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I am using R to run a logistic regression to analyze how a categorical variable ("population") correlates with a binary variable ("response") and am having some trouble interpreting the results (shown below).

Only the p-value for the intercept is significant. As I understand it, the intercept deals with what would happen if all x=0. This seems important since one of my variables in binary. Doesn't it imply that there is a correlation between the variables because we're saying that if all were from the same population we'd be able to make some predictions about the response? But if this is true, how is it possible that the other p-value is not significant.

call: glm(formula = response ~ population, family = binomial, data = sfpa9)

Deviance Residuals: 
    Min       1Q   Median       3Q      Max  
-1.4929  -1.2388   0.8918   0.9482   1.1173  

Coefficients:
                  Estimate Std. Error z value Pr(>|z|)   
(Intercept)         0.7167     0.2662   2.692   0.0071 **
populationSurface  -0.5736     0.3777  -1.519   0.1289   
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

(Dispersion parameter for binomial family taken to be 1)

    Null deviance: 160.68  on 119  degrees of freedom
Residual deviance: 158.35  on 118  degrees of freedom
AIC: 162.35

Number of Fisher Scoring iterations: 4` 
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Each coefficient significance test is shown for the null hypothesis that the value of a coefficient equals 0. For logistic regression, the intercept is the log-odds of your response binary variable when the population variable is at its reference level (which is coded as 0, not named here). A 0 value for the intercept would mean even odds (odds = 1), or probability of 0.5, for response levels when the population variable is at its reference level.

Thus your result for the intercept simply means that the log-odds of response are significantly different from 0 (or from even odds) when the population variable is at its reference level. It says nothing directly about how differences in values of the population variable are associated with differences in the response variable.

The result for the Surface level coefficient of the population variable suggests that these two levels of population do not differ significantly in log-odds of response.

For future reference, with one categorical response variable and one categorical predictor variable a chi-square test on the corresponding contingency table would be the classic approach. Then you would just get a single p value indicating whether there is a significant association between the two variables and there wouldn't be a p value for an intercept to add to potential confusion. Logistic regression is most helpful when you have continuous or multiple predictors, although there's no harm in setting up your data as a logistic regression.

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