Chapter 11: Inferential methods in Regression and Correlation

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Chapter 11: Inferential methods in Regression and Correlation. http://jonfwilkins.blogspot.com/2011_08_01_archive.html. Example: distribution of y.

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http://jonfwilkins.blogspot.com/2011_08_01_archive.html

Example: distribution of y

The relationship between age and change in systolic blood pressure (BP, mm Hg) after 24 hours in response to a particular treatment has a linear regression equation of y = 20.11 – 0.526x + e with σ = 6.52.

• What is the mean value of y when x = 30? x = 50? x = 70?
• What is the standard deviation of y when x = 30? x = 50? x = 70?
Example: Estimating  and 

The cetane number is a critical property in specifying the ignition quality of a fuel used in a diesel engine. Determination of this number for a biodiesel fuel is expensive and time-consuming. Therefore a way of predicting this number is wanted. The data on the next slide is x = iodine value (g) and y = cetane number for a sample of 14 biofuels. The iodine value is the amount of iodine necessary to saturate a sample of 100g of oil.

• What are the point estimates of  and ?
• What is a point estimate of the true average cetane number whose iodine value is 100?
Example: Estimating  and  (cont)

What are the point estimates of  and ?

Example: Estimating  and  (cont)

b) What is a point estimate of the true average cetane number whose iodine value is 100?

Example: Estimating  and 

The cetane number is a critical property in specifying the ignition quality of a fuel used in a diesel engine. Determination of this number for a biodiesel fuel is expensive and time-consuming. Therefore a way of predicting this number is wanted. The data on the next slide is x = iodine value (g) and y = cetane number for a sample of 14 biofuels. The iodine value is the amount of iodine necessary to saturate a sample of 100g of oil.

c) Find the point estimate of the error standard deviation, σ.

d) What proportion of the observed variation in y can be attributed to the simple linear regression relationship between x and y?

Example: Estimating  and  (cont)

c) Find the point estimate of the error standard deviation, σ.

d) What proportion of the observed variation in y can be attributed to the simple linear regression relationship between x and y?

Example: Estimating  and  (SAS)

The REG Procedure

Model: MODEL1

Dependent Variable: cetane

Number of Observations Used 14

Number of Observations with Missing Values 1

Analysis of Variance

Sum of Mean

Source DF Squares Square F Value Pr > F

Model 1 298.25443 298.25443 45.35 <.0001

Error 12 78.91986 6.57665

Corrected Total 13 377.17429

Root MSE 2.56450 R-Square 0.7908

Dependent Mean 55.65714 Adj R-Sq 0.7733

CoeffVar 4.60767

Parameter Estimates

Parameter Standard

Variable DF Estimate Error t Value Pr > |t|

Intercept 1 75.21243 2.98363 25.21 <.0001

iodine 1 -0.20939 0.03109 -6.73 <.0001

Example: CI

The cetane number is a critical property in specifying the ignition quality of a fuel used in a diesel engine. Determination of this number for a biodiesel fuel is expensive and time-consuming. Therefore a way of predicting this number is wanted. The data on the next slide is x = iodine value (g) and y = cetane number for a sample of 14 biofuels. The iodine value is the amount of iodine necessary to saturate a sample of 100g of oil.

e) What is the 95% CI for the true slope?

Example: Output (SAS)

The SAS System 09:20 Thursday, November 10, 2011 3

The REG Procedure

Model: MODEL1

Dependent Variable: cetane

Number of Observations Used 14

Number of Observations with Missing Values 1

Analysis of Variance

Sum of Mean

Source DF Squares Square F Value Pr > F

Model 1 298.25443 298.25443 45.35 <.0001

Error 12 78.91986 6.57665

Corrected Total 13 377.17429

Root MSE 2.56450 R-Square 0.7908

Dependent Mean 55.65714 Adj R-Sq 0.7733

CoeffVar 4.60767

Parameter Estimates

Parameter Standard

Variable DF Estimate Error t Value Pr > |t| 95% Confidence Limits

Intercept 1 75.21243 2.98363 25.21 <.0001 68.71165 81.71321

iodine 1 -0.20939 0.03109 -6.73 <.0001 -0.27713 -0.14164

Sxx = 6802.7693

Example: Hypothesis test

The cetane number is a critical property in specifying the ignition quality of a fuel used in a diesel engine. Determination of this number for a biodiesel fuel is expensive and time-consuming. Therefore a way of predicting this number is wanted. The data on the next slide is x = iodine value (g) and y = cetane number for a sample of 14 biofuels. The iodine value is the amount of iodine necessary to saturate a sample of 100g of oil.

f) Is the model useful (that is, is there a useful linear relationship between x and y)?

