1、化驗結果診斷模型問題重述與分析人們到醫院就診時，通常要化驗一些指標來協助醫生的診斷。本題給出了人們是否患某種疾病時通常要化驗的幾種指標以及其檢驗值。表1是確診病例的化驗結果，其中130號病例是已經確診為患該種疾病的化驗結果；3160號病例是已經確診為健康人的結果。表2是某些就診人員的化驗結果，但未確診其是否患有該種疾病。根據已知數據，需要解答如下問題：1）問題：根據表1中的數據，提出一種簡便的判別方法，判別屬于患者或健康人的方法，并檢驗你提出方法的正確性。分析：根據表1當中60個化驗結果，將Zn、CuFe、CaMgK、Na看成是七個指標，則前30個為該疾病患者的指標值，后30個為健康人的指標值

2、，可以將這些數據進行標準化處理，再采用主成分分析方法，將多個指標轉化為幾個綜合指標，當給定一個患者的各指標值時，可以算出各綜合指標的得分，當這些得分滿足一定條件時，如根據正負值可以判定為健康或疾病。2）問題：按照（1）提出的方法，對表2中的15名就診人員的化驗結果進行判別，判定他們是患該種疾病的病人還是健康人。分析：由（1）中已有的綜合指標，根據給定的15名就診人員的指標值計算出綜合指標的得分，以此判斷他們的健康狀況。3）問題：能否根據表1的數據特征，確定哪些指標是影響人們患該疾病的關鍵或主要因素，以便減少化驗的指標。并根據你給出的結果，重復2的工作。分析：為了確定哪些指標是影響該疾病的主要因

3、素，則需要確定出哪些因素在判別中起的權重最大，可以考慮采取回歸模型，通過去除一些變量，然后比較各組的顯著性與正確率，正確率最高的那組中的變量即為影響該疾病的主要因素。、模型假設1）假設醫院化驗設備先進，化驗過程科學可靠，化驗結果真實可信，確診情況（有病/健康人）符合實際。2）在解決本題過程中，所有的化驗結果只是針對該類疾病檢驗，并不考慮其他疾病的影響。3）本文所建模型的檢驗結果只是作為醫生為病人診斷的一個參考，醫生為問診人員作出最終判定還需考慮其他因素，但與本題求解無關。1/13二、符號說明X1LLLLLLLLLLLLLLZn的含量x2LLLLLLLLLLLLLLCu的含量x3LLLLLLLL

4、LLLLLLFe勺含量X4LLLLLLLLLLLLLLCa的含量X5LLLLLLLLLLLLLLMg的含量X6LLLLLLLLLLLLLLK的含量x7LLLLLLLLLLLLLLNa的含量三、模型建立與求解（一）問題一的求解：模型一：1、數據“標準化”題目已2&出了60為確診病例的化驗結果以及診斷結果，但是60個病例中各元素的含量的呈無規律性。所以我們需要對原始數據進行處理，首先對其進行標準化分析：用向量X=（Xi,X2,X3,X4,X5,X6,X7）'表示每個就診人員的化驗結果,則X=（Xi，X2，X3，X4，X5,X6,X7）表示第a病人的化驗結果。將每個指“標準化”，即做

5、如下變換:XjE(Xj)j(varXj)1/211,712”xE(Xj)n其中E(Xj)X,varXj1標準化的數據見附錄2、主成分分析對標準化的數據運用SPS球件進行主成分分析，結果如表1、表2:2/13解釋的總方差成份初始特征值提取平方和載入合計力差的累積合計方差的累積13.129,14.70244.7023.12944.70244.70221.973:>8.19272.8941.97328.19272.8943.72310.32783.2214.5708.14791.3685.2844.05295.4206.2042.91298.3327.1171.668100.000由表1可以看

6、出，前兩個主成分y1，y2的方差和占全部方差的比例為72.894%,我們就選取yi為第一主成分，y2為第二主成分，基本上保留了原來7個指標的信息，這樣得到了2個新指標。SPSSB件得到的這成分系數矩陣如表2:表2:成份矩陣a成份12x1.453-.538x2.852.293x3.682.195x4.898-.051x5.941.094x6-.206.856x7-.005.904由表2得到前2個主成分y,y2的線性組合為：yi=0.453xi0.852x20.682x30.898x40.941x50.206x60.005x7y2=0.538x10.293x20.195x30.051x40.094

7、x50.856x60.904x7(4.1)3、模型驗證3/13將60個就診人員的化驗結果帶入（4.1）式得到結果如表3,我們的判別標準為：第一主成分為正值表示健康，為負值表示患病。表3:病例號第一主成分第二主成分正誤判標志（正=0;誤=1）1-1.491420.2399702-1.14428-0.1898303-1.281340.0753804-1.496641.2768505-0.5735-1.8200906-1.58142-1.3215907-1.22966-1.9461608-1.761880.8638609-2.16782-0.99101010-1.700220.09606011-3.

