Interpreting coefficients: standardized variables, converted from log-odds scale

Disclaimer: I’m a beginner, little formal training in statistics.

I’ve run the model quoted in part here, and now have a couple basic questions on the interpretation of the coefficients. The model output for the four key parameters is below:

                 mean se_mean   sd  2.5%   25%   50%   75% 97.5% n_eff Rhat
base_survival[2] -1.54    0.01 0.22 -1.94 -1.69 -1.54 -1.40 -1.12   730    1
beta_release[1]  -0.08    0.00 0.08 -0.23 -0.13 -0.07 -0.03  0.07  1106    1
beta_release[2]  -0.22    0.00 0.05 -0.32 -0.25 -0.22 -0.18 -0.11   765    1
beta_release[3]   0.33    0.00 0.08  0.17  0.28  0.33  0.39  0.49  1283    1

Where base_survival[2] is the intercept for probability of survival, beta_release[1] is standardized pumping rate (in CFS), beta_release[2] is temperature in C, centered at the mean temp (15.7 degrees), and beta_release[3] is standardized turbidity. I have two questions so far about the interpretation of these coefficients on the probability scale:

  1. Before I convert from log-odds to probability by taking the inverse logit (called below as the logistic() function from the rethinking package) of the coefficient value, can the sign of the coefficient tell me anything intuitively? Especially since the sign often switches from negative to positive and vice-versa after conversion, I haven’t been able to wrap my head around what a negative or positive relationship with survival on the scale of log-odds really means.

  2. For the standardized variables, does the relationship scale in both directions of the standard deviation? For example, for pumping rate, the resulting change in survival on the probability scale would be:

logistic(-1.54) - logistic(-1.54 - 0.08) = 0.177 - 0.165 = 0.012

… for each increase in 1 standard deviation of pumping rate. Does this mean that if pumping rate were to decrease by two standard deviations, the change in survival would then be:

logistic(-1.54) - logistic(-1.54 + (-0.08*-2)) = 0.177 - 0.201 = -0.024


Thanks so much @betanalpha for encouraging me to post what I fear might be very basic questions here. Any help is much appreciated.

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Hey, my PhD lab was this one: so I’m going to assume this is actually the setting I think it is, where people have asked you how altering release schedules from a dam kills or don’t kill things (or similar). In that case, and in most other similar settings, the pumping rate, temperature, and turbidity are going to co-vary. On their own your coefficients are likely biologically meaningless, particularly if they were derived from data where these co-variates do co-vary. My suggestion is to come up with a set of plausible scenarios where each scenario consists of a plausible combination of pumping rate, temperature, and turbidity. For each scenario calculate the survival over some relevant chunk of time (per sample) and then summarize the samples (I like using quantiles). That gives you a broad picture. Then for each scenario re-do the calculation with small changes in covariates (say half a degree higher temp and half a degree lower). That gives you some idea (that people really want) of what the effect of each covariate is locally (around the scenario) and that might make sense without considering covariance because small independent changes can occur in otherwise correlated biological covariates. This can even make sense in the usual applied-stats “effect of a change in co-variate” way if you think of it on a small enough interval that the change is linear.

I try not to think specifically about what a parameter means but maybe somebody else is willing to think through that for this model. I think it’s less relevant to your audience even if they think they want it.

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An important property of the logistic function is that it is monotonic so that if the logit_probability increases then the probability will increase. Consequently the signs of the slopes indicate whether the corresponding covariate increases or decreases survival probability. For example, beta_release[2] is far enough away from zero that we can safely call it negative, which implies that if you increase temperature while holding everything else constant then logit_probability and hence probability will decrease. This is consistent with your logic in (2).

Of course, if the covariates all covary then this “hold everything else constant” interpretation may not have much applicability in practice!

Note that the sign indicates only in what direction things will change but not by how much they will change. The nonlinearity of the logistic function makes that trickier to interpret. In particular, the baseline determines the overall magnitude of each contribution so you can’t just take the logistic of individual terms and try to interpret them independently.

Thanks, that makes sense! Re: biological applicability, this is definitely not the final model, I’m just trying to understand the interpretation of coefficients in a classical sense. I appreciate @sakrejda’s suggestions for making it a more biologically relevant model.

@betanalpha, one more question - what would the back-transformation to the probability scale be if one of the parameters (say temperature) had been log-transformed instead of centered?

I’m not talking about changing the model at all. This is an issue with what sort of output to produce from any given model. Ecological data typically violates the"hold everything else constant" assumption so interpreting coefficients individually makes no sense.

@sakrejda can you point me to an example (a publication or online source, maybe) of the type of analysis you described? I understand the reasoning, but It would be helpful to see how it’s been applied in other systems.

You want to be careful and try to not think too much about transforming back and forth because the transformations are nonlinear which frustrates common intuition. Because log is monotonic, however, then what you can do is follow the same logic above when discussing directions of effects. If the slope is negative then then

  • Delta T -> + Delta log(T) -> - Delta logit_prob -> - Delta p.

But how the magnitude of Delta T transforms into the magnitude of Delta p is much harder to interpret.

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