Data transformations1 / 29

Dr. Lucy D'Agostino McGowan

 Species AreaGo to RStudio Cloud and open Species Area
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Steps for modeling

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Steps for modeling

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Dr. Lucy D'Agostino McGowan

Conditions for simple linear regressionLinearity
Zero Mean
Constant Variance
Independence
Random
Normality
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Conditions for simple linear regression

What can we do when these conditions aren't met?

Linearity
Zero Mean
Constant Variance
Independence
Random
Normality

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Dr. Lucy D'Agostino McGowan

transformations!7 / 29

Example

Number of MDs and community hospitals for sample of 83 metropolitan areas

City	NumMDs	NumHospitals
Holland-Grand Haven, MI	349	3
Louisville, KY-IN	4042	18
Battle Creek, MI	256	3
Madison, WI	2679	7
Fort Smith, AR-OK	502	8
Sarasota-Bradenton-Venice, FL	2352	7
Anderson, IN	200	2
Honolulu, HI	3478	13
Asheville, NC	1489	5
Winston-Salem, NC	2018	6

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Example

Number of MDs and community hospitals for sample of 83 metropolitan areas

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Choose

Number of MDs and community hospitals for sample of 83 metropolitan areas

$\hat{Number of MDs} = {\hat{β}}_{0} + {\hat{β}}_{1} Number of hospitals$

10 / 29

Fit

Number of MDs and community hospitals for sample of 83 metropolitan areas

lm(NumMDs ~ NumHospitals, data = MetroHealth83)

## 
## Call:
## lm(formula = NumMDs ~ NumHospitals, data = MetroHealth83)
## 
## Coefficients:
##  (Intercept)  NumHospitals  
##       -385.1         282.0

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Fit

Number of MDs and community hospitals for sample of 83 metropolitan areas

Refresher: What is ${\hat{β}}_{0}$ and what does it mean?

lm(NumMDs ~ NumHospitals, data = MetroHealth83)

## 
## Call:
## lm(formula = NumMDs ~ NumHospitals, data = MetroHealth83)
## 
## Coefficients:
##  (Intercept)  NumHospitals  
##       -385.1         282.0

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Fit

Number of MDs and community hospitals for sample of 83 metropolitan areas

Refresher: What is ${\hat{β}}_{1}$ and what does it mean?

lm(NumMDs ~ NumHospitals, data = MetroHealth83)

## 
## Call:
## lm(formula = NumMDs ~ NumHospitals, data = MetroHealth83)
## 
## Coefficients:
##  (Intercept)  NumHospitals  
##       -385.1         282.0

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Assess

Number of MDs and community hospitals for sample of 83 metropolitan areas

What can I use to assess the linearity and constant variance assumptions?

14 / 29

Assess

Number of MDs and community hospitals for sample of 83 metropolitan areas

What can I use to assess the linearity and constant variance assumptions?

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Assess

Number of MDs and community hospitals for sample of 83 metropolitan areas

What do you think?

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Assess

Number of MDs and community hospitals for sample of 83 metropolitan areas

What can I use to assess the normality assumption?

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Assess

Number of MDs and community hospitals for sample of 83 metropolitan areas

What do you think?

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Assess

Number of MDs and community hospitals for sample of 83 metropolitan areas

What do you think?

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Dr. Lucy D'Agostino McGowan

Chooseto stabilize the variance of the response ( yy, in this case NumMDs) across different values of the predictor ( xx, in this case NumHospitals), we can transform yy or xx
20 / 29

Dr. Lucy D'Agostino McGowan

Chooseto stabilize the variance of the response ( yy, in this case NumMDs) across different values of the predictor ( xx, in this case NumHospitals), we can transform yy or xx
typical transformations: √yy
logylog⁡y
x2x2
1/x1/x

20 / 29

Dr. Lucy D'Agostino McGowan

Chooseto stabilize the variance of the response ( yy, in this case NumMDs) across different values of the predictor ( xx, in this case NumHospitals), we can transform yy or xx
typical transformations: √yy
logylog⁡y
x2x2
1/x1/x

For count data, such as the number of doctors or hospitals where the variability increases along with the magnitudes of the variables, a square root transformation is often helpful
20 / 29

Choose

Number of MDs and community hospitals for sample of 83 metropolitan areas

$\hat{\sqrt{Number of MDs}} = {\hat{β}}_{0} + {\hat{β}}_{1} Number of hospitals$

21 / 29

Fit

Number of MDs and community hospitals for sample of 83 metropolitan areas

lm(sqrt(NumMDs) ~ NumHospitals, data = MetroHealth83)

## 
## Call:
## lm(formula = sqrt(NumMDs) ~ NumHospitals, data = MetroHealth83)
## 
## Coefficients:
##  (Intercept)  NumHospitals  
##       14.033         2.915

