Implements piecewise structural equation models.

Implementation of piecewise structural equation modeling (SEM) in R, including estimation of path coefficients and goodness-of-fit statistics.

A formal description of this package can be found at:

```
Lefcheck, Jonathan S. (2015) piecewiseSEM: Piecewise structural equation modeling in R for ecology, evolution, and systematics. Methods in Ecology and Evolution. 7(5): 573-579. DOI: 10.1111/2041-210X.12512
```

Version: 1.2.1 (2016-12-06)

Author: Jon Lefcheck [email protected]

Supported model classes include:

`lm`

, `glm`

, `rq`

, `glm.nb`

, `gls`

, `pgls`

, `merMod`

, `merModLmerTest`

, `lme`

, `glmmPQL`

, `glmmadmb`

, and `glmmTMB`

.

**Some tests of directed separation are non-symmetrical -- the partial slope of a ~ b is not the same as b ~ a -- when the variables are non-linear (i.e., are transformed via a link function when fit to a non-normal distribution). We are currently investigating the phenomenon, but in the interim, the latest version of the package returns the lowest P-value. This the more conservative route. Stay tuned for more updates...**

**This is only a problem if you are fitting generalized linear models!!**

```
# Install latest version from CRAN
install.packages("piecewiseSEM")
# Install dev version from GitHub
# library(devtools)
# install_github("jslefche/piecewiseSEM")
library(piecewiseSEM)
```

```
data(shipley2009)
```

The data is alternately hosted in Ecological Archives E090-028-S1 (DOI: 10.1890/08-1034.1).

The model corresponds to the following hypothesis (Fig. 2, Shipley 2009);

Models are constructed using a mix of the `nlme`

and `lmerTest`

packages, as in the supplements of Shipley 2009.

```
# Load required libraries for linear mixed effects models
library(lme4)
library(nlme)
# Load example data
data(shipley2009)
# Create list of models corresponding to SEM
shipley2009.modlist = list(
lme(DD~lat, random = ~1|site/tree, na.action = na.omit,
data = shipley2009),
lme(Date~DD, random = ~1|site/tree, na.action = na.omit,
data = shipley2009),
lme(Growth~Date, random = ~1|site/tree, na.action = na.omit,
data = shipley2009),
glmer(Live~Growth+(1|site)+(1|tree),
family=binomial(link = "logit"), data = shipley2009)
)
```

`sem.fit`

returns a list of the following:
(1) the missing paths (omitting conditional variables), the estimate, standard error, degrees of freedom, and associated p-values;
(2) the Fisher's C statistic, degrees of freedom, and p-value for the model (derived from a Chi-squared distribution);
(3) the AIC, AICc (corrected for small sample size), the likelihood degrees of freedom, and the model degrees of freedom.

The argument `add.vars`

allows you to specify a vector of additional variables whose causal independence you also wish to test. This is useful if you are comparing nested models. Default is `NULL`

.

The argument `adjust.p`

allows you to adjust the p-values returned by the function based on the the total degrees of freedom for the model (see supplementary material, Shipley 2013). Default is `FALSE`

(uses the d.f. reported in the summary table).

(See "p-values and all that" for a discussion of p-values from mixed models using the `lmer`

package.)

```
sem.fit(shipley2009.modlist, shipley2009)
# Conditional variables have been omitted from output table for clarity (or use argument conditional = T)
# $missing.paths
# missing.path estimate std.error df crit.value p.value
# 1 Date ~ lat + ... -0.0091 0.1135 18 -0.0798 0.9373
# 2 Growth ~ lat + ... -0.0989 0.1107 18 -0.8929 0.3837
# 3 Live ~ lat + ... 0.0305 0.0297 NA 1.0279 0.3040
# 4 Growth ~ DD + ... -0.0106 0.0358 1329 -0.2967 0.7667
# 5 Live ~ DD + ... 0.0272 0.0271 NA 1.0046 0.3151
# 6 Live ~ Date + ... -0.0466 0.0298 NA -1.5622 0.1182
#
# $Fisher.C
# fisher.c df p.value
# 1 11.54 12 0.483
#
# $AIC
# AIC AICc K n
# 1 49.54 50.079 19 1431
```

The missing paths output differs from Table 2 in Shipley 2009. However, running each d-sep model by hand yields the same answers as this function, leading me to believe that updates to the `lme4`

and `nlme`

packages are the cause of the discrepancy. Qualitatively, the interpretations are the same.

