# Logistic regression goodness of fit

I have fitted a logistic regression model showing the probability of flight given distance to a disturbance event. My sample size is 140 observations of which 45 were observed to fly. Distance to the disturbance is the only significant predictor among many explored (biologically relevant interactions as well). Here is the out from the R console

Call:
glm(formula = flight ~ approach_km, family = binomial)

Deviance Residuals:
Min       1Q   Median       3Q      Max
-1.1233  -0.9317  -0.7460   1.3538   1.9400

Coefficients:
Estimate Std. Error z value Pr(>|z|)
(Intercept)  -0.1233     0.3438  -0.359   0.7199
approach_km  -0.4530     0.2225  -2.036   0.0418 *

Null deviance: 175.82  on 139  degrees of freedom
Residual deviance: 171.30  on 138  degrees of freedom
AIC: 175.3

Number of Fisher Scoring iterations: 4


Here is a plot of the predicted probabilities of flight. As you can see there is no point along the distance continuum in which the probability of flight exceeds 0.5. Since this is the case the classification tables will never predict a flight (it predicts 67.9% correctly, ALL THE NO FLIGHT EVENTS!). McFadden's psuedo R2 = 0.026. I am happy with the fit of the model but I am under the belief that I need to include these types of goodness of fit statistics when publishing. I would rather show that within a certain range of the data, say 1 km, there is roughly a 0.40 probability of flight. If there were 20 observations associated with this estimate there would be 8 flight events and 12 non-flight events. Is there a way to show this? How can I write up these results to show my findings without referee's balking at my lack of fit tests? Any thoughts or references would be greatly appreciated.

UPDATE: In response to @Drew75.

Here is the ROC curve. Does this mean that the model does not perform as well I has thought? • Have you thought about an ROC curve? I'd guess that this model only performs slightly better than a random guess. – Drew75 Feb 26 '14 at 19:00
• @Drew75 I added the plot of the ROC curve as an edit to the original post – marcellt Feb 26 '14 at 23:38
• @marcelit, do you know how to interpret the ROC curve? It supports the previous graph, saying that your model doesn't predict much better than a random guess. – Drew75 Feb 27 '14 at 6:49

## 1 Answer

You should think of when your predicted probability crosses .301 (the share of flight events) not 0.5. If the probability is higher than .301 then it is higher than a random guess provided the distribution of the data, even if it is not as high as 0.5. i.e. you learn something.