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Background: I'm analyzing correlation between two behavioural types (boldness and aggression). Boldness values are continuous (range: 2 to 1195) and it's unit of measurement is in seconds (latencies). Aggression are ordinal ranks (ranked from 1 to 6, where 1<2<3 and so on..). Higher boldness scores mean the individual is less bold because it takes longer to resume activity. Higher aggression score means the individual is more aggressive. First, I ran a bayesian bivariate regression (MCMCglmm) to partition the variances into among- and within- individual differences, and then checked for covariance between boldness and aggression (the two response variables). I got a significant positive correlation which actually means that boldness and aggression are negatively correlated because increasing boldness values are deemed less bold.

Next, to make the results more intuitive, I reversed the boldness scores so that higher boldness scores indicate higher boldness. Since the boldness range is from 2 to 1195, I subtracted each boldness value from 1197 so that the new variable (lets call it Rboldness) also has a range from 2 to 1195 but in the reverse direction. The variances between boldness and Rboldness is the same, but the mean has changed. The distribution also changed from a right skew (boldness) to left skew (Rboldness) after log transformation.

The Question: The problem is that the strength of the correlation reduces from 0.5 to 0.3 after reversing the boldness scores (Rboldness). Being a purist, I'm not very comfortable in using ordinal ranks as a response variable, so I also analysed the data by collapsing the aggression ranks (1 to 6) into two meaningful and relevant categories (ordinal scores 1-4 coded as 0 and scores 5&6 coded as 1) and ran a binomial bivariate mixed model. The binomial model resulted in a significant correlation coefficient (Figure 1) before reversing boldness scores (r = 0.6) but the correlation becomes insignificant (Figure 2) after reversing the boldness values (Rboldness). Can anyone please inform why the correlation becomes insignificant after reversing one continuous variable? Many thanks in advance! enter image description here

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  • $\begingroup$ It looks like you have a nonlinear association and non constant variance. Correlation only tells you about a linear relationship so it's to be expected that this will happen. $\endgroup$ – Robert Long Jun 1 at 9:39
  • $\begingroup$ Thanks @RobertLong. I'm not quite sure what you mean by "non constant variance"? The variances for boldness scores and the reversed boldness scores (Rboldness) is the same. Do you mean heteroskedasticity and, if so, any tips on how to deal with it? $\endgroup$ – Bharat Jun 1 at 9:50
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    $\begingroup$ Yes, if you draw imginary vertical lines you can see the spread of the data differs a lot accross the range. I would look at transforming the variables (perhaps with a log transform) or maybe using a linear regression model with transformed variables and/or nonlinear terms. $\endgroup$ – Robert Long Jun 1 at 9:54
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    $\begingroup$ If you were originally using $\log(\text{Boldness})$ then you might prefer to move to $\log\left(\frac{2 \times 1195}{\text{Boldness}}\right)$, i.e. setting Rboldness as $\frac{2 \times 1195}{\text{Boldness}}$, to try to preserve your linear caclulations $\endgroup$ – Henry Jun 1 at 10:15
  • $\begingroup$ Thank You for the pointer Robert Long. @Henry, your suggestion has solved the problem! Boldness and Rboldness now give the exact same correlation coefficient but with different signs! Thank you very much! $\endgroup$ – Bharat Jun 1 at 15:21
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Following suggestions from @Robert Long and @Henry (see the comments above), the issue is now solved. Apparently, reversing Boldness scores as I mentioned in my original question also reversed it's skew and so log transformation didn't help for Rboldness because it was negatively skewed. Therefore the model assumptions of linearity and heteroskedasticity were violated and so the non-significant correlation. However, reversing boldness scores by division as suggested by Henry made Rboldness positively skewed, and log transformations helped in reducing the skewness of the data. Subsequently, the model's assumptions were not violated, and the correlation coefficient was similar when analysed with the model consisting the un-reversed Boldness scores, but opposite in sign.

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