Path coefficients can be either unstandardized or standardized (centered and scaled in units of standard deviation of the mean, or scaled by the range the data). Default is `none`

. The function returns a `data.frame`

sorted by increasing significance.

```
sem.coefs(shipley2009.modlist, shipley2009)
# response predictor estimate std.error p.value
# 1 DD lat -0.8354736 0.119422385 0 ***
# 2 Date DD -0.4976475 0.004933274 0 ***
# 3 Growth Date 0.3007147 0.026631405 0 ***
# 4 Live Growth 0.3478536 0.058415948 0 ***
sem.coefs(shipley2009.modlist, shipley2009, standardize = "scale")
# response predictor estimate std.error p.value
# 1 DD lat -0.7014051 0.100258794 0 ***
# 2 Date DD -0.6281367 0.006226838 0 ***
# 3 Growth Date 0.3824224 0.033867469 0 ***
# 4 Live Growth 0.3478536 0.058415948 0 ***
# Warning message:
# In get.scaled.data(modelList, data, standardize) :
# One or more responses not modeled to a normal distribution: keeping response(s) on original scale!
```

Note the error indicating that one of the responses (`Live`

) cannot be scaled because it would violate the distributional assumptions, so only the predictors have been scaled.

We can plot a rudimentary path diagram of the SEM using `sem.plot`

which reports the coefficients, above:

```
sem.plot(shipley2009.modlist, shipley2009)
```

`lavaan`

Generate variance-covariance based SEM from the list of linear mixed models. The resulting object can be treated like any other model object constructed using the package `lavaan`

.

```
(lavaan.model = sem.lavaan(shipley2009.modlist, shipley2009))
# lavaan (0.5-18) converged normally after 27 iterations
#
# Used Total
# Number of observations 1431 1900
#
# Estimator ML
# Minimum Function Test Statistic 38.433
# Degrees of freedom 6
# P-value (Chi-square) 0.000
```

The output shows that the variance-covariance SEM is a worse fit, indicating that a hierarchical piecewise approach is justified given the hierarchical structure of the data.

One might be interested in the partial effects of one variable on another given covariates in the SEM. The function `partial.resid`

returns a `data.frame`

of the partial residuals of `y ~ x`

and plots the partial effect (if `plotit = T`

).

```
# Load model package
library(nlme)
# Load data from Shipley (2013)
data(shipley2013)
# Create list of structured equations
shipley2013.modlist = list(
lme(x2~x1, random = ~x1 | species, data = shipley2013),
lme(x3~x2, random = ~x2 | species, data = shipley2013),
lme(x4~x2, random = ~x2 | species, data = shipley2013),
lme(x5~x3+x4, random = ~x3+x4 | species, data = shipley2013)
)
# Get partial residuals of x3 on x5 conditional on x4
resids.df = partial.resid(x5 ~ x3, shipley2013.modlist, list(lmeControl(opt = "optim")))
```

Return R^{2} and AIC values for component models in the SEM.

```
sem.model.fits(shipley2009.modlist)
# Class Family Link N Marginal Conditional
# 1 lme gaussian identity 1431 0.4766448 0.6932571
# 2 lme gaussian identity 1431 0.4083328 0.9838487
# 3 lme gaussian identity 1431 0.1070265 0.8364736
# 4 glmerMod binomial logit 1431 0.5589205 0.6291980
```

Generate model predictions from new data.

```
# Create new data for predictions
shipley2009.new = data.frame(
lat = rnorm(length(shipley2009$lat), mean(shipley2009$lat, na.rm = T),
sd(shipley2009$lat, na.rm = T)),
DD = rnorm(length(shipley2009$DD), mean(shipley2009$DD, na.rm = T),
sd(shipley2009$DD, na.rm = T)),
Date = rnorm(length(shipley2009$Date), mean(shipley2009$Date, na.rm = T),
sd(shipley2009$Date, na.rm = T)),
Growth = rnorm(length(shipley2009$Growth), mean(shipley2009$Growth, na.rm = T),
sd(shipley2009$Growth, na.rm = T))
)
# Generate predictions
head(sem.predict(shipley2009.modlist, shipley2009.new))
```

- Bug fix: use of
`standardize = "scale"`

in`sem.coefs`

with`cbind`

in`glmer`

- Warning: use of
`poly`

polynomials in model formula - Feature addition: support for models of class
`rq`

