Examining Lack Of Fit In Linear Regression 11.5

Front 1

In our study of linear regression, we have been concerned with how well a linear regression model
y =b0 + b1x + e fits, but only from an intuitive standpoint. We could examine a __________ of the data to see whether it looked _________ and we could test whether the slope differed from _____;

Back 1

scatterplot, linear, 0

Front 2

Pictures (or graphs) are always a good starting point for examining ________.First, use a scatterplot of y versus x. Second, a plot of residuals yi -yˆi versus predicted values yˆi may give an indication of the following problems:

1.Outliers or erroneous observations. In examining the residual plot, your eye will naturally be drawn to data points with unusually high (in absolute value) residuals.

2. Violation of the assumptions. For the model y = b0 + b1x +e, we have assumed a _______ relation between y and the dependent variable x, and independent, _________ distributed errors with a ____________ variance.

Back 2

lack of fit, linear , normally, constant

Front 3

The residual plot for a model and data set that has none of these apparent problems would look much like the plot in Figure 11.15. Note from this plot that there are no extremely ________ residuals (and hence no apparent __________) and there is no trend in the residuals to indicate that the linear model is inappropriate.

Back 3

large, outliers

Front 4

A check of the constant variance assumption can be addressed in the y versus x scatterplot or with a plot of the residuals (yi -yˆi) versus xi. For example, a pattern of residuals as shown in Figure 11.17 indicates __________(all of the same degree) error variances across values of x; Figure 11.18 indicates that the error variances increase with increasing values of x.

Back 4

homogeneous,

Front 5

How can we test for the apparent lack of fit of the linear regression model in Example 11.10? When there is more than one ___________ per level of the independent variable, we can conduct a test for __________ of the fitted model by partitioning SS (Residuals) into two parts, one pure experimental error and the other lack of fit.

Back 5

observation, lack of fit

Front 6

Let yij denote the response for the jth observation at the ith level of the independent variable. Then, if there are ni observations at the ith level of the independent variable, the quantity ∑j(yij-y.bari)^2 provides a measure of what we will call _________________. This sum of squares has ni - 1 degrees of freedom.

Back 6

pure experimental error

Front 7

Similarly, for each of the other levels of x, we can compute a sum of squares due to _____________. The pooled sum of squares SSPexp=∑ij(yij-y.bari)^2 called the sum of squares for pure experimental error, has a ∑i(ni -1) degrees of freedom.

Back 7

pure experimental error

Front 8

With SSLack representing the remaining portion of ______, we have
SS(Residuals) =SSPexp (due to pure experimental error) + SSlack ( due to lack of fit)
SSR = SSPexp + SSlack

Back 8

SSE

Front 9

SSR = SSPexp + SSlack
If SS(Residuals) is based on n - 2 degrees of freedom in the linear regression model, then ________ will have df =n -2-∑i(ni - 1).

SSPexp (due to pure experimental error)
SSlack ( due to lack of fit)

Back 9

SSLack

Front 10

Under the null hypothesis that our model is correct, we can form independent estimates of σ^2ε, the model error variance, by dividing SSPexp and SSLack by their respective _____________; these estimates are called mean squares and are denoted by MSPexp and MSLack, respectively.

SSPexp (due to pure experimental error)
SSlack ( due to lack of fit)

Back 10

degrees of freedom

Front 11

A Test for Lack of Fit in Linear Regression

H0: A linear regression model is appropriate.
Ha: A linear regression model is not appropriate.

T.S.: F = MSLack / MSPexp

where
MSexp = SSPexp/ ∑i(ni -1) = ∑ij(yij-y.bari)^2 / ∑i(ni-1)
and
MSlack = (SS(Residual) - SSPexp)/ (n-2-∑(n -1) )

R.R.: For specified value of a, reject H0 (the adequacy of the model) if the computed value of F __________ the table value for df1=n-2-∑i(ni-1) and df2 =∑i(n1-1).

Conclusion: If the F test is significant, this indicates that the linear regression model is _________. A nonsignificant result indicates that there is ____________ evidence to suggest that the linear regression model is _________________.


SSPexp (due to pure experimental error)
SSlack ( due to lack of fit)

Back 11

exceeds, inadequate, insufficient, inappropriate

Front 12

The calculation of SSPexp can be obtained by using the One-Way ANOVA command in a software package. Using the theory from Chapter 8, designate the levels of the independent variable x as the levels of a treatment. The sum of squares error from this output is the value of _________. This concept is illustrated using the output from Minitab given here.

SSPexp= sum of squares due to pure experimental error

Back 12

SSPexp
sum of squares due to pure experimental error

Front 13

The F statistic for the test of lack of fit is
F =MSLack / MSPexp = 760.5 / 34.06 = 22.33
Using df1 =1, df2 =6, and α= .05, we will reject H0 if F > 5.99.
Because the computed value of F exceeds 5.99, we ______H0 and conclude that there is significant _________ for a linear regression model.

Back 13

reject , lack of fit

Front 14

To summarize: In situations for which there is more than one __________ at one or more levels of x, it is possible to conduct a formal test for ____________ of the linear regression model. This test should precede any inferences made using the fitted linear regression line. If the test for lack of fit is_________, some higher-order polynomial in x may be more appropriate.

Back 14

y-value, lack of fit, significant

Front 15

A scatterplot of the data and a_____________ from the linear regression line should help in selecting the appropriate model. More information on the selection of an appropriate model will be discussed along with multiple regression (Chapters 12 and 13).
If the F test for lack of fit is ______ significant, proceed with inferences based on the fitted linear regression line.

Back 15

residual plot, not