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Chapter 12: Multiple Regression and Model Building

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Chapter 12: Multiple Regression and Model Building

- Introduced the straight-line model relating a dependent variable y to an independent variable x
- Estimated the parameters of the straight-line model using least squares
- Assesses the model estimates
- Used the model to estimate a value of y given x

McClave: Statistics, 11th ed. Chapter 12: Multiple Regression and Model Building

- Introduce a multiple-regression model to relate a variable y to two or more x variables
- Present multiple regression models with both quantitative and qualitative independent variables
- Assess how well the multiple regression model fits the sample data
- Show how analyzing the model residuals can help detect problems with the model and the necessary modifications

McClave: Statistics, 11th ed. Chapter 12: Multiple Regression and Model Building

McClave: Statistics, 11th ed. Chapter 12: Multiple Regression and Model Building

- Analyzing a Multiple-Regression Model
Step 1: Hypothesize the deterministic portion of the model by choosing the independent variables x1, x2, … , xk.

Step 2: Estimate the unknown parameters 0, 1, 2, … , k .

Step 3: Specify the probability distribution of and estimate the standard deviation of this distribution.

McClave: Statistics, 11th ed. Chapter 12: Multiple Regression and Model Building

- Analyzing a Multiple-Regression Model
Step 4: Check that the assumptions about are satisfied; if not make the required modifications to the model.

Step 5: Statistically evaluate the usefulness of the model.

Step 6: If the model is useful, use it for prediction, estimation and other purposes.

McClave: Statistics, 11th ed. Chapter 12: Multiple Regression and Model Building

- Assumptions about the Random Error
- The mean is equal to 0.
- The variance is equal to 2.
- The probability distribution is a normal distribution.
- Random errors are independent of one another.

McClave: Statistics, 11th ed. Chapter 12: Multiple Regression and Model Building

A First-Order Model in Five Quantitative Independent Variables

where x1, x2, … , xk are all quantitative variables that are not functions of other independent variables.

McClave: Statistics, 11th ed. Chapter 12: Multiple Regression and Model Building

A First-Order Model in Five Quantitative Independent Variables

The parameters are estimated by finding the values for the ‘s that minimize

McClave: Statistics, 11th ed. Chapter 12: Multiple Regression and Model Building

A First-Order Model in Five Quantitative Independent Variables

The parameters are estimated by finding the values for the ‘s that minimize

Only a truly talented mathematician (or geek) would choose to solve the necessary system of simultaneous linear equations by hand. In practice, computers are left to do the complicated calculation required by multiple regression models.

McClave: Statistics, 11th ed. Chapter 12: Multiple Regression and Model Building

- A collector of antique clocks hypothesizes that the auction price can be modeled as

McClave: Statistics, 11th ed. Chapter 12: Multiple Regression and Model Building

- Based on the data in Table 12.1, the least squares prediction equation, the equation that minimizes SSE, is

McClave: Statistics, 11th ed. Chapter 12: Multiple Regression and Model Building

- Based on the data in Table 12.1, the least squares prediction equation, the equation that minimizes SSE, is

The estimate for 1 is interpreted as the expected change in y given a one-unit change in x1 holding x2 constant

The estimate for 2 is interpreted as the expected change in y given a one-unit change in x2 holding x1 constant

McClave: Statistics, 11th ed. Chapter 12: Multiple Regression and Model Building

- Based on the data in Table 12.1, the least squares prediction equation, the equation that minimizes SSE, is

Since it makes no sense to sell a clock of age 0 at an auction with no bidders, the intercept term has no meaningful interpretation in this example.

McClave: Statistics, 11th ed. Chapter 12: Multiple Regression and Model Building

One-Tailed Test

Two-Tailed Test

Test of an Individual Parameter Coefficient in the Multiple Regression Model

McClave: Statistics, 11th ed. Chapter 12: Multiple Regression and Model Building

Test of the Parameter Coefficient on the Number of Bidders

McClave: Statistics, 11th ed. Chapter 12: Multiple Regression and Model Building

Test of the Parameter Coefficient on the Number of Bidders

Since t* > t, reject the null hypothesis.

McClave: Statistics, 11th ed. Chapter 12: Multiple Regression and Model Building

A 100(1-)% Confidence Interval for a Parameter

McClave: Statistics, 11th ed. Chapter 12: Multiple Regression and Model Building

A 100(1-)% Confidence Interval for 1

McClave: Statistics, 11th ed. Chapter 12: Multiple Regression and Model Building

A 100(1-)% Confidence Interval for 1

Holding the number of bidders constant, the result above tells us that we can be 90% sure that the auction price will rise between $11.20 and $14.28 for each 1-year increase in age.

