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logit - interpreting coefficients as probabilities
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I seem to be missing some vital piece of information. I am aware that the coefficient of logistic regression are in log(odds), called logit scale. Therefore to interpret them, exp(coef) is taken and yields OR, the odds ratio.
If $beta_1 = 0.012$ the interpretation is as follows: For one unit increase in the covariate $X_1$, the log odds ratio is 0.012 - which does not provide meaningful information as it is.
Exponentiation yields that that for one unit increase in the covariate $X_1$, the odds ratio is 1.012 ($exp(0.012)=1.012$), or $Y=1$ is 1.012 more likely than $Y=0$.
But I would like to express the coefficient as percentage. According to Gelman and Hill in Data Analysis Using Regression and Multilevel/Hierarchical Models, pg 111:
The coefficients β can be exponentiated and treated as multiplicative
effects."
Such that if β1=0.012, then "the expected multiplicative increase is
exp(0.012)=1.012, or a 1.2% positive difference ...
However, according to my scripts
$$textODDS = fracp1-p
$$
and the inverse logit formula states
$$
P=fracOR1+OR=frac1.0122.012= 0.502$$
Which i am tempted to intrepret as if the covariate increases by one unit the probability of Y=1 increases by 50% - which i assume is wrong, but i do not understand why.
How can logit coeifficient be interpreted in terms of probabilities?
probability logistic binary-data logit odds-ratio
edited Aug 24 at 20:06
Ben Bolker
20.5k15583
asked Aug 24 at 16:16
user1607
1199
add a comment |Â
up vote
4
down vote
favorite
I seem to be missing some vital piece of information. I am aware that the coefficient of logistic regression are in log(odds), called logit scale. Therefore to interpret them, exp(coef) is taken and yields OR, the odds ratio.
If $beta_1 = 0.012$ the interpretation is as follows: For one unit increase in the covariate $X_1$, the log odds ratio is 0.012 - which does not provide meaningful information as it is.
Exponentiation yields that that for one unit increase in the covariate $X_1$, the odds ratio is 1.012 ($exp(0.012)=1.012$), or $Y=1$ is 1.012 more likely than $Y=0$.
But I would like to express the coefficient as percentage. According to Gelman and Hill in Data Analysis Using Regression and Multilevel/Hierarchical Models, pg 111:
The coefficients β can be exponentiated and treated as multiplicative
effects."
Such that if β1=0.012, then "the expected multiplicative increase is
exp(0.012)=1.012, or a 1.2% positive difference ...
However, according to my scripts
$$textODDS = fracp1-p
$$
and the inverse logit formula states
$$
P=fracOR1+OR=frac1.0122.012= 0.502$$
Which i am tempted to intrepret as if the covariate increases by one unit the probability of Y=1 increases by 50% - which i assume is wrong, but i do not understand why.
How can logit coeifficient be interpreted in terms of probabilities?
probability logistic binary-data logit odds-ratio
edited Aug 24 at 20:06
Ben Bolker
20.5k15583
asked Aug 24 at 16:16
user1607
1199
(1) You seem to conflate the odds and the odds ratio: they are different things. (2) Be a little careful with your arithmetic. You're dealing with small changes, so you need sufficient precision to express them. For 1.012/2.012 I obtain 0.5030 (to four significant figures), which--as a relative change compared to 0.50--is 50% greater than your number! (3) We have several good threads on interpreting logistic regression coefficients and ORs. Why don't you search for them and check them out?
– whuber♦
Aug 24 at 16:34
1
@whuber thank you. I did search some more and found the answers. I have summarised my finding in the answer below. Hopefully it will be helpful to some other users also!
– user1607
Aug 24 at 18:07
add a comment |Â
up vote
4
down vote
favorite
up vote
4
down vote
favorite
I seem to be missing some vital piece of information. I am aware that the coefficient of logistic regression are in log(odds), called logit scale. Therefore to interpret them, exp(coef) is taken and yields OR, the odds ratio.
