正交旋轉二次回歸設計與RsReg第七節響應面分析當試驗中考察的指標宜于用多元二次回歸方程來擬合因素與指標的函數關系，就可以分析回歸方程所反應的曲面形狀，如果得到的曲面是凸面(像山丘)或凹面(像山谷)這類簡單曲面，那么預測的最佳指標值(極大值或極小值)可以從所估計的曲面上獲得；如果曲面很復雜，或者預測的最佳點遠離所考察因素的試驗范圍，那么可以通過嶺嵴分析來確定重新進行試驗的方向.這就是應用較廣，頗有實用價值的響應面分析法(ResponseSurfaceAnalysis).第六章回歸分析第六章回歸分析第六章回歸分析如果穩定點不是理想點就要進一步作嶺嵴分析,請看示意圖和例子演示第六章回歸分析RSREG的SAS過程二次型回歸響應面分析SAS簡單程序如下:datarubber;inputyx1x2@@;cards;數(略);procsort;byx1x2;/*對自變量x1x2進行sort由小到大排序*/procrsreg;modelf=td/lackfit;/*選項lackfit要求對回歸模型執行不適合度檢定(lack-of-fittest),預先應先對自變量進行sort由小到大排序*/run;第六章回歸分析PROCRSREG<options>;

（options:data=SASdataset,指明回歸所用數據集Out=SASdataset,指明回歸分析所得輸出的數據集）MODELresponses=independents</options>;

指定模型，

響應變量=自變量/選項（options:lackfif,要求回歸模型運行不適合檢定.若選用此項,則須先將數據集內的自變量由小到大排序。Nooptimal,停止尋求二項式反應面分析所需的臨界值Covar=n,指定前n個變量為共變量，所以它們只以一次式類型進入回歸模型里。L95，輸出95%置信區間的下界。U95，輸出95%置信區間的上界。）二次型回歸RIDGE<options>;脊嶺分析（options:CENTER=uncoded-factor-values給出脊嶺分析的初始值。MAX，輸出脊嶺分析的最大響應值。MIN，輸出脊嶺分析的最小響應值。RADIUS=coded-radii，脊嶺分析的距離。例如，radius=mtonbyj）WEIGHTvariable;（給指定的變量加以權重。）IDvariables;指定名稱變量。BYvariables;指定要獨立分析的變量，此選項須要數據集以由小到大排序。二次型回歸程序：dataa;inputnx1-x2y@@;cards;1-1-176.52-1177.631-178.041179.550080.360080.070079.780079.891.414078.410-1.414075.61101.41478.5120-1.41477.0;proc

rsregdata=a;modely=x1x2;ridgemax;idn;run;二次型回歸結果1：TheRSREGProcedureCodingCoefficientsfortheIndependentVariablesFactorSubtractedoffDividedbyx101.414000x201.414000ResponseSurfaceforVariabley響應變量的均值ResponseMean78.408333RootMSE0.372059R-Square0.9671變異系數CoefficientofVariation0.4745二次型回歸TypeISumRegressionDFofSquaresR-SquareFValuePr>F線性項Linear29.5571510.378534.520.0005方項Quadratic214.8214490.587053.530.0001交叉項Crossproduct10.0400000.00160.290.6102TotalModel524.4186000.967135.280.0002SumofResidualDFSquaresMeanSquareTotalError60.8305660.138428（總均方誤差參數估計與檢驗二次型回歸EigenvectorsEigenvaluesx1x2-1.9239350.1298960.991528-2.7001280.991528-0.129896Stationarypointisamaximum.TheRSREGProcedure嶺嵴分析EstimatedRidgeofMaximumResponseforVariabley編碼半徑CodedEstimatedStandardUncodedFactorValuesRadiusResponseErrorx1x20.079.9499210.186029000.180.0808990.1851200.1150410.0822160.280.1654920.1826510.2214990.1758230.380.2048610.1794250.3188260.2798140.480.2001070.1768740.4070400.3927110.580.1522150.1770920.4866480.5128580.680.0620270.1826420.5584660.6386690.779.9302410.1960010.6234430.7687800.879.7574260.2188260.6825270.9020920.979.5440340.2516370.7365881.0377621.079.2904300.2940900.7863901.175154典型分析二次型回歸例2

1971年John組織作一試驗要求達某一種難聞的化學氣味最小，設表示Odor一種難聞的化學氣味，T設表示溫度（Temperature）,R設表示氣體比（Gas-LiquidRatio）,H設表示容器高度（PackingHeight）,數據如下：編號溫度T氣體比R容器高度H化學氣味Odor1400.346621200.34393400.744341200.74495400.525861200.52177400.56-581200.56-409800.326510800.72711800.364312800.76-2213800.54-3114800.54-3515800.54-26SAS程序：title'ResponseSurfacewithaSimpleOptimum';datasmell;

inputOdorTRH@@;

label

T="Temperature"

R="Gas-LiquidRatio"

H="PackingHeight";

datalines;

6640.3439120.344340.7449120.74

5840.5217120.52-540.56-40120.56

6580.32780.724380.36-2280.76

-3180.54-3580.54-2680.54

procrsregdata=smell;

modelOdor=TRH/lackfit;

run;二次型回歸結果輸出TheRSREGProcedure

CodingCoefficientsfortheIndependentVariables

FactorSubtractedoffDividedby

T80.00000040.000000R0.5000000.200000H4.0000002.000000

ResponseSurfaceforVariableOdor

ResponseMean15.200000RootMSE22.478508R-Square0.8820CoefficientofVariation147.8849

