The zero effect size means that there is no effect by

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not. The zero effect size means that there is no effect by the factor. With the effect size and the range of the confidence interval, The confidence interval does depend on the sample size. So by simply increasing the sample size, you can narrow the confidence interval, and will be able to claim that the effect exists at some point (again this might require you to recruit hundreds or thousands of participants though). The effect size itself is independent of the sample size, and increasing the sample size does not contribute to improving the size of the effect. Thus, looking at both the effect size and its confidence interval is important to correctly interpret the result [Houle TT].
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Size of the p-value is related to the effect size or the observed association or difference. A very large study may result in very small p-value based on a small difference of effect that may not be important when translated into clinical practice. Therefore, important to look at the effect size and confidence intervals [Bland M]. References: 1. Rothman KJ. A show of confidence. N Engl J Med. 1978; 299:1362–3. 2. Ware JH, Delgado F, Donnelly C, Ingelfinger JA. P values. 3. Rothman KJ. Modern Epidemiology. Boston 4. Houle TT. Importance of effect sizes for the accumulation of knowledge. Anesthesiology. 5. Bland M, Peacock J. Interpreting statistics with confidence.
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