- Bug fix: error in scaling of transformed variables in
`get.scaled.data`

- Bug fix:
`sem.basis.set`

failed to remove variables regressed against their interaction - Bug fix: multiple levels of
`grouping.vars`

in`sem.fit`

- Feature addition: new function
`acyclic`

to test for acyclic DAGs - Feature addition: new function
`rsquared`

for easier calling of`sem.model.fits`

- Bug fix: order of adjacency matrix in
`get.sort.dag`

- Feature addition: model class
`glmmTMB`

- Bug fix: inclusion of transformed variables as additional boxes in
`sem.plot`

- Bug fix: specification of invalid models in
`endogenous.reverse`

- Feature addition: implemented two-column binding as response in GLM(M)s
- Bug fix: memory allocation error in
`sem.model.fits`

- Bug fix: inclusion of entries in the basis set without corresponding models in the model list in
`endogenous.reverse`

- Bug fix: crossed random effects in
`get.random.formula`

not returning correct random slopes - Bug fix:
`gls`

models in`sem.missing.paths`

could not update basis model

- Feature addition: report of intercepts in
`sem.coefs`

- Bug fix: warning with random slopes not present as fixed effects in
`sem.model.fits`

- Feature addition: new function
`endogenous.reverse`

to assist with independence claims among intermediate endogenous variables not fit to a normal distribution - Bug fix: negative binomial models fail to return R2s in
`sem.model.fits`

- Bug fix: Failed to index correct P-value in
`sem.missing.paths`

when categorical variables are present - Bug fix: intermediate endogenous variables not properly removed from the basis set in
`sem.missing.paths`

- Feature addition: significance indicators for P-value outputs
- Bug fix: remove random effects from
`get.scaled.data`

- Feature addition: Use
`KRmodcomp`

from`pbkrtest`

instead of`get_LB_dff`

to calculate P-values for`lmer`

models - Bug fix: transformations for scaled coefficients no longer fails in
`sem.coefs`

- Bug fix: all interactions now show up in
`sem.coefs`

when scaled - Bug fix: random slopes as fixed effects not registering properly in
`sem.model.fits`

- Bug fix: added \dontrun{} to all help files

- Reduced time it takes examples to run
- Stop: if duplicate responses are detected in the model list
- Temporary bug fix: returns lowest P-value for when d-sep tests include non-linear intermediate endogenous variables, only if family is not Gaussian
- Bug fix: reversed order of conditioning variables so they come first
- Bug fix: removed entries from the basis set that attempt to predict an interaction that does not appear in the model list

- Feature addition: rudimentary plotting using
`sem.plot`

- Feature addition: support for
`glmmadmb`

models - Feature addition:
`pbkrtest`

to reliably extract p-values from`lme4`

models *Major Bug Fix:*incorrect basis set specified by`ggm::topSort`

; introduced`sort.dag`

function, removed dependency on`ggm`

*Major Bug Fix:*`sem.basis.set`

now treats transformed variables as untransformed, resolving duplicate vars in the basis set- Bug fix: resolved switched interactions (e.g., x1:x2 vs x2:x1) leading to duplicate vars in the basis set using
`get.dag`

- Bug fix: fixed long standing bug with
`lmer`

models not returning p-values

- Bug fix: correlated errors among exogenous variables in
`sem.coefs`

- Bug fix: correctly scale data from
`pgls`

models using`get.scaled.data`

- Bug fix: corrected basis set for models created with
`gls`

- Feature addition: helper function
`get.scaled.model`

to get model from scaled coefficients - Feature addition: (partial) correlations in sem.coefs
- Bug fix: passing of correlated errors to lavaan
- Bug fix: standardized coefficients for variables transformed in model formula

- Bug fix: get.model.control updated for latest versions of lme4 and nlme
- Bug fix: hand compute interactions in
`sem.lavaan`

using argument`compute.int = TRUE`

- Feature addition: function
`get.scaled.data`

to handle transformed variables in`sem.coefs`

when standardize = "scale"

- Feature addition: added additional plotting arguments for partial.residuals
- Feature addition: AICc and delta AIC to sem.model.fits
- Bug fix: Offsets in model formula treated as predictors in d-sep tests

- Feature addition: AICc and delta AIC to sem.model.fits
- Bug fix: issue with interactions in basis set and d-sep tests in sem.missing.paths
- Bug fix: issue with fixed intercept models and calculating the basis set
- Bug fix: issue with offset() variables in sem.missing.paths
- Bug fix: transformed responses for partial residuals
- Bug fix: duplicated values in basis set (function DAG in ggm package)