二次型回歸TypeISumRegressionDFofSquaresR-SquareFValuePr>F

Linear37143.250.33374.710.0641Quadratic3114450.53467.550.0264Crossproduct3293.500.01370.190.8965TotalModel9188820.88204.150.0657

Sumof

ResidualDFSquaresMeanSquareFValuePr>F

LackofFit32485.750000828.58333340.750.0240PureError240.66666720.333333TotalError52526.416667505.283333

二次型回歸TheRSREGProcedureParameterEstimateStandardfromCodedParameterDFEstimateErrortValuePr>|t|Data

Intercept1568.958333134.6098164.230.0083-30.666667T1-4.1020831.489024-2.750.0401-12.125000R1-1345.833333335.220685-4.010.0102-17.000000H1-22.16666729.780489-0.740.4902-21.375000T*T10.0200520.0073112.740.040732.083333R*T11.0312501.4049070.730.49598.250000R*R11195.833333292.4546654.090.009547.833333H*T10.0187500.1404910.130.89901.500000H*R1-4.37500028.098135-0.160.8824-1.750000H*H11.5208332.9245470.520.62526.083333二次型回歸TheRSREGProcedure（響應曲面點典型分析）CanonicalAnalysisofResponseSurfaceBasedonCodedData

CriticalValue（穩定點）Factor（編碼）Coded（非編碼）UncodedLabel（注釋）T0.12191384.876502TemperatureR0.1995750.539915Gas-LiquidRatioH1.7705257.541050PackingHeight

（穩定點最小預測值）Predictedvalueatstationarypoint:-52.024631

二次型回歸EigenvectorsEigenvaluesTRH（特征值）（特征向量）48.8588070.2380910.971116-0.01569031.1034610.970696-0.2373840.0373996.037732-0.0325940.0241350.999177

特征值全大于0，故曲面有最小值Stationarypointisaminimum.二次型回歸可以作二個因素的響應面圖(固定其它因素),E62的響應面圖如下(作圖程序參見SAS操作入門):第六章回歸分析一．正交、旋轉回歸設計步驟（一）．確定試驗的因素及各因素的試驗水平的上下限A、B、C．…..（二）對試驗水平的上下限編碼因素試驗的下限————》-1因素試驗的上限————》1，例設因素A分為A1,A2兩水平[A1,A2];中點:Z=(A1+A2)/2半間隔:H=(A2-A1)/2編碼值:Xi=(Ai-Z)/H編碼值在[-1,1]上.A1=225,A2=375Z=(225+375)/2=300H=(375-225)/2=75X2=(375-300)/75=1X1=(225-300)/75=-1X0=(300-300)/75=0對1.682=(At-300)/75可求得At=426（三）設計試驗1.一般p個變量的組合設計由下列N個點：，，

——2水平（+1和-1）的全因子試驗的試驗點個數-----分布在p個坐標軸上的星號點，它們與中心點的距離稱為星號臂——在各變量都取零水平的中心點的重復試驗次數。它可以只做一次，也可以重復二次或多次。值表二次回歸正交2345(實施)11.001.4762.002.3921.1601.6502.1982.5831.3171.8312.3902.7741.4752.0002.5802.9551.6062.1642.7703.1461.7422.3252.9503.3171.8732.4813.1403.4982.0002.6333.3103.6692.1232.7823.4903.83102.2432.9283.664.003.二次回歸旋轉設計正交通用方案正交N通用N正交通用1241.4141613852381.68223209634162.000363112745322.37859117155(）162.0003632106dataExperiment;inputx1x2Yt1t2;datalines;8017076.5–1-18018077.0–1190170781-19018079.5118517579.9008517580.3008517580008517579.7008517579.80092.0717578.41.414077.9317575.6–1.414085182.0778.501.41485167.93770–1.414;procrsregdata=Experiment;modelY=x1x2;run;TheRSREGProcedureCodingCoefficientsfortheIndependentVariablesFactorSubtractedoffDividedbyx185.0000007.070000x2175.0000007.070000ResponseSurfaceforVariableYResponseMean78.476923RootMSE0.266290R-Square0.9827CoefficientofVariation0.3393

TypeISumRegressionDFofSquaresR-SquareFValuePr>FLinear210.0429550.349470.81<.0001Quadratic217.9537490.6246126.59<.0001Crossproduct10.2500000.00873.530.1025TotalModel528.2467030.982779.67<.0001SumofResidualDFSquaresMeanSquareTotalError70.4963730.070910

ParameterEstimateStandardfromCodedParameterDFEstimateErrortValuePr>|t|DataIntercept1-1430.688438152.851334-9.36<.000179.939955x117.8088651.1578236.740.00031.407001x2113.2717451.4846068.94<.00010.728497x1*x11-0.0550580.004039-13.63<.0001-2.752067x2*x110.0100000.0053261.880.10250.499849x2*x21-0.0400530.004039-9.92<.0001-2.002067SumofFactorDFSquaresMeanSquareFValuePr>Fx1321.3440087.114669100.33<.0001x239.3452513.11508443.93<.0001

TheRSREGProcedureCanonicalAnalysisofResponseSurfaceBasedonCodedDataCriticalValueFactorCodedUncodedx10.27526986.946152x20.216299176.529233Predictedvalueatstationarypoint:80.212393Eigenvectors