# Test if models are different

Source:`R/test_bf.R`

, `R/test_likelihoodratio.R`

, `R/test_performance.R`

, and 2 more
`test_performance.Rd`

Testing whether models are "different" in terms of accuracy or explanatory power is a delicate and often complex procedure, with many limitations and prerequisites. Moreover, many tests exist, each coming with its own interpretation, and set of strengths and weaknesses.

The `test_performance()`

function runs the most relevant and appropriate
tests based on the type of input (for instance, whether the models are
*nested* or not). However, it still requires the user to understand what the
tests are and what they do in order to prevent their misinterpretation. See
the *Details* section for more information regarding the different tests
and their interpretation.

## Usage

```
test_bf(...)
# S3 method for default
test_bf(..., reference = 1, text_length = NULL)
test_likelihoodratio(..., estimator = "ML")
test_lrt(..., estimator = "ML")
test_performance(..., reference = 1)
test_vuong(...)
test_wald(...)
```

## Arguments

- ...
Multiple model objects.

- reference
This only applies when models are non-nested, and determines which model should be taken as a reference, against which all the other models are tested.

- text_length
Numeric, length (number of chars) of output lines.

`test_bf()`

describes models by their formulas, which can lead to overly long lines in the output.`text_length`

fixes the length of lines to a specified limit.- estimator
Applied when comparing regression models using

`test_likelihoodratio()`

. Corresponds to the different estimators for the standard deviation of the errors. Defaults to`"OLS"`

for linear models,`"ML"`

for all other models (including mixed models), or`"REML"`

for linear mixed models when these have the same fixed effects. See 'Details'.

## Details

### Nested vs. Non-nested Models

Model's "nesting" is an important concept of models comparison. Indeed, many
tests only make sense when the models are *"nested",* i.e., when their
predictors are nested. This means that all the predictors of a model are
contained within the predictors of a larger model (sometimes referred to as
the encompassing model). For instance, `model1 (y ~ x1 + x2)`

is
"nested" within `model2 (y ~ x1 + x2 + x3)`

. Usually, people have a list
of nested models, for instance `m1 (y ~ 1)`

, `m2 (y ~ x1)`

,
`m3 (y ~ x1 + x2)`

, `m4 (y ~ x1 + x2 + x3)`

, and it is conventional
that they are "ordered" from the smallest to largest, but it is up to the
user to reverse the order from largest to smallest. The test then shows
whether a more parsimonious model, or whether adding a predictor, results in
a significant difference in the model's performance. In this case, models are
usually compared *sequentially*: m2 is tested against m1, m3 against m2,
m4 against m3, etc.

Two models are considered as *"non-nested"* if their predictors are
different. For instance, `model1 (y ~ x1 + x2)`

and `model2 (y ~ x3 + x4)`

.
In the case of non-nested models, all models are usually compared
against the same *reference* model (by default, the first of the list).

Nesting is detected via the `insight::is_nested_models()`

function.
Note that, apart from the nesting, in order for the tests to be valid,
other requirements have often to be the fulfilled. For instance, outcome
variables (the response) must be the same. You cannot meaningfully test
whether apples are significantly different from oranges!

### Estimator of the standard deviation

The estimator is relevant when comparing regression models using
`test_likelihoodratio()`

. If `estimator = "OLS"`

, then it uses the same
method as `anova(..., test = "LRT")`

implemented in base R, i.e., scaling
by n-k (the unbiased OLS estimator) and using this estimator under the
alternative hypothesis. If `estimator = "ML"`

, which is for instance used
by `lrtest(...)`

in package **lmtest**, the scaling is done by n (the
biased ML estimator) and the estimator under the null hypothesis. In
moderately large samples, the differences should be negligible, but it
is possible that OLS would perform slightly better in small samples with
Gaussian errors. For `estimator = "REML"`

, the LRT is based on the REML-fit
log-likelihoods of the models. Note that not all types of estimators are
available for all model classes.

### REML versus ML estimator

When `estimator = "ML"`

, which is the default for linear mixed models (unless
they share the same fixed effects), values from information criteria (AIC,
AICc) are based on the ML-estimator, while the default behaviour of `AIC()`

may be different (in particular for linear mixed models from **lme4**, which
sets `REML = TRUE`

). This default in `test_likelihoodratio()`

intentional,
because comparing information criteria based on REML fits requires the same
fixed effects for all models, which is often not the case. Thus, while
`anova.merMod()`

automatically refits all models to REML when performing a
LRT, `test_likelihoodratio()`

checks if a comparison based on REML fits is
indeed valid, and if so, uses REML as default (else, ML is the default).
Set the `estimator`

argument explicitely to override the default behaviour.