McClave: Statistics, 11th ed. Chapter 12: Multiple Regression and Model Building

Reject H 0 for i

Do Not Reject H 0 for i

There may be no relationship between y and xi

Type II error occurred

The relationship between y and xi is more complex than a straight-line relationship

- Evidence of a linear relationship between y and xi

McClave: Statistics, 11th ed. Chapter 12: Multiple Regression and Model Building

- The multiple coefficient of determination, R2,measures how much of the overall variation in y is explained by the least squares prediction equation.

McClave: Statistics, 11th ed. Chapter 12: Multiple Regression and Model Building

- High values of R2 suggest a good model, but the usefulness of R2falls as the number of observations becomes close to the number of parameters estimated.

McClave: Statistics, 11th ed. Chapter 12: Multiple Regression and Model Building

Ra2 adjusts for the number of observations and the number of parameter estimates. It will always have a value no greater than R2.

McClave: Statistics, 11th ed. Chapter 12: Multiple Regression and Model Building

McClave: Statistics, 11th ed. Chapter 12: Multiple Regression and Model Building

Rejecting the null hypothesis means that something in your model helps explain variations in y, but it may be that another model provides more reliable estimates and predictions.

McClave: Statistics, 11th ed. Chapter 12: Multiple Regression and Model Building

A collector of antique clocks hypothesizes that the auction price can be modeled as

McClave: Statistics, 11th ed. Chapter 12: Multiple Regression and Model Building

A collector of antique clocks hypothesizes that the auction price can be modeled as

Something in the model is useful, but the F-test can’t tell us which x-variables are individually useful.

McClave: Statistics, 11th ed. Chapter 12: Multiple Regression and Model Building

- Checking the Utility of a Multiple-Regression Model
- Use the F-test to conduct a test of the adequacy of the overall model.
- Conduct t-tests on the “most important” parameters.
- Examine Ra2 and 2s to evaluate how well the model fits the data.

McClave: Statistics, 11th ed. Chapter 12: Multiple Regression and Model Building

- The model of antique clock prices can be used to predict sale prices for clocks of a certain age with a particular number of bidders.
- What is the mean sale price for all 150-year-old clocks with 10 bidders?

McClave: Statistics, 11th ed. Chapter 12: Multiple Regression and Model Building

The average value of all clocks with these characteristics can be found by using the statistical software to generate a confidence interval. (See Figure 12.7)

In this case, the confidence interval indicates that we can be 95% sure that the average price of a single 150-year-old clock sold at auction with 10 bidders will be between $1,154.10 and $1,709.30.

- What is the mean auction sale price for a single 150-year-old clock with 10 bidders?

McClave: Statistics, 11th ed. Chapter 12: Multiple Regression and Model Building

McClave: Statistics, 11th ed. Chapter 12: Multiple Regression and Model Building

- What is the mean sale price for a single 50-year-old clock with 2 bidders?

McClave: Statistics, 11th ed. Chapter 12: Multiple Regression and Model Building

- What is the mean sale price for a single 50-year-old clock with 2 bidders?

Since 50 years-of-age and 2 bidders are both outside of the range of values in our data set, any prediction using these values would be unreliable.

McClave: Statistics, 11th ed. Chapter 12: Multiple Regression and Model Building

- In some cases, the impact of an independent variable xi on y will depend on the value of some other independent variable xk.
- Interaction models include the cross-products of independent variables as well as the first-order values.

McClave: Statistics, 11th ed. Chapter 12: Multiple Regression and Model Building

McClave: Statistics, 11th ed. Chapter 12: Multiple Regression and Model Building

- In the antique clock auction example, assume the collector has reason to believe that the impact of age (x1) on price (y) varies with the number of bidders (x2) .
- The model is now
y = 0 + 1x1 + 2x2 + 3x1x2 + .

McClave: Statistics, 11th ed. Chapter 12: Multiple Regression and Model Building

McClave: Statistics, 11th ed. Chapter 12: Multiple Regression and Model Building

- In the antique clock auction example, assume the collector has reason to believe that the impact of age (x1) on price (y) varies with the number of bidders (x2) .
- The model is now
y = 0 + 1x1 + 2x2 + 3x1x2 + .

McClave: Statistics, 11th ed. Chapter 12: Multiple Regression and Model Building

The MINITAB results are reported

in Figure 12.11 in the text.

