Regression StatisticsMultiple R 0.941073R Square 0.885618

Adjusted R Square 0.828426Standard Error 0.2431Observations 10

ANOVA  df SS MS F

Regression 3 2.745414 0.915138 15.4852

Residual 6 0.354586 0.059098Total 9 3.1    

  Coefficients Standard Error t Stat P-value

Intercept 0.345097 0.530667 0.650308 0.53958

TradeEx 0.254822 0.085555 2.978446 0.024686

Use 0.132492 0.140426 0.943501 0.381848

Range 0.458519 0.123186 3.72216 0.009827

Nonlinear Relationships

Nonlinear relationships can be modeled by including a variable that is a nonlinear function of an independent variable.

For example it is usually assumed that health care expenditures increase at an increasing rate as people age.

Nonlinear Relationships

In that case you might try including age squared into the model:Health expend = 500 + (5)Age + (.5)AgeSQ

Age Health Expend10 60020 80030 110040 1500

Nonlinear Relationships

If the dependent variable increases at a decreasing rate as the independent variable rises you might want to include the square root of the independent variable.

If you are unsure of the nature of the relationship you can use dummy variables for different ranges of values of the independent variable.

Non-continuous Relationships

If the relationship between the dependent variable and an independent variable is non-continuous a slope dummy variable can be used to estimate two sets of coefficients and intercepts for the independent variable.

For example, if natural gas usage is not affected by temperature when the temperature rises above 60 degrees, we could have:Gas usage = b0 + b1(GT60) + b2(Temp) + b3(GT60)(Temp)

Non-continuous Relationships

Note that at temperatures above 60 degrees the net effect of a 1 degree increase in temperature on gas usage is -0.056 (-.866+.810)

  CoefficientsStandard Error t Stat P-value

Intercept 53.002 2.415 21.95 7.48E-18

GT60 -46.623 16.682 -2.79 0.0098

Temp -0.866 0.0595 -14.56 1.02E-13

(GT60)(Temp) 0.810 0.255 3.18 0.0039

Interaction Terms

You can try to control for interactions between two variables by including a variable that is the product of two independent variables.

For example, assume we were estimating the salaries of baseball players. If there was a premium paid to players that were both good fielders and good hitters, we might want to include an interaction term for hitting and fielding in the model.

Standardized Coefficients Unstandardized Standardized

Coefficients Coefficients B Std. Error Beta t Sig.

(Constant) -14.485 4.038 -3.587 .000 Weight -.007 .000 -.706 -14.177 .000 Year .761 .050 .360 15.262 .000 Cylinders -.074 .232 -.016 -.320 .749a Dependent Variable: MPG

When the regression model is estimated after standardizing the values of the dependent and independent variables. Used to compare the magnitude of the effects of the independent variables.

Standardized Residuals

iyy

yy

ii

hss

s

yy

ii

ii

1

ˆ

ˆ

ˆ

Where s is the standard error of estimate and hi is the leverage of observation i. Leverage is determined by the difference between the value of the independent variables and their means.

Standardized Residuals

The random deviation in the value of y, e, is assumed to be normally distributed. Looking at the standardized residuals gives some indication if that is true. Values should lie within 2 standard deviations of 0. Values greater than 2 may indicate the presence of outliers.

Standardized Residuals

ObservationPredicted

Rating ResidualsStandard Residuals

1 4.085043 -0.08504 -0.428452 3.534748 -0.03475 -0.175063 3.346196 0.153804 0.774874 3.617428 -0.11743 -0.59165 3.295368 0.204632 1.0309436 3.616226 -0.11623 -0.585557 2.877459 0.122541 0.6173658 2.663425 0.336575 1.6956749 2.638991 -0.13899 -0.70024

10 2.325116 -0.32512 -1.63795