### Tests Description

**Bayes factor for Model Comparison**-`test_bf()`

: If all models were fit from the same data, the returned`BF`

shows the Bayes Factor (see`bayestestR::bayesfactor_models()`

) for each model against the reference model (which depends on whether the models are nested or not). Check out this vignette for more details.**Wald's F-Test**-`test_wald()`

: The Wald test is a rough approximation of the Likelihood Ratio Test. However, it is more applicable than the LRT: you can often run a Wald test in situations where no other test can be run. Importantly, this test only makes statistical sense if the models are nested.Note: this test is also available in base R through the

`anova()`

function. It returns an`F-value`

column as a statistic and its associated p-value.**Likelihood Ratio Test (LRT)**-`test_likelihoodratio()`

: The LRT tests which model is a better (more likely) explanation of the data. Likelihood-Ratio-Test (LRT) gives usually somewhat close results (if not equivalent) to the Wald test and, similarly, only makes sense for nested models. However, maximum likelihood tests make stronger assumptions than method of moments tests like the F-test, and in turn are more efficient. Agresti (1990) suggests that you should use the LRT instead of the Wald test for small sample sizes (under or about 30) or if the parameters are large.Note: for regression models, this is similar to

`anova(..., test="LRT")`

(on models) or`lmtest::lrtest(...)`

, depending on the`estimator`

argument. For**lavaan**models (SEM, CFA), the function calls`lavaan::lavTestLRT()`

.For models with transformed response variables (like

`log(x)`

or`sqrt(x)`

),`logLik()`

returns a wrong log-likelihood. However,`test_likelihoodratio()`

calls`insight::get_loglikelihood()`

with`check_response=TRUE`

, which returns a corrected log-likelihood value for models with transformed response variables. Furthermore, since the LRT only accepts nested models (i.e. models that differ in their fixed effects), the computed log-likelihood is always based on the ML estimator, not on the REML fits.**Vuong's Test**-`test_vuong()`

: Vuong's (1989) test can be used both for nested and non-nested models, and actually consists of two tests.The

**Test of Distinguishability**(the`Omega2`

column and its associated p-value) indicates whether or not the models can possibly be distinguished on the basis of the observed data. If its p-value is significant, it means the models are distinguishable.The

**Robust Likelihood Test**(the`LR`

column and its associated p-value) indicates whether each model fits better than the reference model. If the models are nested, then the test works as a robust LRT. The code for this function is adapted from the**nonnest2**package, and all credit go to their authors.

## References

Vuong, Q. H. (1989). Likelihood ratio tests for model selection and non-nested hypotheses. Econometrica, 57, 307-333.

Merkle, E. C., You, D., & Preacher, K. (2016). Testing non-nested structural equation models. Psychological Methods, 21, 151-163.

## See also

`compare_performance()`

to compare the performance indices of
many different models.

