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Thread: Determining whether the exponent of an allometric equation = 3

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    Determining whether the exponent of an allometric equation = 3




    I have derived several allometric (exponential) equations in the form Y=bX^a
    These relate body mass of wildebeest (Y) to linear body measurements (X).

    If the shape of a wildebeest didn't change as it grows then the exponent (a) of the allometric equation would be 3 (as Volume is proportional to length^3). This would indicate isometry.

    Animals do change shape and so the difference between the value of a and 3 is interesting.

    How could I show that a is significantly different from 3? So far I have been able to calculate the 95% CIs of (a) and see whether they include 3 but I'm sure there must be a better way.

    I have sample sizes of between 36 and 42.
    Body mass ranges from 15 to 250kg.
    The r^2 of my equations is usually greater than 0.9
    I derived the equations by simple least squares regression on (log)mass against (log) linear measurement and then back transformed them to give power relationships.

    Sometimes I have derived two equations for the same relationship from two different populations (say male and female equations for the relationship between girth and body mass). Is there a formal way to determine whether the two equations, or sets of data are significantly different?

    Thanks for taking the time to read my question and to think about it.

    Gnu

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    Re: Determining whether the exponent of an allometric equation = 3

    Quote Originally Posted by Gnu View Post
    Sometimes I have derived two equations for the same relationship from two different populations (say male and female equations for the relationship between girth and body mass). Is there a formal way to determine whether the two equations, or sets of data are significantly different?
    To help answer this question first, consider combining the two populations together and adding an interaction term. For example, when examining male and female, add a dichotomous variable for gender to your log-transformed model and also include the interaction between gender and log linear measurement. If the interaction is significant, then you could conclude that the association between log mass and log linear measurement differs with respect to gender.

    Quote Originally Posted by Gnu View Post
    How could I show that a is significantly different from 3? So far I have been able to calculate the 95% CIs of (a) and see whether they include 3 but I'm sure there must be a better way.
    If you want a p-value then you will probably have to add a specific test. Most stat software will provide results for testing the null hypothesis that beta ("a" in your case) equal to zero by default. You want to test H0: beta = 3 (or equivalently H0: beta - 3 = 0). You will probably need to specify that in whatever software you are using. Alternately, given that you have a 95% confidence interval, you could always say p<0.05 and leave it at that...if 3 is not contained in the interval.

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    Re: Determining whether the exponent of an allometric equation = 3

    Thank you Mostater for your useful advice.

    I am using SPSS and playing with linear general linear models to try to understand how to add interaction terms. Then I just need to learn which results to ask for and how to interpret them. Wish me luck.

    I'll look at testing H0: beta = 3 as well and see whether SPSS can test that for me.

    Cheers
    Gnu

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    The effect of Season and Sex

    Hello again

    I have been playing with SPSS (Version 22). I have been doing Univariate General Linear models using the 'Analyze' - 'General linear model' - ' Univariate' tabs.

    I am trying to find out if the sex of a wildebeest or the season that an animal is measured in affects the relationship between Body mass and the product of three linear body measurements (Body Length, Shoulder Height and Girth)

    I have plotted the relationship between the log of Body Mass (Mb) and the log of the product of Body length, Girth and Shoulder height (LxGxSh). I like LxGxSh as an independent variable because it contains three different measurements and should be more robust than just using one.

    I have then tried to determine whether sex (male or female) and season (wet, mid and dry) have an effect on the relationship. I did this as suggested by mostater:

    consider combining the two populations together and adding an interaction term. For example, when examining male and female, add a dichotomous variable for gender to your log-transformed model and also include the interaction between gender and log linear measurement. If the interaction is significant, then you could conclude that the association between log mass and log linear measurement differs with respect to gender.
    Well I hope I did what he suggested. I plotted the interactions for the following combinations of indeterminate variables:
    (LxgxSh) alone
    (Sex) alone
    (LxGxSh) and (sex)
    The interaction term (LxGxSh * Sex) alone
    (LxGxSh) and the interaction term (LxGxSh * Sex)
    (sex) and the interaction term (LxGxSh*Sex)

    I then repeated the analysis using Season instead of Sex.

    I have attached the .sav and .spv files converted to .docx and .xlx files

    Unfortunately I am not sure how to interpret them properly, here is what I see, am I looking at it correctly?

    Here is a summary of what I considered important information from the outputs.

    The indeterminate variables used in the regression: R^2, Significance of the variables, the values of B(the slope of the equation).

