# 11 Comparing Models with Resampling

Once we create two or more models, the next step is to compare them to understand which one is best. In some cases, comparisons might be within-model, where the same model might be evaluated with different features or preprocessing methods. Alternatively, between-model comparisons, such as when we compared linear regression and random forest models in Chapter 10, are the more common scenario.

In either case, the result is a collection of resampled summary statistics (e.g., RMSE, accuracy, etc.) for each model. In this chapter, we’ll first demonstrate how workflow sets can be used to fit multiple models. Then, we’ll discuss important aspects of resampling statistics. Finally, we’ll look at how to formally compare models (using either hypothesis testing or a Bayesian approach).

## 11.1 Creating Multiple Models with Workflow Sets

In Section 7.5 we described the idea of a workflow set where different preprocessors and/or models can be combinatorially generated. In Chapter 10, we used a recipe for the Ames data that included an interaction term as well as spline functions for longitude and latitude. To demonstrate more with workflow sets, let’s create three different linear models that add these preprocessing steps incrementally; we can test whether these additional terms improve the model results. We’ll create three recipes then combine them into a workflow set:

library(tidymodels)
tidymodels_prefer()

basic_rec <-
recipe(Sale_Price ~ Neighborhood + Gr_Liv_Area + Year_Built + Bldg_Type +
Latitude + Longitude, data = ames_train) %>%
step_log(Gr_Liv_Area, base = 10) %>%
step_other(Neighborhood, threshold = 0.01) %>%
step_dummy(all_nominal_predictors())

interaction_rec <-
basic_rec %>%
step_interact( ~ Gr_Liv_Area:starts_with("Bldg_Type_") )

spline_rec <-
interaction_rec %>%
step_ns(Latitude, Longitude, deg_free = 50)

preproc <-
list(basic = basic_rec,
interact = interaction_rec,
splines = spline_rec
)

lm_models <- workflow_set(preproc, list(lm = linear_reg()), cross = FALSE)
lm_models
#> # A workflow set/tibble: 3 × 4
#>   wflow_id    info             option    result
#>   <chr>       <list>           <list>    <list>
#> 1 basic_lm    <tibble [1 × 4]> <opts[0]> <list [0]>
#> 2 interact_lm <tibble [1 × 4]> <opts[0]> <list [0]>
#> 3 splines_lm  <tibble [1 × 4]> <opts[0]> <list [0]>

We’d like to resample each of these models in turn. To do so, we will use a purrr-like function called workflow_map(). This function takes an initial argument of the function to apply to the workflows, followed by options to that function. We also set a verbose argument that will print the progress as well as a seed argument that makes sure that each model uses the same random number stream as the others.

lm_models <-
lm_models %>%
workflow_map("fit_resamples",
# Options to workflow_map():
seed = 1101, verbose = TRUE,
# Options to fit_resamples():
resamples = ames_folds, control = keep_pred)
#> i 1 of 3 resampling: basic_lm
#> ✔ 1 of 3 resampling: basic_lm (2.4s)
#> i 2 of 3 resampling: interact_lm
#> ✔ 2 of 3 resampling: interact_lm (2.5s)
#> i 3 of 3 resampling: splines_lm
#> ✔ 3 of 3 resampling: splines_lm (3.6s)
lm_models
#> # A workflow set/tibble: 3 × 4
#>   wflow_id    info             option    result
#>   <chr>       <list>           <list>    <list>
#> 1 basic_lm    <tibble [1 × 4]> <opts[2]> <rsmp[+]>
#> 2 interact_lm <tibble [1 × 4]> <opts[2]> <rsmp[+]>
#> 3 splines_lm  <tibble [1 × 4]> <opts[2]> <rsmp[+]>

Notice that the option and result columns are now populated. The former includes the options to fit_resamples() that were given (for reproducibility), and the latter column contains the results produced by fit_resamples().

