ^it^erate^(^#^)^ ^adoonly^ ^adapt^ ^rob^ust ^clu^ster^(^varname^)^ ^l^evel^(^#^)^ ^eform^ ^allc^ ^tr^ace ^nolo^g ^nodis^play ^do^ts > ]
where family is and link is ^gau^ssian ^id^entity ^poi^sson ^log^ ^gam^ma ^rec^iprocal ^bin^omial ^logi^t ^pro^bit ^cll^ (complementary log-log) ^ll^ (log-log) ^olo^git (o stands for ordinal) ^opr^obit ^ocl^l ^mlo^git ^spr^obit (scaled probit) ^sop^robit
and clist is of the form #[^-^#][^,^ #[^-^#] ...]
^gllamm^ shares the features of all estimation commands; see help @est@.
^gllamm^ typed without arguments redisplays previous results.
Predictions of the latent variables or random effects (and many other quantities) can be obtained using @gllapred@ and the models can be simulated using @gllasim@
Description -----------
^gllamm^ estimates ^G^eneralized ^L^inear ^L^atent ^A^nd ^M^ixed ^M^odels. These models include multilevel (hierarchical) regression models with an arbitrary number of levels, generalized linear mixed models, multilevel factor models and some types of latent class models. We refer to the random effects (random intercepts, slopes or coefficients), factors, etc. as latent variables or random effects.
If the latent variables are assumed to be multivariate normal, ^gllamm^ uses Gauss-Hermite quadrature, or adaptive quadrature if the ^adapt^ option is also specified. Adaptive quadrature can be considerably more accurate than ordinary quadrature, see first reference at the bottom of this help file. With the ^ip(^f^)^ option, the latent variables are specified as discrete with freely estimated probabilities (masses) and locations.
More information on the models is available from
http://www.gllamm.org
Options --------
(a) Structure of the model ---------------------------------------------------------------------------
^i(^varlist^)^ gives the variables that define the hierarchical, nested clusters, from the lowest level (finest clusters) to the highest level, e.g. i(pupil class school).
^noconstant^ omits the constant term from the fixed effects equation.
^offset(^varname^)^ specifies a variable to be added to the linear predictor without estimating a corresponding regression coefficient (e.g. log exposure for Poisson regression).
^nrf(^#^,^...^,^#^)^ specifies the number of random effects for each level, i.e., for each variable in ^i(^varlist^)^. The default is nrf(1,...,1).
^eqs(^eqnames^)^ specifies equation names (defined before running gllamm) for the linear predictors multiplying the latent variables; see help @eq_g@ > . If required, constants should be explicitly included in the equation definitions using variables equal to 1. If the option is not used, the latent variables are assumed to be random intercepts and only one random effect is allowed per level. The first lambda coefficient is set to one unless the ^frload()^ option is specified. The other coefficients are estimated together with the (co)variance(s) of the random effect(s).
^frload(^#^,^...^,^#^)^ lists the latent variables for which the first factor loading should be freely estimated along with the other factor loadings. It is up to the user to define appropriate constraints to identify the model. Here the latent variables are referred to as 1 2 3 etc. in the order in which they are defined by the ^eqs()^ option.
^ip(^sting^)^ if string is g, Gaussian quadrature points are used and if string is f, the mass-points are freely estimated. The default is Gaussian quadrature. The ^ip(^f^)^ option causes nip-1 locations to be estimated, the nipth mass being determined by setting the mean location to 0 so that an intercept can be included in the fixed effects equation. The ^ip(^fn^)^ option can be used to set the last mass to 0 instead of to the mean. If string is m, spherical quadrature rules are used for multidimensional integrals.
^nip(^#^,^...^,^#^)^ specifies the number of integration points or masses to be used for each integral or summation. When quadrature is used, a value may be given for each random effect. When freely estimated masses are used, a value may be given for each level of the model. If only one argument is given, the same number of integration points will be used for each summation. Combined with the ^ip(m)^ option, ^nip()^ specifies the degree of the approximation instead of the number of points. Only the following degrees are available: for two random effects, 5, 7, 9, 11, 15 and for more than two random effects 5, 7.
^peqs(^eqname^)^ can be used with the ^ip(^f^)^ or ^ip(^fn^)^ options to allow the (prior) latent class probabilities to depend on covariates. The model for the latent class probabilities is a multinomial logit model with the last latent class as reference category. A constant is automatically included in addition to the covariates specified in the equation command; see help @eq_g@.