- First release to CRAN

- Added vignette
- Fixed issue with design matrix including omitted observations and dropped levels in sem.model.fits
- Incorporated interactions into partial.resid
- Fixed bug with lme residuals in partial.resid and predict.sem
- Modified sem.coefs to return NA for standardized interactions

- Added functions get.dag and get.basis.set
- Fixed issue with interactions in sem.missing.paths
- Fixed issue in sem.missing.paths where only 1 missing path returned NA instead of p-value
- Removed argument filter.exog = TRUE
- Fixed error for add.vars in get.dag
- Fixed issue with transformed variables and sem.coefs
- Fixed issue with lme4 models and sem.model.fits
- Fixed issue with incorrect independence claims in get.basis.set
- Fixed (nlme) merging error in partial.resid
- Fixed issue with glmmPQL and get.random.structure
- Added new citation (arXiv)

- Implemented workaround in sem.basis.set where output changed based on order of variables in formula with interactions
- Saturated models now return AIC (and AICc) values (with warning)
- Added standard errors on predictions for mixed models in predict.sem()
- Fixed typo in output for sem.fisher.c with incorrect df
- Fixed error with lme4 and gls models in partial.resid()
- Fixed error with lmerTest returning "merMod" objects
- New function predict.sem returns model predictions
- Fixed "unsupported model class" error for merModTest

- Major revisions and annotations to all functions to improve transparency and efficiency
- Included new helper functions get.random.formula() and get.model.control()
- Now reports df for all model types except glmer
- Included support for "gls" and phylogenetically independent contrast ("pgls") models
- Fixed deparse error in get.random.struture()
- Improved handling of gls() models

- Fixed issue with transformed corr.errors in get.basis.set
- get.sem.coefs now does range standardization
- get.sem.coefs now returns single table with significance values
- Improved handling of uneven observations in get.partial.resid
- get.partial.resid now accepts a single model in addition to a list of models
- get.lavaan.sem now supports additional arguments from lavaan

- Added new function to return R^2 and AIC values for component models in SEM
- get.partial.resid now returns residuals plot with fitted line

- get.missing.paths now returns more information for d-sep tests
- get.basis.set drops independence claims in the basis set where interactions predict their corresponding main effects
- get.fisher.c now reports k df and get.aic now reports K and n df
- Fixed rounding error that misreported Fisher's C
- Allow user-specified rounding using
`sig=`

argument in get.sem.fit, get.fisher.c, and get.aic

- Fixed minor bug in get.partial.resid where random effects threw an error for lme models
- Fixed interaction bug in filter.exogenous which was returning incompatible models from the basis set
- Fixed bug in get.sem.coefs where missing values returned NA for corr.errors
- Fixed issue with transformed/untransformed variables predicting one another in the basis set
- Fixed interaction bug in get.missing.paths, filter.exogenous, get.partial.resiod
- Now can supply single model control or list

- Fixed bug for transformed variables in get.partial.resid
- Can now specify an equation as input into get.partial.resid
- Fixed bug for transformed interaction terms in structured equations for get.basis.set
- Fixed bug for transformed variables when also specifying raw variables in corr.error
- Added model control arguments to get.sem.fit and get.partial.resid

- Added optional switch to display conditional variables in get.missing.paths
- Removed logLik df from get.aic output

- Incorporated multilevel data into get.sem.fit
- Fixed error when length(basis.set) = 1 and .progressBar = T
- Expanded interaction term in get.basis.set to include 20 letters instead of 10, and run them backwards
- Renamed variable names in filter.exogenous to better reflect what they are

- Incorporated transformed variables and multiple interactions into get.sem.coefs
- Fixed but with add.vars; many functions now require the user to supply a data.frame

- Standardized argument order across functions
- Improved handling of correlated errors in get.sem.coefs
- Incorporated correlated errors into get.sem.lavaan

- Fixed return of standardized coefficients in get.sem.coefs

- Added argument to define and quantify correlated errors
- Added new function get.partial.resid

- Added new function get.basis.set (replaced dag.updated)
- Improved handling of interactions in the basis set
- Improved handling of p-value rownames in get.missing.paths
- get.aic now returns df

- Added new functions get.aic, get.fisher.c, get.sem.coefs, get.lavaan.sem
- Added README.md
- Completed all existing .Rd files
- Updated .progressBar as to not conflict with existing function progressBar
- Altered handling of model formulae to collapse to a single character string

- Initial build of package
- Added CHANGELOG