Deriving the odds ratio of a 3-way interaction logistic regression model
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Suppose a logistic regression model has three binary explanatory variables $x_1$, $x_2$ and $x_3$ used
to estimate the probability of success. This model includes all three main effects, the three $2$-
way interactions, and the one $3$-way interaction.
Derive the odds ratio to examine the odds of a success across two levels of one of the
explanatory variables, say $x_1$, and derive its variance.
I assume you start by:
$log(frac{pi}{1-pi}) = beta_1x_1 + beta_2x_2 + beta_3x_3 +beta_4x_1x_2 + beta_5x_1x_3 + beta_6x_2x_3 + beta_7x_1x_2x_3$
and then taking exponetials on both sides:
$frac{pi}{1-pi} = e^{beta_1x_1 + beta_2x_2 + beta_3x_3 +beta_4x_1x_2 + beta_5x_1x_3 + beta_6x_2x_3 + beta_7x_1x_2x_3}$
However I am not sure how to proceed.
probability statistics regression logistic-regression
asked Nov 21 at 3:24
Alex Chavez
440620
add a comment |
up vote
2
down vote
favorite
Suppose a logistic regression model has three binary explanatory variables $x_1$, $x_2$ and $x_3$ used
to estimate the probability of success. This model includes all three main effects, the three $2$-
way interactions, and the one $3$-way interaction.
Derive the odds ratio to examine the odds of a success across two levels of one of the
explanatory variables, say $x_1$, and derive its variance.
I assume you start by:
$log(frac{pi}{1-pi}) = beta_1x_1 + beta_2x_2 + beta_3x_3 +beta_4x_1x_2 + beta_5x_1x_3 + beta_6x_2x_3 + beta_7x_1x_2x_3$
and then taking exponetials on both sides:
$frac{pi}{1-pi} = e^{beta_1x_1 + beta_2x_2 + beta_3x_3 +beta_4x_1x_2 + beta_5x_1x_3 + beta_6x_2x_3 + beta_7x_1x_2x_3}$
However I am not sure how to proceed.
probability statistics regression logistic-regression
asked Nov 21 at 3:24
Alex Chavez
440620
add a comment |
up vote
2
down vote
favorite
up vote
2
down vote
favorite
Suppose a logistic regression model has three binary explanatory variables $x_1$, $x_2$ and $x_3$ used
to estimate the probability of success. This model includes all three main effects, the three $2$-
way interactions, and the one $3$-way interaction.
Derive the odds ratio to examine the odds of a success across two levels of one of the
explanatory variables, say $x_1$, and derive its variance.
I assume you start by:
$log(frac{pi}{1-pi}) = beta_1x_1 + beta_2x_2 + beta_3x_3 +beta_4x_1x_2 + beta_5x_1x_3 + beta_6x_2x_3 + beta_7x_1x_2x_3$
and then taking exponetials on both sides:
$frac{pi}{1-pi} = e^{beta_1x_1 + beta_2x_2 + beta_3x_3 +beta_4x_1x_2 + beta_5x_1x_3 + beta_6x_2x_3 + beta_7x_1x_2x_3}$
However I am not sure how to proceed.
probability statistics regression logistic-regression
asked Nov 21 at 3:24
Alex Chavez
440620
Suppose a logistic regression model has three binary explanatory variables $x_1$, $x_2$ and $x_3$ used
to estimate the probability of success. This model includes all three main effects, the three $2$-
way interactions, and the one $3$-way interaction.
Derive the odds ratio to examine the odds of a success across two levels of one of the
explanatory variables, say $x_1$, and derive its variance.
I assume you start by:
$log(frac{pi}{1-pi}) = beta_1x_1 + beta_2x_2 + beta_3x_3 +beta_4x_1x_2 + beta_5x_1x_3 + beta_6x_2x_3 + beta_7x_1x_2x_3$
and then taking exponetials on both sides:
$frac{pi}{1-pi} = e^{beta_1x_1 + beta_2x_2 + beta_3x_3 +beta_4x_1x_2 + beta_5x_1x_3 + beta_6x_2x_3 + beta_7x_1x_2x_3}$
However I am not sure how to proceed.
probability statistics regression logistic-regression
probability statistics regression logistic-regression
asked Nov 21 at 3:24
Alex Chavez
440620
asked Nov 21 at 3:24
Alex Chavez
440620
edited Nov 24 at 6:49
asked Nov 21 at 3:24
Alex Chavez
440620
asked Nov 21 at 3:24
Alex Chavez
440620
asked Nov 21 at 3:24
Alex Chavez
440620
440620
add a comment |
add a comment |
1 Answer
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Think of odds ratio as, keeping all else constant what difference does change by 1 in this variable do.