8、453138.3533012-1.790410.81834013-0.42078-0.98245014-1.80823-0.49929015-2.032760.68222016-1.44607-0.50165017-2.014460.70872018-0.85694-0.3205019-1.061552.25198020-3.601883.81686021-3.60084.08631022-1.470191.01608023-0.89605-1.51287024-4.295350.05951025-1.54492-1.80111026-2.12941.19027-3.17737-0.35639

9、028-2.924070.77313029-4.096574.48592030-0.27663-1.237290311.32902-1.89114032-0.22283-0.92755133-0.59422-1.621041340.49476-0.728640350.16255-1.646280360.60017-1.801560370.49891-1.12276038-1.10205-1.6646814/13390.904982.1258604017.600424.835980411.86782-0.258560421.34229-1.57748043-0.188020.541531441.

10、42776-1.53630452.81954-0.52070460.40085-2.066580471.1396-0.992880481.622951.903360496.182691.173070503.73368-0.482520514.016640.792220522.24093-0.429760531.013090.4541054-0.23626-1.035621551.9522-2.265040561.389-1.78010573.45006-1.066840582.94115-1.719420590.06593-1.093790600.45845-0.900860由表3可以看出，前

11、30個就診人員的第一主成分均為負值，判定為患病，后30個就診人員的第一主成分大致上為正值，判定為健康，正確率為91.6667%（二）問題二的求解由模型一得到前兩個主成分的線性組合為：y1=0.453xi0.852x20.682x30.898x40.941x50.206x60.005x7y2=0.538xi0.293x20.195x30.051x40.094x50.856x60.904x7將15名待診人員的化驗結果帶入上式得:表4:病例號第一主成分第二主成分61-5.20057841P1.4897649562-3.70917326-1.1761822263-3.33417236-0.185212

12、1964-2.468497761-144950972-0.3016820166-0.010136984.8819197667-1.53876026P0.882981368-0.4575774-0.70249665690.7083415-1.078355575/13703.5182755-0.25750581711.40770589:-0.63732864722.04054597-1.29664076732.69524135-0.85109768743.119459511-1.0433454975-0.220201161.44700283用第一主成分來判定化驗結果，由表

13、4可知，15名待診人員中有8名患有該疾病，7名健康。(三)模型一的改進：模型二：Logistic回歸模型問題一的模型的正確率為91.6667%,因此考慮正確率更高的其他模型，且模型一中忽略了第二主成分的作用，故解釋時有較大誤差。以Y=0表示健康，Y=1表示不健康，考慮的因變量為一個二元變量，且只取0與1兩個值，因變量取1的概率p為要研究的對象，且lnPbobe3*7是為，X7的線性函1P數，故考慮采用Logistic線性回歸模型。對附錄一中白數據運用SPSS1彳TLogistic回歸分析得表5:表5方程中的變量Bs.e,WalsdfSig.Exp(B)步驟1ax1.48948.943.0001

14、.9921.630x2.347276.987.0001.9991.415x3-1.479160.310.0001.993.228x4-.0887.972.0001.991.916x5.02162.318.00011.0001.021x6.234109.431.0001.9981.264x7.01532.972.00011.0001.016常量33.4707350.783.0001.9963.435E14由表5可以看出Xi,X7這7個變量都是顯著的，因而最終的回歸方程為:eX)(33.4700.489x10.347x21.479x30.088x40.021x50.234x60.015x7)P1e

15、)p(33.4700.489x10.347x21.479x30.088x40.021x50.234x60.015x7)根據以上公式，我們可以將這個模型計算出來的p應用于實際病例的判別。只要給出某一個受檢者的化驗結果，就能應用此計算公式算出其患病幾率，我們以0.5為參照，當p>0.5時表示該受檢者患病，當p<0.5時表示該受檢者健康。具體數據如下：6/13病號x1x2x3x4x5x6x7是否健康判別結果116615.824.570011217951311218515.731.57011251844271131939.8:25.9541163128P642111415914.239.7