If $beta_1 = 0.012$ the interpretation is as follows: For one unit increase in the covariate $X_1$, the log odds ratio is 0.012 - which does not provide meaningful information as it is.
Exponentiation yields that that for one unit increase in the covariate $X_1$, the odds ratio is 1.012 ($exp(0.012)=1.012$), or $Y=1$ is 1.012 more likely than $Y=0$.
But I would like to express the coefficient as percentage. According to Gelman and Hill in Data Analysis Using Regression and Multilevel/Hierarchical Models, pg 111:
The coefficients β can be exponentiated and treated as multiplicative
effects."
Such that if β1=0.012, then "the expected multiplicative increase is
exp(0.012)=1.012, or a 1.2% positive difference ...
However, according to my scripts
$$textODDS = fracp1-p
$$
and the inverse logit formula states
$$
P=fracOR1+OR=frac1.0122.012= 0.502$$
Which i am tempted to intrepret as if the covariate increases by one unit the probability of Y=1 increases by 50% - which i assume is wrong, but i do not understand why.
How can logit coeifficient be interpreted in terms of probabilities?
probability logistic binary-data logit odds-ratio
edited Aug 24 at 20:06
Ben Bolker
20.5k15583
asked Aug 24 at 16:16
user1607
1199
I seem to be missing some vital piece of information. I am aware that the coefficient of logistic regression are in log(odds), called logit scale. Therefore to interpret them, exp(coef) is taken and yields OR, the odds ratio.
If $beta_1 = 0.012$ the interpretation is as follows: For one unit increase in the covariate $X_1$, the log odds ratio is 0.012 - which does not provide meaningful information as it is.
Exponentiation yields that that for one unit increase in the covariate $X_1$, the odds ratio is 1.012 ($exp(0.012)=1.012$), or $Y=1$ is 1.012 more likely than $Y=0$.
But I would like to express the coefficient as percentage. According to Gelman and Hill in Data Analysis Using Regression and Multilevel/Hierarchical Models, pg 111:
The coefficients β can be exponentiated and treated as multiplicative
effects."
Such that if β1=0.012, then "the expected multiplicative increase is
exp(0.012)=1.012, or a 1.2% positive difference ...
However, according to my scripts
$$textODDS = fracp1-p
$$
and the inverse logit formula states
$$
P=fracOR1+OR=frac1.0122.012= 0.502$$
Which i am tempted to intrepret as if the covariate increases by one unit the probability of Y=1 increases by 50% - which i assume is wrong, but i do not understand why.
How can logit coeifficient be interpreted in terms of probabilities?
probability logistic binary-data logit odds-ratio
edited Aug 24 at 20:06
Ben Bolker
20.5k15583
asked Aug 24 at 16:16
user1607
1199
edited Aug 24 at 20:06
Ben Bolker
20.5k15583
edited Aug 24 at 20:06
Ben Bolker
20.5k15583
edited Aug 24 at 20:06
Ben Bolker
20.5k15583
20.5k15583
asked Aug 24 at 16:16
user1607
1199
asked Aug 24 at 16:16
user1607
1199
asked Aug 24 at 16:16
user1607
1199
1199
(1) You seem to conflate the odds and the odds ratio: they are different things. (2) Be a little careful with your arithmetic. You're dealing with small changes, so you need sufficient precision to express them. For 1.012/2.012 I obtain 0.5030 (to four significant figures), which--as a relative change compared to 0.50--is 50% greater than your number! (3) We have several good threads on interpreting logistic regression coefficients and ORs. Why don't you search for them and check them out?
– whuber♦
Aug 24 at 16:34
1
@whuber thank you. I did search some more and found the answers. I have summarised my finding in the answer below. Hopefully it will be helpful to some other users also!
