Binary files /tmp/tmpbaWbaI/dfzecUzuwL/lme4-1.1-23/build/partial.rdb and /tmp/tmpbaWbaI/TSO4DPDBci/lme4-1.1-26/build/partial.rdb differ Binary files /tmp/tmpbaWbaI/dfzecUzuwL/lme4-1.1-23/build/vignette.rds and /tmp/tmpbaWbaI/TSO4DPDBci/lme4-1.1-26/build/vignette.rds differ diff -Nru lme4-1.1-23/debian/changelog lme4-1.1-26/debian/changelog --- lme4-1.1-23/debian/changelog 2020-05-30 19:13:29.000000000 +0000 +++ lme4-1.1-26/debian/changelog 2020-12-01 16:40:35.000000000 +0000 @@ -1,8 +1,18 @@ -lme4 (1.1-23-1build1) groovy; urgency=medium +lme4 (1.1-26-1) unstable; urgency=medium - * No-change rebuild against r-api-4.0 + * New upstream release - -- Graham Inggs Sat, 30 May 2020 19:13:29 +0000 + -- Dirk Eddelbuettel Tue, 01 Dec 2020 10:40:35 -0600 + +lme4 (1.1-25-1) unstable; urgency=medium + + * New upstream release + + * debian/control: Set Build-Depends: to current R version + * debian/control: Switch to virtual debhelper-compat (= 11) + * debian/compat: Removed + + -- Dirk Eddelbuettel Fri, 23 Oct 2020 21:11:30 -0500 lme4 (1.1-23-1) unstable; urgency=medium diff -Nru lme4-1.1-23/debian/compat lme4-1.1-26/debian/compat --- lme4-1.1-23/debian/compat 2019-03-09 13:11:51.000000000 +0000 +++ lme4-1.1-26/debian/compat 1970-01-01 00:00:00.000000000 +0000 @@ -1 +0,0 @@ -9 diff -Nru lme4-1.1-23/debian/control lme4-1.1-26/debian/control --- lme4-1.1-23/debian/control 2020-04-09 21:11:10.000000000 +0000 +++ lme4-1.1-26/debian/control 2020-12-01 16:40:23.000000000 +0000 @@ -2,7 +2,7 @@ Section: gnu-r Priority: optional Maintainer: Dirk Eddelbuettel -Build-Depends: debhelper (>= 10), dh-r, r-base-dev (>= 3.6.3), r-cran-matrix (>= 1.1-0), r-cran-lattice, r-cran-nlme, r-cran-mass, r-cran-rcppeigen (>= 0.3.2.0.2-2), r-cran-minqa (>= 1.2.2-2), r-cran-nloptr, r-cran-boot, r-cran-statmod +Build-Depends: debhelper-compat (= 11), dh-r, r-base-dev (>= 4.0.3), r-cran-matrix (>= 1.1-0), r-cran-lattice, r-cran-nlme, r-cran-mass, r-cran-rcppeigen (>= 0.3.2.0.2-2), r-cran-minqa (>= 1.2.2-2), r-cran-nloptr, r-cran-boot, r-cran-statmod Standards-Version: 4.5.0 Vcs-Browser: https://salsa.debian.org/edd/r-cran-lme4 Vcs-Git: https://salsa.debian.org/edd/r-cran-lme4.git diff -Nru lme4-1.1-23/debian/source.lintian-overrides lme4-1.1-26/debian/source.lintian-overrides --- lme4-1.1-23/debian/source.lintian-overrides 1970-01-01 00:00:00.000000000 +0000 +++ lme4-1.1-26/debian/source.lintian-overrides 2020-12-01 16:40:35.000000000 +0000 @@ -0,0 +1,4 @@ +lme4 source: source-is-missing inst/doc/lmerperf.html line length is 14886 characters (>512) +lme4 source: source-is-missing inst/doc/lmerperf.html line length is 31270 characters (>512) +lme4 source: source-is-missing inst/doc/lmerperf.html line length is 2545 characters (>512) +lme4 source: source-is-missing inst/doc/lmerperf.html line length is 31952 characters (>512) \ No newline at end of file diff -Nru lme4-1.1-23/DESCRIPTION lme4-1.1-26/DESCRIPTION --- lme4-1.1-23/DESCRIPTION 2020-04-07 10:00:21.000000000 +0000 +++ lme4-1.1-26/DESCRIPTION 2020-12-01 00:50:10.000000000 +0000 @@ -1,5 +1,5 @@ Package: lme4 -Version: 1.1-23 +Version: 1.1-26 Title: Linear Mixed-Effects Models using 'Eigen' and S4 Authors@R: c( person("Douglas","Bates", role="aut", @@ -20,7 +20,10 @@ person("Gabor", "Grothendieck", role="ctb"), person("Peter", "Green", role="ctb", comment=c(ORCID="0000-0002-0238-9852")), - person("John", "Fox", role="ctb") + person("John", "Fox", role="ctb"), + person("Alexander", "Bauer", role="ctb"), + person("Pavel N.", "Krivitsky", role=c("ctb","cph"), comment=c(ORCID="0000-0002-9101-3362", + "shared copyright on simulate.formula")) ) Contact: LME4 Authors Description: Fit linear and generalized linear mixed-effects models. @@ -33,8 +36,8 @@ nlme (>= 3.1-123), minqa (>= 1.1.15), nloptr (>= 1.0.4), statmod Suggests: knitr, rmarkdown, PKPDmodels, MEMSS, testthat (>= 0.8.1), - ggplot2, mlmRev, optimx (>= 2013.8.6), gamm4, pbkrtest, HSAUR2, - numDeriv, car, dfoptim + ggplot2, mlmRev, optimx (>= 2013.8.6), gamm4, pbkrtest, HSAUR3, + numDeriv, car, dfoptim, mgcv VignetteBuilder: knitr LazyData: yes License: GPL (>= 2) @@ -42,7 +45,7 @@ BugReports: https://github.com/lme4/lme4/issues Encoding: UTF-8 NeedsCompilation: yes -Packaged: 2020-04-06 00:39:09 UTC; bolker +Packaged: 2020-11-30 20:47:41 UTC; bolker Author: Douglas Bates [aut] (), Martin Maechler [aut] (), Ben Bolker [aut, cre] (), @@ -54,7 +57,10 @@ Fabian Scheipl [ctb] (), Gabor Grothendieck [ctb], Peter Green [ctb] (), - John Fox [ctb] + John Fox [ctb], + Alexander Bauer [ctb], + Pavel N. Krivitsky [ctb, cph] (, + shared copyright on simulate.formula) Maintainer: Ben Bolker Repository: CRAN -Date/Publication: 2020-04-07 10:00:21 UTC +Date/Publication: 2020-12-01 00:50:10 UTC diff -Nru lme4-1.1-23/inst/doc/lme4/inst/doc/Doxyfile lme4-1.1-26/inst/doc/lme4/inst/doc/Doxyfile --- lme4-1.1-23/inst/doc/lme4/inst/doc/Doxyfile 2012-11-17 18:14:27.000000000 +0000 +++ lme4-1.1-26/inst/doc/lme4/inst/doc/Doxyfile 1970-01-01 00:00:00.000000000 +0000 @@ -1,1551 +0,0 @@ -# Doxyfile 1.6.3 - -# This file describes the settings to be used by the documentation system -# doxygen (www.doxygen.org) for a project -# -# All text after a hash (#) is considered a comment and will be ignored -# The format is: -# TAG = value [value, ...] -# For lists items can also be appended using: -# TAG += value [value, ...] -# Values that contain spaces should be placed between quotes (" ") - -#--------------------------------------------------------------------------- -# Project related configuration options -#--------------------------------------------------------------------------- - -# This tag specifies the encoding used for all characters in the config file -# that follow. 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Doxygen will use this -# information to generate all constant output in the proper language. -# The default language is English, other supported languages are: -# Afrikaans, Arabic, Brazilian, Catalan, Chinese, Chinese-Traditional, -# Croatian, Czech, Danish, Dutch, Esperanto, Farsi, Finnish, French, German, -# Greek, Hungarian, Italian, Japanese, Japanese-en (Japanese with English -# messages), Korean, Korean-en, Lithuanian, Norwegian, Macedonian, Persian, -# Polish, Portuguese, Romanian, Russian, Serbian, Serbian-Cyrilic, Slovak, -# Slovene, Spanish, Swedish, Ukrainian, and Vietnamese. - -OUTPUT_LANGUAGE = English - -# If the BRIEF_MEMBER_DESC tag is set to YES (the default) Doxygen will -# include brief member descriptions after the members that are listed in -# the file and class documentation (similar to JavaDoc). -# Set to NO to disable this. - -BRIEF_MEMBER_DESC = YES - -# If the REPEAT_BRIEF tag is set to YES (the default) Doxygen will prepend -# the brief description of a member or function before the detailed description. -# Note: if both HIDE_UNDOC_MEMBERS and BRIEF_MEMBER_DESC are set to NO, the -# brief descriptions will be completely suppressed. - -REPEAT_BRIEF = YES - -# This tag implements a quasi-intelligent brief description abbreviator -# that is used to form the text in various listings. 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Use a value of 0 to hide initializers completely. -# The appearance of the initializer of individual variables and defines in the -# documentation can be controlled using \showinitializer or \hideinitializer -# command in the documentation regardless of this setting. - -MAX_INITIALIZER_LINES = 30 - -# Set the SHOW_USED_FILES tag to NO to disable the list of files generated -# at the bottom of the documentation of classes and structs. If set to YES the -# list will mention the files that were used to generate the documentation. - -SHOW_USED_FILES = YES - -# If the sources in your project are distributed over multiple directories -# then setting the SHOW_DIRECTORIES tag to YES will show the directory hierarchy -# in the documentation. The default is NO. - -SHOW_DIRECTORIES = NO - -# Set the SHOW_FILES tag to NO to disable the generation of the Files page. -# This will remove the Files entry from the Quick Index and from the -# Folder Tree View (if specified). The default is YES. - -SHOW_FILES = YES - -# Set the SHOW_NAMESPACES tag to NO to disable the generation of the -# Namespaces page. -# This will remove the Namespaces entry from the Quick Index -# and from the Folder Tree View (if specified). The default is YES. - -SHOW_NAMESPACES = YES - -# The FILE_VERSION_FILTER tag can be used to specify a program or script that -# doxygen should invoke to get the current version for each file (typically from -# the version control system). Doxygen will invoke the program by executing (via -# popen()) the command , where is the value of -# the FILE_VERSION_FILTER tag, and is the name of an input file -# provided by doxygen. Whatever the program writes to standard output -# is used as the file version. See the manual for examples. - -FILE_VERSION_FILTER = - -# The LAYOUT_FILE tag can be used to specify a layout file which will be parsed by -# doxygen. The layout file controls the global structure of the generated output files -# in an output format independent way. The create the layout file that represents -# doxygen's defaults, run doxygen with the -l option. You can optionally specify a -# file name after the option, if omitted DoxygenLayout.xml will be used as the name -# of the layout file. - -LAYOUT_FILE = - -#--------------------------------------------------------------------------- -# configuration options related to warning and progress messages -#--------------------------------------------------------------------------- - -# The QUIET tag can be used to turn on/off the messages that are generated -# by doxygen. Possible values are YES and NO. If left blank NO is used. - -QUIET = NO - -# The WARNINGS tag can be used to turn on/off the warning messages that are -# generated by doxygen. Possible values are YES and NO. If left blank -# NO is used. - -WARNINGS = YES - -# If WARN_IF_UNDOCUMENTED is set to YES, then doxygen will generate warnings -# for undocumented members. If EXTRACT_ALL is set to YES then this flag will -# automatically be disabled. - -WARN_IF_UNDOCUMENTED = YES - -# If WARN_IF_DOC_ERROR is set to YES, doxygen will generate warnings for -# potential errors in the documentation, such as not documenting some -# parameters in a documented function, or documenting parameters that -# don't exist or using markup commands wrongly. - -WARN_IF_DOC_ERROR = YES - -# This WARN_NO_PARAMDOC option can be abled to get warnings for -# functions that are documented, but have no documentation for their parameters -# or return value. If set to NO (the default) doxygen will only warn about -# wrong or incomplete parameter documentation, but not about the absence of -# documentation. - -WARN_NO_PARAMDOC = NO - -# The WARN_FORMAT tag determines the format of the warning messages that -# doxygen can produce. The string should contain the $file, $line, and $text -# tags, which will be replaced by the file and line number from which the -# warning originated and the warning text. Optionally the format may contain -# $version, which will be replaced by the version of the file (if it could -# be obtained via FILE_VERSION_FILTER) - -WARN_FORMAT = "$file:$line: $text " - -# The WARN_LOGFILE tag can be used to specify a file to which warning -# and error messages should be written. If left blank the output is written -# to stderr. - -WARN_LOGFILE = - -#--------------------------------------------------------------------------- -# configuration options related to the input files -#--------------------------------------------------------------------------- - -# The INPUT tag can be used to specify the files and/or directories that contain -# documented source files. You may enter file names like "myfile.cpp" or -# directories like "/usr/src/myproject". Separate the files or directories -# with spaces. - -INPUT = ../../src - -# This tag can be used to specify the character encoding of the source