There are a few convenience functions for workflow sets, including collect_metrics() to collate the performance statistics. We can filter() to any specific metric we are interested in:

collect_metrics(lm_models) %>%
filter(.metric == "rmse")
#> # A tibble: 3 × 9
#>   wflow_id    .config             preproc model .metric .esti…¹   mean     n std_err
#>   <chr>       <chr>               <chr>   <chr> <chr>   <chr>    <dbl> <int>   <dbl>
#> 1 basic_lm    Preprocessor1_Mode… recipe  line… rmse    standa… 0.0803    10 0.00264
#> 2 interact_lm Preprocessor1_Mode… recipe  line… rmse    standa… 0.0799    10 0.00272
#> 3 splines_lm  Preprocessor1_Mode… recipe  line… rmse    standa… 0.0785    10 0.00282
#> # … with abbreviated variable name ¹​.estimator

What about the random forest model from the previous chapter? We can add it to the set by first converting it to its own workflow set then binding rows. This requires that, when the model was resampled, the save_workflow = TRUE option was set in the control function.

four_models <-
as_workflow_set(random_forest = rf_res) %>%
bind_rows(lm_models)
four_models
#> # A workflow set/tibble: 4 × 4
#>   wflow_id      info             option    result
#>   <chr>         <list>           <list>    <list>
#> 1 random_forest <tibble [1 × 4]> <opts[0]> <rsmp[+]>
#> 2 basic_lm      <tibble [1 × 4]> <opts[2]> <rsmp[+]>
#> 3 interact_lm   <tibble [1 × 4]> <opts[2]> <rsmp[+]>
#> 4 splines_lm    <tibble [1 × 4]> <opts[2]> <rsmp[+]>

The autoplot() method, with output in Figure 11.1, shows confidence intervals for each model in order of best to worst. In this chapter, we’ll focus on the coefficient of determination (a.k.a. $$R^2$$) and use metric = "rsq" in the call to set up our plot:

library(ggrepel)
autoplot(four_models, metric = "rsq") +
geom_text_repel(aes(label = wflow_id), nudge_x = 1/8, nudge_y = 1/100) +
theme(legend.position = "none")

From this plot of $$R^2$$ confidence intervals, we can see that the random forest method is doing the best job and there are minor improvements in the linear models as we add more recipe steps.

Now that we have 10 resampled performance estimates for each of the four models, these summary statistics can be used to make between-model comparisons.

## 11.2 Comparing Resampled Performance Statistics

Considering the preceding results for the three linear models, it appears that the additional terms do not profoundly improve the mean RMSE or $$R^2$$ statistics for the linear models. The difference is small, but it might be larger than the experimental noise in the system, i.e., considered statistically significant. We can formally test the hypothesis that the additional terms increase $$R^2$$.

Before making between-model comparisons, it is important for us to discuss the within-resample correlation for resampling statistics. Each model was measured with the same cross-validation folds, and results for the same resample tend to be similar.

In other words, there are some resamples where performance across models tends to be low and others where it tends to be high. In statistics, this is called a resample-to-resample component of variation.

To illustrate, let’s gather the individual resampling statistics for the linear models and the random forest. We will focus on the $$R^2$$ statistic for each model, which measures correlation between the observed and predicted sale prices for each house. Let’s filter() to keep only the $$R^2$$ metrics, reshape the results, and compute how the metrics are correlated with each other.

rsq_indiv_estimates <-
collect_metrics(four_models, summarize = FALSE) %>%
filter(.metric == "rsq")

rsq_wider <-
rsq_indiv_estimates %>%
select(wflow_id, .estimate, id) %>%
pivot_wider(id_cols = "id", names_from = "wflow_id", values_from = ".estimate")

corrr::correlate(rsq_wider %>% select(-id), quiet = TRUE)
#> # A tibble: 4 × 5
#>   term          random_forest basic_lm interact_lm splines_lm
#>   <chr>                 <dbl>    <dbl>       <dbl>      <dbl>
#> 1 random_forest        NA        0.887       0.888      0.889
#> 2 basic_lm              0.887   NA           0.993      0.997
#> 3 interact_lm           0.888    0.993      NA          0.987
#> 4 splines_lm            0.889    0.997       0.987     NA

These correlations are high, and indicate that, across models, there are large within-resample correlations. To see this visually in Figure 11.2, the $$R^2$$ statistics are shown for each model with lines connecting the resamples:

rsq_indiv_estimates %>%
mutate(wflow_id = reorder(wflow_id, .estimate)) %>%
ggplot(aes(x = wflow_id, y = .estimate, group = id, color = id)) +
geom_line(alpha = .5, lwd = 1.25) +
theme(legend.position = "none")

If the resample-to-resample effect was not real, there would not be any parallel lines. A statistical test for the correlations evaluates whether the magnitudes of these correlations are not simply noise. For the linear models:

rsq_wider %>%
with( cor.test(basic_lm, splines_lm) ) %>%
tidy() %>%
select(estimate, starts_with("conf"))
#> # A tibble: 1 × 3
#>   estimate conf.low conf.high
#>      <dbl>    <dbl>     <dbl>
#> 1    0.997    0.987     0.999

The results of the correlation test (the estimate of the correlation and the confidence intervals) show us that the within-resample correlation appears to be real.

