Does anyone know how to find correlation between 3 variables?

Due to growing concerns about the replication crisis in the scientific community in recent years, many scientists and statisticians have proposed abandoning the concept of statistical significance and null hypothesis significance testing procedure (NHSTP). For example, the international journal Basic and Applied Social Psychology (BASP) has officially banned the NHSTP (p-values, t-values, and F-values) and confidence intervals since 2015 [1]. Cumming [2] proposed ‘New Statistics’ that mainly includes (1) abandoning the NHSTP, and (2) using the estimation of effect size (ES).

The t-test, especially the two-sample t-test, is the most commonly used NHSTP. Therefore, abandoning the NHSTP means abandoning the two-sample t-test. In my opinion, the two-sample t-test can be misleading; it may not provide a valid solution to practical problems. To understand this, consider a well-posted example that is originally given in a textbook of Roberts [3]. Two manufacturers, denoted by A and B, are suppliers for a component. We are concerned with the lifetime of the component and want to choose the manufacturer that affords the longer lifetime. Manufacturer A supplies 9 units for lifetime test. Manufacturer B supplies 4 units. The test data give the sample means 42 and 50 hours, and the sample standard deviations 7.48 and 6.87 hours, for the units of manufacturer A and B respectively. Roberts [3] discussed this example with a two-tailed t-test and concluded that, at the 90% level, the samples afford no significant evidence in favor of either manufacturer over the other. Jaynes [4] discussed this example with a Bayesian analysis. He argued that our common sense tell us immediately, without any calculation, the test data constitutes fairly substantial (although not overwhelming) evidence in favor of manufacturer B.

For this example, in order to choose between the two manufacturers, what we really care about is (1) how likely the lifetime of manufacturer B’s components (individual units) is greater than the lifetime of manufacturer A’s components? and (2) on average, how much the lifetime of manufacturer B’s components is greater than the lifetime of manufacturer A’s components? However, according to Roberts’ two-sample t-test, the difference between the two manufacturers’ components is labeled as “insignificant”. This label does not answer these two questions. Moreover, the true meaning of the p-value associated with Roberts’ t-test is not clear.

I recently visited this example [5]. I calculated the exceedance probability (EP), i.e. the probability that the lifetime of manufacturer B’s components (individual units) is greater than the lifetime of manufacturer A’s components. The result is EP(XB>XA)=77.8%. In other words, the lifetime of manufacturer B’s components is greater than the lifetime of manufacturer A’s components at an odds of 3.5:1. I also calculated the relative mean effect size (RMES). The result is RMES=17.79%. That is, the mean lifetime of manufacturer B’s components is greater than the mean lifetime of manufacturer A’s component by 17.79%. Based on the values of the EP and RMES, we should have a preference of manufacturer B. In my opinion, the meaning of exceedance probability (EP) is clear without confusion; a person even not trained in statistics can understand it. The exceedance probability (EP) analysis, in conjunction with the relative mean effect size (RMES), provides the valid solution to this example.

[1] Trafimow D and Marks M 2015 Editorial Basic and Applied Social Psychology 37 1-2

[2] Cumming G 2014 The New Statistics Psychological Science 25(1)DOI: 10.1177/0956797613504966

[3] Roberts N A 1964 Mathematical Methods in Reliability Engineering McGraw-Hill Book Co. Inc. New York

[4] Jaynes E T 1976 Confidence intervals vs Bayesian intervals in Foundations of Probability Theory, Statistical Inference and Statistical Theories of Science, eds. Harper and Hooker, Vol. II, 175-257, D. Reidel Publishing Company Dordrecht-Holland

I have already read that the test requires and also that the test does not require equal variances of the samples tested.

For example:

I am using Prism to carry out some two-way ANOVA tests. I have a series of fairly small datasets that I want to analyse the same way. But a few of the datasets do not pass the Shapiro-Wilk normality test. Reading the Prism Guide, it seems like I would be justified in still carrying out an ANOVA, seeing as the Q-Q plots looks like they do not deviate too far a normal distribution.

However I am struggling to find examples in the literature of this being done. I am wondering how I would justify this in a thesis/paper and if the following would be acceptable:

"Before undergoing an ANOVA, the each dataset was tested for Gaussian distribution using the Shapiro-Wilk normality test (ɑ = 0.05). Although not all datasets passed, majority of datasets in the series were normally distributed. Of those that failed, Q-Q was assessed and no major violations were detected. As statistical tests are generally robust to mild violations, and to maintain consistency across datasets, two-way ANOVA was carried out."

Although may manuals and websites state that ANOVA is robust there don't seem to be any peer reviewed references for two- or three-way ANOVA but I can find a couple of references for one-way and RM ANOVA (PMID 36695847 & 29048317). If someone could supply a reference to justify my use that would be great.