SAS-LOGISTIC回归.doc

LOGISTIC回归二、Logit回归模型除这三个特殊点之外，还应有一个自然的要求，就是的极限存在，至少随的增加而变化的速率应该越来越慢，而不能象线性模型那样直来直去成比例增长。以住房收入模型而言， 当收入为10时，有住房的可能性是0.0607；当收入提高到20时，有住房的可能性为1.1087，已超过100%；当收入为30时，则为2.1567，等等。显然，这个模型需要改进。oX1图 A改进的目标可以用图A表示。如果有一个这样的模型函数，则它满足，同时变化速率在起始阶段比较慢，中期越来越快，到后期又越来越缓，比较符合实际。怎样找到这样一个函数呢?函数具有此性质原来是 如果改进为 则，并且在时变化越来越缓。记，则 这就得到了我们需要的Logit模型函数，原来是对它取了对数，故名Log it。这个函数不是与呈线性关系，而是与呈线性关系。当时， 。与的关系曲线正是上图表示的形曲线。将自变量扩充为多元，加上随机项，就得到一般的Logit回归模型： 如果我们从这个模型中得到的估计，就可以估计出第个样本有(或无)的可能性。但是又产生一个新问题，我们如何得到呢? 如果从原来的二值选择数据出发，我们连回归模型都建立不起来。因为二值选择或，无法取对数。原来数据是从纯粹的个体出发的。我们可以改从小范围的个体出发建立数据，即将自变量按某些值分成若干组。比如分成组。每组里有个样品，取有时是个，则取无时是个，于是以这样的数据可以代入Logit回归模型。例1The data, taken fromCox and Snell (1989, pp. 1011), consist of the number, r, of ingots not ready forrolling, out of n tested, for a number of combinations of heating time and soakingtime.data ingots;input Heat Soak r n ;datalines;7 1.0 0 10 14 1.0 0 31 27 1.0 1 56 51 1.0 3 137 1.7 0 17 14 1.7 0 43 27 1.7 4 44 51 1.7 0 17 2.2 0 7 14 2.2 2 33 27 2.2 0 21 51 2.2 0 17 2.8 0 12 14 2.8 0 31 27 2.8 1 22 51 4.0 0 17 4.0 0 9 14 4.0 0 19 27 4.0 1 16;proc logistic data=ingots;model r/n=Heat Soak;run;分析两个因素heat,soak影响铸铁成功加执时间Heat浸处理时间Soak没准备好的次数（失败次数）r试验次数n7101014103127115651131371.7017141.7043271.7444511.70172.207142.2233272.2021512.20172.8012142.8031272.8122514017409144019274116HeatSoakNotReadyFreq71010141031144019272.20215111371.7017141.704327111272.81151101072.207142.212271055272.8021511.70172.8012142.2031271.71427411512.2017409142.8031271.704027401551401data ingots;input Heat Soak NotReady Freq ;datalines;7 1.0 0 10 14 1.0 0 31 14 4.0 0 19 27 2.2 0 21 51 1.0 1 37 1.7 0 17 14 1.7 0 43 27 1.0 1 1 27 2.8 1 1 51 1.0 0 107 2.2 0 7 14 2.2 1 2 27 1.0 0 55 27 2.8 0 21 51 1.7 0 17 2.8 0 12 14 2.2 0 31 27 1.7 1 4 27 4.0 1 1 51 2.2 0 17 4.0 0 9 14 2.8 0 31 27 1.7 0 40 27 4.0 0 15 51 4.0 0 1;proc logistic data=ingots descending;model NotReady = Soak Heat;freq Freq;run;例研究癌免疫(remiss=1有免疫,，因素变量六个，细胞cell，smear infil li blast tempremisscellsmearinfilliblasttemp10.80.830.661.91.10.99610.90.360.321.40.740.99200.80.880.70.80.1760.982010.870.870.71.0530.98610.90.750.681.30.5190.98010.650.650.60.5190.98210.950.970.9211.230.99200.950.870.831.91.3541.02010.450.450.80.3220.99900.950.360.340.501.03800.850.390.330.70.2790.98800.70.760.531.20.1460.98200.80.460.370.40.381.00600.20.390.080.80.1140.99010.90.91.11.0370.99110.840.841.92.0641.0200.650.420.270.50.1141.014010.750.7511.3221.00400.50.440.220.60.1140.99110.630.631.11.0720.986010.330.330.40.1761.0100.90.930.840.61.5911.02110.580.5810.5311.00200.950.320.31.60.8860.988110.60.61.70.9640.99110.690.690.90.3980.986010.730.730.70.3980.986data Remission;input remiss cell smear infil li blast temp;label remiss=Complete Remission;datalines;1 .8 .83 .66 1.9 1.1 .9961 .9 .36 .32 1.4 .74 .9920 .8 .88 .7 .8 .176 .9820 1 .87 .87 .7 1.053 .9861 .9 .75 .68 1.3 .519 .980 1 .65 .65 .6 .519 .9821 .95 .97 .92 1 1.23 .9920 .95 .87 .83 1.9 1.354 1.020 1 .45 .45 .8 .322 .9990 .95 .36 .34 .5 0 1.0380 .85 .39 .33 .7 .279 .9880 .7 .76 .53 1.2 .146 .9820 .8 .46 .37 .4 .38 1.0060 .2 .39 .08 .8 .114 .990 1 .9 .9 1.1 1.037 .991 1 .84 .84 1.9 2.064 1.020 .65 .42 .27 .5 .114 1.0140 1 .75 .75 1 1.322 1.0040 .5 .44 .22 .6 .114 .991 1 .63 .63 1.1 1.072 .9860 1 .33 .33 .4 .176 1.010 .9 .93 .84 .6 1.591 1.021 1 .58 .58 1 .531 1.0020 .95 .32 .3 1.6 .886 .9881 1 .6 .6 1.7 .964 .991 1 .69 .69 .9 .398 .9860 1 .73 .73 .7 .398 .986;proc logistic data=Remission descending;model remiss=cell smear infil li blast temp/selection=backward;output out=pred p=phat lower=lcl upper=ucl predprobs=(individual crossvalidate);run;proc print data=pred;run; The LOGISTIC Procedure Model Information Data Set SAS.SSS Response Variable remiss Complete Remission Number of Response Levels 2 Model binary logit Optimization Technique Fishers scoring Number of Observations Read 27 Number of Observations Used 27 Response Profile Ordered Total Value remiss Frequency 1 1 9 2 0 18 Probability modeled is remiss=1. Model Convergence Status Convergence criterion (GCONV=1E-8) satisfied. Model Fit Statistics Intercept Intercept and Criterion Only Covariates AIC 36.372 35.751 SC 37.668 44.822 -2 Log L 34.372 21.751 Testing Global Null Hypothesis: BETA=0 Test Chi-Square DF Pr ChiSq Likelihood Ratio 12.6211 6 0.0495 Score 9.4609 6 0.1493 Wald 4.5302 6 0.6053 The LOGISTIC Procedure Analysis of Maximum Likelihood Estimates Standard Wald Parameter DF Estimate Error Chi-Square Pr ChiSq Intercept 1 58.0385 71.2364 0.6638 0.4152 cell 1 24.6613 47.8376 0.2658 0.6062 smear 1 19.2933 57.9499 0.1108 0.7392 infil 1 -19.6009 61.6814 0.1010 0.7507 li 1 3.8960 2.3371 2.7789 0.0955 blast 1 0.1511 2.2786 0.0044 0.9471 temp 1 -87.4337 67.5735 1.6742 0.1957 Odds Ratio Estimates Point 95% Wald Effect Estimate Confidence Limits cell 999.999 999.999 smear 999.999 999.999 infil 0.001 999.999 li 49.203 0.504 999.999 blast 1.163 0.013 101.191 temp 0.001 999.999 Association of Predicted Probabilities and Observed Responses Percent Concordant 88.3 Somers D 0.765 Percent Discordant 11.7 Gamma 0.765 Percent Tied 0.0 Tau-a 0.353 Pairs 162 c 0.883 Step 5. Effect temp is removed: Model Convergence Status Convergence criterion (GCONV=1E-8) satisfied. Model Fit Statistics Intercept Intercept and Criterion Only Covariates AIC 36.372 30.073 SC 37.668 32.665 -2 Log L 34.372 26.073 The LOGISTIC Procedure Testing Global Null Hypothesis: BETA=0 Test Chi-Square DF Pr ChiSq Likelihood Ratio 8.2988 1 0.0040 Score 7.9311 1 0.0049 Wald 5.9594 1 0.0146 Residual Chi-Square Test Chi-Square DF Pr ChiSq 3.1174 5 0.6819 NOTE: No (additional) effects met the 0.05 significance level for removal from the model. Analysis of Maximum Likelihood Estimates Standard Wald Parameter DF Estimate Error Chi-Square Pr ChiSq Intercept 1 -3.7771 1.3786 7.5064 0.0061 li 1 2.8973 1.1868 5.9594 0.0146 Odds Ratio Estimates Point 95% Wald Effect Estimate Confidence Limits li 18.124 1.770 185.563 Association of Predicted Probabilit