files -# that doxygen parses. Internally doxygen uses the UTF-8 encoding, which is -# also the default input encoding. Doxygen uses libiconv (or the iconv built -# into libc) for the transcoding. See http://www.gnu.org/software/libiconv for -# the list of possible encodings. - -INPUT_ENCODING = UTF-8 - -# If the value of the INPUT tag contains directories, you can use the -# FILE_PATTERNS tag to specify one or more wildcard pattern (like *.cpp -# and *.h) to filter out the source-files in the directories. If left -# blank the following patterns are tested: -# *.c *.cc *.cxx *.cpp *.c++ *.java *.ii *.ixx *.ipp *.i++ *.inl *.h *.hh *.hxx -# *.hpp *.h++ *.idl *.odl *.cs *.php *.php3 *.inc *.m *.mm *.py *.f90 - -FILE_PATTERNS = - -# The RECURSIVE tag can be used to turn specify whether or not subdirectories -# should be searched for input files as well. Possible values are YES and NO. -# If left blank NO is used. - -RECURSIVE = NO - -# The EXCLUDE tag can be used to specify files and/or directories that should -# excluded from the INPUT source files. This way you can easily exclude a -# subdirectory from a directory tree whose root is specified with the INPUT tag. - -EXCLUDE = - -# The EXCLUDE_SYMLINKS tag can be used select whether or not files or -# directories that are symbolic links (a Unix filesystem feature) are excluded -# from the input. - -EXCLUDE_SYMLINKS = NO - -# If the value of the INPUT tag contains directories, you can use the -# EXCLUDE_PATTERNS tag to specify one or more wildcard patterns to exclude -# certain files from those directories. Note that the wildcards are matched -# against the file with absolute path, so to exclude all test directories -# for example use the pattern */test/* - -EXCLUDE_PATTERNS = - -# The EXCLUDE_SYMBOLS tag can be used to specify one or more symbol names -# (namespaces, classes, functions, etc.) that should be excluded from the -# output. The symbol name can be a fully qualified name, a word, or if the -# wildcard * is used, a substring. Examples: ANamespace, AClass, -# AClass::ANamespace, ANamespace::*Test - -EXCLUDE_SYMBOLS = - -# The EXAMPLE_PATH tag can be used to specify one or more files or -# directories that contain example code fragments that are included (see -# the \include command). - -EXAMPLE_PATH = - -# If the value of the EXAMPLE_PATH tag contains directories, you can use the -# EXAMPLE_PATTERNS tag to specify one or more wildcard pattern (like *.cpp -# and *.h) to filter out the source-files in the directories. If left -# blank all files are included. - -EXAMPLE_PATTERNS = - -# If the EXAMPLE_RECURSIVE tag is set to YES then subdirectories will be -# searched for input files to be used with the \include or \dontinclude -# commands irrespective of the value of the RECURSIVE tag. -# Possible values are YES and NO. If left blank NO is used. - -EXAMPLE_RECURSIVE = NO - -# The IMAGE_PATH tag can be used to specify one or more files or -# directories that contain image that are included in the documentation (see -# the \image command). - -IMAGE_PATH = - -# The INPUT_FILTER tag can be used to specify a program that doxygen should -# invoke to filter for each input file. Doxygen will invoke the filter program -# by executing (via popen()) the command , where -# is the value of the INPUT_FILTER tag, and is the name of an -# input file. Doxygen will then use the output that the filter program writes -# to standard output. -# If FILTER_PATTERNS is specified, this tag will be -# ignored. - -INPUT_FILTER = - -# The FILTER_PATTERNS tag can be used to specify filters on a per file pattern -# basis. -# Doxygen will compare the file name with each pattern and apply the -# filter if there is a match. -# The filters are a list of the form: -# pattern=filter (like *.cpp=my_cpp_filter). See INPUT_FILTER for further -# info on how filters are used. If FILTER_PATTERNS is empty, INPUT_FILTER -# is applied to all files. - -FILTER_PATTERNS = - -# If the FILTER_SOURCE_FILES tag is set to YES, the input filter (if set using -# INPUT_FILTER) will be used to filter the input files when producing source -# files to browse (i.e. when SOURCE_BROWSER is set to YES). - -FILTER_SOURCE_FILES = NO - -#--------------------------------------------------------------------------- -# configuration options related to source browsing -#--------------------------------------------------------------------------- - -# If the SOURCE_BROWSER tag is set to YES then a list of source files will -# be generated. Documented entities will be cross-referenced with these sources. -# Note: To get rid of all source code in the generated output, make sure also -# VERBATIM_HEADERS is set to NO. - -SOURCE_BROWSER = YES - -# Setting the INLINE_SOURCES tag to YES will include the body -# of functions and classes directly in the documentation. - -INLINE_SOURCES = NO - -# Setting the STRIP_CODE_COMMENTS tag to YES (the default) will instruct -# doxygen to hide any special comment blocks from generated source code -# fragments. Normal C and C++ comments will always remain visible. - -STRIP_CODE_COMMENTS = YES - -# If the REFERENCED_BY_RELATION tag is set to YES -# then for each documented function all documented -# functions referencing it will be listed. - -REFERENCED_BY_RELATION = YES - -# If the REFERENCES_RELATION tag is set to YES -# then for each documented function all documented entities -# called/used by that function will be listed. - -REFERENCES_RELATION = YES - -# If the REFERENCES_LINK_SOURCE tag is set to YES (the default) -# and SOURCE_BROWSER tag is set to YES, then the hyperlinks from -# functions in REFERENCES_RELATION and REFERENCED_BY_RELATION lists will -# link to the source code. -# Otherwise they will link to the documentation. - -REFERENCES_LINK_SOURCE = YES - -# If the USE_HTAGS tag is set to YES then the references to source code -# will point to the HTML generated by the htags(1) tool instead of doxygen -# built-in source browser. The htags tool is part of GNU's global source -# tagging system (see http://www.gnu.org/software/global/global.html). You -# will need version 4.8.6 or higher. - -USE_HTAGS = NO - -# If the VERBATIM_HEADERS tag is set to YES (the default) then Doxygen -# will generate a verbatim copy of the header file for each class for -# which an include is specified. Set to NO to disable this. - -VERBATIM_HEADERS = YES - -#--------------------------------------------------------------------------- -# configuration options related to the alphabetical class index -#--------------------------------------------------------------------------- - -# If the ALPHABETICAL_INDEX tag is set to YES, an alphabetical index -# of all compounds will be generated. Enable this if the project -# contains a lot of classes, structs, unions or interfaces. - -ALPHABETICAL_INDEX = YES - -# If the alphabetical index is enabled (see ALPHABETICAL_INDEX) then -# the COLS_IN_ALPHA_INDEX tag can be used to specify the number of columns -# in which this list will be split (can be a number in the range [1..20]) - -COLS_IN_ALPHA_INDEX = 5 - -# In case all classes in a project start with a common prefix, all -# classes will be put under the same header in the alphabetical index. -# The IGNORE_PREFIX tag can be used to specify one or more prefixes that -# should be ignored while generating the index headers. - -IGNORE_PREFIX = - -#--------------------------------------------------------------------------- -# configuration options related to the HTML output -#--------------------------------------------------------------------------- - -# If the GENERATE_HTML tag is set to YES (the default) Doxygen will -# generate HTML output. - -GENERATE_HTML = YES - -# The HTML_OUTPUT tag is used to specify where the HTML docs will be put. -# If a relative path is entered the value of OUTPUT_DIRECTORY will be -# put in front of it. If left blank `html' will be used as the default path. - -HTML_OUTPUT = . - -# The HTML_FILE_EXTENSION tag can be used to specify the file extension for -# each generated HTML page (for example: .htm,.php,.asp). If it is left blank -# doxygen will generate files with .html extension. - -HTML_FILE_EXTENSION = .html - -# The HTML_HEADER tag can be used to specify a personal HTML header for -# each generated HTML page. If it is left blank doxygen will generate a -# standard header. - -HTML_HEADER = - -# The HTML_FOOTER tag can be used to specify a personal HTML footer for -# each generated HTML page. If it is left blank doxygen will generate a -# standard footer. - -HTML_FOOTER = - -# The HTML_STYLESHEET tag can be used to specify a user-defined cascading -# style sheet that is used by each HTML page. It can be used to -# fine-tune the look of the HTML output. If the tag is left blank doxygen -# will generate a default style sheet. Note that doxygen will try to copy -# the style sheet file to the HTML output directory, so don't put your own -# stylesheet in the HTML output directory as well, or it will be erased! - -HTML_STYLESHEET = - -# If the HTML_TIMESTAMP tag is set to YES then the footer of each generated HTML -# page will contain the date and time when the page was generated. Setting -# this to NO can help when comparing the output of multiple runs. - -HTML_TIMESTAMP = YES - -# If the HTML_ALIGN_MEMBERS tag is set to YES, the members of classes, -# files or namespaces will be aligned in HTML using tables. If set to -# NO a bullet list will be used. - -HTML_ALIGN_MEMBERS = YES - -# If the HTML_DYNAMIC_SECTIONS tag is set to YES then the generated HTML -# documentation will contain sections that can be hidden and shown after the -# page has loaded. For this to work a browser that supports -# JavaScript and DHTML is required (for instance Mozilla 1.0+, Firefox -# Netscape 6.0+, Internet explorer 5.0+, Konqueror, or Safari). - -HTML_DYNAMIC_SECTIONS = NO - -# If the GENERATE_DOCSET tag is set to YES, additional index files -# will be generated that can be used as input for Apple's Xcode 3 -# integrated development environment, introduced with OSX 10.5 (Leopard). -# To create a documentation set, doxygen will generate a Makefile in the -# HTML output directory. Running make will produce the docset in that -# directory and running "make install" will install the docset in -# ~/Library/Developer/Shared/Documentation/DocSets so that Xcode will find -# it at startup. -# See http://developer.apple.com/tools/creatingdocsetswithdoxygen.html for more information. - -GENERATE_DOCSET = NO - -# When GENERATE_DOCSET tag is set to YES, this tag determines the name of the -# feed. A documentation feed provides an umbrella under which multiple -# documentation sets from a single provider (such as a company or product suite) -# can be grouped. - -DOCSET_FEEDNAME = "Doxygen generated docs" - -# When GENERATE_DOCSET tag is set to YES, this tag specifies a string that -# should uniquely identify the documentation set bundle. This should be a -# reverse domain-name style string, e.g. com.mycompany.MyDocSet. Doxygen -# will append .docset to the name. - -DOCSET_BUNDLE_ID = org.doxygen.Project - -# If the GENERATE_HTMLHELP tag is set to YES, additional index files -# will be generated that can be used as input for tools like the -# Microsoft HTML help workshop to generate a compiled HTML help file (.chm) -# of the generated HTML documentation. - -GENERATE_HTMLHELP = NO - -# If the GENERATE_HTMLHELP tag is set to YES, the CHM_FILE tag can -# be used to specify the file name of the resulting .chm file. You -# can add a path in front of the file if the result should not be -# written to the html output directory. - -CHM_FILE = - -# If the GENERATE_HTMLHELP tag is set to YES, the HHC_LOCATION tag can -# be used to specify the location (absolute path including file name) of -# the HTML help compiler (hhc.exe). If non-empty doxygen will try to run -# the HTML help compiler on the generated index.hhp. - -HHC_LOCATION = - -# If the GENERATE_HTMLHELP tag is set to YES, the GENERATE_CHI flag -# controls