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This question already has an answer here:

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.

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marked as duplicate by Scortchi Mar 4 '16 at 23:45

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    $\begingroup$ The change is one-unit on the same scale as your explanatory variable. Thus a one-unit change in your case is 1 hour. The effect is the same all along the scale (ie. from 1 to 2 hours or from 4 to 5 hours) $\endgroup$ – ekstroem Mar 4 '16 at 22:33
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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.)
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    $\begingroup$ Thanks for the detailed answer about interpreting the probability. However, what I am really trying to understand is the definition of a 'one unit' change in terms of a continuous predictor variable. Is it fractional? or an integer as @ekstroem suggested? $\endgroup$ – user86569 Mar 7 '16 at 16:18
  • $\begingroup$ When referring to "one unit of change" in the predictor one refers at adding or subtracting $1$ from a continuous predictor variable in the scale that this variable is recorded (eg. hours in this case). Notice that the change to the predicted proportions is not quantized in the same way as the $\log(\text{odd})$ transform is not linear. $\endgroup$ – usεr11852 Mar 7 '16 at 17:38
  • $\begingroup$ Makes sense now - @usεr11852 I appreciate your help. $\endgroup$ – user86569 Mar 7 '16 at 17:50

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