– user1607
Aug 24 at 18:07
add a comment |Â
(1) You seem to conflate the odds and the odds ratio: they are different things. (2) Be a little careful with your arithmetic. You're dealing with small changes, so you need sufficient precision to express them. For 1.012/2.012 I obtain 0.5030 (to four significant figures), which--as a relative change compared to 0.50--is 50% greater than your number! (3) We have several good threads on interpreting logistic regression coefficients and ORs. Why don't you search for them and check them out?
– whuber♦
Aug 24 at 16:34
1
@whuber thank you. I did search some more and found the answers. I have summarised my finding in the answer below. Hopefully it will be helpful to some other users also!
– user1607
Aug 24 at 18:07
(1) You seem to conflate the odds and the odds ratio: they are different things. (2) Be a little careful with your arithmetic. You're dealing with small changes, so you need sufficient precision to express them. For 1.012/2.012 I obtain 0.5030 (to four significant figures), which--as a relative change compared to 0.50--is 50% greater than your number! (3) We have several good threads on interpreting logistic regression coefficients and ORs. Why don't you search for them and check them out?
– whuber♦
Aug 24 at 16:34
(1) You seem to conflate the odds and the odds ratio: they are different things. (2) Be a little careful with your arithmetic. You're dealing with small changes, so you need sufficient precision to express them. For 1.012/2.012 I obtain 0.5030 (to four significant figures), which--as a relative change compared to 0.50--is 50% greater than your number! (3) We have several good threads on interpreting logistic regression coefficients and ORs. Why don't you search for them and check them out?
– whuber♦
Aug 24 at 16:34
1
1
@whuber thank you. I did search some more and found the answers. I have summarised my finding in the answer below. Hopefully it will be helpful to some other users also!
– user1607
Aug 24 at 18:07
@whuber thank you. I did search some more and found the answers. I have summarised my finding in the answer below. Hopefully it will be helpful to some other users also!
– user1607
Aug 24 at 18:07
add a comment |Â
3 Answers
3
active
oldest
votes
up vote
4
down vote
accepted
These odds ratios are the exponential of the corresponding regression coefficient:
$$textodds ratio = e^hatbeta$$
For example, if the logistic regression coefficient is $hatbeta=0.25$ the odds ratio is $e^0.25 = 1.28$.
The odds ratio is the multiplier that shows how the odds change for a one-unit increase in the value of the X. The odds ratio increases by a factor of 1.28. So if the initial odds ratio was, say 0.25, the odds ratio after one unit increase in the covariate becomes $0.25 times 1.28$.
Another way to try to interpret the odds ratio is to look at the fractional part and interpret it as a percentage change. For example, the odds ratio of 1.28 corresponds to a 28% increase in the odds for a 1-unit increase in the corresponding X.
In case we are dealing with an decreasing effect (OR < 1), for example odds ratio = 0.94, then there is a 6% decrease in the odds for a 1-unit increase in the corresponding X.
The formula is:
$$ textPercent Change in the Odds = left( textOdds Ratio - 1 right) times 100 $$
answered Aug 24 at 18:04
user1607
1199
+1: good explanation.
– whuber♦
Aug 24 at 18:23
add a comment |Â
up vote
1
down vote
Part of the problem is that you're taking a sentence from Gelman and Hill out of context. Here's a Google books screenshot:
Note that the heading says "Interpreting Poisson regression coefficients" (emphasis added). Poisson regression uses a logarithmic link, in contrast to logistic regression, which uses a logit (log-odds) link. The interpretation of exponentiated coefficients as multiplicative effects only works for a log-scale coefficients (or, at the risk of muddying the waters slightly, for logit-scale coefficients if the baseline risk is very low ...)
Everyone would like to be able to quote effects of treatments on probabilities in a simple, universal scale-independent way, but this is basically impossible: this is why there are so many tutorials on interpreting odds and log-odds circulating in the wild, and why epidemiologists spend so much time arguing about relative risk vs. odds ratios vs ...
answered Aug 24 at 20:11
Ben Bolker
20.5k15583
add a comment |Â
up vote
1
down vote
If you want to interpret in terms of the percentages, then you need the y-intercept ($beta_0$). Taking the exponential of the intercept gives the odds when all the covariates are 0, then you can multiply by the odds-ratio of a given term to determine what the odds would be when that covariate is 1 instead of 0.
