Compute and extract model parameters. See the documentation for your object's class:
Correlations, t-tests, ... (htest, pairwise.htest)
ANOVAs (aov, anova, afex, ...)
Regression models (lm, glm, survey, ...)
Additive models (gam, gamm, ...)
Zero-inflated models (hurdle, zeroinfl, zerocount)
Multinomial, ordinal and cumulative link models (bracl, multinom, mlm, ...)
Other special models (model.avg, betareg, glmx, ...)
Mixed models (lme4, nlme, glmmTMB, afex, ...)
Bayesian tests (BayesFactor)
Bayesian models (rstanarm, brms, MCMCglmm, blavaan, ...)
PCA and FA (psych)
CFA and SEM (lavaan)
Cluster models (k-means, ...)
Meta-Analysis via linear (mixed) models (rma, metaplus, metaBMA, ...)
Hypothesis testing (glht, PMCMRplus)
Robust statistical tests (WRS2)
model_parameters(model, ...)parameters(model, ...)
Statistical Model.
Arguments passed to or from other methods. Non-documented
arguments are digits, p_digits, ci_digits and
footer_digits to set the number of digits for the output.
group can also be passed to the print() method. See details
in print.parameters_model() and 'Examples' in
model_parameters.default().
A data frame of indices related to the model's parameters.
Standardization is based on effectsize::standardize_parameters(). In case
of standardize = "refit", the data used to fit the model will be
standardized and the model is completely refitted. In such cases, standard
errors and confidence intervals refer to the standardized coefficient. The
default, standardize = "refit", never standardizes categorical predictors
(i.e. factors), which may be a different behaviour compared to other R
packages or other software packages (like SPSS). To mimic behaviour of SPSS
or packages such as lm.beta, use standardize = "basic".
refit: This method is based on a complete model re-fit with a
standardized version of the data. Hence, this method is equal to
standardizing the variables before fitting the model. It is the "purest" and
the most accurate (Neter et al., 1989), but it is also the most
computationally costly and long (especially for heavy models such as Bayesian
models). This method is particularly recommended for complex models that
include interactions or transformations (e.g., polynomial or spline terms).
The robust (default to FALSE) argument enables a robust standardization
of data, i.e., based on the median and MAD instead of the mean and
SD. See standardize() for more details.
Note that standardize_parameters(method = "refit") may not return
the same results as fitting a model on data that has been standardized with
standardize(); standardize_parameters() used the data used by the model
fitting function, which might not be same data if there are missing values.
see the remove_na argument in standardize().
posthoc: Post-hoc standardization of the parameters, aiming at
emulating the results obtained by "refit" without refitting the model. The
coefficients are divided by the standard deviation (or MAD if robust) of
the outcome (which becomes their expression 'unit'). Then, the coefficients
related to numeric variables are additionally multiplied by the standard
deviation (or MAD if robust) of the related terms, so that they correspond
to changes of 1 SD of the predictor (e.g., "A change in 1 SD of x is
related to a change of 0.24 of the SD of y). This does not apply to binary
variables or factors, so the coefficients are still related to changes in
levels. This method is not accurate and tend to give aberrant results when
interactions are specified.
basic: This method is similar to method = "posthoc", but treats all
variables as continuous: it also scales the coefficient by the standard
deviation of model's matrix' parameter of factors levels (transformed to
integers) or binary predictors. Although being inappropriate for these cases,
this method is the one implemented by default in other software packages,
such as lm.beta::lm.beta().
smart (Standardization of Model's parameters with Adjustment,
Reconnaissance and Transformation - experimental): Similar to method = "posthoc" in that it does not involve model refitting. The difference is
that the SD (or MAD if robust) of the response is computed on the relevant
section of the data. For instance, if a factor with 3 levels A (the
intercept), B and C is entered as a predictor, the effect corresponding to B
vs. A will be scaled by the variance of the response at the intercept only.
As a results, the coefficients for effects of factors are similar to a Glass'
delta.
pseudo (for 2-level (G)LMMs only): In this (post-hoc) method, the
response and the predictor are standardized based on the level of prediction
(levels are detected with performance::check_heterogeneity_bias()): Predictors
are standardized based on their SD at level of prediction (see also
datawizard::demean()); The outcome (in linear LMMs) is standardized based
on a fitted random-intercept-model, where sqrt(random-intercept-variance)
is used for level 2 predictors, and sqrt(residual-variance) is used for
level 1 predictors (Hoffman 2015, page 342). A warning is given when a
within-group variable is found to have access between-group variance.
Throughout the parameters package, we decided to label the residual
degrees of freedom df_error. The reason for this is that these degrees
of freedom not always refer to the residuals. For certain models, they refer
to the estimate error - in a linear model these are the same, but in - for
instance - any mixed effects model, this isn't strictly true. Hence, we
think that df_error is the most generic label for these degrees of
freedom.
There are different ways of approximating the degrees of freedom depending
on different assumptions about the nature of the model and its sampling
distribution. The ci_method argument modulates the method for computing degrees
of freedom (df) that are used to calculate confidence intervals (CI) and the
related p-values. Following options are allowed, depending on the model
class:
Classical methods:
Classical inference is generally based on the Wald method. The Wald approach to inference computes a test statistic by dividing the parameter estimate by its standard error (Coefficient / SE), then comparing this statistic against a t- or normal distribution. This approach can be used to compute CIs and p-values.
