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Regression Analysis

Modeling Relationships

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Regression Analysis

Regression Analysis is a study of the relationship between a set of independent variables and the dependent variable.

Errorxxy ...22110

DependentVariable

IndependentVariables

The Linear Equation representing the ‘true’ or population relationship:

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Variables

Dependent Variable: Also called the predicted variable. Its value depends on, or can be predicted by the independent variables.

Independent Variables: Also called the predictor variables. These can be measured directly, and are used to predict the dependent (or to simply understand it better).

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Modeling Process

Define Goal To study the impact of various factors on individual health

Choose y Lung Capacity, measured in cc.

List possible Xs Minutes of Exercise per day, # of days/week of exercise, ethnicity, gender, age, height, altitude at which lived.

Collect Data Primary, Secondary sources

Preliminary Analyses Univariate, bivariate

Build Regression Model How is y related to all the Xs?

Evaluate Model How good is the model at predicting y?

Implement/Monitor Create DSS, monitor, update

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The Data

Y X1 X2 X3 X4 X5

Lung Capacity

(cc) Gender Height Smoker Exercise Age

5673 1 69.5 0 25 47

5632 1 70.1 0 24 67

5712 1 68.2 0 26 36

5723 1 70.9 0 26 68

5484 1 71.9 1 20 58

5308 1 69.2 1 15 19

5133 1 71.9 1 0 40

A portion of the data is shown below. See Spreadsheet for all data.

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Preliminary Analyses

 

Lung Capacity

(cc) Gender Height Smoker Exercise Age

Mean 5325.60 0.50 68.23 0.39 21.35 46.42

Stdev 410.48 0.50 3.45 0.49 8.91 13.98

Min 4233.71 0.00 58.93 0.00 0.00 19.00

Max 6261.00 1.00 76.61 1.00 40.29 82.14

The table below shows some descriptive statistics for each variable. What basic statements about our data can we make from this?

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Capacity by Gender, Smoking

    Gender    

Smoker Data Female MaleGrand

Total

Non-Smoker Average of Lung Capacity (cc)5427.6

75662.2

2 5546.87

  StdDev of Lung Capacity (cc) 256.41 284.71 293.75

  Count of Smoker 30.00 31.00 61.00

Smoker Average of Lung Capacity (cc)4837.4

55129.0

5 4979.51

  StdDev of Lung Capacity (cc) 273.74 297.51 318.12

  Count of Smoker 20.00 19.00 39.00

Total Average of Lung Capacity (cc)5191.5

85459.6

1 5325.60

Total StdDev of Lung Capacity (cc) 391.51 387.93 410.48

Total Count of Smoker 50.00 50.00 100.00Does there appear to be a relationship between, Smoking, Gender, and Lung Capacity?

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Distributions

Lung Capacity (cc.)

0

10

20

30

40

4400 4800 5200 5600 6000 More

Capacity in cc, up to number shown

Fre

qu

ency

0.00%20.00%

40.00%60.00%80.00%

100.00%120.00%

Height Distribution

0

10

20

30

40

50

60 64 68 72 76 More

Height in Inches

Fre

qu

en

cy

0.00%

20.00%

40.00%

60.00%

80.00%

100.00%

120.00%

Distribution of Exercise Time

0

5

10

15

20

25

30

5 10 15 20 25 30 35 More

Minutes of exercise per day

Fre

qu

ency

0.00%

20.00%

40.00%

60.00%

80.00%

100.00%

120.00%

Distribution of Age

0

5

1015

20

25

30

20 30 40 50 60 70 80 More

Age in years

Fre

qu

ency

0.00%

20.00%

40.00%

60.00%80.00%

100.00%

120.00%

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Bivariate Analysis – Matrix Plot

Lung Capacity (cc)

Exercise

Age

Height

726660 40200 906030

6000

5000

4000

72

66

6040

20

0

Matrix Plot of Lung Capacity (cc), Height, Exercise, Age

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Capacity distribution by Gender, Smoking

Lung Capacity (cc)

Frequency

640060005600520048004400

10

8

6

4

2

0

Mean StDev N5192 391.5 505460 387.9 50

Gender01

Histogram of Lung Capacity (cc)Normal

Lung Capacity (cc)

Frequency

60005600520048004400

18

16

14

12

10

8

6

4

2

0

Mean StDev N5547 293.7 614980 318.1 39

Smoker0

1

Histogram of Lung Capacity (cc)Normal

Men have a larger lung capacity than women, on average.

Non-Smokers have a larger lung capacity than smokers on average. What about the variance?

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Simple Regression

How well can exercise time alone predict the lung capacity?

Lung Capacity and Exercise Time y = 28.71x + 4712.5

R2 = 0.3881

0

1000

2000

3000

4000

5000

6000

7000

0 10 20 30 40 50

Minutes of exercise per day

Lu

ng

Cap

acit

y in

cc.

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Multiple Regression

How do all the Xs together help predict y?

SUMMARY OUTPUT

Regression Statistics

Multiple R 0.8798341

R Square 0.7741081

Adjusted R Square 0.7620926

Standard Error 200.21

Observations 100

  CoefficientsStandard

Error t Stat P-value

Intercept 1662.3965 475.1456634 3.498709192 0.000716253

Gender 202.3282 41.86861042 4.832456809 5.23607E-06

Height 50.3468 7.08207335 7.109058989 2.24959E-10

Smoker -278.9711 52.71395448 -5.292169492 7.88193E-07

Exercise 11.2949 2.991170972 3.776112614 0.000279023

Age -0.1174 1.462303258 -0.080303367 0.936166702

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Final Model

SUMMARY OUTPUT

Regression Statistics

Multiple R 0.879825

R Square 0.774093

Adjusted R Square 0.764581

Standard Error 199.164

Observations 100

  Coefficients Standard Error t Stat P-value

Intercept 1656.937 467.7903 3.54205 0.000617

Gender 202.104 41.55695 4.86332 4.57E-06

Height 50.359 7.043082 7.150271 1.78E-10

Smoker -279.025 52.43341 -5.3215 6.85E-07

Exercise 11.259 2.943494 3.825342 0.000234

y 1656.937 + 202.104 * Gender + 50.359 * Height – 279.025 * Smoker + 11.259 * Exercise

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Prediction Exercise

1. Predict the lung capacity for a non-smoking female who does not exercise, and is 66 inches tall, based on the model above.

2. What would be the predicted value if she smoked?

3. What would it be for a male in both the above cases?