The inverse logit transform above can be applied to the odds to give the percent chance of $Y=1$.
So when all $x=0$:
$p(Y=1) = frace^beta_01+e^beta_0$
and if $x_1=1$ (and any other covariates are 0) then:
$p(Y=1) = frac e^(beta_0 + beta_1) 1+ e^(beta_0 + beta_1)$
and those can be compared. But notice that the effect of $x_1$ is different depending on $beta_0$, it is not a constant effect like in linear regression, only constant on the log-odds scale.
Also notice that your estimate of $beta_0$ will depend on how the data was collected. A case-control study where equal number of subjects with $Y=0$ and $Y=1$ are selected, then their value of $x$ is observed can give a very different $beta_0$ estimate than a simple random sample, and the interpretation of the percentage(s) from the first could be meaningless as interpretations of what would happen in the second case.
answered Aug 24 at 20:40
Greg Snow
38.5k157121
add a comment |Â
3 Answers
3
active
oldest
votes
3 Answers
3
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
4
down vote
accepted
These odds ratios are the exponential of the corresponding regression coefficient:
$$textodds ratio = e^hatbeta$$
For example, if the logistic regression coefficient is $hatbeta=0.25$ the odds ratio is $e^0.25 = 1.28$.
The odds ratio is the multiplier that shows how the odds change for a one-unit increase in the value of the X. The odds ratio increases by a factor of 1.28. So if the initial odds ratio was, say 0.25, the odds ratio after one unit increase in the covariate becomes $0.25 times 1.28$.
Another way to try to interpret the odds ratio is to look at the fractional part and interpret it as a percentage change. For example, the odds ratio of 1.28 corresponds to a 28% increase in the odds for a 1-unit increase in the corresponding X.
In case we are dealing with an decreasing effect (OR < 1), for example odds ratio = 0.94, then there is a 6% decrease in the odds for a 1-unit increase in the corresponding X.
The formula is:
$$ textPercent Change in the Odds = left( textOdds Ratio - 1 right) times 100 $$
answered Aug 24 at 18:04
user1607
1199
+1: good explanation.
– whuber♦
Aug 24 at 18:23
add a comment |Â
up vote
4
down vote
accepted
These odds ratios are the exponential of the corresponding regression coefficient:
$$textodds ratio = e^hatbeta$$
For example, if the logistic regression coefficient is $hatbeta=0.25$ the odds ratio is $e^0.25 = 1.28$.
The odds ratio is the multiplier that shows how the odds change for a one-unit increase in the value of the X. The odds ratio increases by a factor of 1.28. So if the initial odds ratio was, say 0.25, the odds ratio after one unit increase in the covariate becomes $0.25 times 1.28$.
Another way to try to interpret the odds ratio is to look at the fractional part and interpret it as a percentage change. For example, the odds ratio of 1.28 corresponds to a 28% increase in the odds for a 1-unit increase in the corresponding X.
In case we are dealing with an decreasing effect (OR < 1), for example odds ratio = 0.94, then there is a 6% decrease in the odds for a 1-unit increase in the corresponding X.
The formula is:
$$ textPercent Change in the Odds = left( textOdds Ratio - 1 right) times 100 $$
answered Aug 24 at 18:04
user1607
1199
+1: good explanation.
– whuber♦
Aug 24 at 18:23
add a comment |Â
up vote
4
down vote
accepted
up vote
4
down vote
accepted
These odds ratios are the exponential of the corresponding regression coefficient:
$$textodds ratio = e^hatbeta$$
For example, if the logistic regression coefficient is $hatbeta=0.25$ the odds ratio is $e^0.25 = 1.28$.
