Chapter 11: Inferential methods in Regression and Correlation .

Chapter 11: Inferential methods in Regression and Correlation

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.

a) What is the mean value of y when x = 30? x = 50? x = 70?

b) 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.

a) What are the point estimates of and ?b) What is a point estimate of the true average cetane

number whose iodine value is 100?

Example: Estimating and (cont)x: 132.0 129.0 120.0 113.2 105.0 92.0 84.0y: 46.0 48.0 51.0 52.1 54.0 52.0 59.0x: 83.2 88.4 59.0 80.0 81.5 71.0 69.2y: 58.7 61.6 64.0 61.4 54.6 58.8 58.0

a) What are the point estimates of and ?

Example: Estimating and (cont)

50 60 70 80 90 100 110 120 130 14045

50

55

60

65

Iodine (g)

ceta

ne n

umbe

r

Example: Estimating and (cont)x: 132.0 129.0 120.0 113.2 105.0 92.0 84.0y: 46.0 48.0 51.0 52.1 54.0 52.0 59.0x: 83.2 88.4 59.0 80.0 81.5 71.0 69.2y: 58.7 61.6 64.0 61.4 54.6 58.8 58.0

ix 1307.5 iy 779.22ix 128913.93 2

iy 43745.22i ix y 71347.30


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

50 60 70 80 90 100 110 120 130 14045

50

55

60

65

f(x) = − 0.209387416490154 x + 75.2124319329198

Iodine (g)

ceta

ne n

umbe

r

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?


ix 1307.5 iy 779.22ix 128913.93 2

iy 43745.22i ix y 71347.30

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 Read 15 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 Coeff Var 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: CIThe 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 Read 15 Number of Observations Used 14 Number of Observations with Missing Values 1




Root MSE 2.56450 R-Square 0.7908 Dependent Mean 55.65714 Adj R-Sq 0.7733 Coeff Var 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 testThe cetane number is a critical property in specifying


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






Parameter Estimates



Summary SlideSource df SS MS

Model(Regression) 1 SSR

Error n - 2 SST – b Sxy

Total n - 1 Syy

SSE SSE

dfe n 2

2iˆ(y y)

2

i2yy i

yS y

n

i ixy i i

x yS x y

n xy

xx

Sb

S

Example: ANOVA (SAS) The REG Procedure Model: MODEL1 Dependent Variable: cetane






Parameter Estimates



Example: Hypothesis test for The cetane number is a critical property in specifying


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






Parameter Estimates



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).

x 0.066 0.088 0.120 0.050 0.162 0.186 0.057 0.100y 4.6 11.6 9.5 6.3 13.8 15.4 2.5 11.8x 0.112 0.055 0.154 0.074 0.111 0.140 0.071 0.110y 8.0 7.0 20.6 16.6 9.2 17.9 2.8 13.0

Example: Hypothesis test for (2)

x

y



Using the population correlation coefficient, is this model useful?

ix 1.656 iy 170.62ix 0.196912 2

iy 2253.56i ix y 20.0397

The summary statistics are:

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


Parameter Estimates


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






Parameter Estimates





Example: Multiple Linear RegressionIt 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 + ea) 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 RegressionSuppose 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.

a) 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?

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

xi (oF) 95 82 90 81 99 100 93 95 93 97yi

(MW)214 152 156 129 254 266 210 204 213 150

Example: Polynomial Regression (SAS)data newpower;set power;temp2 = temp*temp; proc reg data=newpower;model load=temp temp2;output out=fit r=res;run; Analysis of Variance

Sum of MeanSource DF Squares Square F Value Pr > F

Model 2 18089 9044.26725 53.88 <.0001Error 7 1175.06549 167.86650Corrected Total 9 19264


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?

line adj r2 se

linear y=-419.8+6.7 temp 0.88 16.18quadratic y=1784-42.4temp+0.27temp2 0.92 12.96

Residual Plots

80 85 90 95 100-30

-10

10

30

Temperature oF

Resi

dual

Interaction Effect

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

y = 31.6244 + 0.01428 x1 – 58.0779 x2 + e

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

Conceptual Understanding

X1

X2

X3

Total Variation of Y

ANOVA table - MRRSource df SS MS

Model(Regression) k SSM

(from data)

Error n – k - 1 SSE(from data)

Total n - 1 SST(from data)

SSE SSE

dfe n k 1

SSM SSM

dfm k

Example: Multiple Linear RegressionIt 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


Parameter Estimates


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



Parameter Estimates



Conceptual Understanding

X1

X2

X3

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



Parameter Estimates



Example: MLR (backwards elimination) (cont)





Parameter Estimates


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


full w/o feedline y = 101.77 – 0.096

speed - 667.97 feed – 472.30 depth

y = 95.89 – 0.095 speed – 500.32 depth

R2 0.7584 0.7223adj R2 0.7222 0.6958ANOVA table

Model 3 2743.83 914.61Error 20 874.13 43.71Total 23 3617.96

Model 2 2613.10 1306.55Error 21 1004.86 47.85Total 23 3617.96


full full - P w/o w/o - PF – test 20.93 <0.0001 27.30 <0.0001t-tests:speed -6.72 <0.0001 -6.40 <0.0001t- tests: feed -1.73 0.0991t-tests:depth -2.92 0.0085 -2.97 0.0073

Chapter 11: Inferential methods in Regression and Correlation .

Documents

cetane number of observations

mean value of y

iodine value g

true average cetane

standard deviation of

iodine necessary

point estimates

biodiesel fuel