if a separate .chi index file is generated (YES) or that -# it should be included in the master .chm file (NO). - -GENERATE_CHI = NO - -# If the GENERATE_HTMLHELP tag is set to YES, the CHM_INDEX_ENCODING -# is used to encode HtmlHelp index (hhk), content (hhc) and project file -# content. - -CHM_INDEX_ENCODING = - -# If the GENERATE_HTMLHELP tag is set to YES, the BINARY_TOC flag -# controls whether a binary table of contents is generated (YES) or a -# normal table of contents (NO) in the .chm file. - -BINARY_TOC = NO - -# The TOC_EXPAND flag can be set to YES to add extra items for group members -# to the contents of the HTML help documentation and to the tree view. - -TOC_EXPAND = NO - -# If the GENERATE_QHP tag is set to YES and both QHP_NAMESPACE and QHP_VIRTUAL_FOLDER -# are set, an additional index file will be generated that can be used as input for -# Qt's qhelpgenerator to generate a Qt Compressed Help (.qch) of the generated -# HTML documentation. - -GENERATE_QHP = NO - -# If the QHG_LOCATION tag is specified, the QCH_FILE tag can -# be used to specify the file name of the resulting .qch file. -# The path specified is relative to the HTML output folder. - -QCH_FILE = - -# The QHP_NAMESPACE tag specifies the namespace to use when generating -# Qt Help Project output. For more information please see -# http://doc.trolltech.com/qthelpproject.html#namespace - -QHP_NAMESPACE = - -# The QHP_VIRTUAL_FOLDER tag specifies the namespace to use when generating -# Qt Help Project output. For more information please see -# http://doc.trolltech.com/qthelpproject.html#virtual-folders - -QHP_VIRTUAL_FOLDER = doc - -# If QHP_CUST_FILTER_NAME is set, it specifies the name of a custom filter to add. -# For more information please see -# http://doc.trolltech.com/qthelpproject.html#custom-filters - -QHP_CUST_FILTER_NAME = - -# The QHP_CUST_FILT_ATTRS tag specifies the list of the attributes of the custom filter to add.For more information please see -# Qt Help Project / Custom Filters. - -QHP_CUST_FILTER_ATTRS = - -# The QHP_SECT_FILTER_ATTRS tag specifies the list of the attributes this project's -# filter section matches. -# Qt Help Project / Filter Attributes. - -QHP_SECT_FILTER_ATTRS = - -# If the GENERATE_QHP tag is set to YES, the QHG_LOCATION tag can -# be used to specify the location of Qt's qhelpgenerator. -# If non-empty doxygen will try to run qhelpgenerator on the generated -# .qhp file. - -QHG_LOCATION = - -# If the GENERATE_ECLIPSEHELP tag is set to YES, additional index files -# will be generated, which together with the HTML files, form an Eclipse help -# plugin. To install this plugin and make it available under the help contents -# menu in Eclipse, the contents of the directory containing the HTML and XML -# files needs to be copied into the plugins directory of eclipse. The name of -# the directory within the plugins directory should be the same as -# the ECLIPSE_DOC_ID value. After copying Eclipse needs to be restarted before the help appears. - -GENERATE_ECLIPSEHELP = NO - -# A unique identifier for the eclipse help plugin. When installing the plugin -# the directory name containing the HTML and XML files should also have -# this name. - -ECLIPSE_DOC_ID = org.doxygen.Project - -# The DISABLE_INDEX tag can be used to turn on/off the condensed index at -# top of each HTML page. The value NO (the default) enables the index and -# the value YES disables it. - -DISABLE_INDEX = NO - -# This tag can be used to set the number of enum values (range [1..20]) -# that doxygen will group on one line in the generated HTML documentation. - -ENUM_VALUES_PER_LINE = 8 - -# The GENERATE_TREEVIEW tag is used to specify whether a tree-like index -# structure should be generated to display hierarchical information. -# If the tag value is set to YES, a side panel will be generated -# containing a tree-like index structure (just like the one that -# is generated for HTML Help). For this to work a browser that supports -# JavaScript, DHTML, CSS and frames is required (i.e. any modern browser). -# Windows users are probably better off using the HTML help feature. - -GENERATE_TREEVIEW = YES - -# By enabling USE_INLINE_TREES, doxygen will generate the Groups, Directories, -# and Class Hierarchy pages using a tree view instead of an ordered list. - -USE_INLINE_TREES = NO - -# If the treeview is enabled (see GENERATE_TREEVIEW) then this tag can be -# used to set the initial width (in pixels) of the frame in which the tree -# is shown. - -TREEVIEW_WIDTH = 250 - -# Use this tag to change the font size of Latex formulas included -# as images in the HTML documentation. The default is 10. Note that -# when you change the font size after a successful doxygen run you need -# to manually remove any form_*.png images from the HTML output directory -# to force them to be regenerated. - -FORMULA_FONTSIZE = 10 - -# When the SEARCHENGINE tag is enabled doxygen will generate a search box for the HTML output. The underlying search engine uses javascript -# and DHTML and should work on any modern browser. Note that when using HTML help (GENERATE_HTMLHELP), Qt help (GENERATE_QHP), or docsets (GENERATE_DOCSET) there is already a search function so this one should -# typically be disabled. For large projects the javascript based search engine -# can be slow, then enabling SERVER_BASED_SEARCH may provide a better solution. - -SEARCHENGINE = NO - -# When the SERVER_BASED_SEARCH tag is enabled the search engine will be implemented using a PHP enabled web server instead of at the web client using Javascript. Doxygen will generate the search PHP script and index -# file to put on the web server. The advantage of the server based approach is that it scales better to large projects and allows full text search. The disadvances is that it is more difficult to setup -# and does not have live searching capabilities. - -SERVER_BASED_SEARCH = NO - -#--------------------------------------------------------------------------- -# configuration options related to the LaTeX output -#--------------------------------------------------------------------------- - -# If the GENERATE_LATEX tag is set to YES (the default) Doxygen will -# generate Latex output. - -GENERATE_LATEX = NO - -# The LATEX_OUTPUT tag is used to specify where the LaTeX docs will be put. -# If a relative path is entered the value of OUTPUT_DIRECTORY will be -# put in front of it. If left blank `latex' will be used as the default path. - -LATEX_OUTPUT = latex - -# The LATEX_CMD_NAME tag can be used to specify the LaTeX command name to be -# invoked. If left blank `latex' will be used as the default command name. -# Note that when enabling USE_PDFLATEX this option is only used for -# generating bitmaps for formulas in the HTML output, but not in the -# Makefile that is written to the output directory. - -LATEX_CMD_NAME = latex - -# The MAKEINDEX_CMD_NAME tag can be used to specify the command name to -# generate index for LaTeX. If left blank `makeindex' will be used as the -# default command name. - -MAKEINDEX_CMD_NAME = makeindex - -# If the COMPACT_LATEX tag is set to YES Doxygen generates more compact -# LaTeX documents. This may be useful for small projects and may help to -# save some trees in general. - -COMPACT_LATEX = NO - -# The PAPER_TYPE tag can be used to set the paper type that is used -# by the printer. Possible values are: a4, a4wide, letter, legal and -# executive. If left blank a4wide will be used. - -PAPER_TYPE = letter - -# The EXTRA_PACKAGES tag can be to specify one or more names of LaTeX -# packages that should be included in the LaTeX output. - -EXTRA_PACKAGES = - -# The LATEX_HEADER tag can be used to specify a personal LaTeX header for -# the generated latex document. The header should contain everything until -# the first chapter. If it is left blank doxygen will generate a -# standard header. Notice: only use this tag if you know what you are doing! - -LATEX_HEADER = - -# If the PDF_HYPERLINKS tag is set to YES, the LaTeX that is generated -# is prepared for conversion to pdf (using ps2pdf). The pdf file will -# contain links (just like the HTML output) instead of page references -# This makes the output suitable for online browsing using a pdf viewer. - -PDF_HYPERLINKS = NO - -# If the USE_PDFLATEX tag is set to YES, pdflatex will be used instead of -# plain latex in the generated Makefile. Set this option to YES to get a -# higher quality PDF documentation. - -USE_PDFLATEX = YES - -# If the LATEX_BATCHMODE tag is set to YES, doxygen will add the \\batchmode. -# command to the generated LaTeX files. This will instruct LaTeX to keep -# running if errors occur, instead of asking the user for help. -# This option is also used when generating formulas in HTML. - -LATEX_BATCHMODE = YES - -# If LATEX_HIDE_INDICES is set to YES then doxygen will not -# include the index chapters (such as File Index, Compound Index, etc.) -# in the output. - -LATEX_HIDE_INDICES = NO - -# If LATEX_SOURCE_CODE is set to YES then doxygen will include source code with syntax highlighting in the LaTeX output. Note that which sources are shown also depends on other settings such as SOURCE_BROWSER. - -LATEX_SOURCE_CODE = NO - -#--------------------------------------------------------------------------- -# configuration options related to the RTF output -#--------------------------------------------------------------------------- - -# If the GENERATE_RTF tag is set to YES Doxygen will generate RTF output -# The RTF output is optimized for Word 97 and may not look very pretty with -# other RTF readers or editors. - -GENERATE_RTF = NO - -# The RTF_OUTPUT tag is used to specify where the RTF docs will be put. -# If a relative path is entered the value of OUTPUT_DIRECTORY will be -# put in front of it. If left blank `rtf' will be used as the default path. - -RTF_OUTPUT = rtf - -# If the COMPACT_RTF tag is set to YES Doxygen generates more compact -# RTF documents. This may be useful for small projects and may help to -# save some trees in general. - -COMPACT_RTF = NO - -# If the RTF_HYPERLINKS tag is set to YES, the RTF that is generated -# will contain hyperlink fields. The RTF file will -# contain links (just like the HTML output) instead of page references. -# This makes the output suitable for online browsing using WORD or other -# programs which support those fields. -# Note: wordpad (write) and others do not support links. - -RTF_HYPERLINKS = NO - -# Load stylesheet definitions from file. Syntax is similar to doxygen's -# config file, i.e. a series of assignments. You only have to provide -# replacements, missing definitions are set to their default value. - -RTF_STYLESHEET_FILE = - -# Set optional variables used in the generation of an rtf document. -# Syntax is similar to doxygen's config file. - -RTF_EXTENSIONS_FILE = - -#--------------------------------------------------------------------------- -# configuration options related to the man page output -#--------------------------------------------------------------------------- - -# If the GENERATE_MAN tag is set to YES (the default) Doxygen will -# generate man pages - -GENERATE_MAN = NO - -# The MAN_OUTPUT tag is used to specify where the man pages will be put. -# If a relative path is entered the value of OUTPUT_DIRECTORY will be -# put in front of it. If left blank `man' will be used as the default path. - -MAN_OUTPUT = man - -# The MAN_EXTENSION tag determines the extension that is added to -# the generated man pages (default is the subroutine's section .3) - -MAN_EXTENSION = .3 - -# If the MAN_LINKS tag is set to YES and Doxygen generates man output, -# then it will generate one additional man file for each entity -# documented in the real man page(s). These additional files -# only source the real man page, but without them the man command -# would be unable to find the correct page. The default is NO. - -MAN_LINKS = NO - -#--------------------------------------------------------------------------- -# configuration options