##############################################################
#               AULA 22 - Testes de aderencia                #
##############################################################

rm(list = ls()); cat('\014')

##############################################################
###           ###  
### EXEMPLO 1 ###
###           ###
#################


# Passo 1: Especificar as hipoteses H0 e Ha
##############################################################

# Amostra: frequencia observada
o = c(22, 52, 26)


# Passo 2: Especificar a estatistica do teste e sua
# distribuicao, sob H0
##############################################################

# Parametros
p = c(0.25, 0.5, 0.25)
n = sum(o)

# Frequencia esperada
e = n*p
e

## [1] 25 50 25

Q2 = sum( (o - e)^2 / e )
Q2
## [1] 0.48


# Passo 3: Fixar o nivel de significancia do teste (α)
##############################################################

# erro tipo 1 = alfa
alfa = 0.05

# numero de par´ametros envolvidos
k = 3

# hipoteses de uma cauda
# valor de qui-quadrado para o erro tipo 1
qc = round(qchisq(1 - alfa, k - 1), 3)
qc

## [1] 5.991


# Passo 4: Calcular o p-valor (ou a regiao critica do teste)
##############################################################

valorp = round(1 - pchisq(Q2, 2), 4)
valorp

## [1] 0.7866


# Passo 5: Decidir entre H0 e Ha, comparando o valor p com α
# (ou verificando se a estatistica do teste pertence ou nao 
# a regiao critica)
##############################################################

chisq.test(o, p = e/100)
##
## Chi-squared test for given probabilities
##
## data: o
## X-squared = 0.48, df = 2, p-value = 0.7866

# OU

chisq.test(o, p = e, rescale.p = TRUE)
##
## Chi-squared test for given probabilities
##
## data: o
## X-squared = 0.48, df = 2, p-value = 0.7866



##############################################################

rm(list = ls()); cat('\014')

##############################################################
###           ###  
### EXEMPLO 2 ###
###           ###
#################


# Passo 1: Especificar as hipoteses H0 e Ha
##############################################################

# Frequencia observada
o = c(2, 5, 16, 42, 69, 51, 32, 23, 9, 1)

# tamanho amostra
n = sum(o)

# Classes
cl = seq(9.5, 19.5, 1)


# Ponto medio
x = c()
for(i in 1:length(o))
{
  x[i] = (cl[i] + cl[i+1])/2
}

# Media amostral
xbarra = sum(x*o)/n

# variancia amostral
aux = (x-xbarra)^2
s2  = sum(o*aux)/(n-1)

# Desvio padrao
s = sqrt(s2)

# Frequencia esperada
e = c()
for(i in 1:length(o))
{
  e[i] = round(n*(pnorm(cl[i+1], xbarra, s) - pnorm(cl[i], xbarra, s)), 2)
}
e

## [1] 1.54 6.52 19.25 39.65 57.02 57.27 40.16 19.67 6.72 1.60

obs = c(o[1]+o[2], o[3:8], o[9]+o[10])
esp = c(e[1]+e[2], e[3:8], e[9]+e[10])

cbind(obs, esp)

## obs esp
## [1,]  7  8.06
## [2,] 16 19.25
## [3,] 42 39.65
## [4,] 69 57.02
## [5,] 51 57.27
## [6,] 32 40.16
## [7,] 23 19.67
## [8,] 10  8.32


# Passo 2: Especificar a estatistica do teste e sua
# distribuicao, sob H0
##############################################################

# Estatistica do teste
Q2 = round(sum( (obs - esp)^2 / esp ), 4)
Q2

## [1] 6.5918


# Passo 3: Fixar o nivel de significancia do teste (α)
##############################################################

# erro tipo 1 = alfa
alfa = 0.04

# numero de par´ametros envolvidos
k = 8
q = 2

# hipoteses de uma cauda
# valor de qui-quadrado para o erro tipo 1
qc = round(qchisq(1 - alfa, k - q - 1), 2)
qc

## [1] 11.64


# Passo 4: Calcular o p-valor (ou a regiao critica do teste)
##############################################################

valorp = round(1 - pchisq(Q2, k - q - 1), 4)
valorp
## [1] 0.2528



##############################################################

rm(list = ls()); cat('\014')

##############################################################
###           ###  
### EXEMPLO 3 ###
###           ###
#################

mu    = 30
sigma = 16
x     = c(2,5,17,21,25,33,34,37,40,56)
n     = length(x)

F10   = 1:10/n
F10_1 = (1:10 - 1) / n

# Funcao acumulada da normal(30,16^2)
Fx = round(pnorm(x, mu, sigma), 3)

aux1 = abs(Fx-F10)
aux2 = abs(Fx-F10_1)

# Tabela com os valores
Tab = cbind(x, F10, F10_1, aux1, aux2)
colnames(Tab) = c("x", "F10", "F10_1", "|Fx - F10|", "|Fx - F10_1|")
Tab

## x F10 F10_1 |Fx - F10| |Fx - F10_1|
## [1,] 2 0.1 0.0 0.060 0.040
## [2,] 5 0.2 0.1 0.141 0.041
## [3,] 17 0.3 0.2 0.092 0.008
## [4,] 21 0.4 0.3 0.113 0.013
## [5,] 25 0.5 0.4 0.123 0.023
## [6,] 33 0.6 0.5 0.026 0.074
## [7,] 34 0.7 0.6 0.101 0.001
## [8,] 37 0.8 0.7 0.131 0.031
## [9,] 40 0.9 0.8 0.166 0.066
## [10,] 56 1.0 0.9 0.052 0.048

# Estatistica do teste
Dn = max(aux1, aux2)
Dn

## [1] 0.166

ks.test(x, "pnorm", 30, 16)

##
## One-sample Kolmogorov-Smirnov test
##
## data: x
## D = 0.16599, p-value = 0.9052
## alternative hypothesis: two-sided