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.

test_bf(...)

# S3 method for default
test_bf(..., text_length = NULL)

test_likelihoodratio(..., estimator = "ML")

performance_lrt(..., estimator = "ML")

test_lrt(..., estimator = "ML")

test_performance(..., reference = 1)

test_vuong(...)

test_wald(...)

## Arguments

... Multiple model objects. 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. Applied when comparing regression models using test_likelihoodratio(). Corresponds to the different estimators for the standard deviation of the errors. If estimator="OLS" (default), 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. 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.

## Value

A data frame containing the relevant indices.

## 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). \cr\cr 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!

### 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().

• 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.

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
#>
#> 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
#>
#> 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

test_vuong(m1, m2, m3) # nonnest2::vuongtest(m1, m2, nested=TRUE)
#> Name | Model |   Omega2 | p (Omega2) |   LR | p (LR)
#> ----------------------------------------------------
#> m1   |    lm |          |            |      |
#> m2   |    lm | 9.54e-04 |      0.935 | 0.15 |  0.919
#> m3   |    lm |     0.02 |      0.081 | 3.41 |  0.099
#> Models were detected as nested and are compared in sequential order.

# 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)
#> Name | Model |      BF | Omega2 | p (Omega2) |    LR | p (LR)
#> -------------------------------------------------------------
#> m1   |    lm |         |        |            |       |
#> m2   |    lm |  > 1000 |   0.19 |     < .001 | -4.57 | < .001
#> m3   |    lm | < 0.001 |   0.12 |     < .001 |  2.51 | 0.006
#> Each model is compared to m1.
test_bf(m1, m2, m3)
#> Bayes Factors for Model Comparison
#>
#>      Model              BF
#> [m2] Petal.Length 2.90e+10
#> [m3] Species      2.00e-06
#>
#> * Against Denominator: [m1] Petal.Width
#> *   Bayes Factor Type: BIC approximation
test_vuong(m1, m2, m3) # nonnest2::vuongtest(m1, m2)
#> 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) |  > 1000 |   0.19 |     < .001 | -4.57 | < .001
#> m3   |      lm(Sepal.Length ~ Species) | < 0.001 |   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)
}
}