related to the XML output -#--------------------------------------------------------------------------- - -# If the GENERATE_XML tag is set to YES Doxygen will -# generate an XML file that captures the structure of -# the code including all documentation. - -GENERATE_XML = NO - -# The XML_OUTPUT tag is used to specify where the XML pages will be put. -# If a relative path is entered the value of OUTPUT_DIRECTORY will be -# put in front of it. If left blank `xml' will be used as the default path. - -XML_OUTPUT = xml - -# The XML_SCHEMA tag can be used to specify an XML schema, -# which can be used by a validating XML parser to check the -# syntax of the XML files. - -XML_SCHEMA = - -# The XML_DTD tag can be used to specify an XML DTD, -# which can be used by a validating XML parser to check the -# syntax of the XML files. - -XML_DTD = - -# If the XML_PROGRAMLISTING tag is set to YES Doxygen will -# dump the program listings (including syntax highlighting -# and cross-referencing information) to the XML output. Note that -# enabling this will significantly increase the size of the XML output. - -XML_PROGRAMLISTING = YES - -#--------------------------------------------------------------------------- -# configuration options for the AutoGen Definitions output -#--------------------------------------------------------------------------- - -# If the GENERATE_AUTOGEN_DEF tag is set to YES Doxygen will -# generate an AutoGen Definitions (see autogen.sf.net) file -# that captures the structure of the code including all -# documentation. Note that this feature is still experimental -# and incomplete at the moment. - -GENERATE_AUTOGEN_DEF = NO - -#--------------------------------------------------------------------------- -# configuration options related to the Perl module output -#--------------------------------------------------------------------------- - -# If the GENERATE_PERLMOD tag is set to YES Doxygen will -# generate a Perl module file that captures the structure of -# the code including all documentation. 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This -# makes dot run faster, but since only newer versions of dot (>1.8.10) -# support this, this feature is disabled by default. - -DOT_MULTI_TARGETS = YES - -# If the GENERATE_LEGEND tag is set to YES (the default) Doxygen will -# generate a legend page explaining the meaning of the various boxes and -# arrows in the dot generated graphs. - -GENERATE_LEGEND = YES - -# If the DOT_CLEANUP tag is set to YES (the default) Doxygen will -# remove the intermediate dot files that are used to generate -# the various graphs. - -DOT_CLEANUP = YES diff -Nru lme4-1.1-23/inst/doc/lme4/inst/doc/lmerperf.R lme4-1.1-26/inst/doc/lme4/inst/doc/lmerperf.R --- lme4-1.1-23/inst/doc/lme4/inst/doc/lmerperf.R 2018-03-27 15:08:03.000000000 +0000 +++ lme4-1.1-26/inst/doc/lme4/inst/doc/lmerperf.R 1970-01-01 00:00:00.000000000 +0000 @@ -1,33 +0,0 @@ -## ----opts, echo = FALSE, message = FALSE--------------------------------- -library("knitr") -knitr::opts_chunk$set( - eval = FALSE -) - -## ----loadpkg,message=FALSE----------------------------------------------- -# library("lme4") - -## ----noderivs,eval=FALSE------------------------------------------------- -# lmer(y ~ service * dept + (1|s) + (1|d), InstEval, -# control = lmerControl(calc.derivs = FALSE)) - -## ----nlminbfit,eval=FALSE------------------------------------------------ -# library("optimx") -# lmer(y ~ service * dept + (1|s) + (1|d), InstEval, -# control = lmerControl(optimizer = "optimx", calc.derivs = FALSE, -# optCtrl = list(method = "nlminb", starttests = FALSE, kkt = FALSE))) - -## ----nlopt,eval=FALSE---------------------------------------------------- -# nlopt <- function(par, fn, lower, upper, control) { -# .nloptr <<- res <- nloptr(par, fn, lb = lower, ub = upper, -# opts = list(algorithm = "NLOPT_LN_BOBYQA", print_level = 1, -# maxeval = 1000, xtol_abs = 1e-6, ftol_abs = 1e-6)) -# list(par = res$solution, -# fval = res$objective, -# conv = if (res$status > 0) 0 else res$status, -# message = res$message -# ) -# } -# lmer(y ~ service * dept + (1|s) + (1|d), InstEval, -# control = lmerControl(optimizer = "nloptwrap", calc.derivs = FALSE)) - diff -Nru lme4-1.1-23/inst/doc/lme4/inst/doc/lmerperf.Rmd lme4-1.1-26/inst/doc/lme4/inst/doc/lmerperf.Rmd --- lme4-1.1-23/inst/doc/lme4/inst/doc/lmerperf.Rmd 2018-02-16 14:15:46.000000000 +0000 +++ lme4-1.1-26/inst/doc/lme4/inst/doc/lmerperf.Rmd 1970-01-01 00:00:00.000000000 +0000 @@ -1,76 +0,0 @@ - - -```{r opts, echo = FALSE, message = FALSE} -library("knitr") -knitr::opts_chunk$set( - eval = FALSE -) -``` - -```{r loadpkg,message=FALSE} -library("lme4") -``` - -# lme4 Performance Tips - -- use control = `[g]lmerControl(calc.derivs = FALSE)` to turn off the time - consuming derivative calculation that is performed after the optmization - is finished, e.g. - -```{r noderivs,eval=FALSE} -lmer(y ~ service * dept + (1|s) + (1|d), InstEval, - control = lmerControl(calc.derivs = FALSE)) -``` -Note that this will disable some of the convergence tests, as well as -(for `glmer` only) making `lme4` use a less -accurate approximation to compute the standard errors of the fixed effects. - -- models that only contain random effects of the form `(1|f)` use better - starting values for the optimization which in tests have cut run time in - certain examples by up to 50% relative to the previous default starting - values. The `InstEval` fit shown above is one such example. - -- `lmer` uses the `bobyqa` optimizer from the `minqa` package by default; - `glmer` uses a combination of Nelder-Mead and `bobyqa`. If you - are specifying the `optimx` package optimizer, note that by - default `optimx` performs - certain time-consuming processing at the beginning and end which can be - turned off as follows (here we have specified the `"nlminb"` method but this - applies to any `optimx` method): - -```{r nlminbfit,eval=FALSE} -library("optimx") -lmer(y ~ service * dept + (1|s) + (1|d), InstEval, - control = lmerControl(optimizer = "optimx", calc.derivs = FALSE, - optCtrl = list(method = "nlminb", starttests = FALSE, kkt = FALSE))) -``` - -- the `nloptr` package supports a variety of algorithms and importantly - supports additional stopping criteria which can stop the optimization - earlier if it believes it has reached the optimum. - For many problems using these stopping - criteria will result in the - same solution or nearly the same solution as the default optimizer but in - less time (up to 50 percent savings have been observed); however, in - some cases it may stop prematurely giving suboptimal results. - (In the example below omit `print_level` if output tracing is not desired and - increase `maxeval` if the optimization requires more than 1000 iterations - and you wish to allow it to proceed.) - -```{r nlopt,eval=FALSE} -nlopt <- function(par, fn, lower, upper, control) { - .nloptr <<- res <- nloptr(par, fn, lb = lower, ub = upper, - opts = list(algorithm = "NLOPT_LN_BOBYQA", print_level = 1, - maxeval = 1000, xtol_abs = 1e-6, ftol_abs = 1e-6)) - list(par = res$solution, - fval = res$objective, - conv = if (res$status > 0) 0 else res$status, - message = res$message - ) -} -lmer(y ~ service * dept + (1|s) + (1|d), InstEval, - control = lmerControl(optimizer = "nloptwrap", calc.derivs = FALSE)) -``` diff -Nru lme4-1.1-23/inst/doc/lme4/inst/doc/lmer.R lme4-1.1-26/inst/doc/lme4/inst/doc/lmer.R --- lme4-1.1-23/inst/doc/lme4/inst/doc/lmer.R 2018-03-27 15:08:49.000000000 +0000 +++ lme4-1.1-26/inst/doc/lme4/inst/doc/lmer.R 1970-01-01 00:00:00.000000000 +0000 @@ -1,390 +0,0 @@ -## ----preliminaries,include=FALSE,cache=FALSE,message=FALSE---------- -options(width=70, show.signif.stars=FALSE, - str=strOptions(strict.width="cut"), - ## prefer empty continuation for reader's cut'n'paste: - continue = " ", #JSS: prompt = "R> ", continue = "+ ", - useFancyQuotes = FALSE) -library("knitr") -library("lme4") -library("ggplot2")# Keeping default theme, nicer "on line": -#JSS theme_set(theme_bw()) -library("grid") -zmargin <- theme(panel.spacing=unit(0,"lines")) -library("lattice") -library("minqa") -opts_chunk$set(engine='R',dev='pdf', fig.width=9, fig.height=5.5, - prompt=TRUE, cache=TRUE, tidy=FALSE, comment=NA) -render_sweave() - -## ----sleep---------------------------------------------------------- -str(sleepstudy) - -## ----sleepPlot, echo=FALSE, fig.scap="Reaction time versus days by subject", fig.cap="Average reaction time versus days of sleep deprivation by subject. Subjects ordered (from left to right starting on the top row) by increasing slope of subject-specific linear regressions.", fig.align='center', fig.pos='tb'---- -## BMB: seemed more pleasing to arrange by increasing slope rather than -## intercept ... -xyplot(Reaction ~ Days | Subject, sleepstudy, aspect = "xy", - layout = c(9, 2), type = c("g", "p", "r"), - index.cond = function(x, y) coef(lm(y ~ x))[2], - xlab = "Days of sleep deprivation", - ylab = "Average reaction time (ms)", - as.table = TRUE) - -## ----outputExample,results="hide"----------------------------------- -fm1 <- lmer(Reaction ~ Days + (Days | Subject), sleepstudy) - -## ----modularExampleFormula, tidy=FALSE, eval=FALSE------------------ -# parsedFormula <- lFormula(formula = Reaction ~ Days + (Days | Subject), -# data = sleepstudy) - -## ----modularExampleObjective, tidy=FALSE, eval=FALSE---------------- -# devianceFunction <- do.call(mkLmerDevfun, parsedFormula) - -## ----modularExampleOptimization, tidy=FALSE, eval=FALSE------------- -# optimizerOutput <- optimizeLmer(devianceFunction) - -## ----modularExampleOutput, tidy=FALSE, eval=FALSE------------------- -# mkMerMod( rho = environment(devianceFunction), -# opt = optimizerOutput, -# reTrms = parsedFormula$reTrms, -# fr = parsedFormula$fr) - -## ----MMformula, eval=FALSE------------------------------------------ -# resp ~ FEexpr + (REexpr1 | factor1) + (REexpr2 | factor2) + ... - -## ----uncorrelatedModel,results="hide"------------------------------- -fm2 <- lmer(Reaction ~ Days + (Days || Subject), sleepstudy) - -## ----setSeed, echo=FALSE-------------------------------------------- -set.seed(2) - -## ----factorToSparseMatrix------------------------------------------- -(f <- gl(3, 2)) -(Ji <- t(as(f, Class = "sparseMatrix"))) - -## ----rawModelMatrix------------------------------------------------- -(Xi <- cbind(1, rep.int(c(-1, 1), 3L))) - -## ----KhatriRao------------------------------------------------------ -(Zi <- t(KhatriRao(t(Ji), t(Xi)))) - -## ----sanityCheck, include=FALSE------------------------------------- -## alternative formulation of Zi (eq:Zi) -rBind( - Ji[1,] %x% Xi[1,], - Ji[2,] %x% Xi[2,], - Ji[3,] %x% Xi[3,], - Ji[4,] %x% Xi[4,], - Ji[5,] %x% Xi[5,], - Ji[6,] %x% Xi[6,]) - -## ----nc, echo=FALSE------------------------------------------------- -nc <- 3 - -## ----template------------------------------------------------------- -(rowIndices <- rep(1:nc, 1:nc)) -(colIndices <- sequence(1:nc)) -(template <- sparseMatrix(rowIndices, colIndices, - x = 1 * (rowIndices == colIndices))) - -## ----thetaFromTemplate---------------------------------------------- -(theta <- template@x) - -## ----thetaSanityCheck, include=FALSE-------------------------------- -lFormula(Reaction ~ (Days + I(Days^2) | Subject), sleepstudy)$reTrms$theta - -## ----nl, echo=FALSE------------------------------------------------- -nl <- 2 - -## ----relativeCovarianceBlock---------------------------------------- -(Lambdati <- .bdiag(rep(list(t(template)), nl))) - -## ----relativeCovarianceBlockIndices--------------------------------- -LindTemplate <- rowIndices + nc * (colIndices - 1) - choose(colIndices, 2) -(Lind <- rep(LindTemplate, nl)) - -## ----newTheta------------------------------------------------------- -thetanew <- round(runif(length(theta)), 1) - -## ----relativeCovarianceBlockUpdate---------------------------------- -Lambdati@x <- thetanew[Lind] - -## ----PLSupdateTheta, eval = FALSE----------------------------------- -# Lambdat@x[] <- mapping(theta) - -## ----PLSupdateThetaWithLind----------------------------------------- -mapping <- function(theta) theta[Lind] - -## ----exampleNewTheta------------------------------------------------ -thetanew <- c(1, -0.1, 2, 0.1, -0.2, 3) - -## ----exampleLindUpdate---------------------------------------------- -Lambdati@x[] <- mapping(thetanew) -Lambdati - -## ----updateL, eval=FALSE-------------------------------------------- -# L <- update(L, Lambdat %*% ZtW, mult = 1) - -## ----PLSsolve, eval = FALSE----------------------------------------- -# cu[] <- as.vector(solve(L, solve(L, Lambdat %*% ZtWy, -# system = "P"), system = "L")) -# RZX[] <- as.vector(solve(L, solve(L, Lambdat %*% ZtWX, -# system = "P"), system = "L")) -# RXtRX <- as(XtWX - crossprod(RZX), "dpoMatrix") -# beta[] <- as.vector(solve(RXtRX, XtWy - crossprod(RZX, cu))) -# u[] <- as.vector(solve(L, solve(L, cu - RZX %*% beta, -# system = "Lt"), system = "Pt")) - -## ----PLSupdateResp, eval = FALSE------------------------------------ -# b[] <- as.vector(crossprod(Lambdat, u)) -# mu[] <- as.vector(crossprod(Zt, b) + X %*% beta + offset) -# wtres <- sqrtW * (y - mu) - -## ----PLScalculateObjective, eval = FALSE---------------------------- -# pwrss <- sum(wtres^2) + sum(u^2) -# logDet <- 2*determinant(L, logarithm = TRUE)$modulus -# if (REML) logDet <- logDet + determinant(RXtRX, -# logarithm = TRUE)$modulus -# attributes(logDet) <- NULL -# profDev <- logDet + degFree * (1 + log(2 * pi * pwrss) - log(degFree)) - -## ----vcovByHand----------------------------------------------------- -RX <- getME(fm1, "RX") -sigma2 <- sigma(fm1)^2 -sigma2 * chol2inv(RX) - -## ----condVarExample, echo=FALSE, eval=FALSE------------------------- -# s2 <- sigma(fm1)^2 -# Lambdat <- getME(fm1, "Lambdat") -# Lambda <- getME(fm1, "Lambda") -# Zt <- getME(fm1, "Zt") -# Z <- getME(fm1, "Z") -# y <- getME(fm1, "y") -# X <- getME(fm1, "X") -# beta <- getME(fm1, "beta") -# L <- getME(fm1, "L") -# (V <- crossprod(solve(L, system = "L")))[1:2, 1:2] -# (s2 * Lambda %*% V %*% Lambdat)[1:2, 1:2] -# attr(ranef(fm1, condVar = TRUE)$Subject, "postVar")[ , , 1] # should be same as Lam %*% V %*% Lamt -# (U <- as.vector((V %*% Lambdat %*% Zt %*% (y - X %*% beta)))) -# getME(fm1, "u") - -## ----updateModel---------------------------------------------------- -fm3 <- update(fm1, . ~ . - (Days | Subject) + (1 | Subject)) -formula(fm3) - -## ----summary1,echo=FALSE-------------------------------------------- -ss <- summary(fm1) -cc <- capture.output(print(ss)) -reRow <- grep("^Random effects", cc) -cat(cc[1:(reRow - 2)], sep = "\n") - -## ----summary1reproduced, eval=FALSE--------------------------------- -# formula(fm1) -# REMLcrit(fm1) -# quantile(residuals(fm1, "pearson", scaled = TRUE)) - -## ----summary2, echo=FALSE------------------------------------------- -feRow <- grep("^Fixed effects", cc) -cat(cc[reRow:(feRow - 2)], sep = "\n") - -## ----summary2reproduced, eval=FALSE--------------------------------- -# print(vc <- VarCorr(fm1), comp = c("Variance", "Std.Dev.")) -# nobs(fm1) -# ngrps(fm1) -# sigma(fm1) - -## ----VarCorr-------------------------------------------------------- -as.data.frame(VarCorr(fm1)) - -## ----summary3, echo=FALSE------------------------------------------- -corRow <- grep("^Correlation", cc) -cat(cc[feRow:(corRow - 2)], sep = "\n") - -## ----summary4, echo=FALSE------------------------------------------- -cat(cc[corRow:length(cc)], sep = "\n") - -## ----diagplot1,fig.keep="none"-------------------------------------- -plot(fm1, type = c("p", "smooth")) - -## ----diagplot2,fig.keep="none"-------------------------------------- -plot(fm1, sqrt(abs(resid(.))) ~ fitted(.), - type = c("p", "smooth")) - -## ----diagplot3,fig.keep="none"-------------------------------------- -qqmath(fm1, id = 0.05) - -## ----ppsim,results="hide"------------------------------------------- -iqrvec <- sapply(simulate(fm1, 1000), IQR) -obsval <- IQR(sleepstudy$Reaction) -post.pred.p <- mean(obsval >= c(obsval, iqrvec)) - -## ----anovaQuadraticModel-------------------------------------------- -fm4 <- lmer(Reaction ~ polyDays[ , 1] + polyDays[ , 2] + - (polyDays[ , 1] + polyDays[ , 2] | Subject), - within(sleepstudy, polyDays <- poly(Days, 2))) -anova(fm4) - -## ----anovaSanityCheck, include=FALSE-------------------------------- -(getME(fm4, "RX")[2, ] %*% getME(fm4, "fixef"))^2 - -## ----anovaManyModels------------------------------------------------ -anova(fm1, fm2, fm3) - -## ----anovaRes,echo=FALSE-------------------------------------------- -fm3ML <- refitML(fm3) -fm2ML <- refitML(fm2) -fm1ML <- refitML(fm1) -ddiff <- deviance(fm3ML) - deviance(fm2ML) -dp <- pchisq(ddiff, 1, lower.tail = FALSE) -ddiff2 <- deviance(fm2ML) - deviance(fm1ML) - -## ----pvaluesHelp, eval=FALSE---------------------------------------- -# help("pvalues") - -## ----compareCI,echo=FALSE,cache=TRUE,message=FALSE,warning=FALSE---- -ccw <- confint(fm1, method = "Wald") -ccp <- confint(fm1, method = "profile", oldNames = FALSE) -ccb <- confint(fm1, method = "boot") - -## ----CIqcomp,echo=FALSE,eval=TRUE----------------------------------- -rtol <- function(x,y) { - abs((x - y) / ((x + y) / 2)) -} -bw <- apply(ccb, 1, diff) -pw <- apply(ccp, 1, diff) -mdiffpct <- round(100 * max(rtol(bw, pw))) - -## ----CIplot,echo=FALSE,eval=FALSE----------------------------------- -# ## obsolete -# ## ,fig.cap="Comparison of confidence intervals",fig.scap="CI comparison" -# tf <- function(x, method) data.frame(method = method, -# par = rownames(x), -# setNames(as.data.frame(x), c("lwr", "upr"))) -# cc.all <- do.call(rbind, mapply(tf, list(ccw, ccp, ccb), -# c("Wald", "profile", "boot"), -# SIMPLIFY = FALSE)) -# ggplot(cc.all, aes(x = 1, ymin = lwr, ymax = upr, colour = method)) + -# geom_linerange(position = position_dodge(width = 0.5)) + -# facet_wrap( ~ par, scale = "free") + -# theme(axis.text.x = element_blank()) + -# labs(x = "") - -## ----profile_calc,echo=FALSE,cache=TRUE----------------------------- -pf <- profile(fm1) - -## ----profile_zeta_plot,fig.cap="Profile zeta plot: \\code{xyplot(prof.obj)}",fig.scap="Profile zeta plot",echo=FALSE,fig.align='center',fig.pos='tb'---- -xyplot(pf) - -## ----profile_density_plot,fig.cap="Profile density plot: \\code{densityplot(prof.obj)}",echo=FALSE,fig.align='center',fig.pos='tb'---- -densityplot(pf) - -## ----profile_pairs_plot,fig.cap="Profile pairs plot: \\code{splom(prof.obj)}",echo=FALSE,fig.height=8,fig.width=8,fig.align='center',fig.pos='htb',out.height='5.5in'---- -splom(pf) - -## ----modularSimulationFormula--------------------------------------- -form <- respVar ~ 1 + (explVar1 + explVar2 | groupFac) - -## ----dataTemplate--------------------------------------------------- -set.seed(1) -dat <- mkDataTemplate(form, - nGrps = 500, - nPerGrp = 20, - rfunc = rnorm) -head(dat) - -## ----parsTemplate--------------------------------------------------- -(pars <- mkParsTemplate(form, dat)) - -## ----simCorr-------------------------------------------------------- -vc <- matrix(c(1.0, 0.5, 0.5, - 0.5, 1.0, 0.0, - 0.5, 0.0, 1.0), 3, 3) - -## ----vcTheta-------------------------------------------------------- -pars$theta[] <- Vv_to_Cv(mlist2vec(vc)) - -## ----varCorrStructure----------------------------------------------- -dat$respVar <- simulate(form, - newdata = dat, - newparams = pars, - family = "gaussian")[[1]] - -## ----graphSims------------------------------------------------------ -formLm <- form -formLm[[3]] <- findbars(form)[[1]] -print(formLm) -cor(t(sapply(lmList(formLm, dat), coef))) - -## ----phiToTheta, cache = FALSE-------------------------------------- -phiToTheta <- function(phi) { - theta5 <- -(phi[2]*phi[3])/phi[4] - c(phi[1:4], theta5, phi[5]) -} - -## ----compute deviance function modular, cache = FALSE--------------- -lf <- lFormula(formula = form, data = dat, REML = TRUE) -devf <- do.call(mkLmerDevfun, lf) - -## ----wrapper modular, cache = FALSE--------------------------------- -devfWrap <- function(phi) devf(phiToTheta(phi)) - -## ----opt modular, cache = FALSE------------------------------------- -opt <- bobyqa(par = lf$reTrms$theta[-5], - fn = devfWrap, - lower = lf$reTrms$lower[-5]) - -## ----varCorr modular, cache = FALSE--------------------------------- -vcEst <- vec2mlist(Cv_to_Vv(phiToTheta(opt$par)))[[1]] -dimnames(vcEst) <- rep(lf$reTrms$cnms, 2) -round(vcEst, 2) -vc - -## ----simulateSplineData,message=FALSE------------------------------- -library("gamm4") -library("splines") -set.seed(1) -n <- 100 -pSimulation <- 3 -pStatistical <- 8 -x <- rnorm(n) -Bsimulation <- ns(x, pSimulation) -Bstatistical <- ns(x, pStatistical) -beta <- rnorm(pSimulation) -y <- as.numeric(Bsimulation %*% beta + rnorm(n, sd = 0.3)) - -## ----splineExampleDataPlot, fig.width=4, fig.height=3,fig.align="center"---- -par(mar = c(4, 4, 1, 1), las = 1, bty = "l") -plot(x, y, las = 1) -lines(x[order(x)], (Bsimulation %*% beta)[order(x)]) - -## ----splineExampleApproximateFormula-------------------------------- -pseudoGroups <- as.factor(rep(1:pStatistical, length = n)) -parsedFormula <- lFormula(y ~ x + (1 | pseudoGroups)) - -## ----splineExampleModifyZt------------------------------------------ -parsedFormula$reTrms <- within(parsedFormula$reTrms, { - Bt <- t(as.matrix(Bstatistical))[] - cnms$pseudoGroups <- "spline" - Zt <- as(Bt, class(Zt)) -}) - -## ----splineExampleRemainingModularSteps----------------------------- -devianceFunction <- do.call(mkLmerDevfun, parsedFormula) -optimizerOutput <- optimizeLmer(devianceFunction) -mSpline <- mkMerMod( rho = environment(devianceFunction), - opt = optimizerOutput, - reTrms = parsedFormula$reTrms, - fr = parsedFormula$fr) -mSpline - -## ----splineExampleFittedModelPlot, fig.width=4, fig.height=3, fig.align="center"---- -xNew <- seq(min(x), max(x), length = 100) -newBstatistical <- predict(Bstatistical, xNew) -yHat <- cbind(1, xNew) %*% getME(mSpline, "fixef") + - newBstatistical %*% getME(mSpline, "u") -par(mar = c(4, 4, 1, 1), las = 1, bty = "l") -plot(x, y) -lines(xNew, yHat) -lines(x[order(x)], (Bsimulation %*% beta)[order(x)],lty = 2) -legend("topright", bty = "n", c("fitted", "generating"), lty = 1:2,col = 1) - diff -Nru lme4-1.1-23/inst/doc/lme4/inst/doc/lmer.Rnw lme4-1.1-26/inst/doc/lme4/inst/doc/lmer.Rnw --- lme4-1.1-23/inst/doc/lme4/inst/doc/lmer.Rnw 2018-03-22 18:09:40.000000000 +0000 +++ lme4-1.1-26/inst/doc/lme4/inst/doc/lmer.Rnw 1970-01-01 00:00:00.000000000 +0000 @@ -1,3033 +0,0 @@ -%\VignetteEngine{knitr::knitr} -%\VignetteDepends{ggplot2} -%\VignetteDepends{gamm4} -%\VignetteIndexEntry{Fitting Linear Mixed-Effects Models using lme4} -\documentclass[nojss]{jss} -\usepackage[T1]{fontenc}% for correct hyphenation and T1 encoding -\usepackage[utf8]{inputenc}% -\usepackage{lmodern}% latin modern font -\usepackage[american]{babel} %% for texi2dvi ~ bug -\usepackage{bm,amsmath,thumbpdf,amsfonts}%,minted} -\usepackage{blkarray} -\usepackage{array} -\newcolumntype{P}[1]{>{\raggedright\arraybackslash}p{#1}} -\newcommand{\matindex}[1]{\mbox{\scriptsize#1}}% Matrix index -\newcommand{\github}{Github} -\DeclareMathOperator{\tr}{tr} -\DeclareMathOperator{\VEC}{vec} -\newcommand{\bmb}[1]{{\color{red} \emph{#1}}} -\newcommand{\scw}[1]{{\color{blue} \emph{#1}}} -\newcommand{\dmb}[1]{{\color{magenta} \emph{#1}}} - -\shortcites{bolker_strategies_2013,sleepstudy,gelman2013bayesian} - -\author{Douglas Bates\\University of Wisconsin-Madison\And - Martin M\"achler\\ETH Zurich\And - Benjamin M. Bolker\\McMaster University\And - Steven C. Walker\\McMaster University -} -\Plainauthor{Douglas Bates, Martin M\"achler, Ben Bolker, Steve Walker} -\title{Fitting Linear Mixed-Effects Models Using \pkg{lme4}} -\Plaintitle{Fitting Linear Mixed-Effects Models using lme4} -\Shorttitle{Linear Mixed Models with lme4} -\Abstract{% - - Maximum likelihood or restricted maximum likelihood (REML) estimates - of the parameters in linear mixed-effects models can be determined - using the \code{lmer} function in the \pkg{lme4} package for - \proglang{R}. As for most model-fitting functions in \proglang{R}, - the model is described in an \code{lmer} call by a formula, in this - case including both fixed- and random-effects - terms. The formula and data together determine a numerical - representation of the model from which the profiled deviance or the - profiled REML criterion can be evaluated as a function of some of - the model parameters. The appropriate criterion is optimized, using - one of the constrained optimization functions in \proglang{R}, to - provide the parameter estimates. We describe the structure of the - model, the steps in evaluating the profiled deviance or REML - criterion, and the structure of classes or types that represents such - a model. Sufficient detail is included to allow specialization of - these structures by users who wish to write functions to fit - specialized linear mixed models, such as models incorporating - pedigrees or smoothing splines, that are not easily expressible in - the formula language used by \code{lmer}.