Example: Hypothesis test (SAS)

The REG Procedure

Model: MODEL1

Dependent Variable: cetane

Number of Observations Used 14

Number of Observations with Missing Values 1

Analysis of Variance

Sum of Mean

Source DF Squares Square F Value Pr > F

Model 1 298.25443 298.25443 45.35 <.0001

Error 12 78.91986 6.57665

Corrected Total 13 377.17429

Root MSE 2.56450 R-Square 0.7908

Dependent Mean 55.65714 Adj R-Sq 0.7733

CoeffVar 4.60767

Parameter Estimates

Parameter Standard

Variable DF Estimate Error t Value Pr > |t|

Intercept 1 75.21243 2.98363 25.21 <.0001

iodine 1-0.209390.03109-6.73<.0001

Example: ANOVA (SAS)

The REG Procedure

Model: MODEL1

Dependent Variable: cetane

Number of Observations Used 14

Number of Observations with Missing Values 1

Analysis of Variance

Sum of Mean

Source DF Squares Square F Value Pr > F

Model 1 298.25443 298.25443 45.35 <.0001

Error 12 78.91986 6.57665

Corrected Total 13 377.17429

Root MSE 2.56450 R-Square 0.7908

Dependent Mean 55.65714 Adj R-Sq 0.7733

CoeffVar 4.60767

Parameter Estimates

Parameter Standard

Variable DF Estimate Error t Value Pr > |t|

Intercept 1 75.21243 2.98363 25.21 <.0001

iodine 1 -0.20939 0.03109 -6.73 <.0001

Example: Hypothesis test for 

The cetane number is a critical property in specifying the ignition quality of a fuel used in a diesel engine. Determination of this number for a biodiesel fuel is expensive and time-consuming. Therefore a way of predicting this number is wanted. The data on the next slide is x = iodine value (g) and y = cetane number for a sample of 14 biofuels. The iodine value is the amount of iodine necessary to saturate a sample of 100g of oil.

g) Is the model useful (that is, is there a useful linear relationship between x and y) using the population correlation coefficient?

Example: ANOVA (SAS)

The REG Procedure

Model: MODEL1

Dependent Variable: cetane

Number of Observations Used 14

Number of Observations with Missing Values 1

Analysis of Variance

Sum of Mean

Source DF Squares Square F Value Pr > F

Model 1 298.25443 298.25443 45.35 <.0001

Error 12 78.91986 6.57665

Corrected Total 13 377.17429

Root MSE 2.56450 R-Square 0.7908

Dependent Mean 55.65714 Adj R-Sq 0.7733

CoeffVar 4.60767

Parameter Estimates

Parameter Standard

Variable DF Estimate Error t Value Pr > |t|

Intercept 1 75.21243 2.98363 25.21 <.0001

iodine 1 -0.20939 0.03109 -6.73<.0001

Example: Hypothesis test for  (2)

In some locations, there is a strong association between concentrations for two different pollutants. The following data consists of the concentrations of x = ozone (ppm) and y = secondary carbon concentration (μg/m3).

Example: Hypothesis test for  (2)

The summary statistics are:

Using the population correlation coefficient, is this model useful?

Example: Hypothesis test for  (2)

Analysis of Variance

Sum of Mean

Source DF Squares Square F Value Pr > F

Model 1 222.47934 222.47934 14.69 0.0018

Error 14 212.05816 15.14701

Corrected Total 15 434.53750

Root MSE 3.89192 R-Square 0.5120

Dependent Mean 10.66250 Adj R-Sq 0.4771

CoeffVar 36.50097

Parameter Estimates

Parameter Standard

Variable DF Estimate Error t Value Pr > |t|

Intercept 1 0.99801 2.70292 0.37 0.7175

x 1 93.37670 24.36448 3.83 0.0018

Example: Hypothesis test for  (2)

The REG Procedure

Model: MODEL1

Dependent Variable: cetane

Number of Observations Used 14

Number of Observations with Missing Values 1

Analysis of Variance

Sum of Mean

Source DF Squares Square F Value Pr > F

Model 1 298.25443 298.25443 45.35 <.0001

Error 12 78.91986 6.57665

Corrected Total 13 377.17429

Root MSE 2.56450 R-Square 0.7908

Dependent Mean 55.65714 Adj R-Sq 0.7733

CoeffVar 4.60767

Parameter Estimates

Parameter Standard

Variable DF Estimate Error t Value Pr > |t|

Intercept 1 75.21243 2.98363 25.21 <.0001

iodine 1 -0.20939 0.03109 -6.73<.0001

Example: Multiple Linear Regression

It is important to know how long a tool will last (min) in the industrial setting. The cutting tool in this study is used to cut a particular type and size of cold-rolled steel. The predictors of interest are x1 = cutting speed (feet/min), x2 = feed rate (in/revolution) and x3 = depth of cut (in). The predicted model is

y = 101.765 – 0.0958 x1 – 667.972 x2 - 472.304 x3 + e

a) What is the mean life of a tool that is being used to cut depths of 0.03 inch at a speed rate of 450 feet/min with a feed rate of 0.01 in/revolution?

b) What is the interpretation of 1 = -0.0958? Of 2 = -667.972? Of 3 = -472.304?

Example: Polynomial Regression

Suppose the mean daily peak load (MW) for a power plant and the maximum outdoor temperature (oF) for a sample of 10 days is given below.