- In the antique clock auction example, assume the collector has reason to believe that the impact of age (x1) on price (y) varies with the number of bidders (x2) .
- The model is now
y = 0 + 1x1 + 2x2 + 3x1x2 + .

McClave: Statistics, 11th ed. Chapter 12: Multiple Regression and Model Building

- The model is now
y = 0 + 1x1 + 2x2 + 3x1x2 + .

McClave: Statistics, 11th ed. Chapter 12: Multiple Regression and Model Building

Once the interaction term has passed the t-test, it is unnecessary to test the individual independent variables.

McClave: Statistics, 11th ed. Chapter 12: Multiple Regression and Model Building

- A quadratic (second-order) model includes the square of an independent variable:
y = 0 + 1x+ 2x2 + .

This allows more complex relationships to be modeled.

McClave: Statistics, 11th ed. Chapter 12: Multiple Regression and Model Building

- A quadratic (second-order) model includes the square of an independent variable:
y = 0 + 1x+ 2x2 + .

1 is the shift parameter and

2 is the rate of curvature.

McClave: Statistics, 11th ed. Chapter 12: Multiple Regression and Model Building

- Example 12.7 considers whether home size (x) impacts electrical usage (y) in a positive but decreasing way.
- The MINITAB results are shown in Figure 12.13.

McClave: Statistics, 11th ed. Chapter 12: Multiple Regression and Model Building

McClave: Statistics, 11th ed. Chapter 12: Multiple Regression and Model Building

- According to the results, the equation that minimizes SSE for the 10 observations is

McClave: Statistics, 11th ed. Chapter 12: Multiple Regression and Model Building

McClave: Statistics, 11th ed. Chapter 12: Multiple Regression and Model Building

- Since 0 is not in the range of the independent variable (a house of 0 ft2?), the estimated intercept is not meaningful.
- The positive estimate on 1indicates a positive relationship, although the slope is not constant (we’ve estimated a curve, not a straight line).
- The negative value on 2indicates the rate of increase in power usage declines for larger homes.

McClave: Statistics, 11th ed. Chapter 12: Multiple Regression and Model Building

- The Global F-Test
- H0: 1= 2= 0
- Ha: At least one of the coefficients ≠ 0
- The test statistic is F = 189.71, p-value near 0.
- Reject H0.

McClave: Statistics, 11th ed. Chapter 12: Multiple Regression and Model Building

- t-Test of 2
- H0: 2= 0
Ha: 2< 0

- The test statistic is t = -7.62, p-value = .0001 (two-tailed).
- The one-tailed test statistic is .0001/2 = .00005
- Reject the null hypothesis.

- H0: 2= 0

McClave: Statistics, 11th ed. Chapter 12: Multiple Regression and Model Building

Complete Second-Order Model with Two Quantitative Independent Variables

E(y) = 0 + 1x1 + 2x2 + 3x1x2 + 4x12 + 5x22

y-intercept

Signs and values of these parameters control the type of surface and the rates of curvature

Changing 1and 2causes the surface to shift along the x1and x2axes

Controls the rotation of the surface

McClave: Statistics, 11th ed. Chapter 12: Multiple Regression and Model Building

McClave: Statistics, 11th ed. Chapter 12: Multiple Regression and Model Building

- Qualitative variables can be included in regression models through the use of dummy variables.
- Assign a value of 0 (the base level) to one category and 1, 2, 3 … to the other categories.

McClave: Statistics, 11th ed. Chapter 12: Multiple Regression and Model Building

A Qualitative Independent Variable with k Levels

where xi is the dummy variable for level i + 1 and

McClave: Statistics, 11th ed. Chapter 12: Multiple Regression and Model Building

- For the golf ball example from Chapter 10, there were four levels (the brands).Testing differences in brands can be done with the model

McClave: Statistics, 11th ed. Chapter 12: Multiple Regression and Model Building

- Brand A is the base level, so 0 represents the mean distance (A) for Brand A, and
1 = B - A

2 = C - A

3 = D - A

McClave: Statistics, 11th ed. Chapter 12: Multiple Regression and Model Building

- Testing that the four means are equal is equivalent to testing the significance of the s:
H0: 1 = 2 = 3 = 0

Ha: At least of one the s ≠ 0

McClave: Statistics, 11th ed. Chapter 12: Multiple Regression and Model Building

- Testing that the four means are equal is equivalent to testing the significance of the s:
H0: 1 = 2 = 3 = 0

Ha: At least of one the s ≠ 0

The test statistic is the F-statistic.

