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    Contribution of multiple correlated effects on global observation




    Hello,

    In a perfect system, I have 0% of signal loss. I found that there are N effects that can contribute to signal loss. I activate each effect individually to assess its contribution to signal loss. Let's say the effect Ei contributes to loss by xi, and so on for all N effects. Then, I activate both effects Ei and Ej, and we found a contribution to signal loss equals to xij, not necessary equals to xi + xj, because there might be some correlation between the two effects. When I activate all N effects simultaneously, I observed a global signal loss xij...n. I am able to find the contribution of any combination of effects. However, I have trouble expressing this mathematically in an efficient way.

    Is is possible to express the contribution of each effect in order to get the non linear relation between the global signal loss and all effects xij...n = fct(xi,xj,...,xn)?

    Thank you!

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    Re: Contribution of multiple correlated effects on global observation

    Hopefully your N effects number is not too big! This seems ripe for using mediation analyses and visualizing in within structural equation modeling.


    You can construct a path analysis that shows the effect each has on the signal loss and also incorporation of the correlation. This visualization will help you write out the process in mathematic terms, where you may state the total effects, natural direct effects, and natural indirect effects. Unsure how the covariances may get incorporated (not experience with that), but that can be seen if you draw out the illustration.
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    Re: Contribution of multiple correlated effects on global observation

    Wouldn't this be a straightforward ANOVA, followed by calculation of epsilon^2 for the % contribution?

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    Re: Contribution of multiple correlated effects on global observation

    Good point Miner. Depends on if you can us ANOVA (given assumptions and their data) and if they want to report covariances as well (which is probably doable).
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    Re: Contribution of multiple correlated effects on global observation

    Thank you for your answers. I'm just a little familiar with ANOVA, so to add to the question I first asked, let's say I have N = 4 effects for simplification (A, B, C and D). I run a simulation with only effect A activated and get a signal loss of xA. I run a simulation with only effect B activated and get a signal loss of xB, and so on with any combinations (A and B; B, C and D; A and D; ...). When I activate all 4 effects, I get a global signal loss, so yes, I would like to know the % of contribution for each effect, and maybe at the same time the correlation between the different effects. Before getting into it, does an ANOVA test really is adequate for this?

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    Re: Contribution of multiple correlated effects on global observation

    In ANOVA you can get partial eta-square values, which I believe are how much of the variability in the loss of signal is explained by the variable controlling for the other variables.


    Correct me if I am wrong on this Miner.
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    Re: Contribution of multiple correlated effects on global observation

    From Cab's description, this sounds like a 2^4 full factorial design, which is ideally suited for a 4-way ANOVA analysis, including all desired interactions. Epsilon ^2 (or eta^2) may easily be calculated using the sum of squares, df, and mse information in the ANOVA table. Epsilon^2 is the unbiased estimate of the % contribution of each factor and interaction in the model.

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    Re: Contribution of multiple correlated effects on global observation

    Thanks again. My final design was a 3-factors each with 3 levels (3^3 full factorial design). Therefore, I ran 100 simulations for each combination (100 x 27 simulations). Performing a 3-way ANOVA with this factorial design told me that all main effects, two-way as well as three-way interactions were highly significant. However, I suspect that I can't trust these results since my simulations sometimes yielded (very) non-normal distributions. In that case, is Friedman test adapted to give me similar interpretations of effects and interactions? Or any other tests?

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    Re: Contribution of multiple correlated effects on global observation

    The normality assumption is in respect to the Model's error term. Is that what you are referring to? You may be able to pull that term out of the models and create Q-Q plots. A hundred seems doable.


    Here is a link to looking at model convergence in logistic regression. It may give you an idea of what you need to do in your model:


    http://blogs.sas.com/content/iml/201...gence-sim.html

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    Re: Contribution of multiple correlated effects on global observation

    In addition, keep in mind the difference between statistical significance and practical significance (effect size). With 100 simulations for each combinations, you could see statistically significant effects that are very small in size.

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    Re: Contribution of multiple correlated effects on global observation

    I came back to this study recently, did some plots but I am still confused.

    You can see the Q-Q plot, box plot and histogram of my data. For the last graph, I first used the lm function of R : result <- lm(signalLoss ~ factor1 * factor2 * factor3), then plot the residuals with the predict function. An Anova test gives highly significant effects for all factors and interactions.

    For particular configurations in the factorial design, the signal loss distribution (I ran 1000 simulations for each configuration) can be very skewed, but I expected that. For some configurations, the distribution is normal. From these plots, I'm not sure what I can do next.
    Attached Images

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    Re: Contribution of multiple correlated effects on global observation

    Are these plots of the raw data or of the residuals?

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    Re: Contribution of multiple correlated effects on global observation

    My apologies, you're right these were from the raw data. Here are plots of the residuals (using the linear model in R described in previous post).
    Attached Images

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    Re: Contribution of multiple correlated effects on global observation


    The residuals appear problematic. You have some normality and heteroskedastic issues present. Is there a potential for non-linear relationships?

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