Compute and extract model parameters. The available options and arguments depend
on the modeling **package** and model `class`

. Follow one of these links to read
the model-specific documentation:

Default method:

`lm`

,`glm`

,**stats**,**censReg**,**MASS**,**survey**, ...Additive models:

**bamlss**,**gamlss**,**mgcv**,**scam**,**VGAM**,`Gam`

,`gamm`

, ...ANOVA:

**afex**,`aov`

,`anova`

, ...Bayesian:

**BayesFactor**,**blavaan**,**brms**,**MCMCglmm**,**posterior**,**rstanarm**,`bayesQR`

,`bcplm`

,`BGGM`

,`blmrm`

,`blrm`

,`mcmc.list`

,`MCMCglmm`

, ...Clustering:

**hclust**,**kmeans**,**mclust**,**pam**, ...Correlations, t-tests, etc.:

**lmtest**,`htest`

,`pairwise.htest`

, ...Meta-Analysis:

**metaBMA**,**metafor**,**metaplus**, ...Mixed models:

**cplm**,**glmmTMB**,**lme4**,**lmerTest**,**nlme**,**ordinal**,**robustlmm**,**spaMM**,`mixed`

,`MixMod`

, ...Multinomial, ordinal and cumulative link:

**brglm2**,**DirichletReg**,**nnet**,**ordinal**,`mlm`

, ...Multiple imputation:

**mice**PCA, FA, CFA, SEM:

**FactoMineR**,**lavaan**,**psych**,`sem`

, ...Zero-inflated and hurdle:

**cplm**,**mhurdle**,**pscl**, ...Other models:

**aod**,**bbmle**,**betareg**,**emmeans**,**epiR**,**ggeffects**,**glmx**,**ivfixed**,**ivprobit**,**JRM**,**lmodel2**,**logitsf**,**marginaleffects**,**margins**,**maxLik**,**mediation**,**mfx**,**multcomp**,**mvord**,**plm**,**PMCMRplus**,**quantreg**,**selection**,**systemfit**,**tidymodels**,**varEST**,**WRS2**,`bfsl`

,`deltaMethod`

,`fitdistr`

,`mjoint`

,`mle`

,`model.avg`

, ...

## Arguments

- 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. If`s_value = TRUE`

, the p-value will be replaced by the S-value in the output (cf.*Rafi and Greenland 2020*).`pd`

adds an additional column with the*probability of direction*(see`bayestestR::p_direction()`

for details).`groups`

can be used to group coefficients. It will be passed to the print-method, or can directly be used in`print()`

, see documentation in`print.parameters_model()`

. Furthermore, see 'Examples' in`model_parameters.default()`

. For developers, whose interest mainly is to get a "tidy" data frame of model summaries, it is recommended to set`pretty_names = FALSE`

to speed up computation of the summary table.

## Note

The `print()`

method has several
arguments to tweak the output. There is also a
`plot()`

-method
implemented in the
**see**-package, and a dedicated
method for use inside rmarkdown files,
`print_md()`

. **For developers**, if
speed performance is an issue, you can use the (undocumented) `pretty_names`

argument, e.g. `model_parameters(..., pretty_names = FALSE)`

. This will
skip the formatting of the coefficient names and make `model_parameters()`

faster.

## Standardization of model coefficients

Standardization is based on `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"`

.

## Standardization Methods

**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**`datawizard::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.

See also package vignette.

## Labeling the Degrees of Freedom

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.

## Confidence intervals and approximation of 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`

,`merMod`

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()`

.

## Statistical inference - how to quantify evidence

There is no standardized approach to drawing conclusions based on the
available data and statistical models. A frequently chosen but also much
criticized approach is to evaluate results based on their statistical
significance (*Amrhein et al. 2017*).

A more sophisticated way would be to test whether estimated effects exceed
the "smallest effect size of interest", to avoid even the smallest effects
being considered relevant simply because they are statistically significant,
but clinically or practically irrelevant (*Lakens et al. 2018, Lakens 2024*).

A rather unconventional approach, which is nevertheless advocated by various
authors, is to interpret results from classical regression models in terms of
probabilities, similar to the usual approach in Bayesian statistics
(*Greenland et al. 2022; Rafi and Greenland 2020; Schweder 2018; Schweder and
Hjort 2003; Vos 2022*).

The **parameters** package provides several options or functions to aid
statistical inference. These are, for example:

`equivalence_test()`

, to compute the (conditional) equivalence test for frequentist models`p_significance()`

, to compute the probability of*practical significance*, which can be conceptualized as a unidirectional equivalence test`p_function()`

, or*consonance function*, to compute p-values and compatibility (confidence) intervals for statistical modelsthe

`pd`

argument (setting`pd = TRUE`

) in`model_parameters()`

includes a column with the*probability of direction*, i.e. the probability that a parameter is strictly positive or negative. See`bayestestR::p_direction()`

for details. If plotting is desired, the`p_direction()`

function can be used, together with`plot()`

.the

`s_value`

argument (setting`s_value = TRUE`

) in`model_parameters()`

replaces the p-values with their related*S*-values (*Rafi and Greenland 2020*)finally, it is possible to generate distributions of model coefficients by generating bootstrap-samples (setting

`bootstrap = TRUE`

) or simulating draws from model coefficients using`simulate_model()`

. These samples can then be treated as "posterior samples" and used in many functions from the**bayestestR**package.