Here F = 43.99, p-value .000.

Hence we reject the null hypothesis

that the golf balls all have the same

mean driving distance.

McClave: Statistics, 11th ed. Chapter 12: Multiple Regression and Model Building

- Testing that the four means are equal is equivalent to testing the significance of the s:
H0: 1 = 2 = 3 = 0

Ha: At least of one the s ≠ 0

The test statistic if the F-statistic.

Here F = 43.99, p-value .000.

Hence we reject the null hypothesis

that the golf balls all have the same

mean driving distance.

Remember that the maximum number of dummy variables is one less than the number of levels for the qualitative variable.

McClave: Statistics, 11th ed. Chapter 12: Multiple Regression and Model Building

- Suppose a first-order model is used to evaluate the impact on mean monthly sales of expenditures in three advertising media: television, radio and newspaper.
- Expenditure, x1, is a quantitative variable
- Types of media, x2 and x3, are qualitative variables (limited to k levels -1)

McClave: Statistics, 11th ed. Chapter 12: Multiple Regression and Model Building

McClave: Statistics, 11th ed. Chapter 12: Multiple Regression and Model Building

McClave: Statistics, 11th ed. Chapter 12: Multiple Regression and Model Building

- Suppose now a second-order model is used to evaluate the impact of expenditures in the three advertising media on sales.
- The relationship between expenditures, x1, and sales, y, is assumed to be curvilinear.

McClave: Statistics, 11th ed. Chapter 12: Multiple Regression and Model Building

In this model, each medium is assumed to have the save impact on sales.

McClave: Statistics, 11th ed. Chapter 12: Multiple Regression and Model Building

In this model, the

intercepts differ but the shapes of the curves

are the same.

McClave: Statistics, 11th ed. Chapter 12: Multiple Regression and Model Building

In this model, the response curve for each media type is different – that is, advertising expenditure and media type interact, at varying rates.

McClave: Statistics, 11th ed. Chapter 12: Multiple Regression and Model Building

- Two models are nested if one model contains all the terms of the second model and at least one additional term. The more complex of the two models is called the complete model and the simpler of the two is called the reduced model.

McClave: Statistics, 11th ed. Chapter 12: Multiple Regression and Model Building

- Recall the interaction model relating the auction price (y) of antique clocks to age (x1) and bidders (x2) :

McClave: Statistics, 11th ed. Chapter 12: Multiple Regression and Model Building

- If the relationship is not constant, a second-order model should be considered:

McClave: Statistics, 11th ed. Chapter 12: Multiple Regression and Model Building

- If the complete model produces a better fit, then the s on the quadratic terms should be significant.
H0: 4 = 5 = 0

Ha: At least one of 4 and 5 is non-zero

McClave: Statistics, 11th ed. Chapter 12: Multiple Regression and Model Building

F-Test for Comparing Nested Models

McClave: Statistics, 11th ed. Chapter 12: Multiple Regression and Model Building

F-Test for Comparing Nested Models

where

SSER = sum of squared errors for the reduced model

SSEC = sum of squared errors for the complete model

MSEC = mean square error (s2) for the complete model

k – g = number of parameters specified in H0

k + 1 = number of parameters in the complete model

n = sample size

Rejection region: F > F, with k – g numerator and n – (k + 1) denominator degrees of freedom.

McClave: Statistics, 11th ed. Chapter 12: Multiple Regression and Model Building

- The growth of carnations (y) is assumed to be a function of the temperature (x1) and the amount of fertilizer (x2).
- The data are shown in Table 12.6 in the text.

McClave: Statistics, 11th ed. Chapter 12: Multiple Regression and Model Building

The growth of carnations (y) is assumed to be a function of the temperature (x1) and the amount of fertilizer (x2).

The complete second order model is

The least squares prediction equation from Table 12.6 is

rounded to

McClave: Statistics, 11th ed. Chapter 12: Multiple Regression and Model Building

The growth of carnations (y) is assumed to be a function of the temperature (x1) and the amount of fertilizer (x2).

To test the significance of the contribution of the interaction and second-order terms, use

H0: 3 = 4 = 5 = 0

Ha: At least one of 3, 4 or 5 ≠ 0

This requires estimating the complete model in reduced

form, dropping the parameters in the null hypothesis.

Results are given in Figure 12.31.

McClave: Statistics, 11th ed. Chapter 12: Multiple Regression and Model Building

McClave: Statistics, 11th ed. Chapter 12: Multiple Regression and Model Building

Reject the null hypothesis: the complete model seems to provide better predictions than the reduced model.

