Chapter14

1 Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc.

Chapter 14

Multiple Regression Models


A general additive multiple regression model, which relates a dependent variable y to k predictor variables x1, x2,…, xk is given by the model equation

y = + 1x1 + 2x2 + … + kxk + e



The random deviation e is assumed to be normally distributed with mean value 0 and variance 2 for any particular values of x1, x2,…, xk.

This implies that for fixed x1, x2,…, xk values, y has a normal distribution with variance 2 and


(mean y value for fixed x1, x2,…, xk values) = + 1x1 + 2x2 + … + kxk


The i’s are called population regression coefficients; each i can be interpreted as the true average change in y when the predictor xi increases by 1 unit and the values of all the other predictors remain fixed.


The deterministic portion

+ 1x1 + 2x2 + … + kxk

is called the population regression function.


The kth degree polynomial regression model y = + 1x + 2x2 + … + kxk + eis a special case of the general multiple regression model with x1 = x, x2 = x2, … , xk = xk.

The population regression function (mean value of y for fixed values of the predictors) is

+ 1x + 2x2 + … + kxk.

Polynomial Regression Models


The most important special case other than simple linear regression (k = 1) is the quadratic regression model

y = + 1x + 2x2.

This model replaces the line y = + x with a parabolic cure of mean values + 1x + 2x2.

If 2 > 0, the curve opens upward, whereas if 2 < 0, the curve opens downward.

Polynomial Regression Models


If the change in the mean y value associated with a 1-unit increase in one independent variable depends on the value of a second independent variable, there is interaction between these two variables. When the variables are denoted by x1 and x2, such interaction can be modeled by including x1x2, the product of the variables that interact, as a predictor variable.

Interaction


Up to now, we have only considered the inclusion of quantitative (numerical) predictor variables in a multiple regression model.

Two other types are very common:

Dichotomous variable: One with just two possible categories coded 0 and 1

Examples

Gender {male, female}

Marriage status {married, not-married}

Qualitative Predictor Variables.


Ordinal variables: Categorical variables that have a natural ordering

Activity level {light, moderate, heavy} coded respectively as 1, 2 and 3

Education level {none, elementary, secondary, college, graduate} coded respectively 1, 2, 3, 4, 5 (or for that matter any 5 consecutive integers}

Qualitative Predictor Variables.


According to the principle of least squares, the fit of a particular estimated regression function a + b1x1 + b2x2 + … + bkxk to the observed data is measured by the sum of squared deviations between the observed y values and the y values predicted by the estimated function:

[y –(a + b1x1 + b2x2 + … + bkxk )]2

Least Square Estimates

The least squares estimates of , 1, 2,…, k are those values of a, b1, b2, … , bk that make this sum of squared deviations as small as possible.


Predicted Values & Residuals

Doing this successively for the remaining observations yields the predicted values

(sometimes referred to as the fitted values or fits).

2 3 ky ,y , ,yˆ ˆ ˆ

The first predicted value is obtained by taking the values of the predictor variables x1, x2,…, xk for the first sample observation and substituting these values into the estimated regression function.

1y


Predicted Values & Residuals

The residuals are then the differences

between the observed and predicted y values.

1 1 2 2 k ky y ,y y , ,y yˆ ˆ ˆ


Sums of Squares

The number of degrees of freedom associated with SSResid is n - (k + 1), because k + 1 df are lost in estimating the k + 1 coefficients , 1, 2,…,k.

The residual (or error) sum of sqyares, SSResid, and total sum of squares, SSTo, are given by

where is the mean of the y observations in the sample.

2 2ˆSSResid= y-y SSTo= y-y

y


Estimate for 2

An estimate of the random deviation variance 2 is given by

and is the estimate of .

2e

SSResids

n - (k + 1)

2e es s


Coefficient of Multiple Determination, R2

The coefficient of multiple determination, R2, interpreted as the proportion of variation in observed y values that is explained by the fitted model, is

2 SSResidR 1

SSTo


Adjusted R2

Generally, a model with large R2 and small se are desirable. If a large number of variables (relative to the number of data points) is used those conditions may be satisfied but the model will be unrealistic and difficult to interpret.