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Assess

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Assess

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Use

Number of MDs and community hospitals for sample of 83 metropolitan areas

$\hat{\sqrt{Number of MDs}} = {\hat{β}}_{0} + {\hat{β}}_{1} Number of hospitals$

$\hat{Number of MDs} = ({\hat{β}}_{0} + {\hat{β}}_{1} Number of hospitals)^{2}$

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Use

$\hat{Number of MDs} = ({\hat{β}}_{0} + {\hat{β}}_{1} Number of hospitals)^{2}$

## 
## Call:
## lm(formula = sqrt(NumMDs) ~ NumHospitals, data = MetroHealth83)
## 
## Coefficients:
##  (Intercept)  NumHospitals  
##       14.033         2.915

##                City NumMDs NumHospitals
## 1 Louisville, KY-IN   4042           18

26 / 29

Use

$\hat{Number of MDs} = ({\hat{β}}_{0} + {\hat{β}}_{1} Number of hospitals)^{2}$

## 
## Call:
## lm(formula = sqrt(NumMDs) ~ NumHospitals, data = MetroHealth83)
## 
## Coefficients:
##  (Intercept)  NumHospitals  
##       14.033         2.915

##                City NumMDs NumHospitals
## 1 Louisville, KY-IN   4042           18

(14.033 + 2.915 * 18)^2

## [1] 4422.649

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Use

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Dr. Lucy D'Agostino McGowan

 Species AreaGo to RStudio Cloud and open Species Area
For each question you work on, set the eval chunk option to TRUE and knit
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Data transformations

`Species Area`

Steps for modeling

Steps for modeling

Conditions for simple linear regression

Conditions for simple linear regression

transformations!

Example

Example

Choose

$\hat{Number of MDs} = {\hat{β}}_{0} + {\hat{β}}_{1} Number of hospitals$

Fit

Fit

Fit

Assess

Assess

Assess

Assess

Assess

Assess

Choose

Choose

Choose

Choose

$\hat{\sqrt{Number of MDs}} = {\hat{β}}_{0} + {\hat{β}}_{1} Number of hospitals$

Fit

Assess

Assess

Use

$\hat{\sqrt{Number of MDs}} = {\hat{β}}_{0} + {\hat{β}}_{1} Number of hospitals$

$\hat{Number of MDs} = ({\hat{β}}_{0} + {\hat{β}}_{1} Number of hospitals)^{2}$

Use

$\hat{Number of MDs} = ({\hat{β}}_{0} + {\hat{β}}_{1} Number of hospitals)^{2}$

Use

$\hat{Number of MDs} = ({\hat{β}}_{0} + {\hat{β}}_{1} Number of hospitals)^{2}$

Use

`Species Area`

`Species Area`

Help

Data transformations

Species Area

Steps for modeling

Steps for modeling

Conditions for simple linear regression

Conditions for simple linear regression

transformations!

Example

Example

Choose

^Number of MDs=^β0+^β1Number of hospitalsNumber of MDs^=β^0+β^1Number of hospitals

Fit

Fit

Fit

Assess

Assess

Assess

Assess

Assess

Assess

Choose

Choose

Choose

Choose

^√Number of MDs=^β0+^β1Number of hospitalsNumber of MDs^=β^0+β^1Number of hospitals

Fit

Assess

Assess

Use

^√Number of MDs=^β0+^β1Number of hospitalsNumber of MDs^=β^0+β^1Number of hospitals

^Number of MDs=(^β0+^β1Number of hospitals)2Number of MDs^=(β^0+β^1Number of hospitals)2

Use

^Number of MDs=(^β0+^β1Number of hospitals)2Number of MDs^=(β^0+β^1Number of hospitals)2

Use

^Number of MDs=(^β0+^β1Number of hospitals)2Number of MDs^=(β^0+β^1Number of hospitals)2

Use

Species Area

Species Area

Help

`Species Area`

$\hat{Number of MDs} = {\hat{β}}_{0} + {\hat{β}}_{1} Number of hospitals$

$\hat{\sqrt{Number of MDs}} = {\hat{β}}_{0} + {\hat{β}}_{1} Number of hospitals$

$\hat{\sqrt{Number of MDs}} = {\hat{β}}_{0} + {\hat{β}}_{1} Number of hospitals$

$\hat{Number of MDs} = ({\hat{β}}_{0} + {\hat{β}}_{1} Number of hospitals)^{2}$

$\hat{Number of MDs} = ({\hat{β}}_{0} + {\hat{β}}_{1} Number of hospitals)^{2}$

$\hat{Number of MDs} = ({\hat{β}}_{0} + {\hat{β}}_{1} Number of hospitals)^{2}$

`Species Area`

`Species Area`