Establishing equivalence bounds in hypothesis testing involves determining a range of effect sizes that are considered practically equivalent or negligible. In your scenario, you are aiming for a small to medium effect size of d = 0.15 for a between-group comparison with a sample size of n = 150 per condition. To calculate the lower and upper equivalence bounds, you can follow these steps:

1. **Determine the Standardized Effect Size**: Convert the desired effect size (d = 0.15) to a standardized effect size. In this case, Cohen's d is commonly used. Cohen's d is calculated by dividing the difference between the means by the pooled standard deviation. However, since you don't have specific means or standard deviations provided, you'll need to make some assumptions or refer to existing literature to estimate the standard deviation or effect size.

2. **Perform a Power Analysis**: Use a power analysis tool, such as G*Power, to estimate the required sample size for your desired effect size of d = 0.15. Input the desired power level (e.g., 80%) and the significance level (e.g., α = 0.05). The power analysis will provide you with the estimated sample size required to detect an effect of that magnitude.

3. **Calculate Equivalence Bounds**: Equivalence bounds are typically defined in terms of a range around zero, representing negligible or practically equivalent effect sizes. In an equivalence test, you aim to demonstrate that the true effect size falls within this range. The specific bounds depend on the desired level of precision and the context of your study.

Since you mentioned a small to medium effect size, you can choose a range around zero that captures effect sizes within that range. For example, you might choose a range of ±0.10 as your equivalence bounds. This means that effect sizes between -0.10 and 0.10 would be considered practically equivalent or negligible.

Alternatively, you can refer to published literature or guidelines specific to your research field that provide recommendations for equivalence bounds. These guidelines are often based on theoretical or practical considerations.

Regarding published papers employing equivalence bound testing in psychology, here are two examples:

1. Lakens, D., McLatchie, N., Isager, P. M., Scheel, A. M., & Dienes, Z. (2018). Improving inferences about null effects with Bayes factors and equivalence tests. Journal of Experimental Psychology: General, 147(4), 534-547.

2. Francis, G. (2013). Replication, statistical consistency, and publication bias. Journal of Mathematical Psychology, 57(5), 153-169.

These papers discuss the use of equivalence testing in psychological research and provide examples of how to apply this approach in specific experimental designs.

Remember to consult with your thesis advisor or statistical expert to ensure that the chosen equivalence bounds and methodology align with the specific requirements and conventions in your research field.

Hi Jhon Paul Ambit

Thank you so much for answering my query! it's really helpful!

If you don't mind me picking your brain further, how do you think Lakens managed to derive an equivalence bounds of .414, from an affect size of .389, based on a sample of 100 per condition for two between group, which I have highlighted in italics in the original post.

Cheers,

Gregory