r-Funktionen für Inferenzstatistik

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Angaben aus Tambour, M., Holt, M., Speyer, A., Christensen, R., & Gram, B. (2018). Manual lymphatic drainage adds no further volume reduction to complete decongestive therapy on breast cancer-related lymphoedema: a multicentre, randomised, single-blind trial. British journal of cancer, 119(10), 1215-1222. doi: 10.1038/s41416-018-0306-4

# EG = T+MLD
n_EG <- 38
m_EG_1 <- -4.2
se_EG_1 <- 1.1
m_EG_7 <- -6.8
se_EG_7 <- 1.2

# CG = T-MLD
n_CG <- 35
m_CG_1 <- -4.8
se_CG_1 <- 1.2
m_CG_7 <- -5.7
se_CG_7 <- 1.2

Standardabweichung aus SE berechnen

Formel zur Berechnung von SE:

\(SE = \frac{s}{\sqrt{n}}\), daraus folgt \(s = SE \times \sqrt{n}\)

s_EG_1 <- se_EG_1 * sqrt(n_EG)
s_CG_1 <- se_CG_1 * sqrt(n_CG)
s_EG_7 <- se_EG_7 * sqrt(n_EG)
s_CG_7 <- se_CG_7 * sqrt(n_CG)

Vertrauensintervall für Mittelwertsdifferenz berechnen

Formel zur Berechnung des Vertrauensintervalls:

\[CI = (\bar{x_1} - \bar{x_2} \pm z \times SE_{x_1 - x_2})\] Formel zur Berechnung des Standardfehlers SE für die Mittelwertsdifferenz:

\[SE_{x_1 - x_2} = \sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}\]

CI_diff <- function(x1, x2, s1, s2, n1, n2, ci = .95){
  SE_diff <- sqrt((s1^2/n1) + (s2^2/n2))
  quantile <- abs(qnorm((1 - ci)/2))
  ME <- quantile * SE_diff
  CI_diff <- round((x2 - x1) + c(-1, 1) * ME, 4)
  out <- tibble(x1 = x1, 
                s1 = s1, 
                n1 = n1, 
                x2 = x2, 
                s2 = s2, 
                n2 = n2, 
                diff = (x2-x1), 
                ci_lo = CI_diff[1], 
                ci_up = CI_diff[2])
  return(out)
}

month1 <- CI_diff(m_EG_1, m_CG_1, s_EG_1, s_CG_1, n_EG, n_CG)
month7 <- CI_diff(m_EG_7, m_CG_7, s_EG_7, s_CG_7, n_EG, n_CG)

result <- bind_rows(month1, month7)
result <- result %>% 
  add_column(Monat = c(1, 7), .before = "x1")
kable(result)

Monat	x1	s1	n1	x2	s2	n2	diff	ci_lo	ci_up
1	-4.2	6.780855	38	-4.8	7.099296	35	-0.6	-3.7906	2.5906
7	-6.8	7.397297	38	-5.7	7.099296	35	1.1	-2.2262	4.4262

p-Wert berechnen

Formel zur Berechnung von z für die Berechnung des p-Wertes

\[z_p = \frac{x_1 - x_2}{SE_{x_1 - x_2}}\]

p_value <- function(x1, x2, s1, s2, n1, n2){
  SE_diff <- sqrt((s1^2/n1) + (s2^2/n2))
  z_p <- (x2 - x1)/SE_diff
  p <- 2 * pnorm(abs(z_p), lower.tail = FALSE)
  out2 <- tibble(
    z = z_p,
    "p-Wert" = p
    )
  return(out2)
}

p1 <- p_value(m_EG_1, m_CG_1, s_EG_1, s_CG_1, n_EG, n_CG)
p7 <- p_value(m_EG_7, m_CG_7, s_EG_7, s_CG_7, n_EG, n_CG)
p <- bind_rows(p1, p7)


result <- bind_cols(result, p)
kable(result)

Monat	x1	s1	n1	x2	s2	n2	diff	ci_lo	ci_up	z	p-Wert
1	-4.2	6.780855	38	-4.8	7.099296	35	-0.6	-3.7906	2.5906	-0.3685771	0.7124430
7	-6.8	7.397297	38	-5.7	7.099296	35	1.1	-2.2262	4.4262	0.6481812	0.5168677

Cohen’s d berechnen

\[d = \frac{x_2 - x_1}{s_1,_2}\]

\[s_1,_2 = \sqrt{\frac{s_1^2 \times n_1 + s_2^2 \times n_2}{n_1 + n_2}}\]

cohen_d <- function(x1, x2, s1, s2, n1, n2){
  s_paired <- sqrt((s1^2 * n1 + s2^2 * n2)/(n1 + n2))
  print(s_paired)
  d <- (x2 - x1)/s_paired
  return(d)
}

d1 <- cohen_d(m_EG_1, m_CG_1, s_EG_1, s_CG_1, n_EG, n_CG)

## [1] 6.935357

d7 <- cohen_d(m_EG_7, m_CG_7, s_EG_7, s_CG_7, n_EG, n_CG)

## [1] 7.255947

d <- tibble("Cohen's d" = round(c(d1, d7), 4))
d

## # A tibble: 2 x 1
##   `Cohen's d`
##         <dbl>
## 1     -0.0865
## 2      0.152

result <- bind_cols(result, d)
kable(result, 
      digits = 4, 
      caption = "Prozentuale Abnahme des Volumens des betroffenen Armes, 1 = EG, 2 = CG")

Prozentuale Abnahme des Volumens des betroffenen Armes, 1 = EG, 2 = CG

Monat	x1	s1	n1	x2	s2	n2	diff	ci_lo	ci_up	z	p-Wert	Cohen’s d
1	-4.2	6.7809	38	-4.8	7.0993	35	-0.6	-3.7906	2.5906	-0.3686	0.7124	-0.0865
7	-6.8	7.3973	38	-5.7	7.0993	35	1.1	-2.2262	4.4262	0.6482	0.5169	0.1516