• What is the estimated regression line using a quadratic regression model (besides the equation of the line, include the values of adj. r2 and se?
• Using the line, predict the required peak power if the temperature is 98 oF?
Example: Polynomial Regression (SAS)

datanewpower;

set power;

temp2 = temp*temp;

procreg data=newpower;

output out=fit r=res;

run;

Analysis of Variance

Sum of Mean

Source DF Squares Square F Value Pr > F

Model 2 18089 9044.26725 53.88 <.0001

Error 7 1175.06549 167.86650

Corrected Total 9 19264

Root MSE 12.95633 R-Square 0.9390

Dependent Mean 194.80000 Adj R-Sq 0.9216

CoeffVar 6.65109

Parameter Estimates

Parameter Standard

Variable DF Estimate Error t Value Pr > |t|

Intercept 1 1784.18833 944.12303 1.89 0.1007

temp 1 -42.38624 21.00079 -2.02 0.0833

temp2 1 0.27216 0.11634 2.34 0.0519

Example: Polynomial Regression (cont)

b) Using the line, predict the required peak power if the temperature is 98 oF?

I love statistics!

Thank you for not eating me!

Example: Multiple RegressionQualitative Predictors

A study is conducted to determine the effects of x1 = company size and x2 = the presence (1) or absence (0) of a safety program on y = the number of work hours lost due to work-related accidents (thousands). 20 companies with no active safety programs were randomly chosen and 20 companies with active safety programs were randomly chosen. The SAS file (qualpred.txt) is on the class notes web site. The estimated regression line is

ŷ = 31.6244 + 0.01428 x1 – 58.0779 x2 + e

What are the interpretations of 1 = 0.01428 and 2 = -58.0779?

Conceptual Understanding

X3

X1

X2

Total Variation of Y

Example: Multiple Linear Regression

It is important to know how long a tool will last (min) in the industrial setting. The cutting tool in this study is used to cut a particular type and size of cold-rolled steel. The predictors of interest are x1 = cutting speed (feet/min), x2 = feed rate (in/revolution) and x3 = depth of cut (in).

a) Is there a useful linear relationship between the cutting tool lifetime and the predictors?

Example: MLR (cont)

Analysis of Variance

Sum of Mean

Source DF Squares Square F Value Pr > F

Model 3 2743.82814 914.60938 20.93<.0001

Error 20 874.13019 43.70651

Corrected Total 23 3617.95833

Root MSE 6.61109 R-Square 0.7584

Dependent Mean 38.54167 Adj R-Sq 0.7222

CoeffVar 17.15310

Parameter Estimates

Parameter Standard

Variable DF Estimate Error t Value Pr > |t|

Intercept 1 101.76536 8.33310 12.21 <.0001

speed 1 -0.09578 0.01426 -6.72 <.0001

feed 1 -667.97241 386.23081 -1.73 0.0991

depth 1 -472.30426 161.81434 -2.92 0.0085

Example: MLR (cont)

Analysis of Variance

Sum of Mean

Source DF Squares Square F Value Pr > F

Model 3 2743.82814 914.60938 20.93 <.0001

Error 20 874.13019 43.70651

Corrected Total 23 3617.95833

Root MSE 6.61109 R-Square 0.7584

Dependent Mean 38.54167 Adj R-Sq 0.7222

CoeffVar 17.15310

Parameter Estimates

Parameter Standard

Variable DF Estimate Error t Value Pr > |t|

Intercept 1 101.76536 8.33310 12.21 <.0001

speed 1 -0.09578 0.01426 -6.72 <.0001

feed 1 -667.97241 386.23081 -1.73 0.0991

depth 1 -472.30426 161.81434 -2.92 0.0085

Conceptual Understanding

X3

X1

X2

Total Variation of Y

Example: MLR (backwards elimination)

Analysis of Variance

Sum of Mean

Source DF Squares Square F Value Pr > F

Model 3 2743.82814 914.60938 20.93 <.0001

Error 20 874.13019 43.70651

Corrected Total 23 3617.95833

Root MSE 6.61109 R-Square 0.7584

Dependent Mean 38.54167 Adj R-Sq 0.7222

CoeffVar 17.15310

Parameter Estimates

Parameter Standard

Variable DF Estimate Error t Value Pr > |t|

Intercept 1 101.76536 8.33310 12.21 <.0001

speed 1 -0.09578 0.01426 -6.72 <.0001

feed 1 -667.97241 386.23081 -1.73 0.0991

depth 1 -472.30426 161.81434 -2.92 0.0085

Example: MLR (backwards elimination) (cont)

Analysis of Variance

Sum of Mean

Source DF Squares Square F Value Pr > F

Model 2 2613.09992 1306.54996 27.30 <.0001

Error 21 1004.85841 47.85040

Corrected Total 23 3617.95833

Root MSE 6.91740 R-Square 0.7223

Dependent Mean 38.54167 Adj R-Sq 0.6958

CoeffVar 17.94784

Parameter Estimates

Parameter Standard

Variable DF Estimate Error t Value Pr > |t|

Intercept 1 95.88869 7.96137 12.04 <.0001

speed 1 -0.09543 0.01492 -6.40 <.0001

depth 1 -500.32482 168.46077 -2.97 0.0073