What effect does the extra correlation have on our analysis? Consider the variance of a difference of two variables:

$\operatorname{Var}[X - Y] = \operatorname{Var}[X] + \operatorname{Var}[Y] - 2 \operatorname{Cov}[X, Y]$

The last term is the covariance between two items. If there is a significant positive covariance, then any statistical test of this difference would be critically under-powered comparing the difference in two models. In other words, ignoring the resample-to-resample effect would bias our model comparisons towards finding no differences between models.

This characteristic of resampling statistics will come into play in the next two sections.

Before making model comparisons or looking at the resampling results, it can be helpful to define a relevant practical effect size. Since these analyses focus on the $$R^2$$ statistics, the practical effect size is the change in $$R^2$$ that we would consider to be a realistic difference that matters. For example, we might think that two models are not practically different if their $$R^2$$ values are within $$\pm 2$$%. If this were the case, differences smaller than 2% are not deemed important even if they are statistically significant.

Practical significance is subjective; two people can have very different ideas on the threshold for importance. However, we’ll show later that this consideration can be very helpful when deciding between models.

## 11.3 Simple Hypothesis Testing Methods

We can use simple hypothesis testing to make formal comparisons between models. Consider the familiar linear statistical model:

$y_{ij} = \beta_0 + \beta_1x_{i1} + \ldots + \beta_px_{ip} + \epsilon_{ij}$

This versatile model is used to create regression models as well as being the basis for the popular analysis of variance (ANOVA) technique for comparing groups. With the ANOVA model, the predictors ($$x_{ij}$$) are binary dummy variables for different groups. From this, the $$\beta$$ parameters estimate whether two or more groups are different from one another using hypothesis testing techniques.

In our specific situation, the ANOVA can also make model comparisons. Suppose the individual resampled $$R^2$$ statistics serve as the outcome data (i.e., the $$y_{ij}$$) and the models as the predictors in the ANOVA model. A sampling of this data structure is shown in Table 11.1.

Table 11.1: Model performance statistics as a data set for analysis.
Y = rsq model X1 X2 X3 id
0.8108 basic_lm 0 0 0 Fold01
0.8134 interact_lm 1 0 0 Fold01
0.8615 random_forest 0 1 0 Fold01
0.8217 splines_lm 0 0 1 Fold01
0.8045 basic_lm 0 0 0 Fold02
0.8103 interact_lm 1 0 0 Fold02

The X1, X2, and X3 columns in the table are indicators for the values in the model column. Their order was defined in the same way that R would define them, alphabetically ordered by model.

For our model comparison, the specific ANOVA model is:

$y_{ij} = \beta_0 + \beta_1x_{i1} + \beta_2x_{i2} + \beta_3x_{i3} + \epsilon_{ij}$

where

• $$\beta_0$$ is the estimate of the mean $$R^2$$ statistic for the basic linear models (i.e., without splines or interactions),

• $$\beta_1$$ is the change in mean $$R^2$$ when interactions are added to the basic linear model,

• $$\beta_2$$ is the change in mean $$R^2$$ between the basic linear model and the random forest model, and

• $$\beta_3$$ is the change in mean $$R^2$$ between the basic linear model and one with interactions and splines.

From these model parameters, hypothesis tests and p-values are generated to statistically compare models, but we must contend with how to handle the resample-to-resample effect. Historically, the resample groups would be considered a block effect and an appropriate term was added to the model. Alternatively, the resample effect could be considered a random effect where these particular resamples were drawn at random from a larger population of possible resamples. However, we aren’t really interested in these effects; we only want to adjust for them in the model so that the variances of the interesting differences are properly estimated.

Treating the resamples as random effects is theoretically appealing. Methods for fitting an ANOVA model with this type of random effect could include the linear mixed model or a Bayesian hierarchical model (shown in the next section).

