Regression Bounded Between -1 and 1

I am performing a meta-analysis on the response of rodent abundance to clear-cut logging. I have data from multiple sites, across multiple years, and for different species of rodents, and am using these data to compare the abundance of rodents in clear-cuts to the abundance of rodents in un-logged nearby 'control' sites.

The predictor variable is the size of the clearcut (e.g, 1ha, 2ha, etc.). The response variable is calculated as the proportional difference in abundance, between the treatment (i.e., clear-cut) site, $N_T$, and the control site, $N_C$, as follows: $$R = \frac{(N_T-N_C)}{(N_T+N_C)}$$ This response variable is a common choice in meta-analyses, and results in a variable that is bounded between -1 and 1.

I am wondering if there is a regression model that can model this relationship, something similar to a logistic regression model. A simple linear regression could work, but I'm wondering if there are other choices. If possible, I would prefer to use a model that does not require me to transform the data.

• Since transformations are straightforward and powerful methods, could you explain why you prefer not to apply a transformation? This seems like an artificial restriction that can only hamper your ability to perform the analysis. – whuber Dec 29 '16 at 21:04
• Mainly for consistency with previously-published meta-analyses in the same field that don't use transformations. That being said, if the only option is to use a transformation, I will go with that. I'm just curious if there's a non-transformation option. – ZombiePlan37 Dec 29 '16 at 21:07

You can always use beta regression (Ferrari and Cribari-Neto, 2004). It's a model for response variable bounded in $(0, 1)$, but you can easily transform your variable by taking $\frac{Y+1}{2}$ (I know you said you do not want to transform, but it's a really basic transformation).

Moreover, such model still makes perfect sense since what you are estimating is mean of non-standard beta distribution parametrized by mean $\mu_i$ and precision $\phi$. The standard beta regression model can be used for variable with any $(a,b)$ bounds by using above transformation

$$g(\mu_i) = x_i^{T}\beta \\ \tfrac{y_i-a}{b-a} \sim \mathcal{B}(\mu_i, \phi)$$

where $g$ is a link function (e.g. logistic function). Such model is equivalent to

$$h(\mu_i) = x_i^{T}\beta \\ y_i \sim \mathcal{B}_{a,b}(\mu_i, \phi)$$

where $\mathcal{B}_{a,b}$ is is beta distribution bounded in $(a,b)$ and $h$ is a link function used for mapping from given range (e.g. logistic, or hyperbolic tangent functions together re-scaling and shifting if needed).

Ferrari, S., & Cribari-Neto, F. (2004). Beta regression for modelling rates and proportions. Journal of Applied Statistics, 31(7), 799-815.

• @berg2294 logistic regression can be used for binary data or proportions of successes in samples of known size, beta regression is used for continous bounded data -- those are different problems. – Tim Dec 29 '16 at 21:05
• See this Cross Validated page for ways to proceed if the response variables include values of 0 or 1. See this page if a mixed model including random effects is needed. – EdM Dec 29 '16 at 21:09
• An important note about using beta regression is that I believe it is required that $y_i \in (a, b)$ (where $a$ and $b$ are the bounds, in this case -1, 1), rather than $y_i \in [a,b]$. From the OP's question, it's not quite clear if that's the case? I would think $N_T$ is discrete and could easily be 0. – Cliff AB Dec 29 '16 at 23:41
• The main problem I see with applying Beta regression is that if $N_T+N_C$ varies considerably over the data, then the model will need to allow the precision $\phi$ to vary, too. After all, low $N_T+N_C$ means poor precision in the responses and high $N_T+N_C$ means good precision. The ability to capture such heteroscedasticity directly by means of transforming the responses back to the original, raw, $(N_C,N_T)$ pairs is one advantage of returning to the original data. Another advantage is it will allow modeling of multiple $N_T$ values associated with any given $N_C$ (control observation). – whuber Aug 29 '17 at 15:25
• @whuber what would you suggest as an alternative? – Tim Aug 29 '17 at 17:17