Numeric Summary¶
The Numeric Summary is used to display all numerical information associated with the surface-fit for the current equation.
This summary includes goodness of fit criteria, coefficient standard errors and confidence limits for the fitted parameters, volume below the surface, function extrema, the fitting method(s) used, an analysis of variance, exponential scaling constants if used in a basis function, and data table statistics.
The Numeric Summary is toggled on and off by the Numeric button in the Surface-Fit Graph window. The window size and position you choose for the Numeric Summary is automatically saved across sessions.
The Numeric Summary uses the TableCurve 3D Text Viewer to display, modify, and print the information.
Equation Fitted and Rank in List¶
Rank 3 Eqn 522 Fourier Series Bivariate Order 2x3 Goodness of Fit Statistics r2 Coef Det DF Adj r2 Fit Std Err F-value 0.9979528134 0.9969871594 0.2919274448 1096.8193301
Fitted Parameters and Comfidence Limits¶
Parm Value Std Error t-value 99.90% Confidence Limits P>|t| a 74.30517552 1.770059367 41.97891714 68.1453585 80.46499253 0.00000 b 0.986823301 0.3797288 2.598758114 -0.33463516 2.308281762 0.01204 c -0.34202761 0.388577111 -0.88020524 -1.69427824 1.010223026 0.38265 d -5.24346418 2.087734632 -2.51155684 -12.508793 2.021864621 0.01504 e -8.94715852 2.28242333 -3.92002588 -16.8900051 -1.0043119 0.00025 f 1.043002929 1.111378732 0.938476597 -2.8246016 4.910607456 0.35218 g -0.80256585 1.246099619 -0.64406235 -5.13899986 3.533868157 0.52226 h 0.132899921 0.274113322 0.48483569 -0.82101605 1.086815894 0.62975 i 0.37897685 0.261764335 1.447778781 -0.53196458 1.289918276 0.15346 j -0.02660108 0.139694928 -0.19042266 -0.51274025 0.459538093 0.84969 k 0.323133031 0.138188611 2.338347776 -0.15776415 0.80403021 0.02310 l -0.42886429 0.356204615 -1.20398297 -1.66845845 0.810729867 0.23385 m 1.044709382 0.406204822 2.571878338 -0.36888579 2.458304552 0.01290 n 0.158723083 0.129754637 1.223255578 -0.29282382 0.610269983 0.22654 o -0.08908836 0.506974193 -0.17572563 -1.85336153 1.675184818 0.86117 p 0.18937369 0.534998108 0.353970765 -1.67242288 2.051170255 0.72474 q -0.84354929 0.225581907 -3.73943683 -1.62857565 -0.05852293 0.00045 r 0.076935016 0.2327349 0.330569314 -0.7329838 0.886853836 0.74225 s -0.06560973 0.101605264 -0.64573159 -0.41919665 0.287977189 0.52119 t -0.12476745 0.105965018 -1.17744003 -0.49352634 0.243991433 0.24418 u 3.777106659 1.447000418 2.610301014 -1.25846331 8.812676631 0.01168 v 5.497115185 0.472363492 11.63746833 3.853287452 7.140942919 0.00000 aa 2.056483625 0.475719828 4.322888188 0.400975824 3.711991426 0.00007 ab 0.054626318 0.215629076 0.253334659 -0.69576414 0.805016772 0.80097 ac 0.396870188 0.218126386 1.819450616 -0.36221092 1.155951297 0.07439
Surface Minimum and Maximum, Volume (Optional)¶
X at Fn Zmin Y at Fn Zmin Fn Zmin 24.064490901 44.817517889 55.218704542 X at Fn Zmax Y at Fn Zmax Fn Zmax 4 10 74.879049048
Analysis of Variance (ANOVA)¶
Procedure GaussElim r2 Coef Det DF Adj r2 Fit Std Err 0.9979528134 0.9969871594 0.2919274448 Source Sum of Squares DF Mean Square F Statistic P>F Regr 2243.3456 24 93.472734 1096.82 0.00000 Error 4.6019682 54 0.085221633 Total 2247.9476 78
Data Statistics¶
Description: Sample Data for Peak X Variable: Drug A Xmin: 4 Xmax: 40 Xrange: 36 Xmean: 21.772151899 Xstd: 11.453304715 Y Variable: Drug B Ymin: 10 Ymax: 80 Yrange: 70 Ymean: 44.556962025 Ystd: 22.859519049 Z Variable: Response Zmin: 55.8 Zmax: 74.8 Zrange: 19 Zmean: 68.120253165 Zstd: 5.3684114001
The Numeric Summary offers four options which enable you to control the amount of information presented:
· Probabilities - adds probability values to coefficient summary and analysis of variance table
· Volume and Extrema - adds function volume (if selected in Review Preferences) and measured function extrema
· Analysis of Variance - adds ANOVA table
· Data Table Statistics - adds basic data table statistics and titles
Pure Error and Lack of Fit¶
When there are multiple Y observations at a specific X value, the variance in these Y values can be said to consist of pure or random error. When such repeats or replicates are present in a data set, it is possible to separate the portion of the variance attributable to pure error from that portion associated with the fitted model. This expanded analysis of variance will be automatically included in the numeric summary when such replicates are present and all weights remain set at unity.
The r² Attainable is the maximum r² that can be achieved with any model. The Pure Err sum of squares will reflect that portion of the sum of squared residuals attributable to pure error in these repeat observations. The Lack Fit sum of squares is simply the difference between the overall sum of squared residuals and this pure error sum of squares. When this lack of fit F-statistic is significant (>>1), the model may be inadequate. In such cases, the residuals should be checked closely for systematic trends which would confirm the insufficiency of the model.
Volume and Function Extrema¶
The volume below the fitted surface is computed using a Gaussian Quadrature double integration from the lowest active X value in the data set to the highest active X value, and from the lowest active Y value in the data set to the highest active Y value. The precision attained in the volume computation is also shown. The volume is only added if selected in the Review Preferences. Because the volume computation can be very time consuming on non-converging surfaces (those with singularities), you should activate this volume option only if the information is of some value.
The function extrema are from a two-dimensional minimization using Brent's algorithm, the bracketing consisting of this same range of X and Y values.