Adjusted R2

To sort out this problem, sometimes computer packages compute a quantity called the adjusted R2,

2 n 1 SSResidadjusted R 1

n (k 1) SSTo

Notice that when a large number of variables are used to build the model, this value will be substantially lower than R2 and give a better indication of usability of the model.

Copyright © 2005 Brooks/Cole, a division of Thomson Learning, Inc.

F DistributionsF distributions are similar to a Chi-Square distributions, but have two parameters, dfden and dfnum.


The F Test for Model Utility

The regression sum of squares denoted by SSReg is defined by

SSREG = SSTo - SSresid


The F Test for Model Utility

When all k i’s are zero in the model

y = + 1x1 + 2x2 + … + kxk + e

And when the distribution of e is normal with mean 0 and variance 2 for any particular values of x1, x2,…, xk, the statistic

has an F probability distribution based on k numerator df and n - (K+ 1) denominator df

SSRegrkF

SSResidn (k 1)


The F Test for Utility of the Model y = + 1x1 + 2x2 + … + kxk + e

Null hypothesis: H0: 1 = 2 = … = k =0

(There is no useful linear relationship between y and any of the predictors.)

Alternate hypothesis: Ha: At least one among 1, 2, … , k is

not zero(There is a useful linear relationship between y and at least one of the predictors.)


The F Test for Utility of the Model y = + 1x1 + 2x2 + … + kxk + e

Test statistic: SSRegrkF

SSResidn (k 1)

where SSreg = SST0 - SSresid.

An alternate formula: 2

2

RkF

(1 R )n (k 1)

where SSreg = SST0 - SSresid.


The F Test Utility of the Model y = + 1x1 + 2x2 + … + kxk + e

The test is upper-tailed, and the information in the Table of Values that capture specified upper-tail F curve areas is used to obtain a bound or bounds on the P-value using numerator df = k and denominator df = n - (k + 1).

Assumptions: For any particular combination of predictor variable values, the distribution of e, the random deviation, is normal with mean 0 and constant variance.


Example

A number of years ago, a group of college professors teaching statistics met at an NSF program and put together a sample student research project.

They attempted to create a model to explain lung capacity in terms of a number of variables.

Specifically,

Numerical variables: height, age, weight, waist

Categorical variables: gender, activity level and smoking status.


Example

They managed to sample 41 subjects and obtain/measure the variables.

There was some discussion and many felt that the calculated variable (height)(waist)2 would be useful since it would likely be proportional to the volume of the individual.

The initial regression analysis performed with Minitab appears on the next slide.


ExampleLinear Model with All Numerical Variables

The regression equation isCapacity = - 13.0 - 0.0158 Age + 0.232 Height - 0.00064 Weight - 0.0029 Chest + 0.101 Waist -0.000018 hw2

40 cases used 1 cases contain missing values

Predictor Coef SE Coef T PConstant -13.016 2.865 -4.54 0.000Age -0.015801 0.007847 -2.01 0.052Height 0.23215 0.02895 8.02 0.000Weight -0.000639 0.006542 -0.10 0.923Chest -0.00294 0.06491 -0.05 0.964Waist 0.10068 0.09427 1.07 0.293hw2 -0.00001814 0.00001761 -1.03 0.310

S = 0.5260 R-Sq = 78.2% R-Sq(adj) = 74.2%


Example

The only coefficient that appeared to be significant and the 5% level was the height. Since the P-value for the coefficient on the age was very close to 5% (5.2%) it was decided that a linear model with the two independent variables height and age would be calculated.

The resulting model is on the next slide.


ExampleLinear Model with variables: Height & Age

The regression equation isCapacity = - 10.2 + 0.215 Height - 0.0133 Age


Predictor Coef SE Coef T PConstant -10.217 1.272 -8.03 0.000Height 0.21481 0.01921 11.18 0.000Age -0.013322 0.005861 -2.27 0.029

S = 0.5073 R-Sq = 77.2% R-Sq(adj) = 76.0%

Notice that even though the R2 value decreases slightly, the adjusted R2 value actually increases. Also note that the coefficient on Age is now significant at 5%.


