II, III, IV:
The 3 properties can be checked by plotting the residuals versus
(a)time order
(b)fitted value
(c)the covariate
The typical satisfactory plots are as follows:
Three typical unsatisfactory plots are as follows:
(i)
(ii)
(iii)
(a)versus time order:
Suppose the time order of the data is known. The data is
.
, . Suppose the postulated model is
The residual plot (i): nonconstant variance.
As have nonconstant variances and the variances are increasing (i.e., ), then (corresponding to and the “estimate” of ) would be more stable than (corresponding to ). That is, would fluctuate less rapidly than . Thus, the residual plot would look like (i).
Remedy: use weighted least square!!
The residual plot (ii): a first order term in time is missing.
Suppose the true model is
Thus,
Therefore, the residuals would be increasing with the time and the residual plot would look like (ii).
Remedy:add a first order term in time.
The residual plot (iii): the first and second order terms in time are missing.
The reason is similar to the one in (i)..
Remedy:add the first order and second order terms in time.
(b)versus fitted value :
Why , but not :
Therefore, and are correlated. If we plot versus , then a linear trend might be due to the positive correlation of and rather than the violation of the assumptions about the random error. On the other hand, since and are uncorrelated, a unsatisfactory residual plot might do imply the violation of the assumptions.
Derivation of :
Let
and .
Since
and
then
Thus,
Derivation of :
Sicne
thus,
The residual plot (i): nonconstant variance.
The reason is similar to the one given in (a).
Remedy:use weighted least square or transformation of .
Note:
Transformation of the response can stabilize the variance (make the variance of the transformed constant). We will discuss in next section.
The residual plot (ii): errors in analysis, some extra terms highly correlated to theother terms are missing or wrongful omission of .
Suppose the postulated model is
and the true model is
,
Then,
and
.
If and is correlated to (since is correlated to some covariates in the postulated model), then the residual plot will be like (ii).
Remedy:rechecking the analysis process, adding some extra terms to the model or adding back to the model.
The residual plot (iii): some extra terms (might be second order terms of existing variables or some variables not in the model) are missing or unequal variance
Remedy:adding extra terms to the model or transformation of .
(c)versus the covariate :
The residual plot (i): nonconstant variance.
The reason is similar to the one given in (a).
Remedy:use weighted least square or transformation of .
The residual plot (ii): errors in analysis or some extra terms highly correlated to are missing.
Suppose the true model
,
and the covariate is correlated to Z. Thus, heuristically,
.
As and is positively correlated to Z, the residual plot would look like (ii).
Remedy:rechecking the analysis process or adding some extra terms to the model.
The residual plot (iii): some extra terms are missing or the variance of are not equal.
Remedy:adding extra terms (second order terms of the existing variables or the terms not in the original model) to the model or transformation of
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