Odds and odds ratios in logistic regression I am having difficulties understanding one logistic regression explanation. The logistic regression is between temperature and fish which die or do not die. 
The slope of a logistic regression is 1.76.
Then the odds that fish die increase by a factor of exp(1.76) = 5.8. 
In other words, the odds  that fish die increase by a factor of 5.8 for each change of 1 degree Celsius in temperature.


*

*Because 50% fish die in 2012, a 1 degree Celsius increase on 2012 temperature would raise the fish die occurrence to 82%.

*A 2 degree Celsius increase on 2012 temperature would raise the fish die occurrence to 97%.

*A 3 degree Celsius increase -> 100% fish die.
How do we calculate 1, 2 and 3? (82%, 97% and 100%)
 A: If the regression coefficient of your logistic regression is 1.76 on the logit-scale, then the odds ratio for 1 unit increase in temperature is $\mathrm{OR_{+1}}=\exp(\beta) = \exp(1.76)\approx 5.81$, as you already stated. The odds ratio for an increase in temperature for $a$ degrees is $\mathrm{OR_{+a}}=\exp(\beta\times a)$. In your case, $a$ is 2 and 3, respectively. So the odds ratios for an increase of 2 and 3 degrees are: $\mathrm{OR_{+2}}=\exp(1.76\times 2)\approx 33.78$ and $\mathrm{OR_{+3}}=\exp(1.76\times 3)\approx 196.37$. If in 2012 50% of the fish die, the baseline odds of dying are $0.5/(0.5-1)=1$. The odds ratio for 1 degree increase in temperature is 5.8 and thus, the odds of dying are $5.8\times 1$ (i.e. the odds ratio multiplied by the baseline odds) compared to fish without the increase in temperature. The odds can now be converted to probability by: $5.8/(5.8+ 1)\approx 0.853$. The same is true for an increase of 2 and 3 degrees: $33.78/(33.78+1)\approx 0.971$ and $196.37/(196.37+1)\approx 0.995$.
A: The odds is not the same as the probability. The odds is the number of "successes" (deaths) per "failure" (continue to live), while the probability is the proportion of "successes". I find it instructive to compare how one would estimate these two: An estimate of the odds would be the ratio of the number of successes over the number of failures, while an estimate of the probability would be the ratio of the number of success over the total number of observations.
Odds and probabilities are both ways of quantifying how likely an event is, so it is not surprising that there is a one to one relation between the two. You can turn a probability ($p$) into an odds ($o$) using the following formula: $o=\frac{p}{1-p}$. You can turn an odds into a probability like so: $p = \frac{o}{1+o}$.
So to come back to your example:


*

*The baseline probability is .5, so you would expect to find 1 failure per success, i.e. the baseline odds is 1. This odds is multiplied by a factor 5.8, so then the odds would become 5.8, which you can transform back to a probability as: $\frac{5.8}{1+5.8}\approx.85$ or 85%

*A two degree change in temperature is association with a change in the odds of death by a factor $5.8^2=33.6$. So the baseline odds is still 1, which means the new odds would be 33.6, i.e. you would expect 33.6 dead fishes for every live fish, or the probability of finding a dead fish is $\frac{33.6}{1+33.6} \approx .97$

*A three degree change in temperatue leads to a new odds of death of $1\times 5.8^3\approx195$. So the probability of finding a dead fish = $\frac{195}{1+195}\approx.99$