A simple and fast method for comparing two models at a time is to use the differences in $$R^2$$ values as the outcome data in the ANOVA model. Since the outcomes are matched by resample, the differences do not contain the resample-to-resample effect and, for this reason, the standard ANOVA model is appropriate. To illustrate, this call to lm() tests the difference between two of the linear regression models:

compare_lm <-
rsq_wider %>%
mutate(difference = splines_lm - basic_lm)

lm(difference ~ 1, data = compare_lm) %>%
tidy(conf.int = TRUE) %>%
select(estimate, p.value, starts_with("conf"))
#> # A tibble: 1 × 4
#>   estimate   p.value conf.low conf.high
#>      <dbl>     <dbl>    <dbl>     <dbl>
#> 1  0.00913 0.0000256  0.00650    0.0118

# Alternatively, a paired t-test could also be used:
rsq_wider %>%
with( t.test(splines_lm, basic_lm, paired = TRUE) ) %>%
tidy() %>%
select(estimate, p.value, starts_with("conf"))
#> # A tibble: 1 × 4
#>   estimate   p.value conf.low conf.high
#>      <dbl>     <dbl>    <dbl>     <dbl>
#> 1  0.00913 0.0000256  0.00650    0.0118

We could evaluate each pair-wise difference in this way. Note that the p-value indicates a statistically significant signal; the collection of spline terms for longitude and latitude do appear to have an effect. However, the difference in $$R^2$$ is estimated at 0.91%. If our practical effect size were 2%, we might not consider these terms worth including in the model.

We’ve briefly mentioned p-values already, but what actually are they? From Wasserstein and Lazar (2016): “Informally, a p-value is the probability under a specified statistical model that a statistical summary of the data (e.g., the sample mean difference between two compared groups) would be equal to or more extreme than its observed value.”

In other words, if this analysis were repeated a large number of times under the null hypothesis of no differences, the p-value reflects how extreme our observed results would be in comparison.

## 11.4 Bayesian Methods

We just used hypothesis testing to formally compare models, but we can also take a more general approach to making these formal comparisons using random effects and Bayesian statistics . While the model is more complex than the ANOVA method, the interpretation is more simple and straight-forward than the p-value approach. The previous ANOVA model had the form:

$y_{ij} = \beta_0 + \beta_1x_{i1} + \beta_2x_{i2} + \beta_3x_{i3} + \epsilon_{ij}$

where the residuals $$\epsilon_{ij}$$ are assumed to be independent and follow a Gaussian distribution with zero mean and constant standard deviation of $$\sigma$$. From this assumption, statistical theory shows that the estimated regression parameters follow a multivariate Gaussian distribution and, from this, p-values and confidence intervals are derived.

A Bayesian linear model makes additional assumptions. In addition to specifying a distribution for the residuals, we require prior distribution specifications for the model parameters ( $$\beta_j$$ and $$\sigma$$ ). These are distributions for the parameters that the model assumes before being exposed to the observed data. For example, a simple set of prior distributions for our model might be:

\begin{align} \epsilon_{ij} &\sim N(0, \sigma) \notag \\ \beta_j &\sim N(0, 10) \notag \\ \sigma &\sim \text{exponential}(1) \notag \end{align}

These priors set the possible/probable ranges of the model parameters and have no unknown parameters. For example, the prior on $$\sigma$$ indicates that values must be larger than zero, are very right-skewed, and have values that are usually less than 3 or 4.

Note that the regression parameters have a pretty wide prior distribution, with a standard deviation of 10. In many cases, we might not have a strong opinion about the prior beyond it being symmetric and bell shaped. The large standard deviation implies a fairly uninformative prior; it is not overly restrictive in terms of the possible values that the parameters might take on. This allows the data to have more of an influence during parameter estimation.

Given the observed data and the prior distribution specifications, the model parameters can then be estimated. The final distributions of the model parameters are combinations of the priors and the likelihood estimates. These posterior distributions of the parameters are the key distributions of interest. They are a full probabilistic description of the model’s estimated parameters.

To adapt our Bayesian ANOVA model so that the resamples are adequately modeled, we consider a random intercept model. Here, we assume that the resamples impact the model only by changing the intercept. Note that this constrains the resamples from having a differential impact on the regression parameters $$\beta_j$$; these are assumed to have the same relationship across resamples. This model equation is:

$y_{ij} = (\beta_0 + b_{i}) + \beta_1x_{i1} + \beta_2x_{i2} + \beta_3x_{i3} + \epsilon_{ij}$

This is not an unreasonable model for resampled statistics which, when plotted across models as in Figure 11.2, tend to have fairly parallel effects across models (i.e., little cross-over of lines).