The odds ratio is the multiplier that shows how the odds change for a one-unit increase in the value of the X. The odds ratio increases by a factor of 1.28. So if the initial odds ratio was, say 0.25, the odds ratio after one unit increase in the covariate becomes $0.25 times 1.28$.
Another way to try to interpret the odds ratio is to look at the fractional part and interpret it as a percentage change. For example, the odds ratio of 1.28 corresponds to a 28% increase in the odds for a 1-unit increase in the corresponding X.
In case we are dealing with an decreasing effect (OR < 1), for example odds ratio = 0.94, then there is a 6% decrease in the odds for a 1-unit increase in the corresponding X.
The formula is:
$$ textPercent Change in the Odds = left( textOdds Ratio - 1 right) times 100 $$
answered Aug 24 at 18:04
user1607
1199
These odds ratios are the exponential of the corresponding regression coefficient:
$$textodds ratio = e^hatbeta$$
For example, if the logistic regression coefficient is $hatbeta=0.25$ the odds ratio is $e^0.25 = 1.28$.
The odds ratio is the multiplier that shows how the odds change for a one-unit increase in the value of the X. The odds ratio increases by a factor of 1.28. So if the initial odds ratio was, say 0.25, the odds ratio after one unit increase in the covariate becomes $0.25 times 1.28$.
Another way to try to interpret the odds ratio is to look at the fractional part and interpret it as a percentage change. For example, the odds ratio of 1.28 corresponds to a 28% increase in the odds for a 1-unit increase in the corresponding X.
In case we are dealing with an decreasing effect (OR < 1), for example odds ratio = 0.94, then there is a 6% decrease in the odds for a 1-unit increase in the corresponding X.
The formula is:
$$ textPercent Change in the Odds = left( textOdds Ratio - 1 right) times 100 $$
answered Aug 24 at 18:04
user1607
1199
answered Aug 24 at 18:04
user1607
1199
answered Aug 24 at 18:04
user1607
1199
answered Aug 24 at 18:04
user1607
1199
1199
+1: good explanation.
– whuber♦
Aug 24 at 18:23
add a comment |Â
+1: good explanation.
– whuber♦
Aug 24 at 18:23
+1: good explanation.
– whuber♦
Aug 24 at 18:23
+1: good explanation.
– whuber♦
Aug 24 at 18:23
add a comment |Â
up vote
1
down vote
Part of the problem is that you're taking a sentence from Gelman and Hill out of context. Here's a Google books screenshot:
Note that the heading says "Interpreting Poisson regression coefficients" (emphasis added). Poisson regression uses a logarithmic link, in contrast to logistic regression, which uses a logit (log-odds) link. The interpretation of exponentiated coefficients as multiplicative effects only works for a log-scale coefficients (or, at the risk of muddying the waters slightly, for logit-scale coefficients if the baseline risk is very low ...)
Everyone would like to be able to quote effects of treatments on probabilities in a simple, universal scale-independent way, but this is basically impossible: this is why there are so many tutorials on interpreting odds and log-odds circulating in the wild, and why epidemiologists spend so much time arguing about relative risk vs. odds ratios vs ...
answered Aug 24 at 20:11
Ben Bolker
20.5k15583
add a comment |Â
up vote
1
down vote
Part of the problem is that you're taking a sentence from Gelman and Hill out of context. Here's a Google books screenshot:
Note that the heading says "Interpreting Poisson regression coefficients" (emphasis added). Poisson regression uses a logarithmic link, in contrast to logistic regression, which uses a logit (log-odds) link. The interpretation of exponentiated coefficients as multiplicative effects only works for a log-scale coefficients (or, at the risk of muddying the waters slightly, for logit-scale coefficients if the baseline risk is very low ...)
