Skip to main content
Cross Validated

Return to Question

deleted 16 characters in body
Source Link
kjetil b halvorsen ♦
kjetil b halvorsen ♦
  • 82.8k
  • 32
  • 201
  • 663

The appendix of the paper of McPherson et al (1982) (see screenshot below) contains a derivation of the Systematic Component of Variation (SCV). I understand the derivation with exception of the first step. Here are the premises:

$O_i$: observed cases in region i
$E_i$: expected cases in region i
$\lambda_i$: multiplicative factor associated with region i ($O_i=\lambda_i*E_i$)

Now the following assumptions have been made:

$O_i$ is approximately Poisson distributed with mean $\lambda_iE_i$
$\lambda_i$ is considered as a random variable with expected value $1$ and variance $\sigma^2$.

From these the following formula is concluded:

var($O_i$) = $E_i^2\sigma^2$ + $E_i$

I tried to find out how to get the formula by the given premises and assumptions and didn't succeed. Any idea? Thanks for help.

Screenshot of McPherson's derivation:

enter image description here

The appendix of the paper of McPherson et al (1982) (see screenshot below) contains a derivation of the Systematic Component of Variation (SCV). I understand the derivation with exception of the first step. Here are the premises:

$O_i$: observed cases in region i
$E_i$: expected cases in region i
$\lambda_i$: multiplicative factor associated with region i ($O_i=\lambda_i*E_i$)

Now the following assumptions have been made:

$O_i$ is approximately Poisson distributed with mean $\lambda_iE_i$
$\lambda_i$ is considered as a random variable with expected value $1$ and variance $\sigma^2$.

From these the following formula is concluded:

var($O_i$) = $E_i^2\sigma^2$ + $E_i$

I tried to find out how to get the formula by the given premises and assumptions and didn't succeed. Any idea? Thanks for help.

Screenshot of McPherson's derivation:

enter image description here

The appendix of the paper of McPherson et al (1982) (see screenshot below) contains a derivation of the Systematic Component of Variation (SCV). I understand the derivation with exception of the first step. Here are the premises:

$O_i$: observed cases in region i
$E_i$: expected cases in region i
$\lambda_i$: multiplicative factor associated with region i ($O_i=\lambda_i*E_i$)

Now the following assumptions have been made:

$O_i$ is approximately Poisson distributed with mean $\lambda_iE_i$
$\lambda_i$ is considered as a random variable with expected value $1$ and variance $\sigma^2$.

From these the following formula is concluded:

var($O_i$) = $E_i^2\sigma^2$ + $E_i$

I tried to find out how to get the formula by the given premises and assumptions and didn't succeed. Any idea?

Screenshot of McPherson's derivation:

enter image description here

Corrected spelling
Source Link
edit approved Dec 16, 2016 at 14:40
Tumbledown
edit approved Dec 16, 2016 at 14:40
Tumbledown
  • 909
  • 2
  • 10
  • 23

The appendix of the paper of McPherson et al (1982) (see screenshot below) contains a derivation of the systematic component variationSystematic Component of Variation (SCV). I understand the derivation with exception of the first step. Here are the premises:

$O_i$: observed cases in region i
$E_i$: expected cases in region i
$\lambda_i$: multiplicative factor associated with region i ($O_i=\lambda_i*E_i$)

Now the following assumptions have been donemade:

$O_i$ is approximately Poisson distributed with mean $\lambda_iE_i$
$\lambda_i$ is considered as a random variable with expected value $1$ and variance $\sigma^2$.

From these the following formula is concluded:

var($O_i$) = $E_i^2\sigma^2$ + $E_i$

I tried to find out how to get the formula by the given premissespremises and assumptions and didn't succeed. Any idea? Thanks for help.

Screenshot of McPherson's derivation:

enter image description here

The appendix of the paper of McPherson et al (1982) (see screenshot below) contains a derivation of the systematic component variation (SCV). I understand the derivation with exception of the first step. Here are the premises:

$O_i$: observed cases in region i
$E_i$: expected cases in region i
$\lambda_i$: multiplicative factor associated with region i ($O_i=\lambda_i*E_i$)

Now the following assumptions have been done:

$O_i$ is approximately Poisson distributed with mean $\lambda_iE_i$
$\lambda_i$ is considered as a random variable with expected value $1$ and variance $\sigma^2$.