## Examples

```
# Nested Models
# -------------
m1 <- lm(Sepal.Length ~ Petal.Width, data = iris)
m2 <- lm(Sepal.Length ~ Petal.Width + Species, data = iris)
m3 <- lm(Sepal.Length ~ Petal.Width * Species, data = iris)
test_performance(m1, m2, m3)
#> Name | Model | BF | Omega2 | p (Omega2) | LR | p (LR)
#> ------------------------------------------------------------
#> m1 | lm | | | | |
#> m2 | lm | 0.007 | 9.54e-04 | 0.935 | 0.15 | 0.919
#> m3 | lm | 0.037 | 0.02 | 0.081 | 3.41 | 0.099
#> Models were detected as nested and are compared in sequential order.
test_bf(m1, m2, m3)
#> Bayes Factors for Model Comparison
#>
#> Model BF
#> [m2] Petal.Width + Species 0.007
#> [m3] Petal.Width * Species 2.64e-04
#>
#> * Against Denominator: [m1] Petal.Width
#> * Bayes Factor Type: BIC approximation
test_wald(m1, m2, m3) # Equivalent to anova(m1, m2, m3)
#> Name | Model | df | df_diff | F | p
#> -------------------------------------------
#> m1 | lm | 148 | | |
#> m2 | lm | 146 | 2.00 | 0.08 | 0.927
#> m3 | lm | 144 | 2.00 | 1.66 | 0.195
#> Models were detected as nested and are compared in sequential order.
# Equivalent to lmtest::lrtest(m1, m2, m3)
test_likelihoodratio(m1, m2, m3, estimator = "ML")
#> # Likelihood-Ratio-Test (LRT) for Model Comparison (ML-estimator)
#>
#> Name | Model | df | df_diff | Chi2 | p
#> ------------------------------------------
#> m1 | lm | 3 | | |
#> m2 | lm | 5 | 2 | 0.15 | 0.926
#> m3 | lm | 7 | 2 | 3.41 | 0.182
# Equivalent to anova(m1, m2, m3, test='LRT')
test_likelihoodratio(m1, m2, m3, estimator = "OLS")
#> # Likelihood-Ratio-Test (LRT) for Model Comparison (OLS-estimator)
#>
#> Name | Model | df | df_diff | Chi2 | p
#> ------------------------------------------
#> m1 | lm | 3 | | |
#> m2 | lm | 5 | 2 | 0.15 | 0.927
#> m3 | lm | 7 | 2 | 3.31 | 0.191
if (require("CompQuadForm")) {
test_vuong(m1, m2, m3) # nonnest2::vuongtest(m1, m2, nested=TRUE)
# Non-nested Models
# -----------------
m1 <- lm(Sepal.Length ~ Petal.Width, data = iris)
m2 <- lm(Sepal.Length ~ Petal.Length, data = iris)
m3 <- lm(Sepal.Length ~ Species, data = iris)
test_performance(m1, m2, m3)
test_bf(m1, m2, m3)
test_vuong(m1, m2, m3) # nonnest2::vuongtest(m1, m2)
}
#> Loading required package: CompQuadForm
#> Name | Model | Omega2 | p (Omega2) | LR | p (LR)
#> ---------------------------------------------------
#> m1 | lm | | | |
#> m2 | lm | 0.19 | < .001 | -4.57 | < .001
#> m3 | lm | 0.12 | < .001 | 2.51 | 0.006
#> Each model is compared to m1.
# Tweak the output
# ----------------
test_performance(m1, m2, m3, include_formula = TRUE)
#> Name | Model | BF | Omega2 | p (Omega2) | LR | p (LR)
#> ----------------------------------------------------------------------------------------
#> m1 | lm(Sepal.Length ~ Petal.Width) | | | | |
#> m2 | lm(Sepal.Length ~ Petal.Length) | 2.90e+10 | 0.19 | < .001 | -4.57 | < .001
#> m3 | lm(Sepal.Length ~ Species) | 2.00e-06 | 0.12 | < .001 | 2.51 | 0.006
#> Each model is compared to m1.
# SEM / CFA (lavaan objects)
# --------------------------
# Lavaan Models
if (require("lavaan")) {
structure <- " visual =~ x1 + x2 + x3
textual =~ x4 + x5 + x6
speed =~ x7 + x8 + x9
visual ~~ textual + speed "
m1 <- lavaan::cfa(structure, data = HolzingerSwineford1939)
structure <- " visual =~ x1 + x2 + x3
textual =~ x4 + x5 + x6
speed =~ x7 + x8 + x9
visual ~~ 0 * textual + speed "
m2 <- lavaan::cfa(structure, data = HolzingerSwineford1939)
structure <- " visual =~ x1 + x2 + x3
textual =~ x4 + x5 + x6
speed =~ x7 + x8 + x9
visual ~~ 0 * textual + 0 * speed "
m3 <- lavaan::cfa(structure, data = HolzingerSwineford1939)
test_likelihoodratio(m1, m2, m3)
# Different Model Types
# ---------------------
if (require("lme4") && require("mgcv")) {
m1 <- lm(Sepal.Length ~ Petal.Length + Species, data = iris)
m2 <- lmer(Sepal.Length ~ Petal.Length + (1 | Species), data = iris)
m3 <- gam(Sepal.Length ~ s(Petal.Length, by = Species) + Species, data = iris)
test_performance(m1, m2, m3)
}
}
#> Loading required package: mgcv
#> This is mgcv 1.8-41. For overview type 'help("mgcv-package")'.
#>
#> Attaching package: ‘mgcv’
#> The following objects are masked from ‘package:brms’:
#>
#> s, t2
#> The following object is masked from ‘package:mclust’:
#>
#> mvn
#> Name | Model | BF
#> -------------------------
#> m1 | lm |
#> m2 | lmerMod | 3.78e-04
#> m3 | gam | 0.012
#> Each model is compared to m1.
```