`ci()`

attempts to return confidence intervals of model parameters.

## Arguments

- x
A statistical model.

- ci
Confidence Interval (CI) level. Default to

`0.95`

(`95%`

).- dof
Number of degrees of freedom to be used when calculating confidence intervals. If

`NULL`

(default), the degrees of freedom are retrieved by calling`degrees_of_freedom()`

with approximation method defined in`method`

. If not`NULL`

, use this argument to override the default degrees of freedom used to compute confidence intervals.- method
Method for computing degrees of freedom for confidence intervals (CI) and the related p-values. Allowed are following options (which vary depending on the model class):

`"residual"`

,`"normal"`

,`"likelihood"`

,`"satterthwaite"`

,`"kenward"`

,`"wald"`

,`"profile"`

,`"boot"`

,`"uniroot"`

,`"ml1"`

,`"betwithin"`

,`"hdi"`

,`"quantile"`

,`"ci"`

,`"eti"`

,`"si"`

,`"bci"`

, or`"bcai"`

. See section*Confidence intervals and approximation of degrees of freedom*in`model_parameters()`

for further details.- ...
Additional arguments

- component
Model component for which parameters should be shown. See the documentation for your object's class in

`model_parameters()`

or`p_value()`

for further details.- verbose
Toggle warnings and messages.

- iterations
The number of bootstrap replicates. Only applies to models of class

`merMod`

when`method=boot`

.

## 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()`

.

## Examples

```
# \donttest{
library(parameters)
data(Salamanders, package = "glmmTMB")
model <- glmmTMB::glmmTMB(
count ~ spp + mined + (1 | site),
ziformula = ~mined,
family = poisson(),
data = Salamanders
)
ci(model)
#> Parameter CI CI_low CI_high Component
#> 1 (Intercept) 0.95 -0.904868203 0.1834728 conditional
#> 2 sppPR 0.95 -1.741720718 -0.7977701 conditional
#> 3 sppDM 0.95 -0.001515455 0.5395142 conditional
#> 4 sppEC-A 0.95 -0.967367323 -0.1630136 conditional
#> 5 sppEC-L 0.95 0.414693327 0.9167312 conditional
#> 6 sppDES-L 0.95 0.378400449 0.8723050 conditional
#> 7 sppDF 0.95 -0.173790668 0.4038560 conditional
#> 8 minedno 0.95 0.742998151 1.7998569 conditional
#> 9 (Intercept) 0.95 0.255245254 1.3247538 zero_inflated
#> 10 minedno 0.95 -2.460449166 -1.2293688 zero_inflated
ci(model, component = "zi")
#> Parameter CI CI_low CI_high Component
#> 1 (Intercept) 0.95 0.2552453 1.324754 zero_inflated
#> 2 minedno 0.95 -2.4604492 -1.229369 zero_inflated
# }
```