# How do I interpret and report the output of a logistic regression in R on data that are all binary (0,1)?

My minimum adequate model is shown below. My independent variables (e.g. One, Five) represent habitat categories for which species have either been designated to (i.e. the assessment found that they occur there) or not (they are considered not to occur there). My dependent variable groups species as climate change threatened or not. So all variables, independent and dependent, are binary (0,1).

My hypothesis question is as follows:

'Do species for which climate change is a known threat show similarities in terms of habitat associations and is one or more predominant habitat evident compared with species that are not threatened by climate change?'

Specific questions I would like to be answered include:

1. Would calculating an odds ratio be of use to me (considering all my data are binary)?

2. What sort of statements do the output I have (below)/or recommended additional statistics (e.g. odds ratios) allow me to make?

Thanks!

Output:

Deviance Residuals:  Min      1Q    Median   3Q     Max
-0.6900 -0.3589 -0.2550 -0.1931 2.8634

Coefficients:  Estimate Std.Error  z value  Pr(>|z|)
(Intercept)    -2.18672  0.09957  -21.961   < 2e-16 ***
One            -1.23943  0.13614   -9.104   < 2e-16 ***
Five            0.75144  0.17228    4.362  1.29e-05 ***
Seven          -0.66517  0.34117   -1.950   0.05122 .
Fourteen       -0.55602  0.17454   -3.186   0.00144 **

Signif. codes: 0 ‘’ 0.001 ‘’ 0.01 ‘’ 0.05 ‘.’ 0.1 ‘ ’ 1
(Dispersion parameter for binomial family taken to be 1).

• Are the habitat categories mutually exclusive? – Marquis de Carabas Jul 4 '15 at 18:53

You would interpret the regression results as you would any logistic regression model with dummy variables. That is, odds ratios are interpreted as the odds of the category in question relative to the reference category. That said, your model effectively only has one explanatory variable--habitat--so what you show us here is just the binary association between your outcome (threatened/not threatened) and habitat type, i.e. crude ORs. Given that it is likely that other factors affect the species' "threatened status," you need to control for those other factors in your model. Otherwise, the crude ORs are a biased estimate of the effect of habitat type on threatened status.

I do not see a problem with having a model with all binary predictors if the data call for it, though you may run into problems if you have small cell sizes. Some variables are necessarily categorical, e.g. habitat type. Sometimes you may need to categorize continuous variables if the relationship between the continuous variable and the outcome is not linear.

When you fit a model limited dependent variables, such as logit, multinomial logit, ordered logit, probit, tobit, Poisson, etc., the coefficients are not directly interpretable. Therefore, you will need to calculate some transformation of the $\beta$, either odds ratios (as you have done here), marginal effects, predictive margins, etc.

Note also that you can draw conclusions about different habitat types associated with higher or lower likelihood of having threatened status relative to the omitted category, but if you want to know whether habitat type overall is associated with threatened status, you will need to do an F-test of the joint significance of your dummy variables. In a model with just one explanatory variable (as you have here), the model F-test is equivalent to the F-test I just described in the previous sentence.

Finally, you should enter all categories of habitat but one into your model. The omitted category is your reference category, and serves as the group that other habitat categories are compared to with regard to risk of threatened status. If you enter only some categories and not others, the omitted categories will all be lumped together in the reference category.

• Thanks for your comment! I'm not sure I need to control for other factors that may contribute to a "CC threatened" status, however, as I am merely trying to determine which habitats are important for CC threatened species and which for non-CC threatened species. I am unclear whether or not you recommend calculating and reporting odds ratios instead of beta coefficients, and why? Thanks – Hnickolai Jul 4 '15 at 19:13
• Just seen your first question. In answer, yes the habitats are mutually exclusive. – Hnickolai Jul 4 '15 at 19:51
• If there are other factors besides habitat that affect CC status that are related to habitat, failing to control for those factors would make you overstate or understate the association between habitat and CC status. We call this Omitted Variable Bias. Sometimes the significance of a binary association goes away after you control for other factors. What you are doing is fine for exploratory analysis (deciding what to include in a full model), but if you were to write a scientific paper for publication, the binary association may not be enough. – Marquis de Carabas Jul 4 '15 at 20:16
• I have added a couple of sentences about interpreting parameter estimates of limited dependent variable models and dummy variables. – Marquis de Carabas Jul 4 '15 at 20:21
• I feel I haven't described to you well enough the question I wish to answer. My research is based on the IUCN Red List's data for which I have separated species based on whether or not the assessment considers them CC threatened or not. Rather than attempting to comprehensively analyse a range of variables on their importance for determining whether or not species are threatened by CC, I am exploring the importance of different habitats for CC and non-CC threatened species. I will then be discussing the potential reasons for my findings e.g. assessor bias, habitat associations with CC etc. – Hnickolai Jul 4 '15 at 20:45

A useful tool for interpreting the results of a logit is the Zelig package, http://zeligproject.org/, by Gary King et al. It has tools similar to the popular clarify package they made for stata.