1.1 Paired sample t-test

t.test(DV ~ IV, 
       data = simu_long,
       paired = TRUE)

## 
##  Paired t-test
## 
## data:  DV by IV
## t = 3.4454, df = 19, p-value = 0.002711
## alternative hypothesis: true difference in means is not equal to 0
## 95 percent confidence interval:
##  2.225084 9.112426
## sample estimates:
## mean of the differences 
##                5.668755

cohens_d(DV ~ IV, 
       data = simu_long,
       paired = TRUE)

## Cohen's d |       95% CI
## ------------------------
## 0.77      | [0.27, 1.30]

1.2 One sample t-test

simu_one <- simu_long %>% 
  pivot_wider(id_cols = Subject, values_from = DV, names_from = IV) %>% 
  mutate(DV = cond1 - cond2) %>% 
  select(Subject, DV)
# one sample t-test
t.test(simu_one$DV)

## 
##  One Sample t-test
## 
## data:  simu_one$DV
## t = 3.4454, df = 19, p-value = 0.002711
## alternative hypothesis: true mean is not equal to 0
## 95 percent confidence interval:
##  2.225084 9.112426
## sample estimates:
## mean of x 
##  5.668755

(cohen_simu <- cohens_d(simu_one$DV))

## Cohen's d |       95% CI
## ------------------------
## 0.77      | [0.27, 1.30]

Please note that the results of paired sample t-test and those of one sample t-test are exactly the same, including the effect sizes (i.e., Cohen’s d).

1.3 Formula for Cohen’s d

The Cohen’s d for one sample t-test is relatively simple:

\[\begin{align} \tag1 \text{Cohen's}\; d = \frac{mean(data)}{sd(data)} \end{align}\]

where mean denotes the mean and sd denotes the standard deviation.

Following this equation, Cohen’s d for the one sample t-test is mean(simu_one$DV)/sd(simu_one$DV), i.e., 0.77, which is the same as the effect size calculated by cohens_d().

1.4 Formula for t-value

The equation for t-value (in one sample t-test) is \[\begin{align} \tag2 t = \frac{mean(data)}{se(data)} = \frac{mean(data)}{\frac{sd(data)}{\sqrt{N}}} \end{align}\]

where mean denotes the mean, se denotes standard error, sd denotes the standard deviation, and N denotes the number of subjects.

Following this equation, t-value in that one sample t-test is mean(simu_one$DV)/(sd(simu_one$DV)/sqrt(nSubj)), i.e., 3.445, which is the same as that calculated by t.test.

1.5 Relationships between Cohen’s d and t-value

With equation (1) and (2), we can obtain the relationship between Cohen's d and t: \[\begin{align} \tag3 t &= \frac{mean(data)}{sd(data)} \times \sqrt{N} \\ &= \text{Cohen's}\;d \times \sqrt{N} \end{align}\]

2.1 Formula for partial eta squared (based on F-value)

According to these resources (website 1, website 2 and others), partial eta squared can be computed from F-value:

\[\begin{align} \tag4 \text{partial}\ \eta^2 = \frac{F \times df_{\text{effect}}}{F \times df_{\text{effect}} + df_{\text{error}}} \end{align}\]

Since there are only two conditions for paired samples, \[\begin{align} \tag5 df_{\text{effect}} = 1\\ df_{\text{error}} = N-1 \end{align}\]

Thus, \[\begin{align} \tag6 \text{partial}\ \eta^2 = \frac{F}{F + N-1} \end{align}\]

where F is the F-value and N is the number of subjects.

Following equation 6, partial eta squared for the effect is as_tibble(aov_simu$anova_table)$F/(as_tibble(aov_simu$anova_table)$F + as_tibble(aov_simu$anova_table)[1,2]), i.e., 0.384532, which is the same as the effect size calculated by eta_squared.

3.1 From Cohen’s d to partial eta squared

Put equations 3, 6 and 7 together: \[\begin{align} t &= \text{Cohen's}\;d \times \sqrt{N}\\ \\ \text{partial}\:\eta^2 & = \frac{F}{F + N-1} \\ & = \frac{t^2}{t^2 + N - 1} \\ \tag8 & = \frac{{\text{Cohen's}\;d}^2 \times N}{{\text{Cohen's}\;d}^2 \times N + N -1} \end{align}\]

Therefore, the equation for conversion from Cohen’s d to partial eta squared is: \[\begin{align} \tag9 \text{partial}\;\eta^2 = \frac{\text{Cohen's}\;d^2 \times N}{{\text{Cohen's}\;d}^2 \times N + N -1} \end{align}\]

With formula 9, we can calculate partial eta squared from Cohen’s d via cohen_simu$Cohens_d^2 * nSubj /(cohen_simu$Cohens_d^2 * nSubj + nSubj - 1), i.e. 0.38, which is the same as the effect size calculated by eta_squared.

3.2 From partial eta squared to Cohen’s d

The equation for conversion from partial eta squared to Cohen’s d can be obtained from equation 9: \[\begin{align} \text{partial}\:\eta^2 & = \frac{{\text{Cohen's}\;d}^2 \times N}{{\text{Cohen's}\;d}^2 \times N + N -1}\\ ({\text{Cohen's}\;d}^2 \times N + N -1) \times \text{partial}\:\eta^2 & = {\text{Cohen's}\;d}^2 \times N \\ \\ {\text{Cohen's}\;d}^2 \times N \times (\text{partial}\:\eta^2 - 1) & = - (N-1) \times \text{partial}\:\eta^2 \\ \\ {\text{Cohen's}\;d}^2 & = \frac{- (N-1) \times \text{partial}\:\eta^2}{N \times (\text{partial}\:\eta^2 - 1)} \\ & = \frac{(N-1)}{N} \times \frac{\text{partial}\:\eta^2}{(1 - \text{partial}\:\eta^2)} \\ \\ \text{Cohen's}\;d &= \sqrt(\frac{(N-1)}{N} \times \frac{\text{partial}\:\eta^2}{(1 - \text{partial}\:\eta^2)}) \tag{10} \end{align}\]

With formula 10, we can calculate the Cohen’s d from partial eta squared via sqrt((nSubj-1)/nSubj*eta_simu$Eta2_partial/(1-eta_simu$Eta2_partial)), i.e., 0.77, which is the same as the effect size calculated by cohens_d.