If you want to find the odds ratio between x1 = 0 and x1 = 1, you can simply keep all other variables in their base cases and find the ratio between expected odds when x1= 0 and x1 = 1
from the step
$frac{pi}{1-pi} = e^{beta_1x_1 + beta_2x_2 + beta_3x_3 +beta_4x_1x_2 + beta_5x_1x_3 + beta_6x_2x_3 + beta_7x_1x_2x_3}$
set $x_2$ = E($x_2$) and $x_3$ = E($x_3$). When $x_1$ = 1 you have
$frac{pi}{1-pi} = e^{beta_1 + beta_2E(x_2) + beta_3E(x_3) +beta_4E(x_2) + beta_5E(x_3) + beta_6E(x_2)E(x_3) + beta_7E(x_2)E(x_3)}$
when $x_1$ = 0
$frac{pi}{1-pi} = e^{beta_2E(x_2) + beta_3E(x_3) + beta_6E(x_2)E(x_3) }$
dividing the two, you get
odds ratio = $ e^{beta_1 + beta_4E(x_2) + beta_5E(x_3) + beta_7E(x_2)E(x_3) }$
As you can see, all the terms that don't contain the variable x1 simplifies.
In almost all statistical software, the odd ratio is calculated using the "base case", i.e. when all other explanatory variables are 0. This means plugging 0 to E($x_i$) It yields to a much faster runtime and a nicer looking formula of:
odds ratio = $e^{(β_1)}$
Keep in mind, this also ignores the interaction between variables.
For the variance, you can derive it from the final equation using the variance of the coefficients
Hope it helps
Edit: edited for latex
answered Nov 27 at 23:08
Ofya
4948
add a comment |
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1 Answer
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1 Answer
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active
oldest
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active
oldest
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active
oldest
votes
up vote
0
down vote
Think of odds ratio as, keeping all else constant what difference does change by 1 in this variable do.
If you want to find the odds ratio between x1 = 0 and x1 = 1, you can simply keep all other variables in their base cases and find the ratio between expected odds when x1= 0 and x1 = 1
from the step
$frac{pi}{1-pi} = e^{beta_1x_1 + beta_2x_2 + beta_3x_3 +beta_4x_1x_2 + beta_5x_1x_3 + beta_6x_2x_3 + beta_7x_1x_2x_3}$
set $x_2$ = E($x_2$) and $x_3$ = E($x_3$). When $x_1$ = 1 you have
$frac{pi}{1-pi} = e^{beta_1 + beta_2E(x_2) + beta_3E(x_3) +beta_4E(x_2) + beta_5E(x_3) + beta_6E(x_2)E(x_3) + beta_7E(x_2)E(x_3)}$
when $x_1$ = 0
$frac{pi}{1-pi} = e^{beta_2E(x_2) + beta_3E(x_3) + beta_6E(x_2)E(x_3) }$
dividing the two, you get
odds ratio = $ e^{beta_1 + beta_4E(x_2) + beta_5E(x_3) + beta_7E(x_2)E(x_3) }$
As you can see, all the terms that don't contain the variable x1 simplifies.
In almost all statistical software, the odd ratio is calculated using the "base case", i.e. when all other explanatory variables are 0. This means plugging 0 to E($x_i$) It yields to a much faster runtime and a nicer looking formula of:
odds ratio = $e^{(β_1)}$
Keep in mind, this also ignores the interaction between variables.
For the variance, you can derive it from the final equation using the variance of the coefficients
Hope it helps
Edit: edited for latex
answered Nov 27 at 23:08
Ofya
4948
add a comment |
up vote
0
down vote
Think of odds ratio as, keeping all else constant what difference does change by 1 in this variable do.
If you want to find the odds ratio between x1 = 0 and x1 = 1, you can simply keep all other variables in their base cases and find the ratio between expected odds when x1= 0 and x1 = 1
from the step
$frac{pi}{1-pi} = e^{beta_1x_1 + beta_2x_2 + beta_3x_3 +beta_4x_1x_2 + beta_5x_1x_3 + beta_6x_2x_3 + beta_7x_1x_2x_3}$
set $x_2$ = E($x_2$) and $x_3$ = E($x_3$). When $x_1$ = 1 you have
$frac{pi}{1-pi} = e^{beta_1 + beta_2E(x_2) + beta_3E(x_3) +beta_4E(x_2) + beta_5E(x_3) + beta_6E(x_2)E(x_3) + beta_7E(x_2)E(x_3)}$
when $x_1$ = 0
$frac{pi}{1-pi} = e^{beta_2E(x_2) + beta_3E(x_3) + beta_6E(x_2)E(x_3) }$
dividing the two, you get
odds ratio = $ e^{beta_1 + beta_4E(x_2) + beta_5E(x_3) + beta_7E(x_2)E(x_3) }$
As you can see, all the terms that don't contain the variable x1 simplifies.
