I have 4 predictors, and 1 binary response. I fitted a logistic regression model. A strange thing is that all the coefficient of the model are negative. Is that possible? Probably I did something wrong. My interpretation is that the odds ratio of either variable is less than 1. That is, neither variable actually do any good to the response. Even the intercept is negative. Please share your intelligence.
Solved – How to interpret a logistic regression model with all negative coefficient
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Personally, I think it is very difficult for people to grasp what does it mean by certain amount of change in odds...
But it is more clear to directly compare the difference of interest: the p's In your case.
At the current value of your predictors, increase 1 unit (can be any change you specify) in the first predictor means increase of
$logit^{-1}(−14.27+3.32(4)+0.88(7)) - logit^{-1}(−14.27+3.32(3)+0.88(7))$
$\approx0.99-0.86 = 0.13$, i.e., it is a 0.13 increase in probability of being 1.
But of course, this varies as the current values of the predictor changes, a common practice (if you are not interested in this specific predictor value) is to let the current predictor value be at the average, which corresponds the biggest change in probability, then you can say, as the predictor values move towards the end, changes in probability are expected to slow down.
The following bullet points are correct, and equivalent:
Keeping all other predictors constant then, the log odd ratio of survival for having an additional sibling decreases by 0.38 units (what does it mean?)
Keeping all other predictors constant then, the odd ratio of survival for having an additional sibling is 0.68 times lower (
less likely)
To see that they're equivalent, let $r_0$ be an odds ratio, and $r_1$ be the odds ratio with an additional sibling to $r_0$ and all else fixed. Then,
$$ \begin{split} \log r_1 &= \log r_0 - 0.38 \\ r_1 &= \exp\left\{ \log r_0 - 0.38 \right\} \\ &= r_0 \exp(-0.38) \\ &\approx 0.68 r_0. \end{split} $$
As you'll notice, I crossed out "less likely" in your quote above. Reducing an odds ratio by a factor of $x$ is not the same as reducing a probability by a factor of $x$ because an odds ratio is not the same thing as a probability. The odds ratio of survival means, by definition, $p_\text{survival} / p_\text{death}.$
The reason why reducing an odds ratio by a fixed factor can be confusing, is because this does not correspond to reducing a probability by a fixed factor. The factor by which the probability is reduced depends on the original odds ratio.
For example, suppose we reduce an odds ratio by a factor of half. If the odds ratio is $1$ and we reduce it to $1/2$, this corresponds to reducing the probability from $1/2$ to $1/3$, which is a reduction by a factor of $2/3.$ However, if we again reduce the odds ratio by half, from $1/2$ to $1/4$, this corresponds to reducing the probability from $1/3$ to $1/5,$ which is a reduction by a factor of $3/5$, a less severe reduction than before.
Dealing with odds ratios instead of probabilities can be pretty tricky, because we think more intuitively in terms of the probabilities than odds ratios. However, linear models work well with odds ratios because log-odds ratios can fit anywhere on the real line, while probabilities are confined to $[0,1].$
Best Answer
Yes, it is possible. Couple of things here. The direction of your predictors are critical to the interpretation; if they were scaled in the opposite direction, they would be positive. Second, it would seem on the face that your predictors lower the log odds given a unit change, which in itself might be good if say the outcome is death. Also, you might consider centering some of your predictors if the interecpt does not seem interpretable.