correlacion y regresion 2

8/8/2019 Correlacion y Regresion 2

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Correlation & Regression

Do heavier people burn more energy? Does

wine consumption affect cause a decrease inheart disease?

These questions reflect a desire to understand the

relationship between two variables.

What we need:

1. A plot/graph to view the relationship

2. Characteristics to describe

3. Measures of the characteristics4. Method to make inferences about the relationship


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The grapha Scatter Plot

X

YResponse variable

(dependent variable)

Explanatory variable

(independent variable)


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Do heavier people burn more energy?

Response: metabolic rate

Explanatory: weight or mass

Does wine consumption cause a decrease in heart

disease?

Response: death rate from heart diseaseExplanatory: wine consumption


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60504030

2000

1500

1000

Mass(kg)

Rate(cal)

Do heavier people burn more energy?

Lean body mass vs. metabolic rate


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0 1 2 3 4 5 6 7 8 9

100

200

300

Alcoho l

hrt_

deathrate

Is wine good for your heart?

wine consumption vs. heart disease rate (per 100,000)

wine consumption


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Interpretingcharacteristics to look for:

Patterns:

Form (clusters, scatter, linear..)

Direction (positive, negative)

Strength ( how closely points follow form)

Deviations:

Outliers

Interpret the last two scatter plots.


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Options to consider:

Adding a categorical variable


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Scatter plot:

relationship between

quantitative variables

Form: Linear is

probably the most

common form

Strength: We can

measure the strength of

a linear relationship

because our eyes can

deceive us!!!

Strength?

Strength?


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Correlation

measure the direction and strength of a linear relationship

Standardised value of each x

Standardised value of each y

Correlation is an average product of standardised values


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Quantitative variables

Linear relationships

r has no units

r can be between 1 and 1

Positive r =positive association

Negative r =

negative association

0 = no association

r is influenced by outliers

Correlation = r


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Correlations: Mass (kg), Rate (cal)Pearson correlation of Mass(kg) and Rate(cal) = 0.865

P-Value = 0.000

60504030

2000

1500

1000

a g

ae

a

o hea e peop e bu n o e ene g ?

Lean bod a e abo a e

r


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30 40 50 60

1000

1500

2000

a g

ae

a

a e +

Fe ae o

We gh a e abo a e

Correlations: Mass (kg)_F, Rate (cal)_FPearson correlation of Mass(kg)_F and Rate(cal)_F = 0.876

Correlations: Mass (kg)_M, Rate (cal)_M

Pearson correlation of Mass (kg)_M and Rate (cal)_M = 0.592


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Correlations: Alcohol, heart_death ratePearson correlation of Alcohol and hrt_death rate = -0.843

0 1 2 3 4 5 6 7 8 9

100

200

300

Alcohol

h

rt_

thr

a

t

I w ood for ourheart

w econsumpt on s. heart diseaserate per100,000)

wineconsumption


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Correlations: Alcohol Wine consumption, heart death rate

Pearson correlation of Alc Wine consumption and hrt death rate = -0.648

1 2 3 4

150

200

250

300

h

d

eah

ae

hea di ea e dea h a e ine on u p i n

ou lie e o ed

Al wine n u p i n


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Linear relationshipsusing a LINE

0 1 2 3 4 5 6 7 8 9

100

200

300

A l c o h o l

h

rt

thr

t

I ood foryourheart

econsumpt onvs. heart disease rate (per100,000)

ineconsumption

We can summarise an overall linear form with a linethe

best line is called the Regression Line


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9876543210

30 0

20 0

10 0

wi c s m i

t

t

S = 37.8786 R-S q = 71.0 % R-Sq ( j) = 69.3 %

t t = 260.563 - 22.9688w i c s m t

Fitt g ssi li t t vs.wi c s m ti

A regression line describes how a response variable changes as an

explanatory variable changes. We can nowpredicta value of y when

given an x.

What would be the death rate

due to heart disease if the

average daily consumption of

wine was 3 glasses?

191.66 deaths per 100,000


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How do we determine the regression line?

We want the vertical

distances from the

points (observed) to

the line (predicted) to

be as small as

possiblethis means

our error in predicting

y is small.


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Calculating the line

We will use the method of least squares to calculate the line.

Least squares regression is the line that makes the sum of the

squares of the vertical distances as small as possible.

! a bx

b! rsysx

a ! y bx

Equation of the line (read y hat)

b is the slope (rate of change iny whenx

increases)

a is the y intercept (value of y whenxis 0)


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9876543210

30 0

20 0

10 0

wine c ns tion

deat

rate

S = 37.8786 R-Sq = 71.0 % R-S q (ad j ) = 69.3 %

de a t ra te = 260.563 - 22.9688 w ine c on s

t

Fitted regression line deat rate vs.wine cons tion

The regression equation is

death rate = 260.563 - 22.9688 wine consumption

S = 37.8786 R-Sq = 71.0 % R-Sq(adj) = 69.3 %

Analysis of Variance

Source DF SS MS F P

Regression 1 59813.6 59813.6 41.6881 0.000

Error 17 24391.4 1434.8

Total 18 84204.9


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Facts about regression.