For this model configuration, an additional assumption is made for the prior distribution of random effects. A reasonable assumption for this distribution is another symmetric distribution, such as another bell-shaped curve. Given the effective sample size of 10 in our summary statistic data, let’s use a prior that is wider than a standard normal distribution. We’ll use a t-distribution with a single degree of freedom (i.e. $$b_i \sim t(1)$$), which has heavier tails than an analogous Gaussian distribution.

The tidyposterior package has functions to fit such Bayesian models for the purpose of comparing resampled models. The main function is called perf_mod() and it is configured to “just work” for different types of objects:

• For workflow sets, it creates an ANOVA model where the groups correspond to the workflows. Our existing models did not optimize any tuning parameters (see the next three chapters). If one of the workflows in the set had data on tuning parameters, the best tuning parameters set for each workflow is used in the Bayesian analysis. In other words, despite the presence of tuning parameters, perf_mod() focuses on making between-workflow comparisons.

• For objects that contain a single model that has been tuned using resampling, perf_mod() makes within-model comparisons. In this situation, the grouping variables tested in the Bayesian ANOVA model are the submodels defined by the tuning parameters.

• The perf_mod() function can also take a data frame produced by rsample that has columns of performance metrics associated with two or more model/workflow results. These could have been generated by nonstandard means.

From any of these types of objects, the perf_mod() function determines an appropriate Bayesian model and fits it with the resampling statistics. For our example, it will model the four sets of $$R^2$$ statistics associated with the workflows.

The tidyposterior package uses the Stan software for specifying and fitting the models via the rstanarm package. The functions within that package have default priors (see ?priors for more details). The following model uses the default priors for all parameters except for the random intercepts (which follow a t-distribution). The estimation process uses random numbers so the seed is set within the function call. The estimation process is iterative and replicated several times in collections called chains. The iter parameter tells the function how long to run the estimation process in each chain. When several chains are used, their results are combined (assume that this is validated by diagnostic assessments).

library(tidyposterior)
library(rstanarm)

# The rstanarm package creates copious amounts of output; those results
# are not shown here but are worth inspecting for potential issues. The
# option refresh = 0 can be used to eliminate the logging.
rsq_anova <-
perf_mod(
four_models,
metric = "rsq",
prior_intercept = rstanarm::student_t(df = 1),
chains = 4,
iter = 5000,
seed = 1102
)

The resulting object has information on the resampling process as well as the Stan object embedded within (in an element called stan). We are most interested in the posterior distributions of the regression parameters. The tidyposterior package has a tidy() method that extracts these posterior distributions into a tibble:

model_post <-
rsq_anova %>%
# Take a random sample from the posterior distribution
# so set the seed again to be reproducible.
tidy(seed = 1103)

glimpse(model_post)
#> Rows: 40,000
#> Columns: 2
#> $model <chr> "random_forest", "basic_lm", "interact_lm", "splines_lm", "rando… #>$ posterior <dbl> 0.8299, 0.7880, 0.7881, 0.7974, 0.8321, 0.7899, 0.7903, 0.7962, …

The four posterior distributions are visualized in Figure 11.3.

model_post %>%
mutate(model = forcats::fct_inorder(model)) %>%
ggplot(aes(x = posterior)) +
geom_histogram(bins = 50, color = "white", fill = "blue", alpha = 0.4) +
facet_wrap(~ model, ncol = 1)

These histograms describe the estimated probability distributions of the mean $$R^2$$ value for each model. There is some overlap, especially for the three linear models.

There is also a basic autoplot() method for the model results, shown in Figure 11.4, as well as the tidied object that shows overlaid density plots.

autoplot(rsq_anova) +
geom_text_repel(aes(label = workflow), nudge_x = 1/8, nudge_y = 1/100) +
theme(legend.position = "none")

One wonderful aspect of using resampling with Bayesian models is that, once we have the posteriors for the parameters, it is trivial to get the posterior distributions for combinations of the parameters. For example, to compare the two linear regression models, we are interested in the difference in means. The posterior of this difference is computed by sampling from the individual posteriors and taking the differences. The contrast_models() function can do this. To specify the comparisons to make, the list_1 and list_2 parameters take character vectors and compute the differences between the models in those lists (parameterized as list_1 - list_2).

