## 5.10 Teste de Hipóteses

• Seção 3.4 de
• Capítulo 9 de

“Did the sun just explode?” by xkcd.com

### 5.10.1 Fator de Bayes

• Seção 3.4.1 de
• Seção 9.5.1 de

#### 5.10.1.1 Pacote BayesFactor

apresentam o pacote BayesFactor.

library(BayesFactor)
BFManual()

Exemplo 5.13 Exemplo da documentação.

## Sleep data from t test example
sleep2 <- datasets::sleep
plot(extra ~ group, data = sleep2)

## classical paired t test
t.test(x = sleep2$extra[sleep2$group==1],
y = sleep2$extra[sleep2$group==2], paired = TRUE)
##
##  Paired t-test
##
## data:  sleep2$extra[sleep2$group == 1] and sleep2$extra[sleep2$group == 2]
## t = -4.0621, df = 9, p-value = 0.002833
## alternative hypothesis: true mean difference is not equal to 0
## 95 percent confidence interval:
##  -2.4598858 -0.7001142
## sample estimates:
## mean difference
##           -1.58
# classical Wilcoxon (signed rank test (median = 0 versus median > 0)
wilcox.test(x = sleep2$extra[sleep2$group==1],
y = sleep2$extra[sleep2$group==2], paired = TRUE)
##
##  Wilcoxon signed rank test with continuity correction
##
## data:  sleep2$extra[sleep2$group == 1] and sleep2$extra[sleep2$group == 2]
## V = 0, p-value = 0.009091
## alternative hypothesis: true location shift is not equal to 0
## bayesian paired t test
ttestBF(x = sleep2$extra[sleep2$group == 1],
y = sleep2$extra[sleep2$group == 2], paired = TRUE)
## Bayes factor analysis
## --------------
## [1] Alt., r=0.707 : 17.25888 ±0%
##
## Against denominator:
##   Null, mu = 0
## ---
## Bayes factor type: BFoneSample, JZS

Exercício 5.8 Veja a documentação de pingouin.bayesfactor_ttest no link a seguir.
https://pingouin-stats.org/build/html/generated/pingouin.bayesfactor_ttest.html#pingouin.bayesfactor_ttest

Exemplo 5.14 Exemplo Moon and Agression.

# lendo dados

# gráfico
boxplot(ma)

## classical paired t test
t.test(x = ma$Moon, y = ma$Other, paired = TRUE)
##
##  Paired t-test
##
## data:  ma$Moon and ma$Other
## t = 6.4518, df = 14, p-value = 1.518e-05
## alternative hypothesis: true mean difference is not equal to 0
## 95 percent confidence interval:
##  1.623968 3.241365
## sample estimates:
## mean difference
##        2.432667
# classical Wilcoxon (signed rank test (median = 0 versus median > 0)
wilcox.test(x = ma$Moon, y = ma$Other, paired = TRUE)
##
##  Wilcoxon signed rank exact test
##
## data:  ma$Moon and ma$Other
## V = 119, p-value = 0.0001221
## alternative hypothesis: true location shift is not equal to 0
## bayesian paired t test
ttestBF(x = ma$Moon, y = ma$Other, paired = TRUE)
## Bayes factor analysis
## --------------
## [1] Alt., r=0.707 : 1521.194 ±0%
##
## Against denominator:
##   Null, mu = 0
## ---
## Bayes factor type: BFoneSample, JZS

#### 5.10.1.2 Pacote BFpack

apresentam o pacote BFpack.

Exercício 5.9 Veja a vinheta de BFpack em https://cran.r-project.org/package=BFpack.

### 5.10.2 FBST

FBST é o acrônimo para Full Bayesian Significance Test.
- Proposta de para testar hipóteses precisas (sharp hypotheses)
- https://jreduardo.github.io/ce227-bayes/04-fbst.html

#### 5.10.2.1 Pacote fbst

apresenta o pacote de R fbst.

