Solved – Logistic regression: what happens to the coefficients when we switch the labels (0/1) of the binary outcome

With a logistic regression model one models the probability of occurence of a binary event, in your case the probability that it rains. As any other model, your model will have to make assumptions and one of your assumptions is that this probability of rain depends on five explanatory variables, for the ease of notation I will call them $x_i, i=1,2, dots 5$. Furthermore, your model assumes that the probability of rain, given values of the $x_i$ (notation: $P(rain=true|_{x_i})$ has a particular functional $S$-shaped form namely

$P(rain=true|_{x_i})=frac{1}{1+e^{-(beta_0+sum_i beta_i x_i)}}$.

After some manipulations we can transform this to $ln left( frac{P(rain=true|_{x_i})}{1-P(rain=true|_{x_i})} right)=beta_0+sum_i beta_i x_i$. ($ln$ is the natural logarithm)

If $pi$ is the probability of an occurence of an event, then $frac{pi}{1-pi}$ is the odds of the event. For example, if you make a bet with a fair coin, and you win the bet when head turns up, then, as $pi=0.5$, the odds of winning are $frac{0.5}{1-0.5}=1$ or you have as much chance of winning the bet than of losing it. If you make a bet and you win when a die turns up with '1', then the odds are $frac{frac{1}{6}}{1-frac{1}{6}}=frac{1}{5}$ or the odds is 1/5 or you have five times more chance of losing the bet.

From the above it follows that a logistic regression model assumes (a) that the probability of rain is a function of the $x_i$ and (b) that the log of the odds of rain against no rain is linear in the $x_i$.

(Note: further assumptions must be made for estimating the coefficients (e.g. independence of observations)).

Your coefficient of $x_{10}$ is (rounded) 0.64 meaning that if $x_{10}$ increases by one unit then, all other things equal, the log of the odds of rain against no rain will increase by $0.64$.

If the log of the odds increases by $0.64$ then, all other things equal, the odds increase by $e^{0.64}$ (for each unit increase in $x_{10}$).

Maybe good to note with respect to your 'most influential variables': In the above paragraph I said 'change in log-odds for one unit change in $x_i$'. This is important if you want to analyse 'most influential variables', indeed, variables can be espressed in different units: if the coeffient of $x_1$ is 1 and $x_1$ is in kilometer while the coefficient of $x_2$ is 0.1 with $x_2$ in meter then (a) a unit change in $x_1$ (1 km) changes the log-odds by 1 and (b) a unit change in $x_2$ (1 m) changes the log-odds by 0.1.

So in order to assess the impact of variables, an analysis of the magnitude of the coefficients alone is not suffcient, you should take the units of the variables into account (or use standardised variables).

EDIT: I added this after you asked the question in your comment: "what happens if I predict the odds of no rain instead of the odds of rain?"

Obviously it holds that $P(rain=FALSE|_{x_i})=1-P(rain=true|_{x_i})$. So the log-odds of no rain against rain is $ln left( frac{P(rain=FALSE|_{x_i})}{1-P(rain=FALSE|_{x_i})} right) =ln left( frac{1-P(rain=true|_{x_i})}{P(rain=true|_{x_i})} right)=-ln left( frac{P(rain=true|_{x_i})}{1-P(rain=true|_{x_i})} right )$.

(note that $ln left( frac{1}{x} right)=-ln(x)$).

So we find that the odds of 'no rain' against 'rain' is $ln left( frac{P(rain=FALSE|_{x_i})}{1-P(rain=FALSE|_{x_i})} right) = -(beta_0+sum_i beta_i x_i) $. In words: the sign of the coefficients changes.

As @Scortchi has indicated in the comments, one would have serious doubt about the logistic regression if simply switching the class labels would yield a completely different result.

The code below illustrates the 'sign switch' :

# Generate some data: outcome is binary and x is the explanatory #   step 1: generate success probabilities for Bernouilli variables set.seed(1) x<-runif(n=5000, min=-2, max=2) p<-1/(1+exp(-(2*x-1))) #   step 2: generate binary outcome with these probabilities  outcome<-(runif(n=5000,min=0,max=1) <=p)  # Estimate logit using: estimate binary outcome with x as explanatory glm.lr1<-glm(outcome ~ x +1, family=binomial) coef(glm.lr1)  # estimate logit using  **'outcome' SWITCHED** ('!' in front of it) glm.lr2<-glm(!outcome ~ x +1, family=binomial) coef(glm.lr2) 

Note for the results: The way we generated the data, the intercept should be close to -1 and the coefficient of x close to 2 in the first case and with reversed signs in the second case