Engineering Statistics Handbook Episode 5 Part 15 potx

Hidden Structure Revealed Scale of Plot Key The structure in the relationship between the residuals and the load clearly indicates that the functional part of the model is misspecified..

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4.6.1.3 Model Fitting - Initial Model

http://www.itl.nist.gov/div898/handbook/pmd/section6/pmd613.htm (2 of 2) [5/1/2006 10:22:34 AM]

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Residual plots of interest for this model include:

residuals versus the predictor variable

1

residuals versus the regression function values

2

residual run order plot

3

residual lag plot

4

histogram of the residuals

5

normal probability plot

6

A plot of the residuals versus load is shown below

Hidden

Structure

Revealed

Scale of Plot

Key

The structure in the relationship between the residuals and the load clearly indicates that the functional part of the model is misspecified The ability of the residual plot to clearly show this problem, while the plot of the data did not show it, is due to the difference in scale between the plots The curvature in the response is much smaller than the linear trend Therefore the curvature

is hidden when the plot is viewed in the scale of the data When the linear trend is subtracted, however, as it is in the residual plot, the curvature stands out

The plot of the residuals versus the predicted deflection values shows essentially the same structure as the last plot of the residuals versus load For more complicated models, however, this plot can reveal problems that are not clear from plots of the residuals versus the predictor

variables

4.6.1.4 Graphical Residual Analysis - Initial Model

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Residual

Structure

Additional

Diagnostic

Plots

Further residual diagnostic plots are shown below The plots include a run order plot, a lag plot, a histogram, and a normal probability plot Shown in a two-by-two array like this, these plots comprise a 4-plot of the data that is very useful for checking the assumptions underlying the model

Dataplot

4plot

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of Plots

The structure evident in these residual plots also indicates potential problems with different aspects of the model Under ideal circumstances, the plots in the top row would not show any systematic structure in the residuals The histogram would have a symmetric, bell shape, and the normal probability plot would be a straight line Taken at face value, the structure seen here indicates a time trend in the data, autocorrelation of the measurements, and a non-normal distribution of the residuals

It is likely, however, that these plots will look fine once the function describing the systematic relationship between load and deflection has been corrected Problems with one aspect of a regression model often show up in more than one type of residual plot Thus there is currently no clear evidence from the 4-plot that the distribution of the residuals from an appropriate model would be non-normal, or that there would be autocorrelation in the process, etc If the 4-plot still indicates these problems after the functional part of the model has been fixed, however, the possibility that the problems are real would need to be addressed

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4.6.1.5 Interpretation of Numerical Output - Initial Model

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4.6.1.6 Model Refinement

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4.6.1.7 Model Fitting - Model #2

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Indicates

Model

Fits Well

The residuals randomly scattered around zero, indicate that the quadratic is a good function to describe these data There is also no indication of non-constant variability over the range of loads

Plot Also

Indicates

Model

OK

4.6.1.8 Graphical Residual Analysis - Model #2

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This plot also looks good There is no evidence of changes in variability across the range of deflection

No

Problems

Indicated

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All of these residual plots have become satisfactory by simply by changing the functional form of the model There is no evidence in the run order plot of any time dependence in the measurement process, and the lag plot suggests that the errors are independent The histogram and normal probability plot suggest that the random errors affecting the measurement process are normally distributed

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All of the parameters are significantly different from zero, as indicated by the

associated t statistics The 97.5% cut-off for the t distribution with 37 degrees of freedom is 2.026 Since all of the t values are well above this cut-off, we can safely

conclude that none of the estimated parameters is equal to zero.

4.6.1.9 Interpretation of Numerical Output - Model #2

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a

Numerical

Calibration

Value

To solve for the numerical estimate of the load associated with the observed deflection, the observed value substituting in the regression function and the equation is solved for load

Typically this will be done using a root finding procedure in a statistical or mathematical package That is one reason why rough bounds on the value of the load to be estimated are needed

Solving the

Regression

Equation

Which

Solution?

Even though the rough estimate of the load associated with an observed deflection is not necessary to solve the equation, the other reason is to determine which solution to the equation is correct, if there are multiple solutions The quadratic calibration equation, in fact, has two

solutions As we saw from the plot on the previous page, however, there is really no confusion over which root of the quadratic function is the correct load Essentially, the load value must be between 150,000 and 3,000,000 for this problem The other root of the regression equation and the new deflection value correspond to a load of over 229,899,600 Looking at the data at hand, it

is safe to assume that a load of 229,899,600 would yield a deflection much greater than 1.24

+/- What? The final step in the calibration process, after determining the estimated load associated with the

observed deflection, is to compute an uncertainty or confidence interval for the load A single-use 95% confidence interval for the load, is obtained by inverting the formulas for the upper and lower bounds of a 95% prediction interval for a new deflection value These inequalities, shown below, are usually solved numerically, just as the calibration equation was, to find the end points

of the confidence interval For some models, including this one, the solution could actually be obtained algebraically, but it is easier to let the computer do the work using a generic algorithm

The three terms on the right-hand side of each inequality are the regression function ( ), a t-distribution multiplier, and the standard deviation of a new measurement from the process ( ) Regression software often provides convenient methods for computing these quantities for arbitrary values of the predictor variables, which can make computation of the confidence interval end points easier Although this interval is not symmetric mathematically, the asymmetry is very small, so for all practical purposes, the interval can be written as

4.6.1.10 Use of the Model for Calibration

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