Intepreting 'one-unit' Change in Logistic Regression [duplicate]

Question

I have very easy question that I'm hoping someone can assist me with:

I ran an example logistic regression using this R code:

     hours <- c(0.5, 0.75, 1, 1.25, 1.5, 1.75, 1.75, 2, 2.25, 2.5, 2.75, 3, 3.25, 3.5, 4, 4.25, 4.5, 4.75, 5, 5.5)
        pass <- c(0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 1, 1, 1, 1)
        data <- data.frame(hours, pass)
        mylogit <- glm(pass ~ hours, data = data, family = "binomial") #Activates the logistic regression model
        summary(mylogit) #Summary of the model

    Call:
    glm(formula = pass ~ hours, family = "binomial", data = data)

    Deviance Residuals: 
         Min        1Q    Median        3Q       Max  
    -1.70557  -0.57357  -0.04654   0.45470   1.82008  

    Coefficients:
                Estimate Std. Error z value Pr(>|z|)  
    (Intercept)  -4.0777     1.7610  -2.316   0.0206 *
    hours         1.5046     0.6287   2.393   0.0167 *
    ---
    Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

    (Dispersion parameter for binomial family taken to be 1)

        Null deviance: 27.726  on 19  degrees of freedom
    Residual deviance: 16.060  on 18  degrees of freedom
    AIC: 20.06

    Number of Fisher Scoring iterations: 5

    round(exp(cbind(OR = coef(mylogit), confint(mylogit))),3)

               OR 2.5 % 97.5 %
   (Intercept) 0.017 0.000  0.281
    hours       4.503 1.698 23.223

I know that by taking the exponent of the log-odds/coefficients for hours the odds of passing increase by a factor of 4.503 for a one-unit change in hours. However, given that the explanatory variable (hours) is continuous, what is considered a 'one-unit change' i.e. going from 1 to 2 hours as one unit? or from 1.75 to 1.76 hours as one unit? Also, is this interpretation of one-unit the same for regular OLS regression as well? I'm seeking to better understand the rules R applies to creating its regression coefficients.

Answer 1

With logistic regression you analyse the association of a binary outcome with a set of predictors. In particular you are modelling the $\log(\text{odds})$ of that particular outcome. The odds themselves are simply: $\text{odds} = \frac{\text{Pr(of Occurring)}}{\text{Pr(of Not Occurring)}} = \frac{\text{Pr(of Occurring)}}{1 - \text{Pr(of Occurring)}}$ where $\text{Pr}$ refers to proportions (or loosely speaking probability). Therefore the estimated coefficient $\hat{\beta}$ in your model shows the difference in $\log(\text{odds})$ between two subjects that differ by one unit of your predictor (here: hours) when all other predictors are constant (or as in your case just absent).

To find the change in terms of the proportions that are modelled you need to:

1. Get the $\log(\text{odds})$ estimate.
2. Exponentiate it to get the $\text{odds}$.
3. Get the new proportions as: $\text{Pr}_{\text{new}} = \frac{\text{odds}}{1 + \text{odds}} $. (This follows from the equality shown above.)