"wald":
Applies to non-Bayesian models. For linear models, CIs computed using the Wald method (SE and a t-distribution with residual df); p-values computed using the Wald method with a t-distribution with residual df. For other models, CIs computed using the Wald method (SE and a normal distribution); p-values computed using the Wald method with a normal distribution.
"normal"
Applies to non-Bayesian models. Compute Wald CIs and p-values, but always use a normal distribution.
"residual"
Applies to non-Bayesian models. Compute Wald CIs and p-values, but always use a t-distribution with residual df when possible. If the residual df for a model cannot be determined, a normal distribution is used instead.
Methods for mixed models:
Compared to fixed effects (or single-level) models, determining appropriate df for Wald-based inference in mixed models is more difficult. See the R GLMM FAQ for a discussion.
Several approximate methods for computing df are available, but you should
also consider instead using profile likelihood ("profile") or bootstrap ("boot")
CIs and p-values instead.
"satterthwaite"
Applies to linear mixed models. CIs computed using the Wald method (SE and a t-distribution with Satterthwaite df); p-values computed using the Wald method with a t-distribution with Satterthwaite df.
"kenward"
Applies to linear mixed models. CIs computed using the Wald method (Kenward-Roger SE and a t-distribution with Kenward-Roger df); p-values computed using the Wald method with Kenward-Roger SE and t-distribution with Kenward-Roger df.
"ml1"
Applies to linear mixed models. CIs computed using the Wald
method (SE and a t-distribution with m-l-1 approximated df); p-values
computed using the Wald method with a t-distribution with m-l-1 approximated df.
See ci_ml1().
"betwithin"
Applies to linear mixed models and generalized linear mixed models.
CIs computed using the Wald method (SE and a t-distribution with between-within df);
p-values computed using the Wald method with a t-distribution with between-within df.
See ci_betwithin().
Likelihood-based methods:
Likelihood-based inference is based on comparing the likelihood for the maximum-likelihood estimate to the the likelihood for models with one or more parameter values changed (e.g., set to zero or a range of alternative values). Likelihood ratios for the maximum-likelihood and alternative models are compared to a \(\chi\)-squared distribution to compute CIs and p-values.
"profile"
Applies to non-Bayesian models of class glm, polr or glmmTMB.
CIs computed by profiling the likelihood curve for a parameter, using
linear interpolation to find where likelihood ratio equals a critical value;
p-values computed using the Wald method with a normal-distribution (note:
this might change in a future update!)
"uniroot"
Applies to non-Bayesian models of class glmmTMB. CIs
computed by profiling the likelihood curve for a parameter, using root
finding to find where likelihood ratio equals a critical value; p-values
computed using the Wald method with a normal-distribution (note: this
might change in a future update!)
Methods for bootstrapped or Bayesian models:
Bootstrap-based inference is based on resampling and refitting the model to the resampled datasets. The distribution of parameter estimates across resampled datasets is used to approximate the parameter's sampling distribution. Depending on the type of model, several different methods for bootstrapping and constructing CIs and p-values from the bootstrap distribution are available.
For Bayesian models, inference is based on drawing samples from the model posterior distribution.
"quantile" (or "eti")
Applies to all models (including Bayesian models).
For non-Bayesian models, only applies if bootstrap = TRUE. CIs computed
as equal tailed intervals using the quantiles of the bootstrap or
posterior samples; p-values are based on the probability of direction.
See bayestestR::eti().
"hdi"
Applies to all models (including Bayesian models). For non-Bayesian
models, only applies if bootstrap = TRUE. CIs computed as highest density intervals
for the bootstrap or posterior samples; p-values are based on the probability of direction.
See bayestestR::hdi().
"bci" (or "bcai")
Applies to all models (including Bayesian models).
For non-Bayesian models, only applies if bootstrap = TRUE. CIs computed
as bias corrected and accelerated intervals for the bootstrap or
posterior samples; p-values are based on the probability of direction.
See bayestestR::bci().
"si"
Applies to Bayesian models with proper priors. CIs computed as
support intervals comparing the posterior samples against the prior samples;
p-values are based on the probability of direction. See bayestestR::si().
"boot"
Applies to non-Bayesian models of class merMod. CIs computed
using parametric bootstrapping (simulating data from the fitted model);
p-values computed using the Wald method with a normal-distribution)
(note: this might change in a future update!).
For all iteration-based methods other than "boot"
("hdi", "quantile", "ci", "eti", "si", "bci", "bcai"),
p-values are based on the probability of direction (bayestestR::p_direction()),
which is converted into a p-value using bayestestR::pd_to_p().
Note that the interpretation of interaction terms depends on many
characteristics of the model. The number of parameters, and overall
performance of the model, can differ or not between a * b
a : b, and a / b, suggesting that sometimes interaction terms
give different parameterizations of the same model, but other times it gives
completely different models (depending on a or b being factors
of covariates, included as main effects or not, etc.). Their interpretation
depends of the full context of the model, which should not be inferred
from the parameters table alone - rather, we recommend to use packages
that calculate estimated marginal means or marginal effects, such as
modelbased, emmeans or ggeffects. To raise
awareness for this issue, you may use print(...,show_formula=TRUE)
to add the model-specification to the output of the
print() method for model_parameters().
Hoffman, L. (2015). Longitudinal analysis: Modeling within-person fluctuation and change. Routledge.
Neter, J., Wasserman, W., & Kutner, M. H. (1989). Applied linear regression models.
insight::standardize_names() to
rename columns into a consistent, standardized naming scheme.