0
$\begingroup$

So I have a data set, and I'd like to test the influence of a continuous variable (cont) on a categorical (binary) variable (cat) that can be 0 (yes) or 1 (no). I've looked on the internet and binary logistic regression seems to be a good choice. So I plugged this into R:

glm(cat~cont, family = binomial(``logit"))

and got the following results from the summary:

Deviance residuals:
Min      1Q      Median   3Q      Max
-1.0757  -0.9077 -0.7019  1.3583  1.9911

Coefficients:
            Estimate   Std. Error   z value   Pr(>|z|)   
(Intercept)  0.65847    0.58545     1.125     0.26070   
cont        -0.04297    0.01652    -2.602     0.00927 

(Dispersion parameter for binomial family taken to be 1)

Null deviance: 244.35  on 199  degrees of freedom
Residual deviance: 237.11  on 198  degrees of freedom
AIC: 241.11
Number of Fisher Scoring iterations: 4

I can see that the p-value is 0.00927, so I definitely have something significant. But how can I interpret this? I've looked on the internet but haven't found anything useful, but it's possible I just didn't understand it. From what I did understand though, the deviances are really high, but then why do I get something significant? If anyone could help me, or explain how logistic regression works, that'd be great!

$\endgroup$
0
$\begingroup$

Let $cat = Y$ and $cont = X$. Your theoretical model is $$ \mathbb{P}(Y_i=0|X_i=x_i) = \left( 1 + \exp\{-(\beta_0 + \beta_1x_{i}\} \right)^{-1}, $$ the estimated model is $$ \widehat{\mathbb{P}(Y_i=0|X_i=x_i)} = \left( 1 + \exp\{-(-0.66 - 0.043_1x_{i}\} \right)^{-1}. $$ The likelihood ratio test is $$ D(\beta_1, \beta_0|\beta_0)=Dev(\beta_0)-Dev(\beta_0, \beta_1) =2\log\Lambda\xrightarrow{D}\chi^2_{(1)}, $$ hence you a can use the quantiles of $\chi^2_{(1)}$ distribution to test the model's significance, i.e., $$ p.value=\mathbb{P}(\chi^2_{(1)} \ge 6.85) =0.00886, $$
that is very close to the p.value of the Wald ($z$ test), i.e., the square of the $z$ statistic is $2.06 ^2 = 6.76$, that is approximately the value of the $\chi^2$ statistic. Your estimated odds ratio (OR) for $\Delta$ change in $X$ is $$ \widehat{OR} = \exp\{-0.043\Delta x\}. $$

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.