ExampleIn an attempt to determine if incorporating the categorical variables into the model would significantly enhance the it.

Gender was coded as an indicator variable (male = 0 and female = 1),

Smoking was coded as an indicator variable (No = 0 and Yes = 1), and

Activity level (light, moderate, heavy) was coded respectively as 1, 2 and 3.

The resulting Minitab output is given on the next slide.


ExampleLinear Model with categorical variables added

The regression equation isCapacity = - 7.58 + 0.171 Height - 0.0113 Age - 0.383 C-Gender + 0.260 C-Activity - 0.289 C-Smoke


Predictor Coef SE Coef T PConstant -7.584 2.005 -3.78 0.001Height 0.17076 0.02919 5.85 0.000Age -0.011261 0.005908 -1.91 0.066C-Gender -0.3827 0.2505 -1.53 0.137C-Activi 0.2600 0.1210 2.15 0.040C-Smoke -0.2885 0.2126 -1.36 0.185

S = 0.4596 R-Sq = 84.2% R-Sq(adj) = 81.7%


ExampleIt was noted that coefficient for the coded indicator variables gender and smoking were not significant, but after considerable discussion, the group felt that a number of the variables were related.

This, the group felt, was confounding the study. In an attempt to determine a reasonable optimal subgroup of the variables to keep in the study, it was noted that a number of the variables were highly related. Since the study was small, a stepwise regression was run and the variables, Height, Age, Coded Activity, Coded Gender were kept and the following model was obtained.


ExampleLinear Model with Height, Age & Coded Activity

and Gender

The regression equation isCapacity = - 6.93 + 0.161 Height - 0.0137 Age + 0.302 C-Activity - 0.466 C-Gender


Predictor Coef SE Coef T PConstant -6.929 1.708 -4.06 0.000Height 0.16079 0.02454 6.55 0.000Age -0.013744 0.005404 -2.54 0.016C-Activi 0.3025 0.1133 2.67 0.011C-Gender -0.4658 0.2082 -2.24 0.032

S = 0.4477 R-Sq = 83.2% R-Sq(adj) = 81.3%


Example Linear Model with Height, Age & Coded Activity

and Gender

Analysis of Variance

Source DF SS MS F PRegression 4 34.8249 8.7062 43.44 0.000Residual Error 35 7.0151 0.2004Total 39 41.8399

Source DF Seq SSHeight 1 30.9878Age 1 1.3296C-Activi 1 1.5041C-Gender 1 1.0034

Unusual ObservationsObs Height Capacity Fit SE Fit Residual St Resid 4 66.0 2.2000 3.2039 0.1352 -1.0039 -2.35R 23 74.0 5.7000 4.7635 0.2048 0.9365 2.35R 39 70.0 5.4000 4.4228 0.1064 0.9772 2.25R

R denotes an observation with a large standardized residual

The rest of the Minitab output is given below.



and GenderAll of the coefficients in this model were significant at the 5% level and the R2 and adjusted R2 were both fairly large.

This appeared to be a reasonable model for describing lung capacity even though the study was limited by sample size, and measurement limitations due to antique equipment.

Minitab identified 3 outliers (because the standardized residuals were unusually large.

Various plots of the standardized residuals are produced on the next few slides with comments



and Gender

The histogram of the residuals appears to be consistent with the assumption that the residuals are a sample from a normal distribution.

1.00.80.60.40.20.0-0.2-0.4-0.6-0.8-1.0

10

5

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Residual

Fre

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Histogram of the Residuals(response is Capacity)



and Gender

The normality plot also tends to indicate the residuals can reasonably be thought to be a sample from a normal distribution.

10-1

2

1

0

-1

-2

Nor

mal

Sco

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Residual

Normal Probability Plot of the Residuals(response is Capacity)



and Gender

The residual plot also tends to indicate that the model assumptions are not unreasonable, although there would be some concern that the residuals are predominantly positive for smaller fitted lung capacities.

65432

1

0

-1

Fitted Value

Res

idua

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Residuals Versus the Fitted Values(response is Capacity)

Chapter14

Education

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