Everyone would like to be able to quote effects of treatments on probabilities in a simple, universal scale-independent way, but this is basically impossible: this is why there are so many tutorials on interpreting odds and log-odds circulating in the wild, and why epidemiologists spend so much time arguing about relative risk vs. odds ratios vs ...
answered Aug 24 at 20:11
Ben Bolker
20.5k15583
add a comment |Â
up vote
1
down vote
up vote
1
down vote
Part of the problem is that you're taking a sentence from Gelman and Hill out of context. Here's a Google books screenshot:
Note that the heading says "Interpreting Poisson regression coefficients" (emphasis added). Poisson regression uses a logarithmic link, in contrast to logistic regression, which uses a logit (log-odds) link. The interpretation of exponentiated coefficients as multiplicative effects only works for a log-scale coefficients (or, at the risk of muddying the waters slightly, for logit-scale coefficients if the baseline risk is very low ...)
Everyone would like to be able to quote effects of treatments on probabilities in a simple, universal scale-independent way, but this is basically impossible: this is why there are so many tutorials on interpreting odds and log-odds circulating in the wild, and why epidemiologists spend so much time arguing about relative risk vs. odds ratios vs ...
answered Aug 24 at 20:11
Ben Bolker
20.5k15583
Part of the problem is that you're taking a sentence from Gelman and Hill out of context. Here's a Google books screenshot:
Note that the heading says "Interpreting Poisson regression coefficients" (emphasis added). Poisson regression uses a logarithmic link, in contrast to logistic regression, which uses a logit (log-odds) link. The interpretation of exponentiated coefficients as multiplicative effects only works for a log-scale coefficients (or, at the risk of muddying the waters slightly, for logit-scale coefficients if the baseline risk is very low ...)
Everyone would like to be able to quote effects of treatments on probabilities in a simple, universal scale-independent way, but this is basically impossible: this is why there are so many tutorials on interpreting odds and log-odds circulating in the wild, and why epidemiologists spend so much time arguing about relative risk vs. odds ratios vs ...
answered Aug 24 at 20:11
Ben Bolker
20.5k15583
edited Aug 24 at 20:21
answered Aug 24 at 20:11
Ben Bolker
20.5k15583
answered Aug 24 at 20:11
Ben Bolker
20.5k15583
answered Aug 24 at 20:11
Ben Bolker
20.5k15583
20.5k15583
add a comment |Â
add a comment |Â
up vote
1
down vote
If you want to interpret in terms of the percentages, then you need the y-intercept ($beta_0$). Taking the exponential of the intercept gives the odds when all the covariates are 0, then you can multiply by the odds-ratio of a given term to determine what the odds would be when that covariate is 1 instead of 0.
The inverse logit transform above can be applied to the odds to give the percent chance of $Y=1$.
So when all $x=0$:
$p(Y=1) = frace^beta_01+e^beta_0$
and if $x_1=1$ (and any other covariates are 0) then:
$p(Y=1) = frac e^(beta_0 + beta_1) 1+ e^(beta_0 + beta_1)$
and those can be compared. But notice that the effect of $x_1$ is different depending on $beta_0$, it is not a constant effect like in linear regression, only constant on the log-odds scale.
Also notice that your estimate of $beta_0$ will depend on how the data was collected. A case-control study where equal number of subjects with $Y=0$ and $Y=1$ are selected, then their value of $x$ is observed can give a very different $beta_0$ estimate than a simple random sample, and the interpretation of the percentage(s) from the first could be meaningless as interpretations of what would happen in the second case.
answered Aug 24 at 20:40
Greg Snow
38.5k157121
add a comment |Â
up vote
1
down vote
If you want to interpret in terms of the percentages, then you need the y-intercept ($beta_0$). Taking the exponential of the intercept gives the odds when all the covariates are 0, then you can multiply by the odds-ratio of a given term to determine what the odds would be when that covariate is 1 instead of 0.
The inverse logit transform above can be applied to the odds to give the percent chance of $Y=1$.