From these the following formula is concluded:

var($O_i$) = $E_i^2\sigma^2$ + $E_i$

I tried to find out how to get the formula by the given premisses and assumptions and didn't succeed. Any idea? Thanks for help.

Screenshot of McPherson's derivation:

enter image description here

The appendix of the paper of McPherson et al (1982) (see screenshot below) contains a derivation of the Systematic Component of Variation (SCV). I understand the derivation with exception of the first step. Here are the premises:

$O_i$: observed cases in region i
$E_i$: expected cases in region i
$\lambda_i$: multiplicative factor associated with region i ($O_i=\lambda_i*E_i$)

Now the following assumptions have been made:

$O_i$ is approximately Poisson distributed with mean $\lambda_iE_i$
$\lambda_i$ is considered as a random variable with expected value $1$ and variance $\sigma^2$.

From these the following formula is concluded:

var($O_i$) = $E_i^2\sigma^2$ + $E_i$

I tried to find out how to get the formula by the given premises and assumptions and didn't succeed. Any idea? Thanks for help.

Screenshot of McPherson's derivation:

enter image description here

Corrected spelling
Source Link
edit approved Dec 16, 2016 at 14:15
Tumbledown
edit approved Dec 16, 2016 at 14:15
Tumbledown
  • 909
  • 2
  • 10
  • 23

Systematic component variationComponent of Variation

The appendix of the paper of McPherson et al (1982) (see schreenshotscreenshot below) contains a derivation of the systematic component variation SCV(SCV). I understand the derivation with exception of the first step. Here are the premissespremises:

$O_i$: observed cases in region i
$E_i$: expected cases in region i
$\lambda_i$: multiplicative factor associated with region i ($O_i=\lambda_i*E_i$)

Now the following assumptions have been done:

$O_i$ is approximately Poisson distributed with mean $\lambda_iE_i$
$\lambda_i$ is considered as a random variable with expected value $1$ and variance $\sigma^2$.

From these the following formula is concluded:

var($O_i$) = $E_i^2\sigma^2$ + $E_i$

I tried to find out how to get the formula by the given premisses and assumptions and didn't succeed. Any idea? Thanks for help.

SchreenshotScreenshot of PhersonsMcPherson's derivation:

enter image description here

Systematic component variation

The appendix of the paper of McPherson et al (1982) (see schreenshot below) contains a derivation of the systematic component variation SCV. I understand the derivation with exception of the first step. Here are the premisses:

$O_i$: observed cases in region i
$E_i$: expected cases in region i
$\lambda_i$: multiplicative factor associated with region i ($O_i=\lambda_i*E_i$)

Now the following assumptions have been done:

$O_i$ is approximately Poisson distributed with mean $\lambda_iE_i$
$\lambda_i$ is considered as a random variable with expected value $1$ and variance $\sigma^2$.

From these the following formula is concluded:

var($O_i$) = $E_i^2\sigma^2$ + $E_i$

I tried to find out how to get the formula by the given premisses and assumptions and didn't succeed. Any idea? Thanks for help.

Schreenshot of Phersons derivation:

enter image description here

Systematic Component of Variation

The appendix of the paper of McPherson et al (1982) (see screenshot below) contains a derivation of the systematic component variation (SCV). I understand the derivation with exception of the first step. Here are the premises:

$O_i$: observed cases in region i
$E_i$: expected cases in region i
$\lambda_i$: multiplicative factor associated with region i ($O_i=\lambda_i*E_i$)

Now the following assumptions have been done:

$O_i$ is approximately Poisson distributed with mean $\lambda_iE_i$
$\lambda_i$ is considered as a random variable with expected value $1$ and variance $\sigma^2$.

From these the following formula is concluded:

var($O_i$) = $E_i^2\sigma^2$ + $E_i$

I tried to find out how to get the formula by the given premisses and assumptions and didn't succeed. Any idea? Thanks for help.

Screenshot of McPherson's derivation:

enter image description here

Schreenshon
Source Link
giordano
giordano
  • 1.1k
  • 1
  • 12
  • 24
Orthography
Source Link
giordano
giordano
  • 1.1k
  • 1
  • 12
  • 24
Source Link
giordano
giordano
  • 1.1k
  • 1
  • 12
  • 24