How to interpret a large Cohen's d when p-value is non-significant and CI's are close to 0 I have a sample size of 23, with 11 and 12 participants in each group. I conducted t-tests for several continuous dependent variables. All of them are non-significant, but some of them have quite high Cohen's d values (for example 0.6 or above). In addition, the confidence intervals in these cases, while still spanning zero, usually have one case which is very very close to zero.
As an example, one of the tests has a Mann-Whitney U value of 41, a p-value of .11, CIs of -0.00004 and 1.2, and cohen's d of 0.75. How should I interpret this finding? Intuitively I would say it is a power problem, but with medium-large effect sizes I am not sure, since even small samples would theoretically be enough to detect such effects.
I used the jamovi package in R to conduct the tests
ttestIS(data = dat, vars = vars(variable_1, variable_2, variable_3, variable_4, variable_5), group = condition, students = FALSE, mann = TRUE, meanDiff = TRUE, desc = TRUE, effectSize = TRUE, ci = TRUE, plots = TRUE)

Update: I went off the effect size interpretations specified by Cohen (0.2 = small, 0.5 = medium, 0.8 = large). The dependent variables are self-report questionnaire items on a 5 point likert scale. The responses were non-normally distributed, hence the use of Mann-Whitney U tests.
 A: Short answer: You don't.
Since your effect is not significant (you fail to reject the null hypothesis that there is no effect), if you're following the rules of null hypothesis significance testing, you cannot conclude that there is any effect here.
Additional points

*

*You're running 5 tests, without adjusting for multiple comparisons, so the chances of making a type 1 error here are quite high. This makes it even more likely that your large effect size is large just by chance.

*The wide confidence intervals show that even though the effect size might be large (and might even be larger than 0.75), it might also be small, zero, or negative, since your data does not rule out these possibilities.

A: It sounds like you are replying to your own question already. The effect size statistic is independent of your statistical significance. Here a extreme example you can run in R:
data <- matrix(c(1,0,2,0,1.5,0,
                 1.7,1,1.8,1,2.2,1), nrow= 6, ncol = 2, byrow = TRUE)
data <- as.data.frame(data)
data[,2] <- as.factor(data[,2])


effsize::cohen.d(formula = data[,1] ~ data[,2], data = data)
t.test(data[,1] ~ data[,2])


We have only three observations per group, so before even estimating anything we know it is underpowered and a practically useless analysis. Despite this, the output gives a Cohen's d of 4.05, an extremely large effect. This says nothing about our sample size, our power, or its significance, hence there is no conflict in your results. The two-sample t-test gives a p-value of 0.3071, unsurprising given that we just have three samples per group.
You can see Cohen's D statistic just as a standardized difference between means of distributions, and that says little about how stable those distributions are or the quality of your sampling. The image below shows the point: the distributions have great variability and a large overlap, but the distance between their means (the vertical lines) is relatively large, hence the high p-value and high effect size.

Also worth noting what Eoin says on the other answer: you are doing multiple comparisons with no correction for it. This can be seen as a way of p-hacking or torturing the data if all you care about is finding a "significant" result, as it seems to be suggested.
A: 
All of them are non-significant, but some of them have quite high Cohen's d values (for example 0.6 or above)

The fact that the effect size is large doesn't necessarily mean that a test for no-difference will return a tiny p-value. Here's an example:
x1
1.46  1.62  0.77  0.38  0.92  0.36 -0.68 -1.06  0.22  0.89
x2
-1.11  0.38  1.18 -2.25  0.53  0.61 -0.47 -0.45 -0.46 -0.79

If I'm not mistaken, Cohen's d for the difference between mean of x1 and x2 is 0.83 but a t-test returns p=0.08 (not very convincingly different from 0):
t.test(x1, x2)

    Welch Two Sample t-test

data:  x1 and x2
t = 2, df = 20, p-value = 0.08
alternative hypothesis: true difference in means is not equal to 0
95 percent confidence interval:
 -0.099  1.643
sample estimates:
mean of x mean of y 
     0.49     -0.28 