    LxGxSh alone: R^2 = 0.982 Model Significance is 0.000 B=1.037
    Sex alone: R^2 = 0.004, Adj R^2 = 0.019, Model sig. = 0.675 B=0.043
    Sex and LxGxSh: R^2=0.087, Model Sig.=0.000, B(sex)=0.046, B(LGSh)=1.037
    Sex and LxGxSh and (sex*LxGxSh): R^2= 0.986, Model Sig (Model and LxGxSh)=0.00, Model Sig (sex) = 0.219, Model Sig (Sex*LGSh)=0.470.
    LxGxSh and Sex*LxGxSh: R^2 = 0.986, Model sig (LGSh)=0.00, Model Sig. (sex*LGSh) = 0.001, B(LGSh) = 1.017, B(Sex*LGSh)= 0.074.

    From this I concluded that:

    Interpretation 1

    The variable LxGxSh has a significant effect on the Body mass, Sex has a small but significant effect, the interaction between sex and (LxGxSh) is not significant.

    In the model using Sex, LxGxSh and sex*LxGxSh, Why does the sex term have a p value of 0.219? It looks insignificant here.

    When only LxGxSh and (sex*LxGxSh) are used, then the Sex*LxGxSh term has a very low p value (0.001) making it seem significant. Is that because when I leave out the sex term the (sex*LxGxSh) becomes significant?

    Interpretation 2

    Sex has a significant effect on Body mass
    LxGxSh has a significant effect on Body mass

    Sex does not interact with LxGxSh: meaning that it does not affect the relationship between LxGxSh and Body mass.

    Therefore I can use Length Girth and Shoulder height to estimate wildebeest body mass. If I know the sex of the animal that will help to estimate body mass more accurately but the sex of the animal does not actually affect how body mass scales with these three variables.

    Which interpretation is correct if any?
    Do I need to use a different equation to estimate the mass of male and female wildebeest?

    Thanks
    Gnu
    Attached Files
    Last edited by Gnu; 03-20-2015 at 11:58 AM.

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    Re: Determining whether the exponent of an allometric equation = 3

    Hi Gnu,

    I did not see the model in the attached Word document that contained LxGxSh, Sex, and LxGxSh*Sex interaction. Maybe I just missed it. But, based on what you wrote, it looks like there is not a significant interaction (p=0.470). In that case you would conclude that the association between LxGxSh and body mass does not differ between genders. I believe this should answer your original question. (Note: this does not mean that gender does not help explain variation in body mass. There very well may be a significant gender main effect happening but it is hard to determine from the info you provided.)

    It looks like something similar is going on with season. No significant interaction. So season does not influence the association between LxGxSh and body mass. (Note: in the one model that contains LxGxSh and season as independent variables and no interaction, season is signficant with p=0.004. This is similar to the point I was trying to make regarding gender. In this case, the impact of LxGxSh on body mass may not depend on season, but season does help explain variation in body mass.)

    Hope this helps and doesn't add more confusion.

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    Re: Determining whether the exponent of an allometric equation = 3

    Hi again Mostater

    Thank you for having another look and reading what was probably a confusing question. The "LxGxSh, Sex, LxGxSh*Sex" model is in the word doc, I have attached another one with only that model.

    In that case you would conclude that the association between LxGxSh and body mass does not differ between genders
    So just to be clear: An interaction term like LxGxSh*Sex in a multiple linear regression model will describe the effect that one variable has on the relationship between the other variable and the Dependant Variable (Here Body Mass). Is that correct?

    I can't find any useful introductions to using multiple linear regression that actually explain the interaction term in a way that I understand. It is beyond the scope of my basic statistics text book and everything online goes over my head. Can anyone refer me to a good reference that I could understand.

    Thanks for all your help

    Gnu

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    Re: Determining whether the exponent of an allometric equation = 3


    Quote Originally Posted by Gnu View Post
    So just to be clear: An interaction term like LxGxSh*Sex in a multiple linear regression model will describe the effect that one variable has on the relationship between the other variable and the Dependant Variable (Here Body Mass). Is that correct?
    Correct.

    Quote Originally Posted by Gnu View Post
    I can't find any useful introductions to using multiple linear regression that actually explain the interaction term in a way that I understand. It is beyond the scope of my basic statistics text book and everything online goes over my head. Can anyone refer me to a good reference that I could understand.
    Most books that discuss linear regression will probably delve into interactions to some degree. Here are a couple references:
    "Regression Modelling Strategies" by Harrell & "Linear Models" by Searle. I think Searle does a better job at explaining interactions but Harrell's book is very useful all-around when it comes to developing models. Here is a link to a website that goes into detail and contains further references... http://quantpsy.org/interact/interactions.htm

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