n <- 30
sample <- rnorm(n)
alpha1 <- 0.95
t.test(sample, conf.level = alpha1)$conf.int
## [1] -0.3259 0.3943
## attr(,"conf.level")
## [1] 0.95
Увеличим \( \alpha \)
alpha2 <- 0.999
t.test(sample, conf.level = alpha2)$conf.int
## [1] -0.6101 0.6784
## attr(,"conf.level")
## [1] 0.999
получился более широкий доверительный интервал.
Увеличим стандартное отклонение
sample2 <- rnorm(n, 0, 100)
alpha2 <- 0.95
t.test(sample2, conf.level = alpha2)$conf.int
## [1] -14.34 61.11
## attr(,"conf.level")
## [1] 0.95
снова получился более широкий доверительный интервал.
Увеличим размер выборки
n2 <- 1000
sample3 <- rnorm(n2)
alpha2 <- 0.95
t.test(sample3, conf.level = alpha2)$conf.int
## [1] -0.08178 0.04396
## attr(,"conf.level")
## [1] 0.95
Доверительный интервал сузился — чего и следовало ожидать.
Смоделируем выборку 1000 раз, посчитаем доверительный интервал среднего
generate.means <- function(k, n, m = 0, sd = 1) {
sapply(1:k, function(k) {
mean(rnorm(n, m, sd))
})
}
sample.of.means <- generate.means(1000, 30)
hist(sample.of.means)
quantile(sample.of.means, c(0.025, 0.975))
## 2.5% 97.5%
## -0.3692 0.3420
n <- 30
sample1 <- rnorm(n, 1, 1)
sample2 <- rnorm(n, 2, 1)
t.test(sample2, sample1, conf.level = 0.95)$conf.int
## [1] 0.8859 1.8278
## attr(,"conf.level")
## [1] 0.95
Доверительный интервал не включает 0 -> принимаем \( H_1 \).
Теперь увеличим размер выборки…
n <- 1000
sample1 <- rnorm(n, 1, 1)
sample2 <- rnorm(n, 2, 1)
t.test(sample2, sample1, conf.level = 0.95)$conf.int
## [1] 0.9814 1.1525
## attr(,"conf.level")
## [1] 0.95
снова принимаем \( H_1 \), но доверительный интервал на этот раз уже.
n <- 50
sample1 <- rnorm(n, 1, 1)
sample2 <- rnorm(n, 2, 1)
plot(ecdf(sample1), ylim = c(0, 1), main = "Empirical cumulative distributions")
plot(ecdf(sample2), ylim = c(0, 1), col = 2, add = T)
legend("bottomright", c("norm", "exp"), col = c(1, 2), pch = 16)
ks.test(sample1, sample2)
##
## Two-sample Kolmogorov-Smirnov test
##
## data: sample1 and sample2
## D = 0.44, p-value = 9.909e-05
## alternative hypothesis: two-sided
ks.test(sample1, sample2, alternative = "g")
##
## Two-sample Kolmogorov-Smirnov test
##
## data: sample1 and sample2
## D^+ = 0.44, p-value = 6.252e-05
## alternative hypothesis: the CDF of x lies above that of y
ks.test(sample1, sample2, alternative = "l")
##
## Two-sample Kolmogorov-Smirnov test
##
## data: sample1 and sample2
## D^- = 0, p-value = 1
## alternative hypothesis: the CDF of x lies below that of y
Полученные p-value вполне адекватны — p-value двусторонней гипотезы в два раза превышает p-value односторонней
sample1 <- rnorm(n, 1, 2)
sample2 <- rnorm(n, 1, 1)
plot(ecdf(sample1), ylim = c(0, 1), main = "Empirical cumulative distributions")
plot(ecdf(sample2), ylim = c(0, 1), col = 2, add = T)
legend("bottomright", c("norm", "exp"), col = c(1, 2), pch = 16)
ks.test(sample1, sample2)
##
## Two-sample Kolmogorov-Smirnov test
##
## data: sample1 and sample2
## D = 0.4, p-value = 0.0005823
## alternative hypothesis: two-sided
ks.test(sample1, sample2, alternative = "g")
##
## Two-sample Kolmogorov-Smirnov test
##
## data: sample1 and sample2
## D^+ = 0.4, p-value = 0.0003355
## alternative hypothesis: the CDF of x lies above that of y
ks.test(sample1, sample2, alternative = "l")
##
## Two-sample Kolmogorov-Smirnov test
##
## data: sample1 and sample2
## D^- = 0.22, p-value = 0.08892
## alternative hypothesis: the CDF of x lies below that of y
p-value альтернатив слишком малы — мы стремимся принять обе односторонние гипотезы — это странно…
n1 <- 20
n2 <- 10
sample1 <- append(rnorm(n1, 1, 1), runif(n2, 0, 2))
sample2 <- append(rnorm(n1, 2, 1), runif(n2, 0, 2))
plot(ecdf(sample1), ylim = c(0, 1), main = "Empirical cumulative distributions")
plot(ecdf(sample2), ylim = c(0, 1), col = 2, add = T)
legend("bottomright", c("norm", "exp"), col = c(1, 2), pch = 16)
ks.test(sample1, sample2)
##
## Two-sample Kolmogorov-Smirnov test
##
## data: sample1 and sample2
## D = 0.4333, p-value = 0.006548
## alternative hypothesis: two-sided
ks.test(sample1, sample2, alternative = "g")
##
## Two-sample Kolmogorov-Smirnov test
##
## data: sample1 and sample2
## D^+ = 0.4333, p-value = 0.003577
## alternative hypothesis: the CDF of x lies above that of y
ks.test(sample1, sample2, alternative = "l")
