I have constructed a zero-inflated beta regression model in gamlss. I find the output of this model somewhat confusing to interpret and was hoping that someone may be willing to correct my interpretation, fill in the blanks or correct me where I am wrong?
My model describes the impact of several environmental and climatic variables on variation in the maximum abundance count for rabbits. A value of 0 for prop.abun.max represents the maximum abundance count for a particular three-day monitoring period and a value of 1 (re-coded to 0.9999) for prop.abun.max represents an abundance count of 0. Values of prop.abun.max >0 and <1 represent an abundance count that was less than the maximum count for that monitoring period and greater than 0. For ever three day monitoring period I have one maximum abundance count, represented as 0, and two abundance counts that are less than the maximum, represented as their variation away from the maximum, i.e >0 but <=1.
As I have so many 0's in my data and the values of prop.abun.max vary from 0-1, I have implemented a zero-inflated beta regression. My model is below.
Family: c("BEZI", "Zero Inflated Beta")
Call: gamlss(formula = prop.abun.max ~ season + time + temp + last.rain.bom + rain + wind + cloud + re(random = ~1 |
site), sigma.formula = ~1, nu.formula = ~season + time + temp + last.rain.bom + rain + wind + cloud +
re(random = ~1 | site), family = BEZI, data = na.omit(subset3))
Fitting method: RS()
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Mu link function: logit
Mu Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) -0.6400525 0.3356874 -1.907 0.057181 .
seasonAutumn 0.0836078 0.1546704 0.541 0.589075
seasonWinter -0.2218074 0.1958630 -1.132 0.258029
seasonSpring -0.0319850 0.1473012 -0.217 0.828195
time 0.0581566 0.0621388 0.936 0.349807
temp -0.0129982 0.0104319 -1.246 0.213392
last.rain.bom 0.0044248 0.0119330 0.371 0.710954
rain -0.0168664 0.0248624 -0.678 0.497864
windlight 0.0490963 0.1109939 0.442 0.658455
windmedium 0.1446764 0.1686862 0.858 0.391522
windstrong 0.6715118 0.1935037 3.470 0.000569 ***
cloud -0.0008299 0.0013417 -0.619 0.536512
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
------------------------------------------------------------------
Sigma link function: log
Sigma Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 1.55686 0.07311 21.3 <2e-16 ***
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
------------------------------------------------------------------
Nu link function: logit
Nu Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 0.215386 0.635802 0.339 0.7349
seasonAutumn -0.195175 0.312707 -0.624 0.5328
seasonWinter -0.233812 0.384758 -0.608 0.5437
seasonSpring -0.092157 0.291694 -0.316 0.7522
time -0.260762 0.117130 -2.226 0.0265 *
temp -0.012965 0.020335 -0.638 0.5241
last.rain.bom 0.011202 0.023347 0.480 0.6316
rain 0.050998 0.035835 1.423 0.1554
windlight -0.082610 0.213268 -0.387 0.6987
windmedium -0.035651 0.323898 -0.110 0.9124
windstrong 0.020968 0.409759 0.051 0.9592
cloud 0.001351 0.002611 0.518 0.6050
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
------------------------------------------------------------------
NOTE: Additive smoothing terms exist in the formulas:
i) Std. Error for smoothers are for the linear effect only.
ii) Std. Error for the linear terms maybe are not accurate.
------------------------------------------------------------------
No. of observations in the fit: 503
Degrees of Freedom for the fit: 41.62353
Residual Deg. of Freedom: 461.3765
at cycle: 7
Global Deviance: 367.5272
AIC: 450.7743
SBC: 626.4501
From this post I understand that the zero-inflated beta model essentially has three components, one (Mu) that describes the mean of the beta distribution for the interval (0,1), a second (Sigma) that represents the precision (or shape) of the beta distribution, and a third (Nu) that represents the Bernoulli distribution part of the model and the probability of observing a zero value.
Now how to interpret the model output? Both the Mu and Nu components of the model have a logit link, so if we exponentiate these coefficients we get odds ratios. If we exponentiate the significant coefficient in the Nu component of the model, time = -0.012965, we get an odds ratio of 0.77. This suggests that for ever unit increase in time there is a 30% lower likelihood of a zero (maximum abundance in this case).
I now get lost when interpreting the Mu component of the model. For example, in the Mu component of the model, the coefficient of time is 0.0581566, which exponentiates to an odds ratio of 1.06. What is the interpretation of this given that the response is not discrete 0/1, but rather is continuous between the bounds of 0 and 1? Is the interpretation still in reference to the odds/likelihood of a 0/1? Does this mean for every unit increase in time there is a 6% higher likelihood of a 1? How would the significant coefficient for windstrong = 0.6715118 be interpreted, I assume it is interpreted relative to the reference category for wind which is no wind?