# https://cran.r-project.org/package=fbst
library(fbst)

set.seed(57)
grp1 <- rnorm(50,0,1.5)
grp2 <- rnorm(50,0.8,3.2)

p <- as.vector(BayesFactor::ttestBF(x=grp1,y=grp2,
posterior = TRUE, iterations = 3000,
rscale = "medium")[,4])

# flat reference function
res <- fbst(posteriorDensityDraws = p, nullHypothesisValue = 0,
dimensionTheta = 2, dimensionNullset = 1)
summary(res)
## Full Bayesian Significance Test for testing a sharp hypothesis against its alternative:
## Reference function: Flat
## Hypothesis H_0:Parameter= 0 against its alternative H_1
## Bayesian e-value against H_0: 0.92749
## Standardized e-value: 0.02197123
plot(res)

# medium Cauchy C(0,1) reference function
res_med <- fbst(posteriorDensityDraws = p, nullHypothesisValue = 0,
dimensionTheta = 2, dimensionNullset = 1, FUN = dcauchy,
par = list(location = 0, scale = sqrt(2)/2))
summary(res_med)
## Full Bayesian Significance Test for testing a sharp hypothesis against its alternative:
## Reference function: User-defined
## Hypothesis H_0:Parameter= 0 against its alternative H_1
## Bayesian e-value against H_0: 0.9538276
## Standardized e-value: 0.01313566
plot(res_med)

### 5.10.3 Valores-p bayesiano

O valor-p preditivo a posteriori e a probabilidade de direção são reportados na literatura como ‘valores-pr bayesianos’.

### 5.10.4 Combinando ferramentas

#### 5.10.4.1 Comparando proporções

The observed results in this example are that only one of the patients in the control arm responded positively, but in the case arm there were four positive outcomes.

tab <- matrix(c(1,7, 4,4), nrow = 2, byrow = TRUE)
rownames(tab) <- c('C', 'T') # Controle/Tratamento
colnames(tab) <- c('+', '-') # Respondeu positivamente/negativamente
(tab <- as.table(tab))
##   + -
## C 1 7
## T 4 4
# Teste qui-quadrado sem correção de Yates
chisq.test(tab, correct = FALSE)
##
##  Pearson's Chi-squared test
##
## data:  tab
## X-squared = 2.6182, df = 1, p-value = 0.1056
# Teste qui-quadrado com correção de Yates
chisq.test(tab)
##
##  Pearson's Chi-squared test with Yates' continuity correction
##
## data:  tab
## X-squared = 1.1636, df = 1, p-value = 0.2807
# Teste exato de Fisher
fisher.test(tab)
##
##  Fisher's Exact Test for Count Data
##
## data:  tab
## p-value = 0.2821
## alternative hypothesis: true odds ratio is not equal to 1
## 95 percent confidence interval:
##  0.002553456 2.416009239
## sample estimates:
## odds ratio
##  0.1624254
# Fator de Bayes
fh <- function(x,y){
(choose(8,x)*choose(8,y))/(17*choose(16,x+y))
}
fa <- 1/81
bf <- function(x,y){
fh(x,y)/fa
}
BF <- matrix(nrow = 9, ncol = 9)
rownames(BF) <- paste0('x.',0:8)
colnames(BF) <- paste0('y.',0:8)
for(i in 0:8){
for(j in 0:8){
BF[i+1,j+1] <- bf(j,i)
}
}
addmargins(round(BF,3)) # note a questão do arredondamento
##       y.0   y.1   y.2   y.3   y.4   y.5   y.6   y.7   y.8    Sum
## x.0 4.765 2.382 1.112 0.476 0.183 0.061 0.017 0.003 0.000  8.999
## x.1 2.382 2.541 1.906 1.173 0.611 0.267 0.093 0.024 0.003  9.000
## x.2 1.112 1.906 2.052 1.710 1.166 0.653 0.290 0.093 0.017  8.999
## x.3 0.476 1.173 1.710 1.866 1.633 1.161 0.653 0.267 0.061  9.000
## x.4 0.183 0.611 1.166 1.633 1.814 1.633 1.166 0.611 0.183  9.000
## x.5 0.061 0.267 0.653 1.161 1.633 1.866 1.710 1.173 0.476  9.000
## x.6 0.017 0.093 0.290 0.653 1.166 1.710 2.052 1.906 1.112  8.999
## x.7 0.003 0.024 0.093 0.267 0.611 1.173 1.906 2.541 2.382  9.000
## x.8 0.000 0.003 0.017 0.061 0.183 0.476 1.112 2.382 4.765  8.999
## Sum 8.999 9.000 8.999 9.000 9.000 9.000 8.999 9.000 8.999 80.996
(BF_obs <- bf(1,4))
## [1] 0.6108597
sum(BF[BF <= BF_obs])/81 # P-value
## [1] 0.09228027

### Referências

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