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Thread: Beta Regression Coefficients

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    Beta Regression Coefficients




    In a simple model, x is a continuous (normally distributed) variable predicting y. Since y values are proportions ranging from 0 to 1 (0%-100%), simple linear regression may give out-of-bounds estimates for some predicted values (i.e., lower than 1 or higher than 1).

    Therefore, I have decided to use beta regression with boundaries from 0 to 1 (i used betareg() command in betareg R package; the software is however not important). While it is easy to interpret the unstandardized regression parameter from a linear model (see below linear model output: B = 0.126 indicating an increase by 12.6% of y if x rises by 1), I am not sure how to understand, transform, or use the parameters from betareg model to get a meaningful interpretation of the coef (see below - Beta regression output).


    Output for linear regression model: lmMod = lm(formula = y ~ x)
    Code: 
    Coefficients:
                Estimate Std. Error t value Pr(>|t|)    
    (Intercept) -0.57936    0.10849  -5.340 9.57e-07 ***
    x        0.12591    0.01354   9.296 4.07e-14 ***
    Output for beta regression model:betaMod = betareg(formula = y ~ x)
    Code: 
    Coefficients (mean model with logit link):
                Estimate Std. Error z value Pr(>|z|)     
    (Intercept) -4.85712    0.52580  -9.238   <2e-16 *** 
    x            0.56796    0.06498   8.740   <2e-16 ***
    
    Phi coefficients (precision model with identity link):
    Estimate Std. Error z value Pr(>|z|)     
    (phi)    7.686      1.184   6.491 8.54e-11 ***
    How can I interpret the parameter 0.567 in the beta regression output (together with the intercept)? Is there a way how to use 0.567 and get the increase of the absolute value in y (i.e., if x increases by 1, y increases by XX, since y is in %, the interpretation is easy).
    Thank you! M.
    Last edited by Martin Marko; 03-05-2017 at 09:34 AM.
    MM

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    Re: Beta Regression Coefficients

    I had a similar pursuit about 6 months ago. I believe I came across a SAS tech paper from a sas user group that gave a good description. Sorry I am not at my computer right now. Though I will see if I can find it.
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    Re: Beta Regression Coefficients

    It might have been Paper: 335:2011. Looks like they take on a logistic style interpretation.
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    Re: Beta Regression Coefficients

    Thank you a lot for helping,
    logistic interpretation means B1 is log odds, right? So I can use exp(coefficientB1_value) to get "odds" ( = 1.792) which I don't understand at all.

    Perhaps another way to go: I am considering to use the abovementioned simple linear regression and then define the "meaningful" range of its application (like, use linear regression equation to compute the value of x that would predict prob of y = 0 and then estimate upper-bound meaningful value of x that would predict y = 1). Does this make any sense? Not sure, but i really need to know an increase of X changes the value of Y (in %).

    BTW, the relationship Y~X can be seen as linear:


    Thank you,
    MM

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    Re: Beta Regression Coefficients

    The logit model:

    log(p/(1-p) = beta*x

    can be solved to:

    p = exp(beta*x)/(1+exp(beta*x))

    or

    p = 1/(1 + exp(-(beta*x)))

    It gives these numbers:
    Code: 
    # the linear regression model parameter estimates
    a <-   -0.57936 
    b <-    0.12591 
    
    a + b*8
    # [1] 0.42792
    #seems reasonable
    
    a + b*9
    # [1] 0.55383 
    
    
    # the beta-regression model with logit link: 
    alpha <-  -4.85712 
    beta  <-   0.56796 
    
    # log(p/1-p) = xbeta gives
    
    # p =  1/(1-exp(-(alpha + beta*x))) 
    
    p0 =  1/(1+exp(-(alpha + beta*8))) 
    p0
    # [1] 0.4222753
    
    p1 =  1/(1+exp(-(alpha + beta*9))) 
    p1
    # [1] 0.5632887
    
    p1 - p0 
    # [1] 0.1410134   changing from x=8 to x=9 
    
    # compare with the above linear model
    0.55383  -  0.42792
    # 0.12591 
    
    # they are two different models so they don't give exactly the same result
    # but similar results
    But if your original data were 0/1 success/failure then maybe it would be more natural to do the usual logit.

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    Re: Beta Regression Coefficients

    Many thanks for the transformation,
    it was much helpful,

    Best regards,
    MM

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    Re: Beta Regression Coefficients

    Can you post a histogram of your dependent variable values? Linear reg is acceptable given the bulk of values land near 0.5 with minimum dispersion.
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    Re: Beta Regression Coefficients


    Quote Originally Posted by hlsmith View Post
    Can you post a histogram of your dependent variable values? Linear reg is acceptable given the bulk of values land near 0.5 with minimum dispersion.
    Sure,
    just to mention that each data point represents a difficulty parameter of a test item which was estimated on ~200 individuals measure.
    The issue of the linear/beta regression was to model of how theoretical complexity of an item (given by construction) relates to its empirical difficulty.

    M

    Last edited by Martin Marko; 03-06-2017 at 01:38 PM.
    MM

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