^geqs(^eqnames^)^ specifies regressions of latent variables on explanatory vari > ables. The second character of the equation name indicates which latent variable is regressed on the variables used in the equation definition, e.g > . eq f1: a b means that the first latent variable is regressed on a and b (wi > thout a constant); see help @eq_g@.
^bmatrix(^matrix^)^ specifies a matrix B of regression coefficients for the dependence of the latent variables on other latent variables. The matrix must be upper diagonal and have number of rows and columns equal to the total number of random effects.
^nocorrel^ may be used to constrain all correlations to zero if there are several random effects at any of the levels and if these are modeled as multivariate normal.
^constraint(^clist^)^ specifies the constraint numbers of the constraints to be applied. Constraints are defined using the ^constraint^ command; see help @constraint@. To find out the equation names needed to specify the constraints, run gllamm with the noest option.
^weight(^varname^)^ specifies that variables varname1, varname2, etc. contain frequency weights. The suffixes determine at what level each weight applies > . For example, if the level 1 units are subjects, the level 2 units are families, and the result is binary, we can collapse dataset A into dataset B as follows:
A B family subject result family subject result wt1 wt2 1 1 0 1 1 0 2 1 1 2 0 2 3 1 1 2 2 3 1 2 4 0 1 2 2 4 0 3 5 1 3 6 0
The level 1 weight, wt1, indicates that subject 1 in dataset B represents two subjects within family 1 in dataset A, whereas subjects 3 and 4 in dataset B represent single subjects within family 2 in dataset A. The level 2 weight wt2 indicates that family 1 in dataset B represents one family and family 2 represents two families, i.e. all the data for family 2 are replicated once. Collapsing the data in this way can make gllamm run considerably faster.
^pweight(^varname^)^ specifies that variables varname1, varname2, etc. contain sampling weights for levels 1, 2, etc. As far as the estimates and log-likelihood are concerned, the effect of specifying these weights is the same as for frequency weights, but the standard errors will be different. Robust standard errors will automatically be provided. This should be used with caution if the sampling weights apply to units at a lower level than the highest level in the multilevel model. The weights are not rescaled; scaling is the responsibility of the user.
(b) Densities, links, etc. for the response model ------------------------------------------------------------------------------
^family(^families^)^ specifies the families to be used for the conditional densities. The default is ^family(^gauss^)^. Several families may be given in which case the variable allocating families to observations must be given using ^fv(^varname^)^.
^fv(^varname^)^ is required if mixed responses requiring more than a single family of conditional distributions are analyzed. The variable indicates which family applies to which observation. A value of one refers to the first family etc.
^denom(^varname^)^ gives the variable containing the binomial denominator for the responses whose family is specified as binomial. The default denominator is 1.
^s(^eqname^)^ specifies that the log of the standard deviation (or coefficient of variation) at level 1 for normally (or gamma) distributed responses should be given by the linear predictor defined by eqname. This is necessary if the level-1 variance is heteroscedastic. For example, if dummy variables for groups are used, different variances are estimated for different groups.
^link(^link^)^ specifies the links to be used for the conditional densities. If > a single family is specified, the default link is the canonical link. Several links may be given in which case the variable allocating links to observations must be given using ^lv(^varname^)^. This option is currently not available if the ordinal or mlogit links are used. Numerically feasible choices of link depend upon the distributions of the covariates and choice of conditional error and random effects distributions. The sprobit link is only identified in special cases; it may be used for Heckman-type selection models or to model floor or ceiling effects.
^lv(^varname^)^ is the variable whose values indicate which link applies to which observation.
^expanded(^varname varname string^)^ is used together with the mlogit link and specifies that the data have been expanded as illustrated below:
A B choice response altern selected 1 1 1 1 2 1 2 0 1 3 0 2 1 0 2 2 1 2 3 0
where the variable "choice" is the multinomial response (possible values 1,2,3), the "response" labels the original lines of data, "altern" gives the possible responses or alternatives and "selected" is an indicator for the option that was selected. The syntax would be expanded(response selected m) and the variable "altern" would be used as the dependent variable. This expanded form allows the user to use different random effects etc. for different categories of the multinomial response. The third argument is o if one set of coefficients should be estimated for the explanatory variables and m if one set of coefficients is to be estimated for each category of the response except the reference category.
^basecategory^(^#^)^ When the mlogit link is used, this specifies the value of the response to be used as the reference category. This option is ignored if the expanded() option is used with the third argument equal to m.
