I am running a fairly simple Logistic Regression model y= (1[Positive Savings] ,0]) X = (1[Treated Group],0) I got a coefficient of Treated -.64 and OR of .52. My question is how to interpret the meaning of the coefficient?

Is y base 1 and X base 0

My thoughts are that the treatment X 0 is .47% less likely to show positive savings?

Is it always 0 being the base in the binary or categorical?

can I get stats model to give 0- 2 or 0-3 as Odds Ratio as well?

  • 1
    $\begingroup$ Couple of things here: response $y$ has no base level since it's a random variable that follows the Bernoulli distribution. Concerning regressor $x$, the value which brings this variable at base level is $x=0$, but the domain-specific level (e.g. experiment vs control) that's used as your base level has to do with the software you're using. $\endgroup$ – Digio Feb 8 '19 at 21:40

In your model:

$$ y \sim Binomial(n, p) $$ $$logit(p) = \beta_0 + \beta_1 x $$

you get: $$ log{p \over{1-p}} = \beta_0 + \beta_1 x $$ $$ log{~O_{y|x}} = \beta_0 + \beta_1 x $$ and solving for $\beta_1$ gives you: $$ \beta_1 = (\beta_0 + \beta_1) - \beta_0 $$ $$ ~~~~~~~~~~~~~\beta_1 = log{~O_{y|x=1}} - log{~O_{y|x=0}} $$ $$\beta_1 = log{~O_{y|x=1} \over ~O_{y|x=0} } $$ and finally: $$exp(\beta_1) = {O_{treatment} \over O_{control}} $$

Since your OR is in fact $exp(-.64) = 0.53$, you can convert this to a percentage via $(exp(\beta_1)-1) \times 100 = -47$% and conclude that:

The average probability of getting positive savings is 47% lower at level "treatment" than level "control".

If independent variable $x$ was continues you would say:

The average probability of getting positive savings gets 47% lower for every unit increase in $x$.

  • $\begingroup$ Is level control always the lowest binary value? $\endgroup$ – Tinkinc Feb 9 '19 at 3:34
  • $\begingroup$ No. The base level is whatever you set $x=0$ to be. In your case this is control. I also explained this under your question as a comment. $\endgroup$ – Digio Feb 9 '19 at 7:51

The OR is exp(-0.64) ~ 0.53. This is the ratio: odds(Y=1 | X=1) / odds(Y=1 | X=0), where odds(Y=1 | X=x) is P(Y=1 | X=x) / P(Y=0 | X=x).

If X is continuous, then you get the same odds for any one-unit difference in X, e.g. odds(Y=1 | X=2) / odds(Y=1 | X=1) is also ~ 0.53.

If you want the OR for a two-unit difference, just take exp(2 * -0.64).

  • $\begingroup$ Thanks. so my question is X is binary so .53 less likely to have savings than 'non flag group' odds going from x= 0( baseline) to X = 1 which is the target group i was trying to investigate? $\endgroup$ – Tinkinc Feb 8 '19 at 20:48
  • $\begingroup$ The odds is monotone with the probability. So the odds ratio (OR) being less than 1 means that the probability that y=1 when x=1 is less than the probability that y=1 when x=0. If the OR is 1 then the two probabilities are equal. If the OR is greater than 1, then the probability that y=1 when x=1 is greater than the probability that y=1 when x=0. $\endgroup$ – Kerby Shedden Feb 9 '19 at 21:43

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