} - -\Keywords{% - sparse matrix methods, linear mixed models, penalized least squares, - Cholesky decomposition} \Address{ - Douglas Bates\\ - Department of Statistics, University of Wisconsin\\ - 1300 University Ave.\\ - Madison, WI 53706, U.S.A.\\ - E-mail: \email{bates@stat.wisc.edu}\\ - \par\bigskip - Martin M\"achler\\ - Seminar f\"ur Statistik, HG G~16\\ - ETH Zurich\\ - 8092 Zurich, Switzerland\\ - E-mail: \email{maechler@stat.math.ethz.ch}\\ - % URL: \url{http://stat.ethz.ch/people/maechler}\\ - \par\bigskip - Benjamin M. Bolker\\ - Departments of Mathematics \& Statistics and Biology \\ - McMaster University \\ - 1280 Main Street W \\ - Hamilton, ON L8S 4K1, Canada \\ - E-mail: \email{bolker@mcmaster.ca}\\ - \par\bigskip - Steven C. Walker\\ - Department of Mathematics \& Statistics \\ - McMaster University \\ - 1280 Main Street W \\ - Hamilton, ON L8S 4K1, Canada \\ - E-mail: \email{scwalker@math.mcmaster.ca } -} -\newcommand{\thetavec}{{\bm\theta}} -\newcommand{\betavec}{{\bm\beta}} -\newcommand{\Var}{\operatorname{Var}} -\newcommand{\abs}{\operatorname{abs}} -\newcommand{\bLt}{\ensuremath{\bm\Lambda_{\bm\theta}}} -\newcommand{\mc}[1]{\ensuremath{\mathcal{#1}}} -\newcommand{\trans}{\ensuremath{^\top}} % JSS wants \top -\newcommand{\yobs}{\ensuremath{\bm y_{\mathrm{obs}}}} -\newcommand*{\eq}[1]{eqn.~\ref{#1}}% or just {(\ref{#1})} -<>= -options(width=70, show.signif.stars=FALSE, - str=strOptions(strict.width="cut"), - ## prefer empty continuation for reader's cut'n'paste: - continue = " ", #JSS: prompt = "R> ", continue = "+ ", - useFancyQuotes = FALSE) -library("knitr") -library("lme4") -library("ggplot2")# Keeping default theme, nicer "on line": -#JSS theme_set(theme_bw()) -library("grid") -zmargin <- theme(panel.spacing=unit(0,"lines")) -library("lattice") -library("minqa") -opts_chunk$set(engine='R',dev='pdf', fig.width=9, fig.height=5.5, - prompt=TRUE, cache=TRUE, tidy=FALSE, comment=NA) -render_sweave() -@ -\setkeys{Gin}{width=\textwidth} -\setkeys{Gin}{height=3.5in} - -\begin{document} - -A version of this manuscript has been published online in the -\emph{Journal of Statistical Software}, on Oct.\ 2015, with DOI \linebreak[3] -\texttt{10.18637/jss.v067.i01}, -see \url{https://www.jstatsoft.org/article/view/v067i01/}. - -\section{Introduction} -\label{sec:intro} - -The \pkg{lme4} package \citep{lme4} for \proglang{R} \citep{R} provides functions to fit and -analyze linear mixed models, generalized linear mixed models and -nonlinear mixed models. In each of these names, the term ``mixed'' -or, more fully, ``mixed effects'', denotes a model that incorporates -both fixed- and random-effects terms in a linear predictor expression -from which the conditional mean of the response can be evaluated. In -this paper we describe the formulation and representation of linear -mixed models. The techniques used for generalized linear and -nonlinear mixed models will be described separately, in a future -paper. - -At present, the main alternative to \pkg{lme4} for mixed modeling in -\proglang{R} is the \pkg{nlme} package \citep{nlme_pkg}. The main features -distinguishing \pkg{lme4} from \pkg{nlme} are (1) more efficient -linear algebra tools, giving improved performance on large problems; -(2) simpler syntax and more efficient implementation for fitting -models with crossed random effects; (3) the implementation of profile -likelihood confidence intervals on random-effects parameters; and (4) -the ability to fit generalized linear mixed models (although in this -paper we restrict ourselves to linear mixed models). The main -advantage of \pkg{nlme} relative to \pkg{lme4} is a user interface for -fitting models with structure in the residuals (various forms of -heteroscedasticity and autocorrelation) and in the random-effects -covariance matrices (e.g., compound symmetric models). With some extra -effort, the computational machinery of \pkg{lme4} can be used to fit -structured models that the basic \code{lmer} function cannot handle -(see Appendix~\ref{sec:modularExamples}). - - -The development of general software for fitting mixed models remains -an active area of research with many open problems. Consequently, the -\pkg{lme4} package has evolved since it was first released, and -continues to improve as we learn more about mixed models. However, we -recognize the need to maintain stability and backward compatibility of -\pkg{lme4} so that it continues to be broadly useful. In order to -maintain stability while continuing to advance mixed-model -computation, we have developed several additional frameworks that draw -on the basic ideas of \pkg{lme4} but modify its structure or -implementation in various ways. These descendants include the -\mbox{\pkg{MixedModels}} package \citep{MixedModels} in \proglang{Julia} -\citep{Julia}, the -\pkg{lme4pureR} package \citep{lme4pureR} in \proglang{R}, and the \pkg{flexLambda} -development branch of \pkg{lme4}. The current article is largely -restricted to describing the current stable version of the \pkg{lme4} -package (1.1-7), with Appendix~\ref{sec:modularExamples} describing -hooks into the computational machinery that are designed for extension -development. The \pkg{gamm4} \citep{gamm4} and \pkg{blme} \citep{blme, blme2} -packages currently make use of these hooks. - -Another goal of this article is to contrast the approach used by -\pkg{lme4} with previous formulations of mixed models. The -expressions for the profiled log-likelihood and profiled REML -(restricted maximum likelihood) criteria derived in -Section~\ref{sec:profdev} are similar to those presented in -\citet{bates04:_linear} and, indeed, are closely related to -``Henderson's mixed-model equations''~\citep{henderson_1982}. -Nonetheless there are subtle but important changes in the formulation -of the model and in the structure of the resulting penalized least -squares (PLS) problem to be solved (Section~\ref{sec:PLSpureR}). We -derive the current version of the PLS problem -(Section~\ref{sec:plsMath}) and contrast this result with earlier -formulations (Section~\ref{sec:previous_lmm_form}). - -This article is organized into four main sections (Sections~\ref{sec:lFormula}, -\ref{sec:mkLmerDevfun}, \ref{sec:optimizeLmer}, and \ref{sec:mkMerMod}), -each of which corresponds to one of the four largely separate modules -that comprise \pkg{lme4}. Before describing the details of each -module, we describe the general form of the linear mixed model -underlying \pkg{lme4} (Section~\ref{sec:LMMs}); introduce the -\code{sleepstudy} data that will be used as an example throughout -(Section~\ref{sec:sleepstudy}); and broadly outline \pkg{lme4}'s -modular structure (Section~\ref{sec:modular}). - -\subsection{Linear mixed models} -\label{sec:LMMs} - -Just as a linear model is described by the distribution of a -vector-valued random response variable, $\mc{Y}$, whose observed value -is $\yobs$, a linear mixed model is described by the distribution of -two vector-valued random variables: $\mc{Y}$, the response, and -$\mc{B}$, the vector of random effects. In a linear model the -distribution of $\mc Y$ is multivariate normal,%\begin{linenomath} -\begin{equation} - \label{eq:linearmodel} - \mc Y\sim\mc{N}(\bm X\bm\beta+\bm o,\sigma^2\bm W^{-1}), -\end{equation} -where $n$ is the dimension of the response vector, $\bm W$ is a -diagonal matrix of known prior weights, $\bm\beta$ is a -$p$-dimensional coefficient vector, $\bm X$ is an $n\times p$ model -matrix, and $\bm o$ is a vector of known prior offset terms. The -parameters of the model are the coefficients $\bm\beta$ and the scale -parameter $\sigma$. - -In a linear mixed model it is the \emph{conditional} distribution of -$\mc Y$ given $\mc B=\bm b$ that has such a form, -\begin{equation} - \label{eq:LMMcondY} - ( \mc Y|\mc B=\bm b)\sim\mc{N}(\bm X\bm\beta+\bm Z\bm b+\bm o,\sigma^2\bm W^{-1}), % | <- for ESS -\end{equation} -where $\bm Z$ is the $n\times q$ model matrix for the $q$-dimensional -vector-valued random-effects variable, $\mc B$, whose value we are -fixing at $\bm b$. The unconditional distribution of $\mc B$ is also -multivariate normal with mean zero and a parameterized $q\times q$ -variance-covariance matrix, $\bm\Sigma$, -\begin{equation} - \label{eq:LMMuncondB} - \mc B\sim\mc N(\bm0,\bm\Sigma) . -\end{equation} -As a variance-covariance matrix, $\bm\Sigma$ must be positive -semidefinite. It is convenient to express the model in terms of a -\emph{relative covariance factor}, $\bLt$, which is a -$q\times q$ matrix, depending on the \emph{variance-component - parameter}, $\bm\theta$, and generating the symmetric $q\times q$ -variance-covariance matrix, $\bm\Sigma$, according to%\begin{linenomath} -\begin{equation} - \label{eq:relcovfac} - \bm\Sigma_{\bm\theta}=\sigma^2\bLt\bLt\trans , -\end{equation}%\end{linenomath} -where $\sigma$ is the same scale factor as in the conditional -distribution (\ref{eq:LMMcondY}). - -Although Equations~\ref{eq:LMMcondY}, \ref{eq:LMMuncondB}, and -\ref{eq:relcovfac} fully describe the class of linear mixed models -that \pkg{lme4} can fit, this terse description -hides many important details. Before moving on to these details, we -make a few observations: -\begin{itemize} -\item This formulation of linear mixed models allows for a relatively - compact expression for the profiled log-likelihood of $\bm\theta$ - (Section~\ref{sec:profdev}, Equation~\ref{eq:profiledDeviance}). -\item The matrices associated with random effects, $\bm Z$ and $\bLt$, - typically have a sparse structure with a sparsity pattern that - encodes various model assumptions. Sections~\ref{sec:LMMmatrix} and - \ref{sec:CSCmats} provide details on these structures, and how to - represent them efficiently. -\item The interface provided by \pkg{lme4}'s \code{lmer} function - is slightly less general than the model described by Equations~\ref{eq:LMMcondY}, \ref{eq:LMMuncondB}, and \ref{eq:relcovfac}. To - take advantage of the entire range of possibilities, one may use the - modular functions (Sections~\ref{sec:modular} and - Appendix~\ref{sec:modularExamples}) or explore the - experimental \pkg{flexLambda} branch of \pkg{lme4} on \github. -\end{itemize} - -\subsection{Example} -\label{sec:sleepstudy} - -Throughout our discussion of \pkg{lme4}, we will work with a data set -on the average reaction time per day for subjects in a sleep -deprivation study \citep{sleepstudy}. On day 0 the subjects had their -normal amount of sleep. Starting that night they were restricted to 3 -hours of sleep per night. The response variable, \code{Reaction}, -represents average reaction times in milliseconds (ms) on a series of tests given each -\code{Day} to each \code{Subject} (Figure~\ref{fig:sleepPlot}), -% -<>= -str(sleepstudy) -@ - -<>= -## BMB: seemed more pleasing to arrange by increasing slope rather than -## intercept ... -xyplot(Reaction ~ Days | Subject, sleepstudy, aspect = "xy", - layout = c(9, 