^composite^(varname varname varname [more varnames]) specifies that a composite link is used. The first variable is a cluster identifier ("cluster" below) so that linear predictors within the cluster can be combined into a single composite link. The second variable ("ind" below) indicates to which response the composite links defined by the susequent weight variables belong. Observations with ind=0 have a missing link. The remaining variables ("c1" and "c2" below) specify weights for the composite links. The composite link based on the first weight variable will go to where ind=1, etc. Example: Data setup with form of inverse link Interpretation of h_i determined by link() and lv(): composite(cluster ind c1 c2) cluster ind c1 c2 inverse link cluster composite link 1 1 1 0 h_1 1 h_1 - h_2 1 2 -1 1 h_2 1 n_2 + h_3 1 0 0 1 h_3 ==> 1 missing 2 1 1 0 h_4 2 h_4 + h_5 2 2 1 1 h_5 2 h_5 + 2*h_6 2 0 0 2 h_6 2 missing
^thresh(^eqnames^)^ specifies equation(s) for the thresholds for ordinal response(s); see help @eq_g@. One equation is specified for each ordinal response. The purpose of this option is to allow the effects of som > e covariates to be different for different categories of the ordinal variable > rather than assumming a constant effect - the proportional odds assumption if the ologit link is used. Variables used in the model for the thresholds cannot appear in the fixed part of the linear predictor.
^ethresh(^eqnames^)^ is the same as ^thresh(^eqnames^)^ except that a different parameterization is used for the threshold model. To ensure that k_{s-1} <= k_{s}, the model is k_{s} = k_{s-1} + exp(xb), for response categories s=2,...,S.
(c) Starting values -----------------------------------------------------------------------------
^from(^matrix^)^ specifies the matrix to be used for the initial values. Note that the column-names and equation-names have to be correct (see help @matrname@, @matrix@), unless the ^copy^ option is specified. The matrix may be obtained from a previous estimation command using e(b). This is useful if the number of quadrature points needs to be increased or of a new explanatory variable is added. Use the ^skip^ option if the matrix of has extra parameters.
^copy^ and ^skip^ see above.
^long^ may be used with the from(matrix) option when constraints are used to indicate that the matrix of initial values has as many elements as would be needed for the unconstrained model, i.e. more elements than will be estimated.
^lf0(^# #^)^ gives the number of parameters and the log-likelihood for a likelihood ratio test to compare the model to be estimated with a simpler model. A likelihood ratio chi-squared test is only performed if the ^lf0(^# #^)^ option is used.
^gateaux(^min^,^max^,^n^)^ may be used with method ip(f) to increase the number of mass-points by one from a previous solution with parameter estimates specified using from(matrix) and number of parameters and log-likelihood specified by lf0(# #). The program searches for the location of the new mass-point by placing a very small mass at the location given by the first argument and moving it to the second argument in the number of steps specified by the third argument. (If there are several random effects, this search is done in each dimension resulting in a regular grid of search points.) If the maximum increase in likelihood is greater than 0, the location corresponding to this maximum is used as the initial value of the new location, otherwise the program stops. When this happens, it can be shown that for certain models the current solution represents the non-parametric maximum likelihood estimate.
^search(^#^)^ causes the program to search for initial values for the random effects at level 2 (in range 0 to 3). The argument specifies the number of random searches. This option may only be used with ^ip(^g^)^ and when ^fr^om^(^matrix^)^ is not used.
(d) Estimation and output options ------------------------------------------------------------------------------
^noest^ is used to prevent the program from carrying out the estimation. This may be used with the trace option to check that the model is correct and get the information needed to set up a matrix of initial values. Global macros are available that are normally deleted. Particularly useful may be M_initf and M_initr, matrices for the parameters (fixed part and random part respectively).
^eval^ causes the program to simply evaluate the loglikelihood for values passe > d to it using the from(matrix) option.
^init^ causes the program to compute initial estimates of fixed effects only, setting all latent variables to zero. gllamm will be used for estimating initial values even if a Stata command is available for the model (without the init option, gllamm uses Stata commands for initial value > s whenever they are available).
^iterate(^#^)^ specifies the (maximum) number of iterations. With the ^adapt^ option, use of the ^iterate(^#^)^ option will cause ^gllamm^ to skip the "Newton Raphson" iterations usually performed at the end without updating the quadrature locations. ^iterate(^0^)^ is like ^eval^ except that standard errors are computed.
^adoonly^ causes all gllamm to use only ado-code. Gllamm will be faster if if it uses internalised versions of some of the functions available in Stata 7 if updated on or after 26oct2001
^nip(^#^,^...^,^#^)^ when quadrature is used, this specifies the number of quadrature points (integration points) to be used. A value may be given for each random effect. If only one argument is given, the same number of quadrature points will be used for each summation.
