vignettes/indicesEstimationComparison.Rmd
indicesEstimationComparison.Rmd
This vignette can be referred to by citing the package:
One of the main difference between the Bayesian and the frequentist frameworks is that the former returns a probability distribution for each effect (i.e., a model parameter of interest, such as a regression slope) instead of a single value. However, there is still a need and demand - for reporting or use in further analysis - for a single value (point-estimate) that best characterises the underlying posterior distribution.
There are three main indices used in the literature for effect estimation: - the mean - the median - the MAP (Maximum A Posteriori) estimate (roughly corresponding to the mode - the “peak” - of the distribution)
Unfortunately, there is no consensus about which one to use, as no systematic comparison has ever been done.
In the present work, we will compare these three point-estimates of effect with each other, as well as with the widely known beta, extracted from a comparable frequentist model. These comparisons can help us draw bridges and relationships between these two influential statistical frameworks.
We will be carrying out simulation aimed at modulating the following characteristics:
We generated a dataset for each combination of these characteristics, resulting in a total of 2 * 2 * 9 * 1000 = 36000
Bayesian and frequentist models. The code used for generation is available here (please note that it takes usually several days/weeks to complete).
df %>%
select(error, true_effect, outcome_type, Coefficient, Median, Mean, MAP) %>%
gather(estimate, value, -error, -true_effect, -outcome_type) %>%
mutate(temp = as.factor(cut(error, 10, labels = FALSE))) %>%
group_by(temp) %>%
mutate(error_group = round(mean(error), 1)) %>%
ungroup() %>%
filter(value < 6) %>%
ggplot(aes(x = error_group, y = value, fill = estimate, group = interaction(estimate, error_group))) +
# geom_hline(yintercept = 0) +
# geom_point(alpha=0.05, size=2, stroke = 0, shape=16) +
# geom_smooth(method="loess") +
geom_boxplot(outlier.shape = NA) +
theme_modern() +
scale_fill_manual(
values = c("Coefficient" = "#607D8B", "MAP" = "#795548", "Mean" = "#FF9800", "Median" = "#FFEB3B"),
name = "Index"
) +
ylab("Point-estimate") +
xlab("Noise") +
facet_wrap(~ outcome_type * true_effect, scales = "free")
df %>%
select(sample_size, true_effect, outcome_type, Coefficient, Median, Mean, MAP) %>%
gather(estimate, value, -sample_size, -true_effect, -outcome_type) %>%
mutate(temp = as.factor(cut(sample_size, 10, labels = FALSE))) %>%
group_by(temp) %>%
mutate(size_group = round(mean(sample_size))) %>%
ungroup() %>%
filter(value < 6) %>%
ggplot(aes(x = size_group, y = value, fill = estimate, group = interaction(estimate, size_group))) +
# geom_hline(yintercept = 0) +
# geom_point(alpha=0.05, size=2, stroke = 0, shape=16) +
# geom_smooth(method="loess") +
geom_boxplot(outlier.shape = NA) +
theme_modern() +
scale_fill_manual(
values = c("Coefficient" = "#607D8B", "MAP" = "#795548", "Mean" = "#FF9800", "Median" = "#FFEB3B"),
name = "Index"
) +
ylab("Point-estimate") +
xlab("Sample size") +
facet_wrap(~ outcome_type * true_effect, scales = "free")
We fitted a (frequentist) multiple linear regression to statistically test the the predict the presence or absence of effect with the estimates as well as their interaction with noise and sample size.
df %>%
select(sample_size, error, true_effect, outcome_type, Coefficient, Median, Mean, MAP) %>%
pivot_longer(
c(-sample_size, -error, -true_effect, -outcome_type),
names_to = "estimate"
) %>%
glm(true_effect ~ outcome_type / estimate / value, data = ., family = "binomial") %>%
parameters(df_method = "wald") %>%
select(Parameter, Coefficient, p) %>%
filter(
str_detect(Parameter, "outcome_type"),
str_detect(Parameter, ":value")
) %>%
arrange(desc(Coefficient)) %>%
knitr::kable(digits = 2)
Parameter | Coefficient | p |
---|---|---|
outcome_typelinear:estimateMean:value | 10.8 | 0 |
outcome_typelinear:estimateMedian:value | 10.8 | 0 |
outcome_typelinear:estimateMAP:value | 10.7 | 0 |
outcome_typelinear:estimateCoefficient:value | 10.5 | 0 |
outcome_typebinary:estimateMAP:value | 4.4 | 0 |
outcome_typebinary:estimateMedian:value | 4.3 | 0 |
outcome_typebinary:estimateMean:value | 4.2 | 0 |
outcome_typebinary:estimateCoefficient:value | 3.9 | 0 |
This suggests that, in order to delineate between the presence and the absence of an effect, compared to the frequentist’s beta coefficient:
Overall, the median appears to be a safe choice, maintaining a high performance across different types of models.
We will be carrying out another simulation aimed at modulating the following characteristics:
We generated 3 datasets for each combination of these characteristics, resulting in a total of 2 * 2 * 8 * 40 * 9 * 3 = 34560
Bayesian and frequentist models. The code used for generation is avaible here (please note that it takes usually several days/weeks to complete).
df <- read.csv("https://raw.github.com/easystats/circus/master/data/bayesSim_study2.csv")
df %>%
select(iterations, true_effect, outcome_type, beta, Median, Mean, MAP) %>%
gather(estimate, value, -iterations, -true_effect, -outcome_type) %>%
mutate(temp = as.factor(cut(iterations, 5, labels = FALSE))) %>%
group_by(temp) %>%
mutate(iterations_group = round(mean(iterations), 1)) %>%
ungroup() %>%
filter(value < 6) %>%
ggplot(aes(x = iterations_group, y = value, fill = estimate, group = interaction(estimate, iterations_group))) +
geom_boxplot(outlier.shape = NA) +
theme_classic() +
scale_fill_manual(
values = c("beta" = "#607D8B", "MAP" = "#795548", "Mean" = "#FF9800", "Median" = "#FFEB3B"),
name = "Index"
) +
ylab("Point-estimate of the true value 0\n") +
xlab("\nNumber of Iterations") +
facet_wrap(~ outcome_type * true_effect, scales = "free")
df %>%
mutate(warmup = warmup / iterations) %>%
select(warmup, true_effect, outcome_type, beta, Median, Mean, MAP) %>%
gather(estimate, value, -warmup, -true_effect, -outcome_type) %>%
mutate(temp = as.factor(cut(warmup, 3, labels = FALSE))) %>%
group_by(temp) %>%
mutate(warmup_group = round(mean(warmup), 1)) %>%
ungroup() %>%
filter(value < 6) %>%
ggplot(aes(x = warmup_group, y = value, fill = estimate, group = interaction(estimate, warmup_group))) +
geom_boxplot(outlier.shape = NA) +
theme_classic() +
scale_fill_manual(
values = c("beta" = "#607D8B", "MAP" = "#795548", "Mean" = "#FF9800", "Median" = "#FFEB3B"),
name = "Index"
) +
ylab("Point-estimate of the true value 0\n") +
xlab("\nNumber of Iterations") +
facet_wrap(~ outcome_type * true_effect, scales = "free")
Conclusions can be found in the guidelines section article.
If you have any advice, opinion or such, we encourage you to let us know by opening an discussion thread or making a pull request.