Engineering Statistics Handbook Episode 5 Part 15 potx

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] Residual plots of interest for this model include: residuals versus the predictor variable1. residuals versus the regression function values2. residual run order plot3. residual lag plot4. histogram of the residuals5. normal probability plot6. 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 http://www.itl.nist.gov/div898/handbook/pmd/section6/pmd614.htm (2 of 4) [5/1/2006 10:22:34 AM] Similar 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 4.6.1.4. Graphical Residual Analysis - Initial Model http://www.itl.nist.gov/div898/handbook/pmd/section6/pmd614.htm (3 of 4) [5/1/2006 10:22:34 AM] Interpretation 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. 4.6.1.4. Graphical Residual Analysis - Initial Model http://www.itl.nist.gov/div898/handbook/pmd/section6/pmd614.htm (4 of 4) [5/1/2006 10:22:34 AM] 4.6.1.5. Interpretation of Numerical Output - Initial Model http://www.itl.nist.gov/div898/handbook/pmd/section6/pmd615.htm (2 of 2) [5/1/2006 10:22:35 AM] 4.6.1.6. Model Refinement http://www.itl.nist.gov/div898/handbook/pmd/section6/pmd616.htm (2 of 2) [5/1/2006 10:22:35 AM] 4.6.1.7. Model Fitting - Model #2 http://www.itl.nist.gov/div898/handbook/pmd/section6/pmd617.htm (2 of 2) [5/1/2006 10:22:35 AM] Plot 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 http://www.itl.nist.gov/div898/handbook/pmd/section6/pmd618.htm (2 of 4) [5/1/2006 10:22:36 AM] This plot also looks good. There is no evidence of changes in variability across the range of deflection. No Problems Indicated 4.6.1.8. Graphical Residual Analysis - Model #2 http://www.itl.nist.gov/div898/handbook/pmd/section6/pmd618.htm (3 of 4) [5/1/2006 10:22:36 AM] [...]... affecting the measurement process are normally distributed http://www.itl.nist.gov/div898 /handbook/ pmd/section6/pmd618.htm (4 of 4) [5/ 1/2006 10:22:36 AM] 4.6.1.9 Interpretation of Numerical Output - Model #2 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... 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 http://www.itl.nist.gov/div898 /handbook/ pmd/section6/pmd619.htm (2 of 2) [5/ 1/2006 10:22:36 AM] 4.6.1.10 Use of the Model for Calibration Obtaining a Numerical Calibration Value To solve for the numerical estimate of the load associated with the observed... 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... 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... easier Although this interval is not symmetric mathematically, the asymmetry is very small, so for all practical purposes, the interval can be written as http://www.itl.nist.gov/div898 /handbook/ pmd/section6/pmd61a.htm (2 of 3) [5/ 1/2006 10:22:37 AM] . Model http://www.itl.nist.gov/div898 /handbook/ pmd/section6/pmd614.htm (4 of 4) [5/ 1/2006 10:22:34 AM] 4.6.1 .5. Interpretation of Numerical Output - Initial Model http://www.itl.nist.gov/div898 /handbook/ pmd/section6/pmd6 15. htm. Model http://www.itl.nist.gov/div898 /handbook/ pmd/section6/pmd6 15. htm (2 of 2) [5/ 1/2006 10:22: 35 AM] 4.6.1.6. Model Refinement http://www.itl.nist.gov/div898 /handbook/ pmd/section6/pmd616.htm (2 of 2) [5/ 1/2006 10:22: 35 AM] 4.6.1.7. Model Fitting. #2 http://www.itl.nist.gov/div898 /handbook/ pmd/section6/pmd618.htm (4 of 4) [5/ 1/2006 10:22:36 AM] All of the parameters are significantly different from zero, as indicated by the associated t statistics. The 97 .5% cut-off