In almost all statistical software, the odd ratio is calculated using the "base case", i.e. when all other explanatory variables are 0. This means plugging 0 to E($x_i$) It yields to a much faster runtime and a nicer looking formula of:
odds ratio = $e^{(β_1)}$
Keep in mind, this also ignores the interaction between variables.
For the variance, you can derive it from the final equation using the variance of the coefficients
Hope it helps
Edit: edited for latex
answered Nov 27 at 23:08
Ofya
4948
add a comment |
up vote
0
down vote
up vote
0
down vote
Think of odds ratio as, keeping all else constant what difference does change by 1 in this variable do.
If you want to find the odds ratio between x1 = 0 and x1 = 1, you can simply keep all other variables in their base cases and find the ratio between expected odds when x1= 0 and x1 = 1
from the step
$frac{pi}{1-pi} = e^{beta_1x_1 + beta_2x_2 + beta_3x_3 +beta_4x_1x_2 + beta_5x_1x_3 + beta_6x_2x_3 + beta_7x_1x_2x_3}$
set $x_2$ = E($x_2$) and $x_3$ = E($x_3$). When $x_1$ = 1 you have
$frac{pi}{1-pi} = e^{beta_1 + beta_2E(x_2) + beta_3E(x_3) +beta_4E(x_2) + beta_5E(x_3) + beta_6E(x_2)E(x_3) + beta_7E(x_2)E(x_3)}$
when $x_1$ = 0
$frac{pi}{1-pi} = e^{beta_2E(x_2) + beta_3E(x_3) + beta_6E(x_2)E(x_3) }$
dividing the two, you get
odds ratio = $ e^{beta_1 + beta_4E(x_2) + beta_5E(x_3) + beta_7E(x_2)E(x_3) }$
As you can see, all the terms that don't contain the variable x1 simplifies.
In almost all statistical software, the odd ratio is calculated using the "base case", i.e. when all other explanatory variables are 0. This means plugging 0 to E($x_i$) It yields to a much faster runtime and a nicer looking formula of:
odds ratio = $e^{(β_1)}$
Keep in mind, this also ignores the interaction between variables.
For the variance, you can derive it from the final equation using the variance of the coefficients
Hope it helps
Edit: edited for latex
answered Nov 27 at 23:08
Ofya
4948
Think of odds ratio as, keeping all else constant what difference does change by 1 in this variable do.
If you want to find the odds ratio between x1 = 0 and x1 = 1, you can simply keep all other variables in their base cases and find the ratio between expected odds when x1= 0 and x1 = 1
from the step
$frac{pi}{1-pi} = e^{beta_1x_1 + beta_2x_2 + beta_3x_3 +beta_4x_1x_2 + beta_5x_1x_3 + beta_6x_2x_3 + beta_7x_1x_2x_3}$
set $x_2$ = E($x_2$) and $x_3$ = E($x_3$). When $x_1$ = 1 you have
$frac{pi}{1-pi} = e^{beta_1 + beta_2E(x_2) + beta_3E(x_3) +beta_4E(x_2) + beta_5E(x_3) + beta_6E(x_2)E(x_3) + beta_7E(x_2)E(x_3)}$
when $x_1$ = 0
$frac{pi}{1-pi} = e^{beta_2E(x_2) + beta_3E(x_3) + beta_6E(x_2)E(x_3) }$
dividing the two, you get
odds ratio = $ e^{beta_1 + beta_4E(x_2) + beta_5E(x_3) + beta_7E(x_2)E(x_3) }$
As you can see, all the terms that don't contain the variable x1 simplifies.
In almost all statistical software, the odd ratio is calculated using the "base case", i.e. when all other explanatory variables are 0. This means plugging 0 to E($x_i$) It yields to a much faster runtime and a nicer looking formula of:
odds ratio = $e^{(β_1)}$
Keep in mind, this also ignores the interaction between variables.
For the variance, you can derive it from the final equation using the variance of the coefficients
Hope it helps
Edit: edited for latex
answered Nov 27 at 23:08
Ofya
4948
edited Nov 28 at 15:25
answered Nov 27 at 23:08
Ofya
4948
answered Nov 27 at 23:08
Ofya
4948
answered Nov 27 at 23:08
Ofya
4948
4948
add a comment |
add a comment |
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