McClave: Statistics, 11th ed. Chapter 12: Multiple Regression and Model Building

- A parsimonious model is a general linear model with a small number of parameters. In situations where two competing models have essentially the same predictive power (as determined by an F-test), choose the more parsimonious of the two.

McClave: Statistics, 11th ed. Chapter 12: Multiple Regression and Model Building

- A parsimonious model is a general linear model with a small number of parameters. In situations where two competing models have essentially the same predictive power (as determined by an F-test), choose the more parsimonious of the two.

If the models are not nested, the choice is more subjective, based on Ra2, s, and an understanding of the theory behind the model.

McClave: Statistics, 11th ed. Chapter 12: Multiple Regression and Model Building

- It is often unclear which independent variables have a significant impact on y.
- Screening variables in an attempt to identify the most important ones is known as stepwise regression.

McClave: Statistics, 11th ed. Chapter 12: Multiple Regression and Model Building

McClave: Statistics, 11th ed. Chapter 12: Multiple Regression and Model Building

- Stepwise regression must be used with caution
- Many t-tests are conducted, leading to high probabilities of Type I or Type II errors.
- Usually, no interaction or higher-order terms are considered – and reality may not be that simple.

McClave: Statistics, 11th ed. Chapter 12: Multiple Regression and Model Building

- Regression analysis is based on the four assumptions about the random error considered earlier.
- The mean is equal to 0.
- The variance is equal to 2.
- The probability distribution is a normal distribution.
- Random errors are independent of one another.

McClave: Statistics, 11th ed. Chapter 12: Multiple Regression and Model Building

- If these assumptions are not valid, the results of the regression estimation are called into question.
- Checking the validity of the assumptions involves analyzing the residuals of the regression.

McClave: Statistics, 11th ed. Chapter 12: Multiple Regression and Model Building

- A regression residual is defined as the difference between an observed y-valueand its corresponding predicted value:

McClave: Statistics, 11th ed. Chapter 12: Multiple Regression and Model Building

Properties of the Regression Residuals

- The mean of the residuals is equal to 0.
- The standard deviation of the residuals is equal to the standard deviations of the fitted regression model.

McClave: Statistics, 11th ed. Chapter 12: Multiple Regression and Model Building

- If the model is misspecified, the mean of will not equal 0.
- Residual analysis may reveal this problem.
- The home-size electricity usage example illustrates this.

McClave: Statistics, 11th ed. Chapter 12: Multiple Regression and Model Building

- The plot of the first-order model shows a curvilinear residual pattern …

- while the quadratic model shows a more random pattern.

McClave: Statistics, 11th ed. Chapter 12: Multiple Regression and Model Building

A pattern in the residual plot may indicate a problem with the model.

McClave: Statistics, 11th ed. Chapter 12: Multiple Regression and Model Building

- A residual larger than 3s (in absolute value) is considered an outlier.
- Outliers will have an undue influence on the estimates.
1. Mistakenly recorded data

2. An observation that is for some reason truly different from the others

3. Random chance

- Outliers will have an undue influence on the estimates.

McClave: Statistics, 11th ed. Chapter 12: Multiple Regression and Model Building

- A residual larger than 3s (in absolute value) is considered an outlier.
- Leaving an outlier that should be removed in the data set will produce misleading estimates and predictions (#1 & #2 above).
- So will removing an outlier that actually belongs in the data set (#3 above).

McClave: Statistics, 11th ed. Chapter 12: Multiple Regression and Model Building

- Residual plots should be centered on 0 and within ±3s of 0.
- Residual histograms should be relatively bell-shaped.
- Residual normal probability plots should display straight lines.

McClave: Statistics, 11th ed. Chapter 12: Multiple Regression and Model Building

Regression Analysis

is Robust with

respect to (small)

nonnormal errors.

- Slight departures from normality will not seriously harm the validity of the estimates, but as the departure from normality grows, the validity falls.

McClave: Statistics, 11th ed. Chapter 12: Multiple Regression and Model Building

- If the variance of changes as y changes, the constant variance assumption is violated.

McClave: Statistics, 11th ed. Chapter 12: Multiple Regression and Model Building

- A first-order model is used to relate the salaries (y) of social workers to years of experience (x).