1. Clear distinction between the response variable and theexplanatory variable.

2. Correlation and slopea change in one Wofx

corresponds to a change ofrW in y.

3. Least-squares regression line passes through

4. Some variation (spread) in y can be accounted for by

changes in x when there is a linear relationship. The

square of the correlation coefficient is the the fraction of

the variation in y values that is explained by changes in x.

(x,y)

!variation in y due to x

total variation in observed y

= coefficientofdetermination


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9876543210

30 0

20 0

10 0

wine cons tion

deat

rate

S = 37.8786 R-Sq = 71.0 % R-S q (ad j ) = 69.3 %

de a t ra te = 260.563 - 22.9688 w ine c on s

t

Fitted regression line deat rate vs.wine cons tion



S = 37.8786 R-Sq = 71.0 % R-Sq(adj) = 69.3 %

R-sq can have a value between 0 and 1.


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VARIATION OF DEPENDENT Y


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Residuals

the left overs from least-squares regression

Deviations from the overall pattern are important. The deviations

In regression are the scatter of points about the line. The

vertical distances from the line to the points are called residualsand they are the left-over variation after a regression line is fit.

Residual = observedy predictedy

residuals ! y y


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Obs Alcohol hrt_deat Fit SE Fit Residual St Resid1 2.50 211.00 203.14 8.89 7.86 0.21

2 3.90 167.00 170.99 9.23 -3.99 -0.11

3 2.90 131.00 193.95 8.70 -62.95 -1.71

4 2.40 191.00 205.44 8.97 -14.44 -0.39

5 2.90 220.00 193.95 8.70 26.05 0.71

6 0.80 297.00 242.19 11.76 54.81 1.52

7 9.10 71.00 51.55 23.29 19.45 0.65 X

8 0.80 211.00 242.19 11.76 -31.19 -0.87

9 0.70 300.00 244.49 12.00 55.51 1.55

10 7.90 107.00 79.11 19.39 27.89 0.86

11 1.80 167.00 219.22 9.72 -52.22 -1.43

12 1.90 266.00 216.92 9.57 49.08 1.34

13 0.80 227.00 242.19 11.76 -15.19 -0.42

14 6.50 86.00 111.27 15.11 -25.27 -0.73

15 1.60 207.00 223.81 10.06 -16.81 -0.46

16 5.80 115.00 127.34 13.15 -12.34 -0.35

17 1.30 285.00 230.70 10.64 54.30 1.49

18 1.20 199.00 233.00 10.85 -34.00 -0.94

19 2.70 172.00 198.55 8.77 -26.55 -0.72



s = 37.8786 R-Sq = 71.0 % R-Sq(adj) = 69.3 %

The residuals are.

The mean of residuals is always equal to 0


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Residual Plots

9876543210

50

0

-50

Alcohol

Residual

Residuals Versus Alcohol(response is hr

_deat)

Things to look for:

1. A curved pattern means

the relationship is not

linear.

2. Increasing/decreasing

spread about the line

3. Individual points with

large residuals

4. Individual points that areextreme in the x

directionDo we have any influential

points here?


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Ideal residual pattern

Curvaturea linear fit is not

appropriate

Increasing variation


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4321

30 0

25 0

20 0

15 0

C5

C

6

S 40 .0879

-S

42 .0

-S

(

!

j ) 37 .5

C6 28 0 .21 5 -33 .7666 C5

Regressi Pl t

9876543210

30 0

20 0

10 0

wi ec s ti

eat

rate

S

37 .8786 R -S

71 .0

R -S

(adj )

69 .3

de a t"

ra te 26 0 .56 3 -22 .9688 w i#

e c$ #

s% &

t

itted regressi li edeat rate s.wi ec s ti

9876543210

50

0

-50

Alc'

(

'

l

Residual

Residuals VersusAlc l(res ) 0 1 seis 2 rt_deat)

4321

50

0

-50

C5

Res

idual

Res iduals VersusC5(res

3

4

5se is C6 )


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Attention!! Caution!!

1. Correlation and regression describe only linearrelationships

2. R and r-sq are not resistant

3. Do not extrapolate!!! What is extrapolate?

4. Correlations based on averages are too high when

applied to individualsif the data has been averaged,

the values of correlation and regression cannot be used

with un-averaged values. (i.e., average alcohol

consumption per countrynot individuals).

5. Lurking variableslike the male/female variable in theweight vs. energy and the possible Mediterranean

variable in the wine data.

6. Correlation/association is not causation.

correlacion y regresion 2

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