2), type = c("g", "p", "r"), - index.cond = function(x, y) coef(lm(y ~ x))[2], - xlab = "Days of sleep deprivation", - ylab = "Average reaction time (ms)", - as.table = TRUE) -@ -% | - -Each subject's reaction time increases approximately linearly -with the number of sleep-deprived days. -However, subjects also appear to vary in the slopes and intercepts -of these relationships, which suggests a model with random slopes and intercepts. -As we shall see, such a model may be fitted by minimizing the -REML criterion (Equation~\ref{eq:REMLdeviance}) using -<>= -fm1 <- lmer(Reaction ~ Days + (Days | Subject), sleepstudy) -@ -% | -The estimates of the standard deviations of the random effects for the -intercept and the slope are -\Sexpr{round(sqrt(VarCorr(fm1)$Subject[1,1]), 2)} ms % $ -and -\Sexpr{round(sqrt(VarCorr(fm1)$Subject[2,2]), 2)} ms/day. % $ -The fixed-effects coefficients, $\betavec$, are -\Sexpr{round(fixef(fm1)[1], 1)} ms and -\Sexpr{round(fixef(fm1)[2], 2)} ms/day -for the intercept and slope. In this model, one interpretation of -these fixed effects is that they are the estimated population mean values of the -random intercept and slope (Section~\ref{sec:intuitiveFormulas}). - -We have chosen the \code{sleepstudy} example because it is a -relatively small and simple example to illustrate the theory and -practice underlying \code{lmer}. However, \code{lmer} is capable of -fitting more complex mixed models to larger data sets. For example, we -direct the interested reader to \code{RShowDoc("lmerperf", package = - "lme4")} for examples that more thoroughly exercise the -performance capabilities of \code{lmer}. - -\subsection{High-level modular structure} -\label{sec:modular} - -The \code{lmer} function is composed of four largely independent -modules. In the first module, a mixed-model formula is parsed and -converted into the inputs required to specify a linear mixed model -(Section~\ref{sec:lFormula}). The second module uses these inputs to -construct an \proglang{R} function which takes the covariance -parameters, $\bm\theta$, as arguments and returns negative twice the log -profiled likelihood or the REML criterion -(Section~\ref{sec:mkLmerDevfun}). The third module optimizes this -objective function to produce maximum likelihood (ML) or REML -estimates of $\bm\theta$ (Section~\ref{sec:optimizeLmer}). Finally, -the fourth module provides utilities for interpreting the optimized -model (Section~\ref{sec:mkMerMod}). - -\begin{table}[tb] - \centering - \begin{tabular}{lllp{2.1in}} - \hline - Module & & \proglang{R} function & Description \\ - \hline - Formula module & (Section~\ref{sec:lFormula}) & \code{lFormula} & - Accepts a mixed-model formula, data, and other user inputs, and returns - a list of objects required to fit a linear mixed model. \\ - Objective function module & (Section~\ref{sec:mkLmerDevfun}) & - \code{mkLmerDevfun} & - Accepts the results of \code{lFormula} and returns a function to - calculate the deviance (or restricted deviance) as a function of the - covariance parameters, $\bm\theta$.\\ - Optimization module & (Section~\ref{sec:optimizeLmer}) & - \code{optimizeLmer} & - Accepts a deviance function returned by \code{mkLmerDevfun} and - returns the results of the optimization of that deviance function. \\ - Output module & (Section~\ref{sec:mkMerMod}) & \code{mkMerMod} & - Accepts an optimized deviance function and packages the results - into a useful object. \\ - \hline - \end{tabular} - \caption{The high-level modular structure of \code{lmer}.} - \label{tab:modular} -\end{table} - -To illustrate this modularity, we recreate the \code{fm1} object by a -series of four modular steps; the formula module, -<>= -parsedFormula <- lFormula(formula = Reaction ~ Days + (Days | Subject), - data = sleepstudy) -@ -the objective function module, -<>= -devianceFunction <- do.call(mkLmerDevfun, parsedFormula) -@ -the optimization module, -<>= -optimizerOutput <- optimizeLmer(devianceFunction) -@ -and the output module, -<>= -mkMerMod( rho = environment(devianceFunction), - opt = optimizerOutput, - reTrms = parsedFormula$reTrms, - fr = parsedFormula$fr) -@ -% | - -\section{Formula module} -\label{sec:lFormula} - -\subsection{Mixed-model formulas} -\label{sec:formulas} - -Like most model-fitting functions in \proglang{R}, -\code{lmer} takes as its first two arguments a \emph{formula} -specifying the model and the \emph{data} with which to evaluate the -formula. This second argument, \code{data}, is optional but -recommended and is usually the name of an \proglang{R} data frame. -In the \proglang{R} \code{lm} function for fitting linear models, -formulas take the form \verb+resp ~ expr+, -where \code{resp} determines the response variable and \code{expr} -is an expression that specifies the columns of the model matrix. -Formulas for the \code{lmer} function contain special random-effects -terms, -<>= -resp ~ FEexpr + (REexpr1 | factor1) + (REexpr2 | factor2) + ... -@ -where \code{FEexpr} is an expression determining the columns of the -fixed-effects model matrix, $\bm X$, and the random-effects terms, -\code{(REexpr1 | factor1)} and \code{(REexpr2 | factor2)}, determine -both the random-effects model matrix, $\bm Z$ (Section~\ref{sec:mkZ}), -and the structure of the relative covariance factor, -$\bLt$ (Section~\ref{sec:mkLambdat}). In principle, a -mixed-model formula may contain arbitrarily many random-effects terms, -but in practice the number of such terms is typically low. - -\subsection{Understanding mixed-model formulas} -\label{sec:intuitiveFormulas}\label{sec:uncor} - -Before describing the details of how \pkg{lme4} parses mixed-model -formulas (Section~\ref{sec:LMMmatrix}), we provide an informal -explanation and then some examples. Our discussion assumes familiarity -with the standard \proglang{R} modeling paradigm -\citep{Chambers:1993}. - -Each random-effects term is of the form \code{(expr | factor)}. -The expression \code{expr} is evaluated as a linear model formula, -producing a model matrix following the same rules used in standard -\proglang{R} modeling functions (e.g., \code{lm} or \code{glm}). The -expression \code{factor} is evaluated as an \proglang{R} factor. One -way to think about the vertical bar operator is as a special kind of -interaction between the model matrix and the grouping factor. This -interaction ensures that the columns of the model matrix have -different effects for each level of the grouping factor. What makes -this a special kind of interaction is that these effects are modeled -as unobserved random variables, rather than unknown fixed parameters. -Much has been written about important practical and philosophical -differences between these two types of interactions -\citep[e.g., ][]{henderson_1982,gelman2005analysis}. For -example, the random-effects implementation of such interactions can be -used to obtain shrinkage estimates of regression coefficients -\citep[e.g., ][]{1977EfronAndMorris}, or account -for lack of independence in the residuals due to block structure or -repeated measurements \citep[e.g., ][]{laird_ware_1982}. - -Table~\ref{tab:formulas} provides several examples of the -right-hand-sides of mixed-model formulas. The first example, -\code{(1 | g)}, % | -is the simplest possible mixed-model formula, where each level of the -grouping factor, \code{g}, has its own random intercept. The mean and -standard deviation of these intercepts are parameters to be estimated. -Our description of this model incorporates any nonzero mean of the -random effects as fixed-effects parameters. If one wishes to specify -that a random intercept has \emph{a priori} known means, one may use -the \code{offset} function as in the second model in -Table~\ref{tab:formulas}. This model contains no fixed effects, or -more accurately the fixed-effects model matrix, $\bm X$, has zero -columns and $\bm\beta$ has length zero. - -\begin{table}[tb] - \centering - \begin{tabular}{llP{1.5in}} %% see new column type for ragged right - \hline - Formula & Alternative & Meaning \\ - \hline%------------------------------------------------ - \code{(1 | g)} & \code{1 + (1 | g)} - & Random intercept with fixed mean. \\ - \code{0 + offset(o) + (1 | g)} & \code{-1 + offset(o) + (1 | g)} - & Random intercept with \emph{a priori} means. \\ - \code{(1 | g1/g2)} & \code{(1 | g1)+(1 | g1:g2)} % | - & Intercept varying among \code{g1} and \code{g2} within \code{g1}. \\ - \code{(1 | g1) + (1 | g2)} & \code{1 + (1 | g1) + (1 | g2)}. & - Intercept varying among \code{g1} and \code{g2}. \\ - \code{x + (x | g)} & \code{1 + x + (1 + x | g)} & - Correlated random intercept and slope. \\ - \code{x + (x || g)} & \code{1 + x + (1 | g) + (0 + - x | g)} - & Uncorrelated random intercept and slope. \\ - \hline - \end{tabular} - \caption{Examples of the right-hand-sides of mixed-effects model - formulas. The names of grouping factors are denoted \code{g}, - \code{g1}, and \code{g2}, and covariates and \emph{a priori} known - offsets as \code{x} and \code{o}.} - \label{tab:formulas} -\end{table} - -We may also construct models with multiple grouping factors. For -example, if the observations are grouped by \code{g2}, which is nested -within \code{g1}, then the third formula in Table \ref{tab:formulas} -can be used to model variation in the intercept. A common objective -in mixed modeling is to account for such nested (or hierarchical) -structure. However, one of the most useful aspects of \pkg{lme4} is -that it can be used to fit random effects associated with non-nested -grouping factors. For example, suppose the data are grouped by fully -crossing two factors, \code{g1} and \code{g2}, then the fourth formula -in Table \ref{tab:formulas} may be used. Such models are common in -item response theory, where \code{subject} and \code{item} factors are -fully crossed \citep{doran2007estimating}. In addition to varying -intercepts, we may also have varying slopes (e.g., the -\code{sleepstudy} data, Section~\ref{sec:sleepstudy}). The fifth -example in Table~\ref{tab:formulas} gives a model where both the -intercept and slope vary among the levels of the grouping factor. - -\subsubsection{Specifying uncorrelated random effects} -\label{sec:uncor} - -By default, \pkg{lme4} assumes that all coefficients associated with -the same random-effects term are correlated. To specify an -uncorrelated slope and intercept (for example), one may either use -double-bar notation, \code{(x || g)}, or equivalently use multiple -random-effects terms, \code{x + (1 | g) + (0 + x | g)}, as in the -final example of Table~\ref{tab:formulas}. For example, if one -examined the results of model \code{fm1} of the \code{sleepstudy} data -(Section~\ref{sec:sleepstudy}) using \code{summary(fm1)}, one would -see that the estimated correlation between the slope for \code{Days} -and the intercept is fairly low -(\Sexpr{round(attr(VarCorr(fm1)$Subject, "correlation")[2],3)}) % $ -(See Section~\ref{sec:summary} below for more on how to extract the -random-effects covariance matrix.) We may use double-bar notation to -fit a model that excludes a correlation parameter: -<>= -fm2 <- lmer(Reaction ~ Days + (Days || Subject), sleepstudy) -@ - -Although mixed models where the random slopes and intercepts are -assumed independent are commonly used to reduce the complexity of -random-slopes models, they do have one subtle drawback. Models in -which the slopes and intercepts are allowed to have a nonzero -correlation (e.g., \code{fm1}) are invariant to additive shifts of the -continuous predictor (\code{Days} in this case). This invariance -breaks down when the correlation is constrained to zero; any shift in -the predictor will necessarily lead to a change in the estimated -correlation, and in the likelihood and predictions of the model. For -example, we can eliminate the correlation in \code{fm1} simply by -adding an amount equal to the ratio of the estimated -among-subject standard deviations multiplied by the estimated -correlation (i.e., $\sigma_{\text{\small slope}}/\sigma_{\text{\small - intercept}} \cdot \rho_{\text{\small slope:intercept}}$) to the \code{Days} variable. The use -of models such as \code{fm2} should ideally be restricted to cases where the -predictor is measured on a ratio scale (i.e., the zero point on the -scale is meaningful, not just a location defined by convenience or -convention), as is the case here. - -%% <>= -%% sleepstudyShift <- within(sleepstudy, { -%% Days <- Days + (24.74*0.07)/5.92 }) -%% lmer(Reaction ~ Days + (Days | Subject), sleepstudyShift) -%% @ - -\subsection{Algebraic and computational account of mixed-model formulas} -\label{sec:LMMmatrix} - -The fixed-effects terms of a mixed-model formula are parsed to produce -the fixed-effects model matrix, $\bm X$, in the same way that the -\proglang{R} \code{lm} function generates model matrices. However, a -mixed-model formula incorporates $k\ge1$ random-effects terms of the -form \code{(r | f)} as well. % | -These $k$ terms are used to produce the random-effects model matrix, -$\bm Z$ (Equation~\ref{eq:LMMcondY}; Section~\ref{sec:mkZ}), and the -structure of the relative covariance factor, $\bLt$ -(Equation~\ref{eq:relcovfac}; Section~\ref{sec:mkLambdat}), which are -matrices that typically have a sparse structure. We now describe how -one might construct these matrices from the random-effects terms, -considering first a single term, \code{(r | f)}, % | -and then generalizing to multiple terms. Tables~\ref{tab:dim} and -\ref{tab:algebraic} summarize the matrices and vectors that determine -the structure of $\bm Z$ and -$\bLt$. - -\begin{table}[tb] - \centering - \begin{tabular}{lll} - \hline - Symbol & Size \\ - \hline - $n$ & Length of the response vector, $\mc{Y}$ \\ - $p$ & Number of columns of fixed-effects model matrix, $\bm X$ \\ - $q = \sum_i^k q_i$ & Number of columns of random-effects model matrix, $\bm Z$ \\ - $p_i$ & Number of columns of the raw model matrix, $\bm X_i$ \\ - $\ell_i$ & Number of levels of the grouping factor indices, $\bm i_i$ \\ - $q_i = p_i\ell_i$ & Number of columns of the term-wise model matrix, $\bm Z_i$ \\ - $k$ & Number of random-effects terms \\ - $m_i = \binom{p_i+1}{2}$ & Number of covariance parameters for term - $i$ \\ - $m = \sum_i^k m_i$ & Total number of covariance parameters \\ - \hline - \end{tabular} - \caption{Dimensions of linear mixed models. The - subscript $i = 1, \dots, k$ denotes a specific random-effects term.} - \label{tab:dim} -\end{table} - -\begin{table}[tb] - \centering - \begin{tabular}{lll} - \hline - Symbol & Size & Description \\ - \hline - $\bm X_i$ & $n\times p_i$ & Raw random-effects model matrix \\ - $\bm J_i$ & $n\times \ell_i$ & Indicator matrix of grouping factor indices\\ - $\bm X_{ij}$ & $p_i\times 1$ & Column vector containing $j$th row of $\bm X_i$ \\ - $\bm J_{ij}$ & $\ell_i\times 1$ & Column vector containing $j$th row of $\bm J_i$ \\ - $\bm i_i$ & $n$ & Vector of grouping factor indices \\ - $\bm Z_i$ & $n\times q_i$ & Term-wise random-effects model matrix \\ - $\bm\theta$ & $m$ & Covariance parameters \\ - $\bm T_i$ & $p_i\times p_i$ & Lower triangular template matrix \\ - $\bm\Lambda_i$ & $q_i\times q_i$ & Term-wise relative covariance factor \\ - \hline - \end{tabular} - \caption{Symbols used to describe the structure of the random-effects model matrix and the relative covariance factor. The - subscript $i = 1, \dots, k$ denotes a specific random-effects term.} - \label{tab:algebraic} -\end{table} - -The expression, \code{r}, is a linear model formula that evaluates to -an \proglang{R} model matrix, $\bm X_i$, of size $n\times p_i$, called -the \emph{raw random-effects model matrix} for term $i$. A term is -said to be a \emph{scalar} random-effects term when $p_i=1$, otherwise -it is \emph{vector-valued}. For a \emph{simple, scalar} -random-effects term of the form \code{(1 | f)}, $\bm X_i$ is the % | -$n\times 1$ matrix of ones, which implies a random intercept model. - -The expression \code{f} evaluates to an \proglang{R} factor, called -the \emph{grouping factor}, for the term. For the $i$th term, we -represent this factor mathematically with a vector $\bm i_i$ of -\emph{factor indices}, which is an $n$-vector of values from -$1,\dots,\ell_i$.\footnote{In practice, - fixed-effects model matrices and random-effects terms are evaluated - with respect to a \emph{model frame}, ensuring that any expressions - for grouping factors have been coerced to factors and any unused - levels of these factors have been dropped. That is, $\ell_i$, the - number of levels in the grouping factor for the $i$th random-effects - term, is well-defined.} -Let $\bm J_i$ be the $n\times \ell_i$ matrix of -indicator columns for $\bm i_i$. Using the \pkg{Matrix} package -\citep{Matrix_pkg} in -\proglang{R}, we may construct the transpose of $\bm J_i$ from a -factor vector, \code{f}, by coercing \code{f} to a `\code{sparseMatrix}' -object. For example, -<>= -set.seed(2) -@ -<>= -(f <- gl(3, 2)) -(Ji <- t(as(f, Class = "sparseMatrix"))) -@ -When $k>1$ we order the random-effects terms so that -$\ell_1\ge\ell_2\ge\dots\ge\ell_k$; in general, this ordering reduces -``fill-in'' (i.e., the proportion of elements that are zero in the -lower triangle of $\bLt\trans\bm Z\trans\bm W\bm Z\bLt+\bm I$ but not in -the lower triangle of its left Cholesky factor, $\bm L_{\bm\theta}$, -described below in Equation~\ref{eq:blockCholeskyDecomp}). This reduction -in fill-in provides more efficient matrix operations within the -penalized least squares algorithm (Section~\ref{sec:plsMath}). - -\subsubsection{Constructing the random-effects model matrix} -\label{sec:mkZ} - -The $i$th random-effects term contributes $q_i=\ell_ip_i$ columns to -the model matrix $\bm Z$. We group these columns into a matrix, $\bm -Z_i$, which we refer to as the \emph{term-wise model matrix} for the -$i$th term. Thus $q$, the number of columns in $\bm Z$ and the -dimension of the random variable, $\mc{B}$, is -\begin{equation} - \label{eq:qcalc} - q=\sum_{i=1}^k q_i = \sum_{i=1}^k \ell_i\,p_i . -\end{equation} -Creating the matrix $\bm Z_i$ from $\bm X_i$ and $\bm J_i$ is a -straightforward concept that is, nonetheless, somewhat awkward to -describe. Consider $\bm Z_i$ as being further decomposed into -$\ell_i$ blocks of $p_i$ columns. The rows in the first block are the -rows of $\bm X_i$ multiplied by the 0/1 values in the first column of -$\bm J_i$ and similarly for the subsequent blocks. With these -definitions we may define the term-wise random-effects model matrix, -$\bm Z_i$, for the $i$th term as a transposed Khatri-Rao product, -\begin{equation} - \label{eq:Zi} - \bm Z_i = (\bm J_i\trans * \bm X_i\trans)\trans = - \begin{bmatrix} - \bm J_{i1}\trans \otimes \bm X_{i1}\trans \\ - \bm J_{i2}\trans \otimes \bm X_{i2}\trans \\ - \vdots \\ - \bm J_{in}\trans \otimes \bm X_{in}\trans \\ - \end{bmatrix}, -\end{equation} -where $*$ and $\otimes$ are the Khatri-Rao\footnote{Note that the - original definition of the Khatri-Rao product is more general than - the definition used in the \pkg{Matrix} package, which is the - definition we use here.} \citep{khatri1968solutions} and Kronecker -products, and $\bm J_{ij}\trans$ and $\bm X_{ij}\trans$ are row -vectors of the $j$th rows of $\bm J_i$ and $\bm X_i$. These rows -correspond to the $j$th sample in the response vector, $\mc Y$, and -thus $j$ runs from $1, \dots, n$. The \pkg{Matrix} package for -\proglang{R} contains a \code{KhatriRao} function, which can be used -to form $\bm Z_i$. For example, if we begin with a raw model matrix, -<>= -(Xi <- cbind(1, rep.int(c(-1, 1), 3L))) -@ -then the term-wise random-effects -model matrix is, -<>= -(Zi <- t(KhatriRao(t(Ji), t(Xi)))) -@ -<>= -## alternative formulation of Zi (eq:Zi) -rBind( - Ji[1,] %x% Xi[1,], - Ji[2,] %x% Xi[2,], - Ji[3,] %x% Xi[3,], - Ji[4,] %x% Xi[4,], - Ji[5,] %x% Xi[5,], - Ji[6,] %x% Xi[6,]) -@ - -In particular, for a simple, scalar term, $\bm Z_i$ is exactly $\bm -J_i$, the matrix of indicator columns. For other scalar terms, $\bm -Z_i$ is formed by element-wise multiplication of the single column of -$\bm X_i$ by each of the columns of $\bm J_i$. - -Because each $\bm Z_i$ is generated from indicator columns, its -cross-product, $\bm Z_i\trans\bm Z_i$ is block-diagonal consisting of -$\ell_i$ diagonal blocks each of size $p_i$.\footnote{To see this, - note that by the properties of Kronecker products we may write the - cross-product matrix $Z_i\trans Z_i$ as $\sum_{j=1}^n \bm J_{ij} \bm - J_{ij}\trans \otimes \bm X_{ij} \bm X_{ij}\trans$. Because $\bm - J_{ij}$ is a unit vector along a coordinate axis, the cross-product - $\bm J_{ij} \bm J_{ij}\trans$ is a $p_i\times p_i$ matrix of all - zeros except for a single $1$ along the diagonal. Therefore, the - cross-products, $\bm X_{ij} \bm X_{ij}\trans$, will be added to one - of the $\ell_i$ blocks of size $p_i\times p_i$ along the diagonal of - $Z_i\trans Z_i$.} Note that this means that when $k=1$ (i.e., there -is only one random-effects term, and $\bm Z_i = \bm Z$), $\bm -Z\trans\bm Z$ will be block diagonal. These block-diagonal properties -allow for more efficient sparse matrix computations -(Section~\ref{sec:CSCmats}). - -The full random-effects model matrix, $\bm Z$, is constructed from -$k\ge 1$ blocks, -\begin{equation} - \label{eq:Z} - \bm Z = - \begin{bmatrix} - \bm Z_1 & - \bm Z_2 & - \hdots & - \bm Z_k \\ - \end{bmatrix}. -\end{equation} -By transposing Equation~\ref{eq:Z} and substituting in Equation~\ref{eq:Zi}, we -may represent the structure of the transposed random-effects model -matrix as follows, -\begin{equation} - \label{eq:Zt} - \bm Z\trans = - \begin{blockarray}{ccccc} - \text{sample 1} & \text{sample 2} & \hdots & \text{sample } n & \\ - \begin{block}{[cccc]c} - \bm J_{11} \otimes \bm X_{11} & - \bm J_{12} \otimes \bm X_{12} & - \hdots & - \bm J_{1n} \otimes \bm X_{1n} & - \text{term 1} \\ - \bm J_{21} \otimes \bm X_{21} & - \bm J_{22} \otimes \bm X_{22} & - \hdots & - \bm J_{2n} \otimes \bm X_{2n} & - \text{term 2} \\ - \vdots & \vdots & \ddots & \vdots & \vdots \\ - \end{block} - \end{blockarray}. -\end{equation} -Note that the proportion of elements of $Z\trans$ that are structural -zeros is -\begin{equation} - \label{eq:ZtSparsity} - \frac{\sum_{i=1}^k p_i(\ell_i - 1)}{\sum_{i=1}^k p_i} \qquad . -\end{equation} -Therefore, the sparsity of $\bm Z\trans$ increases with the number of -grouping factor levels. As the number of levels is often large in -practice, it is essential for speed and efficiency to take account of -this sparsity, for example by using sparse matrix methods, when -fitting mixed models (Section~\ref{sec:CSCmats}). - -\subsubsection{Constructing the relative covariance factor} -\label{sec:mkLambdat} \label{sec:mkZ} - -The $q\times q$ covariance factor, $\bLt$, is a block diagonal matrix -whose $i$th diagonal block, $\bm\Lambda_i$, is of size -$q_i,i=1,\dots,k$. We refer to $\bm\Lambda_i$ as the \emph{term-wise - relative covariance factor}. Furthermore, $\bm\Lambda_i$ is a -homogeneous block diagonal matrix with each of the $\ell_i$ -lower-triangular blocks on the diagonal being a copy of a $p_i\times -p_i$ lower-triangular \emph{template matrix}, $\bm T_i$. The -covariance parameter vector, $\bm\theta$, of length $m_i -=\binom{p_i+1}{2}$, consists of the elements in the lower triangle of -$\bm T_i,i=1,\dots,k$. To provide a unique representation we require -that the diagonal elements of the $\bm T_i,i=1,\dots,k$ be -non-negative. - -The template, $\bm T_i$, can be constructed from the number $p_i$ -alone. In \proglang{R} code we denote $p_i$ as \code{nc}. For -example, if we set \code{nc <- 3}\Sexpr{nc <- 3}, we could create the -template for term $i$ as, -<>= -nc <- 3 -@ -%% sequence() is equivalent to unlist(lapply(nvec, seq_len)) -%% and (?sequence) ``mainly exists in reverence to the very early history of R'' -%% scw: i like sequence, and in fact i never understood why that -%% statement is there in the help file. -<