Means differs in two paired groups

A notebook made by Paul Amat, amat-design.com

Context and objectives

Is this treatment effective?

Research hypothesis

Statistical hypothesis

H0:

H1:

Data preparation and exploration

Starting session

library(readr)
data <- read_csv("data.csv", show_col_types = FALSE)

Data overview

summary(data$score)

   Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
   6.36   18.75   30.25   30.31   41.45   54.78 

summary(data$score.after)

   Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
  10.73   25.77   36.77   35.66   44.20   59.82 

Mean difference

mean(data$score.after)-mean(data$score)

[1] 5.3547

Parametric test

Conditions

Independent observationss

Normally distributed data of difference vector between groups

Variable of interest

groups.diff <- data$score.after-data$score

Plots

par(mfrow=c(1,2))
hist(groups.diff, freq=FALSE)
curve(dnorm(x, mean=mean(groups.diff), sd=sd(groups.diff)), add=TRUE, col="red")

qqnorm(groups.diff)
qqline(groups.diff, col="red")

Shapiro-Wilk

shapiro.test(groups.diff) # ok in jamovi

    Shapiro-Wilk normality test

data:  groups.diff
W = 0.98985, p-value = 0.1698

Kolmogorov-Smirnov

ks.test(groups.diff, "pnorm", mean(groups.diff), sd(groups.diff)) # ok in jamovi

Warning in ks.test(groups.diff, "pnorm", mean(groups.diff), sd(groups.diff)) :
  ties should not be present for the Kolmogorov-Smirnov test

    One-sample Kolmogorov-Smirnov test

data:  groups.diff
D = 0.042962, p-value = 0.8541
alternative hypothesis: two-sided

No outliers

boxplot(data$score, data$score.after)

Test

t.test(data$score, data$score.after, paired=TRUE) # ok in jamovi

    Paired t-test

data:  data$score and data$score.after
t = -4.005, df = 199, p-value = 8.754e-05
alternative hypothesis: true difference in means is not equal to 0
95 percent confidence interval:
 -7.991207 -2.718193
sample estimates:
mean of the differences 
                -5.3547 

Effect size

if(!require(effsize)) install.packages("effsize")
library(effsize)
(cohentest <- cohen.d(data$score.after, data$score, within = F, paired = T)) #ok jamovi

Cohen's d

d estimate: 0.2831969 (small)
95 percent confidence interval:
    lower     upper 
0.1414240 0.4249699 

Power analysis

Finding power

if(!require(pwr)) install.packages("pwr")
library(pwr)
pwr.t.test(sig.level = 0.05, n = length(data$score)/2, d = cohentest$estimate, type = "paired")

     Paired t test power calculation 

              n = 100
              d = 0.2831969
      sig.level = 0.05
          power = 0.8007871
    alternative = two.sided

NOTE: n is number of *pairs*

Finding N

pwr.t.test(sig.level = 0.05, power = 0.8, d = cohentest$estimate, type = "paired")

     Paired t test power calculation 

              n = 99.80326
              d = 0.2831969
      sig.level = 0.05
          power = 0.8
    alternative = two.sided

NOTE: n is number of *pairs*

Non-parametric test

Conditions

Independent observations

The independence of observations is guaranteed by the recruitment process.

N in each group > 20

length(data$score) > 20

[1] TRUE

length(data$score.after) > 20

[1] TRUE

Test

wilcox.test(data$score, data$score.after, correct = F, paired=TRUE) # ok jamovi

    Wilcoxon signed rank test

data:  data$score and data$score.after
V = 6932.5, p-value = 0.0002076
alternative hypothesis: true location shift is not equal to 0

Wilcoxon rank: to be used when the conditions are not respected and especially in the presence of extreme data.

Effect size

if(!require(effectsize)) install.packages("effectsize")
library(effectsize)

rank_biserial(data$score, data$score.after) # to check

r (rank biserial) |         95% CI
----------------------------------
-0.23             | [-0.33, -0.12]

Results and recommendations