We can compare two of the linear models and visualize the results in Figure 11.5.

rqs_diff <-
contrast_models(rsq_anova,
list_1 = "splines_lm",
list_2 = "basic_lm",
seed = 1104)

rqs_diff %>%
as_tibble() %>%
ggplot(aes(x = difference)) +
geom_vline(xintercept = 0, lty = 2) +
geom_histogram(bins = 50, color = "white", fill = "red", alpha = 0.4)

The posterior shows that the center of the distribution is greater than zero (indicating that the model with splines typically had larger values) but does overlap with zero to a degree. The summary() method for this object computes the mean of the distribution as well as credible intervals, the Bayesian analog to confidence intervals.

summary(rqs_diff) %>%
select(-starts_with("pract"))
#> # A tibble: 1 × 6
#>   contrast               probability    mean   lower  upper  size
#>   <chr>                        <dbl>   <dbl>   <dbl>  <dbl> <dbl>
#> 1 splines_lm vs basic_lm        1.00 0.00914 0.00512 0.0132     0

The probability column reflects the proportion of the posterior that is greater than zero. This is the probability that the positive difference is real. The value is not close to zero, providing a strong case for statistical significance, i.e., the idea that statistically the actual difference is not zero.

However, the estimate of the mean difference is fairly close to zero. Recall that the practical effect size we suggested previously is 2%. With a posterior distribution, we can also compute the probability of being practically significant. In Bayesian analysis, this is a ROPE estimate (for Region Of Practical Equivalence, Kruschke and Liddell (2018)). To estimate this, the size option to the summary function is used:

summary(rqs_diff, size = 0.02) %>%
select(contrast, starts_with("pract"))
#> # A tibble: 1 × 4
#>   contrast               pract_neg pract_equiv pract_pos
#>   <chr>                      <dbl>       <dbl>     <dbl>
#> 1 splines_lm vs basic_lm         0           1         0

The pract_equiv column is the proportion of the posterior that is within [-size, size] (the columns pract_neg and pract_pos are the proportions that are below and above this interval). This large value indicates that, for our effect size, there is an overwhelming probability that the two models are practically the same. Even though the previous plot showed that our difference is likely nonzero, the equivalence test suggests that it is small enough to not be practically meaningful.

The same process could be used to compare the random forest model to one or both of the linear regressions that were resampled. In fact, when perf_mod() is used with a workflow set, the autoplot() method can show the pract_equiv results that compare each workflow to the current best (the random forest model, in this case).

autoplot(rsq_anova, type = "ROPE", size = 0.02) +
geom_text_repel(aes(label = workflow)) +
theme(legend.position = "none")

Figure 11.6 shows us that none of the linear models comes close to the random forest model when a 2% practical effect size is used.

### The effect of the amount of resampling

How does the number of resamples affect these types of formal Bayesian comparisons? More resamples increases the precision of the overall resampling estimate; that precision propagates to this type of analysis. For illustration, additional resamples were added using repeated cross-validation. How did the posterior distribution change? Figure 11.7 shows the 90% credible intervals with up to 100 resamples (generated from 10 repeats of 10-fold cross-validation).26

ggplot(intervals,
aes(x = resamples, y = mean)) +
geom_path() +
geom_ribbon(aes(ymin = lower, ymax = upper), fill = "red", alpha = .1) +
labs(x = "Number of Resamples (repeated 10-fold cross-validation)")

The width of the intervals decreases as more resamples are added. Clearly, going from ten resamples to thirty has a larger impact than going from eighty to 100. There are diminishing returns for using a “large” number of resamples (“large” will be different for different data sets).

## 11.5 Chapter Summary

This chapter described formal statistical methods for testing differences in performance between models. We demonstrated the within-resample effect, where results for the same resample tend to be similar; this aspect of resampled summary statistics requires appropriate analysis in order for valid model comparisons. Further, although statistical significance and practical significance are both important concepts for model comparisons, they are different.

### REFERENCES

Faraway, J. 2016. Extending the Linear Model with R: Generalized Linear, Mixed Effects and Nonparametric Regression Models. CRC press.
Kruschke, J, and T Liddell. 2018. “The Bayesian New Statistics: Hypothesis Testing, Estimation, Meta-Analysis, and Power Analysis from a Bayesian Perspective.” Psychonomic Bulletin and Review 25 (1): 178–206.
McElreath, R. 2020. Statistical Rethinking: A Bayesian Course with Examples in R and Stan. CRC press.
Wasserstein, R, and N Lazar. 2016. “The ASA Statement on p-Values: Context, Process, and pPurpose.” The American Statistician 70 (2): 129–33.

1. The code to generate intervals is available at https://github.com/tidymodels/TMwR/blob/main/extras/ames_posterior_intervals.R.↩︎