Note
The data set
df_stress
is included in thesprtt
package. Thus, the data set is available when the package is loaded.
In the R code sections:

# comment
: is a comment 
function()
: is R code 
#> results of function()
: is console output

Overview
The sprtt
package is a sequential
probability ratio
tests toolbox (sprtt). This vignette
describes an exemplary use case to improve the understanding of the
package and the sequential ttest.
Other recommended vignettes cover:
a general guide, how to use the package.
Use Case
A team of researchers wants to know if the stress of the employees increased in the company itFlow AG over the last year. The Human Resources department has gathered data from a new selfcare tool they implemented for the employees. It is difficult to predict how many employees will participate at the second measurement point (one year after the baseline). The team suggests using a sequential ttest instead of a Students ttest, because the sequential procedure can reduce the required sample size and is stopped right after the decision for one of the hypotheses is made. Furthermore, the sequential ttest is able to gather evidence for both the alternative and the null hypothesis.
Hypothesis
The researchers know that the company has received more orders than in the year before. Thus, they expect an increase in selfreported stress.
Data Analysis
The parameters of the sequential ttest are specified as follows:

The researchers expect an increase in stress but they have no prior knowledge about the expected effect size. Thus, they define the lower limit of a substantial effect of interest.
d < 0.2

They choose the common \(\alpha\) level of .05 and the same level for the \(\beta\) error probability, which leads to a power of .95 (\(1  \beta\)).
alpha < 0.05 power < 0.95

The data are repeated measures, therefore the test is a paired sequential ttest.
paired < TRUE

The researchers expect that the stress will have increased after one year. Thus, they specify that the true difference in means is greater than 0.
(mean(one_year_stress)  mean(baseline_stress)) > 0
alternative < "greater"
The HR department receives the new data piece by piece and passes it directly on to the researchers. The test is performed for the first time and starts with the first two data points.
# first data from the Human Resources department 
# current sample size
n_person < 2
# get data
df < df_stress[1:n_person,]
# print data
df
#> baseline_stress one_year_stress
#> 1 7.175250 7.844337
#> 2 4.918343 5.527191
# sequential ttest
results < seq_ttest(df$one_year_stress,
df$baseline_stress,
alpha = alpha,
power = power,
d = d,
paired = paired,
alternative = alternative,
verbose = FALSE)
# print results: console output
results
#>
#> ***** Sequential Paired ttest *****
#>
#> formula: df$one_year_stress and df$baseline_stress
#> test statistic:
#> loglikelihood ratio = 0.332, decision = continue sampling
#> SPRT thresholds:
#> lower log(B) = 2.944, upper log(A) = 2.944
The decision from the first test is:
results@decision
#> [1] "continue sampling"
As a result, the researchers take one more data point and run the test again.
# new data from the Human Resources department 
# get one more person
n_person < n_person + 1
df < df_stress[1:n_person,]
# print new data
df
#> baseline_stress one_year_stress
#> 1 7.175250 7.844337
#> 2 4.918343 5.527191
#> 3 4.634266 5.783046
# sequential ttest
results < seq_ttest(df$one_year_stress,
df$baseline_stress,
alpha = alpha,
power = power,
d = d,
paired = paired,
alternative = alternative,
verbose = FALSE)
# print results
results@decision
#> [1] "continue sampling"
This process is repeated until the decision is made for one of the two hypotheses. To simulate this sequential process, a whileloop embraces the sequential ttest function in the code below, which will not stop until one of the hypotheses is accepted or the maximum of the data is reached.
# define the starting point
decision < "continue sampling"
n_person < 3
# simulation of the sequential procedure
while(decision == "continue sampling") {
# get the current data
df < df_stress[1:n_person,]
# run the sequential test and save the results
results < seq_ttest(df$one_year_stress,
df$baseline_stress,
alpha = alpha,
power = power,
d = d,
paired = paired,
alternative = alternative)
# save the current desicion
decision < results@decision
# add a new person
n_person < n_person + 1
# break if the maximum of the data is reached
if (n_person > nrow(df_stress)) {
break
}
}
# console output
results
#>
#> ***** Sequential Paired ttest *****
#>
#> formula: df$one_year_stress and df$baseline_stress
#> test statistic:
#> loglikelihood ratio = 2.988, decision = accept H1
#> SPRT thresholds:
#> lower log(B) = 2.944, upper log(A) = 2.944
#> LogLikelihood of the:
#> alternative hypothesis = 2.238
#> null hypothesis = 5.226
#> alternative hypothesis: true difference in means is greater than 0.
#> specified effect size: Cohen's d = 0.2
#> degrees of freedom: df = 47
#> sample estimates:
#> mean difference
#> 0.47931
#> *Note: to get access to the object of the results use the @ or [] instead of the $ operator.
The whileloop comes to an end after 48 data points.
Report Results
# Required results for the report
# likelihood ratio (LR)
LR < round(results@likelihood_ratio, digits = 2)
LR
#> [1] 19.85
# sample size (N) = degrees of freedom +2 (twosamples) or +1 (onesample & paired)
N < results@df + 1
N
#> [1] 48
# baseline stress (M and SD)
mean_t1 < round(mean(df$baseline_stress), digits = 2)
mean_t1
#> [1] 4.99
sd_t1 < round(sd(df$baseline_stress), digits = 2)
sd_t1
#> [1] 1.02
# after one year stress (M and SD)
mean_t2 < round(mean(df$one_year_stress), digits = 2)
mean_t2
#> [1] 5.47
sd_t2 < round(sd(df$one_year_stress), digits = 2)
sd_t2
#> [1] 1.53
# NOT INCLUDED IN THE PACKAGE
# calculate effect size: Cohen´s d
d_results < effsize::cohen.d(df$one_year_stress,
df$baseline_stress,
paired = TRUE)
d < round(d_results$estimate, digits = 2)
d
#> [1] 0.34
# confidence intervall
d_lower < round(d_results$conf.int[[1]], digits = 2)
d_lower
#> [1] 0.11
d_upper < round(d_results$conf.int[[2]], digits = 2)
d_upper
#> [1] 0.57
Starting at N = 2, the test stops sampling at a total sample size of N = 48 with LR_{48} = 19.85. This ratio indicates that the data are about 20 times more likely under H_{1} than under H_{0}. Thus, we accept the alternative hypothesis: The perceived stress at the second measurement (M = 5.47, SD = 1.53) is higher than one year ago at the baseline measurement (M = 4.99, SD = 1.02), Cohen`s d = 0.34, 95% CI [0.11, 0.57].^{1}^{,}^{2}