So when all $x=0$:
$p(Y=1) = frace^beta_01+e^beta_0$
and if $x_1=1$ (and any other covariates are 0) then:
$p(Y=1) = frac e^(beta_0 + beta_1) 1+ e^(beta_0 + beta_1)$
and those can be compared. But notice that the effect of $x_1$ is different depending on $beta_0$, it is not a constant effect like in linear regression, only constant on the log-odds scale.
Also notice that your estimate of $beta_0$ will depend on how the data was collected. A case-control study where equal number of subjects with $Y=0$ and $Y=1$ are selected, then their value of $x$ is observed can give a very different $beta_0$ estimate than a simple random sample, and the interpretation of the percentage(s) from the first could be meaningless as interpretations of what would happen in the second case.
answered Aug 24 at 20:40
Greg Snow
38.5k157121
add a comment |Â
up vote
1
down vote
up vote
1
down vote
If you want to interpret in terms of the percentages, then you need the y-intercept ($beta_0$). Taking the exponential of the intercept gives the odds when all the covariates are 0, then you can multiply by the odds-ratio of a given term to determine what the odds would be when that covariate is 1 instead of 0.
The inverse logit transform above can be applied to the odds to give the percent chance of $Y=1$.
So when all $x=0$:
$p(Y=1) = frace^beta_01+e^beta_0$
and if $x_1=1$ (and any other covariates are 0) then:
$p(Y=1) = frac e^(beta_0 + beta_1) 1+ e^(beta_0 + beta_1)$
and those can be compared. But notice that the effect of $x_1$ is different depending on $beta_0$, it is not a constant effect like in linear regression, only constant on the log-odds scale.
Also notice that your estimate of $beta_0$ will depend on how the data was collected. A case-control study where equal number of subjects with $Y=0$ and $Y=1$ are selected, then their value of $x$ is observed can give a very different $beta_0$ estimate than a simple random sample, and the interpretation of the percentage(s) from the first could be meaningless as interpretations of what would happen in the second case.
answered Aug 24 at 20:40
Greg Snow
38.5k157121
If you want to interpret in terms of the percentages, then you need the y-intercept ($beta_0$). Taking the exponential of the intercept gives the odds when all the covariates are 0, then you can multiply by the odds-ratio of a given term to determine what the odds would be when that covariate is 1 instead of 0.
The inverse logit transform above can be applied to the odds to give the percent chance of $Y=1$.
So when all $x=0$:
$p(Y=1) = frace^beta_01+e^beta_0$
and if $x_1=1$ (and any other covariates are 0) then:
$p(Y=1) = frac e^(beta_0 + beta_1) 1+ e^(beta_0 + beta_1)$
and those can be compared. But notice that the effect of $x_1$ is different depending on $beta_0$, it is not a constant effect like in linear regression, only constant on the log-odds scale.
Also notice that your estimate of $beta_0$ will depend on how the data was collected. A case-control study where equal number of subjects with $Y=0$ and $Y=1$ are selected, then their value of $x$ is observed can give a very different $beta_0$ estimate than a simple random sample, and the interpretation of the percentage(s) from the first could be meaningless as interpretations of what would happen in the second case.
answered Aug 24 at 20:40
Greg Snow
38.5k157121
answered Aug 24 at 20:40
Greg Snow
38.5k157121
answered Aug 24 at 20:40
Greg Snow
38.5k157121
answered Aug 24 at 20:40
Greg Snow
38.5k157121
38.5k157121
add a comment |Â
add a comment |Â
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(1) You seem to conflate the odds and the odds ratio: they are different things. (2) Be a little careful with your arithmetic. You're dealing with small changes, so you need sufficient precision to express them. For 1.012/2.012 I obtain 0.5030 (to four significant figures), which--as a relative change compared to 0.50--is 50% greater than your number! (3) We have several good threads on interpreting logistic regression coefficients and ORs. Why don't you search for them and check them out?
– whuber♦
Aug 24 at 16:34
1
@whuber thank you. I did search some more and found the answers. I have summarised my finding in the answer below. Hopefully it will be helpful to some other users also!
– user1607
Aug 24 at 18:07