##
## Two-sample Kolmogorov-Smirnov test
##
## data: sample1 and sample2
## D^- = 0, p-value = 1
## alternative hypothesis: the CDF of x lies below that of y
На этот раз нормальные p-value
sample1 <- rnorm(n, 1)
sample2 <- rexp(n)
plot(ecdf(sample1), ylim = c(0, 1), main = "Empirical cumulative distributions")
plot(ecdf(sample2), ylim = c(0, 1), col = 2, add = T)
legend("bottomright", c("norm", "exp"), col = c(1, 2), pch = 16)
ks.test(sample1, sample2)
##
## Two-sample Kolmogorov-Smirnov test
##
## data: sample1 and sample2
## D = 0.2, p-value = 0.2719
## alternative hypothesis: two-sided
ks.test(sample1, sample2, alternative = "g")
##
## Two-sample Kolmogorov-Smirnov test
##
## data: sample1 and sample2
## D^+ = 0.18, p-value = 0.1979
## alternative hypothesis: the CDF of x lies above that of y
ks.test(sample1, sample2, alternative = "l")
##
## Two-sample Kolmogorov-Smirnov test
##
## data: sample1 and sample2
## D^- = 0.2, p-value = 0.1353
## alternative hypothesis: the CDF of x lies below that of y
Хм… кажется, снова нечестно — стремимся принять обе односторонние гипотезы
sample1 <- rnorm(n)
sample2 <- rexp(n)
plot(ecdf(sample1), ylim = c(0, 1), main = "Empirical cumulative distributions")
plot(ecdf(sample2), ylim = c(0, 1), col = 2, add = T)
legend("bottomright", c("norm", "exp"), col = c(1, 2), pch = 16)
ks.test(sample1, sample2)
##
## Two-sample Kolmogorov-Smirnov test
##
## data: sample1 and sample2
## D = 0.6, p-value = 1.062e-08
## alternative hypothesis: two-sided
ks.test(sample1, sample2, alternative = "g")
##
## Two-sample Kolmogorov-Smirnov test
##
## data: sample1 and sample2
## D^+ = 0.6, p-value = 1.523e-08
## alternative hypothesis: the CDF of x lies above that of y
ks.test(sample1, sample2, alternative = "l")
##
## Two-sample Kolmogorov-Smirnov test
##
## data: sample1 and sample2
## D^- = 0, p-value = 1
## alternative hypothesis: the CDF of x lies below that of y
Теперь все ок.
Перед применением критерия стоит смотреть на распределения!
Построим хи-квадрат для одной выборки
sample <- rnorm(5000, 0, 100)
h1 <- hist(sample)
breaks <- h1$breaks
counts <- h1$counts
sum(counts < 5)
## [1] 2
p.j <- diff(pnorm(breaks, 0, 100))
n.j <- sum(counts)
chi.sq <- sum((counts - n.j * p.j)^2/(n.j * p.j))
Получили p-value 0.0973.
Смоделируем выборку 1000 раз
n <- 5000
k <- 1000
ps <- sapply(1:k, FUN = function(k) {
sample <- rnorm(n, 0, 100)
h1 <- hist(sample, plot = F)
breaks <- h1$breaks
counts <- h1$counts
p.j <- diff(pnorm(breaks, 0, 100))
while ((sum(counts < 5) > 0 || sum(n * p.j) > 0) && length(counts) > 5) {
nc <- length(counts)
counts <- c(counts[1] + counts[2], counts[3:(nc - 2)], counts[nc - 1] +
counts[nc])
breaks <- breaks[c(-2, -nc)]
p.j <- diff(pnorm(breaks, 0, 100))
}
if (sum(n * p.j) > 0) {
warning("Has bins with less than 5 predicted count")
}
chi.sq <- sum((counts - n * p.j)^2/(n * p.j))
1 - pchisq(chi.sq, length(counts))
})
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
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## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
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## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
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## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
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## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
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## Warning: Has bins with less than 5 predicted count
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## Warning: Has bins with less than 5 predicted count
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## Warning: Has bins with less than 5 predicted count
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## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
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## Warning: Has bins with less than 5 predicted count
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## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
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## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
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## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
## Warning: Has bins with less than 5 predicted count
Мощность критерия: 0.976.
hist(ps, main = "Histogram of p-values")
abline(v = 0.05, col = 2, lty = 5, lwd = 3)