Most of the above shown options or functions derive from methods originally
implemented for Bayesian models (*Makowski et al. 2019*). However, assuming
that model assumptions are met (which means, the model fits well to the data,
the correct model is chosen that reflects the data generating process
(distributional model family) etc.), it seems appropriate to interpret
results from classical frequentist models in a "Bayesian way" (more details:
documentation in `p_function()`

).

## Interpretation of Interaction Terms

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, ggeffects, or
marginaleffects. 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()`

.

## Global Options to Customize Messages and Tables when Printing

The `verbose`

argument can be used to display or silence messages and
warnings for the different functions in the **parameters** package. However,
some messages providing additional information can be displayed or suppressed
using `options()`

:

`parameters_summary`

:`options(parameters_summary = TRUE)`

will override the`summary`

argument in`model_parameters()`

and always show the model summary for non-mixed models.`parameters_mixed_summary`

:`options(parameters_mixed_summary = TRUE)`

will override the`summary`

argument in`model_parameters()`

for mixed models, and will then always show the model summary.`parameters_cimethod`

:`options(parameters_cimethod = TRUE)`

will show the additional information about the approximation method used to calculate confidence intervals and p-values. Set to`FALSE`

to hide this message when printing`model_parameters()`

objects.`parameters_exponentiate`

:`options(parameters_exponentiate = TRUE)`

will show the additional information on how to interpret coefficients of models with log-transformed response variables or with log-/logit-links when the`exponentiate`

argument in`model_parameters()`

is not`TRUE`

. Set this option to`FALSE`

to hide this message when printing`model_parameters()`

objects.

There are further options that can be used to modify the default behaviour for printed outputs:

`parameters_labels`

:`options(parameters_labels = TRUE)`

will use variable and value labels for pretty names, if data is labelled. If no labels available, default pretty names are used.`parameters_interaction`

:`options(parameters_interaction = <character>)`

will replace the interaction mark (by default,`*`

) with the related character.`parameters_select`

:`options(parameters_select = <value>)`

will set the default for the`select`

argument. See argument's documentation for available options.`easystats_html_engine`

:`options(easystats_html_engine = "gt")`

will set the default HTML engine for tables to`gt`

, i.e. the*gt*package is used to create HTML tables. If set to`tt`

, the*tinytable*package is used.`insight_use_symbols`

:`options(insight_use_symbols = TRUE)`

will try to print unicode-chars for symbols as column names, wherever possible (e.g., ω instead of`Omega`

).

## References

Amrhein, V., Korner-Nievergelt, F., and Roth, T. (2017). The earth is flat (p > 0.05): Significance thresholds and the crisis of unreplicable research. PeerJ, 5, e3544. doi:10.7717/peerj.3544

Greenland S, Rafi Z, Matthews R, Higgs M. To Aid Scientific Inference, Emphasize Unconditional Compatibility Descriptions of Statistics. (2022) https://arxiv.org/abs/1909.08583v7 (Accessed November 10, 2022)

Hoffman, L. (2015). Longitudinal analysis: Modeling within-person fluctuation and change. Routledge.

Lakens, D. (2024). Improving Your Statistical Inferences (Version v1.5.1). Retrieved from https://lakens.github.io/statistical_inferences/. doi:10.5281/ZENODO.6409077

Lakens, D., Scheel, A. M., and Isager, P. M. (2018). Equivalence Testing for Psychological Research: A Tutorial. Advances in Methods and Practices in Psychological Science, 1(2), 259–269. doi:10.1177/2515245918770963

Makowski, D., Ben-Shachar, M. S., Chen, S. H. A., and Lüdecke, D. (2019). Indices of Effect Existence and Significance in the Bayesian Framework. Frontiers in Psychology, 10, 2767. doi:10.3389/fpsyg.2019.02767

Neter, J., Wasserman, W., and Kutner, M. H. (1989). Applied linear regression models.

Rafi Z, Greenland S. Semantic and cognitive tools to aid statistical science: replace confidence and significance by compatibility and surprise. BMC Medical Research Methodology (2020) 20:244.

Schweder T. Confidence is epistemic probability for empirical science. Journal of Statistical Planning and Inference (2018) 195:116–125. doi:10.1016/j.jspi.2017.09.016

Schweder T, Hjort NL. Frequentist analogues of priors and posteriors. In Stigum, B. (ed.), Econometrics and the Philosophy of Economics: Theory Data Confrontation in Economics, pp. 285-217. Princeton University Press, Princeton, NJ, 2003

Vos P, Holbert D. Frequentist statistical inference without repeated sampling. Synthese 200, 89 (2022). doi:10.1007/s11229-022-03560-x

## See also

`insight::standardize_names()`

to
rename columns into a consistent, standardized naming scheme.