How to boost the effect size? - use experimental conditions that cause a larger effect (higher doses, longer incubation times, harsher treatments...).

The relative effect size can also be increased by reducing the noise of the measurements, e.g. by using a more robust method, a better instrument, better control of experimental conditions, narrowed inclusion criteria etc.

You note that this all is about your experiment, not about the data. Once you have the data, this is it.

You write that your data violates the normality assumption. Why do you use ANOVA then?

If you are looking for interaction, you must be clear if the effects are additive or multiplicative. If you moder effects as additive but they are actually multiplicative, you will see possible large (and statistically significan) interaction by your (wrong) model which is really not indicative for an interaction of the real variables. Example: females spent about 1 unit of money for electronic toys. Males generally spend about twice as much money for electronic toys than females (i.e. about 2 units). A campain (treatment) should increase the amount of money spent for such toys, and it may have a different effect on males and females. If the treatment will increased the money by a factor of 3, without any interaction, treated males will still spent about twice as much (6 units) as females (3 units), but if you look at absolute values, the effect of sex is one unit and the effect of the treatment is 2 units for mfemales but 4 units for males, indicating a strong interaction (so the interpretation is completely different!). If your interpretation is meaningful depends on if your assumptions about the additivity or multiplicativity of the effects. It's important to think this through.

Not sure what you are asking for or why you think eta^2 is so small, interpreting its magnitude will depend on the context. If you just want a bigger number or don't square it (Rosenthal and Rubin discuss how the un-squared version is easier to interpret sometimes) or multiply it by 2 (then it is twice as big if all you care about is the number). If you want to stay with eta^2, given you have already interpreted the output of these data, you will need to collect a new data set. For these new data, then as Jochen Wilhelm says, reduce the noise in your data by using more reliable instruments.

One solution should be Mixed model. Treatment as a fixed effect and subject as a random effect. You can try a 2X2 cross-over design with Proc Mixed of SAS.

You may try the mixed model approach (see the attachment for details). Besides, you can look at the strategies proposed by Meyvis and Van Osselaer (2018). However, you need to mind the pitfalls reported by Yang et al. (2014).

Meyvis, T., & Van Osselaer, S. M. J. (2018). Increasing the power of your study by increasing the effect size. Journal of Consumer Research, 44(5), 1157–1173. https://doi.org/10.1093/jcr/ucx110

Yang, J., Zaitlen, N. A., Goddard, M. E., Visscher, P. M., & Price, A. L. (2014). Advantages and pitfalls in the application of mixed-model association methods. Nature Genetics, 46(2), 100–106. https://doi.org/10.1038/ng.2876

Good luck,

I agree with the above.

Your effect size is 'given by nature' - it is the estimate in your sample of the true effect of your experimental exposure contrast in the population. So a better treatment, stronger dose, etc. is the only way to increase your ES.

To paraphrase Jochen, you can increase your signal/noise ratio by decreasing variability in measurement (part of the SD is 'true' variability in the phenomenon, but another part of SD can be due to measurement practices or other experimental protocols that can be tightened to minimize this spurious variability).

Then there is the question of the most appropriate statistical model to test your hypothesis and the commenters above have given some excellent guidance.

First, standardised effect size is the effect of interest scaled by a standardiser (usually the SD or variance of variables in the sample). The sample SD is influence by all sorts of factors that may not be of interest or are artefacts of the study design (reliability, range restriction, small sample bias etc). Second, if you are interested in the unstandardised effect (which is usually more useful) it depends on the dose of the effect. So if I can manipulate that dose I can increase effect size. Indeed one can use range restriction to do this for standardised effects (extreme groups analysis - sampling people extreme on your predictors).

The bad news in the present case is that partial eta square already inflates standardised effect size for many designs because it removes variability from the standardiser. This is potentially useful for power estimation but can make the quantity useless for comparing effect sizes between variables or designs. For instance the total variance explained for partial eta squared can often sum to greater than 100%. This is a clue that the statistic is not in general a useful measure of how big an effect is. Use generalised eta-squared if possible.