McClave: Statistics, 11th ed. Chapter 12: Multiple Regression and Model Building

McClave: Statistics, 11th ed. Chapter 12: Multiple Regression and Model Building

- The model seems to provide good predictions, but the residual plot reveals a non-random pattern:
- The residual increases as the estimated mean salary increases, violating the constant variance assumption

McClave: Statistics, 11th ed. Chapter 12: Multiple Regression and Model Building

- Transforming the dependent variable often stabilizes the residual
- Possible transformations of y
- Natural logarithm
- Square root
- sin-1y1/2

McClave: Statistics, 11th ed. Chapter 12: Multiple Regression and Model Building

McClave: Statistics, 11th ed. Chapter 12: Multiple Regression and Model Building

McClave: Statistics, 11th ed. Chapter 12: Multiple Regression and Model Building

McClave: Statistics, 11th ed. Chapter 12: Multiple Regression and Model Building

McClave: Statistics, 11th ed. Chapter 12: Multiple Regression and Model Building

McClave: Statistics, 11th ed. Chapter 12: Multiple Regression and Model Building

McClave: Statistics, 11th ed. Chapter 12: Multiple Regression and Model Building

McClave: Statistics, 11th ed. Chapter 12: Multiple Regression and Model Building

McClave: Statistics, 11th ed. Chapter 12: Multiple Regression and Model Building

Problem 1: Parameter Estimability

If x does not take on a sufficient number of different values, no single unique line can be estimated.

McClave: Statistics, 11th ed. Chapter 12: Multiple Regression and Model Building

Problem 2: Multicollinearity

Multicollinearityexists when two or more of the

independent variables in a regression are correlated.

If xi and xj move together in some way, finding the impact on y of a one-unit change in either of them holding the other constant will be difficult or impossible.

McClave: Statistics, 11th ed. Chapter 12: Multiple Regression and Model Building

Problem 2: Multicollinearity

Multicollinearitycan be detected in various ways.

A simple check is to calculate the correlation coefficients (rij)

for each pair of independent variables in the model.

Any significant rijmay indicate a multicollinearity problem.

- If severe multicollinearity exists, the result may be
- Significant F-values but insignificant t-values
- Signs on s opposite to those expected
- Errors in estimates, standard errors, etc.

McClave: Statistics, 11th ed. Chapter 12: Multiple Regression and Model Building

- The Federal Trade Commission (FTC) ranks cigarettes according to their tar (x1), nicotine (x2), weight in grams (x3) and carbon monoxide (y) content .
- 25 data points (see Table 12.11) are used to estimate the model

McClave: Statistics, 11th ed. Chapter 12: Multiple Regression and Model Building

McClave: Statistics, 11th ed. Chapter 12: Multiple Regression and Model Building

- F = 78.98, p-value < .0001
- t1= 3.97, p-value = .0007
- t2= -0.67, p-value = .5072
- t3= -0.3, p-value = .9735

McClave: Statistics, 11th ed. Chapter 12: Multiple Regression and Model Building

- F = 78.98, p-value < .0001
- t1= 3.97, p-value = .0007
- t2= -0.67, p-value = .5072
- t3= -0.3, p-value = .9735

The negative signs on two variables and the insignificant t-values are suggestive of multicollinearity .

McClave: Statistics, 11th ed. Chapter 12: Multiple Regression and Model Building

- The coefficients of correlation, rij, provide further evidence:
- rtar, nicotine = .9766
- rtar, weight = .4908
- rweight, nicotine = .5002

- Each rij is significantly different from 0 at the = .05 level.

McClave: Statistics, 11th ed. Chapter 12: Multiple Regression and Model Building

- Possible Responses to Problems Created by Multicollinearity in Regression
- Drop one or more correlated independent variables from the model.
- If all the xs are retained,
- Avoid making inferences about the individual parameters from the t-tests.
- Restrict inferences about E(y) and future y values to values of the xs that fall within the range of the sample data.

McClave: Statistics, 11th ed. Chapter 12: Multiple Regression and Model Building

Problem 3: Extrapolation

The data used to estimate the model provide information only on the range of values in the data set. There is no reason to assume that the dependent variable’s response will be the same over a different range of values.

McClave: Statistics, 11th ed. Chapter 12: Multiple Regression and Model Building

Problem 3: Extrapolation

McClave: Statistics, 11th ed. Chapter 12: Multiple Regression and Model Building

Problem 4: Correlated Errors

If the error terms are not independent (a frequent problem in time series), the model tests and prediction intervals are invalid. Special techniques are used to deal with time series models.

McClave: Statistics, 11th ed. Chapter 12: Multiple Regression and Model Building