^adapt^ causes adaptive quadrature to be used instead of ordinary quadrature. This option cannot be used with the ^ip(^f^)^ or ^ip(^f0^)^ options.
^robust^ specifies that the Huber/White/sandwich estimator of the covariance matrix of the parameter estimates is to be used. If a model has been estimated without the ^robust^ option, the robust standard errors can be obtained by simply typing ^gllamm, robust^.
^cluster(^varname^)^ specifies that the highest level units of the GLLAMM model are nested in even higher level clusters where ^varname^ contains the cluster identifier. Robust standard errors will be provided that take this clustering into account. If a model has been estimated without this option, the robust standard errors for clustered data can be obtained using the command ^gllamm, cluster(varname)^.
^level(^#^)^ specifies the confidence level in percent for confidence intervals of the coefficients.
^eform^ causes the expnentiated coefficients and confidence intervals to be displayed.
^allc^ causes all estimated parameters to be displayed in a regression table (including the raw parameters for the random effects) in addition to the usual output.
^trace^ causes more output to be displayed. Before estimation begins, details of the specified model are displayed. In addition, a detailed iteration log is shown including parameter estimates and log-likelihood values for each iteration.
^nolog^ suppresses output for maximum likelihood iterations.
^nodisplay^ suppresses output of the estimates but still shows iteration log unless ^nolog^ is used.
^dots^ causes a dot to be printed (if used together with trace) every time the likelihood evaluation program is called by ml. This helps to assess how long gllamm is likely to take to run and reassures the user that it is making some progress when it is very slow.
Examples --------
(a) 3-level random intercept model ---------------------------------- Some response "resp" and covariate "x" are available for pupils in different schools. "id" is the identifier or label for the pupils and "school" is the identifier for the schools. A linear model with random intercepts at the pupil and school levels can be specified as follows:
. ^gllamm resp x, i(id school) adapt trace^
(b) 2-level random coefficient model - growth curve model --------------------------------------------------------- subjects identified by "id" have been measured repeatedly over time giving responses in "resp". "cons" is a variable equal to 1 and "time" contains the time-points. A model with a random intercept and slope for time is specified as follows:
. ^eq int: cons^ . ^eq slope: time^ . ^gllamm resp time, i(id) nrf(2) eqs(int slope) adapt trace ^
(c) two-parameter logistic item-response model ---------------------------------------------- variable "resp" contains responses to 5 items (e.g. 5 test questions) for each subject. The subject identifier is "id". There are five dummy variables "i1" to "i5" for the items, e.g. "i1" is equal to 1 if the item is item 1 and 0 otherwise.
. ^eq discrim: i1 i2 i3 i4 i5^ . ^gllamm resp i1 i2 i3 i4 i5, link(logit) fam(binom) nocons /*^ ^*/ i(id) eqs(discrim) adapt trace^
Author ------ Sophia Rabe-Hesketh (sophiarh@@berkeley.edu) as part of joint work with Andrew Pickles and Anders Skrondal. We would like to acknowledge Colin Taylor for helping in the early stages of gllamm development. We are also very grateful to Stata Corporation for helping us to speed up gllamm.
Web-page -------- http://www.gllamm.org
References (available from sophiarh@@berkeley.edu) ---------- Rabe-Hesketh, S. and Skrondal, A. (2005). Multilevel and Longitudinal Modeling using Stata. College Station, TX: Stata Press.
Rabe-Hesketh, S., Pickles, A. and Skrondal, S. (2004). GLLAMM Manual. U.C. Berkeley Division of Biostatistics Working Paper Series. Working Paper 160. see http://www.bepress.com/ucbbiostat/paper160/
Rabe-Hesketh, S., Skrondal, A. and Pickles, A. (2005). Maximum likelihood estimation of limited and discrete dependent variable models with nested random effects. Journal of Econometrics 128, 301-323.
Rabe-Hesketh, S., Skrondal, A. and Pickles, A. (2002). Reliable estimation of generalized linear mixed models using adaptive quadrature. The Stata Journal 2, 1-21.
Rabe-Hesketh, S., Skrondal, A. and Pickles, A. (2004). Generalised multilevel structural equation modelling. Psychometrika 69 , 167-190.
Also see --------
Manual: ^[U] 23 Estimation and post-estimation commands^ ^[U] 29 Overview of model estimation in Stata^
On-line: help for @gllapred@, @gllasim@, @ml@, @glm@, @xtreg@, @xtlogit@, @xtpois@, @quadchk@, @test@