16、89699.223972611522616.223.860615270.32181161719.29:9.2930718745.5:25711720113.326.655110149.45306591021546801191728.857.8655175.798.43181:11015611.532.5639107103552111113215.917.757892.413141372111218211.311.376711126467211131869.26137.19582337334710.999999839141628.2327.162510862.44651

17、11151506.6321627140179639111615910.7111.761219098.53901111711716.17.0498895.5136572111818110.14.041437184101542111914620.723.8123212815010921112042.310.39.762993.7439888112128.212.453.137044.1454852112215413.853.36211051607231112317912.217.9113915045.221810.9999998192413.53.3616.813532.651.618210.99

18、9999998251755.8424.980712355.6126112611315.847.362653.6168:62710.9999999942750.511.66.360858.958.913910.999999932878.614.69.742170.81334641129903.278.1762252.3770:852113017828.832.499211270.216910.9999999383121319.136.2222024940168003217013.929.8128522647.9r3300103316213.219.8152116636.2133003420313

19、90.8154416298.9394003516713.114.1227821246.31340103616412.9118.6299319736.3P94.500371671527205626064.6237003815814.437102510144.672.5003913322.8311633401180P899004015613532267471090228810007/13411698308106899.153289004224717.3:8.65255424177.9P373010431668.162.8123325213464900442096.4386.921572887421

20、900451826.49:61.73870432143P367004623515.623.4180616668.8P188004717319.117249729565.8287004815119.7:64.22031403182874004919165.41355361392137P688005022324.486360335397.7479005122120.11553172368150739005221725128.22343373110:494005316422.235.5221228115354900,541738.99361624216103257005520218.6:17.737

21、8522531：67.3005618217.324.8307324650.7P1090'0572112417383642873.5351005824621.593.2211235471.7P1950105916416.138213515264.324000601792135156022647.9330006185.51.713.9950362.3238F762.61621440.715.154779.771218.516385.71.094.279017045.8257.90.999999987641760.5727.331813399.41318.81651927.06132.919

22、69343103P5530661888.2822.61208231131413721671535.8734.8328163264672.516817110.5130.567214547r330.516916213.219.8152116636.21330702031390.8154416298.9394.507116420.1128.9106216147.3134.50.7126521647216713.114.1227821236.596.507316412.918.6299319765.5237.80.7416715127205626044.8r7207515814.43710251011

23、80899.51由上表可以看出，改進后的模型正確率非常高。且15名就診人員中有9人患病，6人健康。（四）問題三的模型建立與求解由模型二中的表5知，xi,X7的各顯著性水平分別為:X1X2X3X4X5X6X70.992；0.9990.9930.9911.0000.9980.9968/13因此我們將各種元素進行組合排除，將XX3,X4分別去掉，比較去掉后該回歸的各個參量的值，以標準誤差和正確率作為評判假設是否合理的依據。再將X1,X2去除，重新建立回歸模型，以相同標準進行評判。同理，分別從Xi,X2,X3,X4,%,X7去除23個量作為一組，進行評判。最后依照相同的方法每兩個、每三個進行分組，均依

24、上述標準評判，具體分組不再贅述。下面僅就回歸后正確率比較大的兩組組給出分析結果。1、去除為？4,%三個變量，由SPS欹件得到的結果見附錄二回歸方程為：eXp(12.2450.118x20.132x30.073x50.022x7)1eXp(12.2450.118x20.132x30.073x50.022x7)x1x2x3x4x5x6x7是否健康判別結果116615.824.570011217951311.00218515.731.570112518442711.0031939.825.9541P163128642:11.00415914.239.789699.223972611.00522616

25、.223.860615270.321810.7161719.299.2930718745.5257：10.87720113.326.655110149.414111I0.95814714.53065910215468011.0091728.857.8655175.798.431811.001015611.532.56391071035521I1.001113215.917.757892.41314137211.001218211.311.376711126467211.00131869.2637.1958233733471I0.04141628.2327.162510862.446511.00

26、151506.632162714017963911.001615910.711.7612r19098.53901I0.981711716.17.0498895.513657211.001818110.14.04143718410154211.001914620.723.81232128150109211.002042.310.39.762993.743988811.002128.212.453.137044.145485211.002215413.853.3621r10516072311.002317912.217.91139M5045.2218:I0.912413.53.3616.81353

27、2.651.618211.00251755.8424.980712355.61261I0.892611315.847.362653.616862711I1.002750.511.66.360858.958.913911.009/132878.614.69.742170.813346411.0029903.278.1762252.3770852:11.003017828.832.499211270.216910.533121319.136.222202494016800.003217013.929.8128522647.933000.073316213.219.81521P16636.21330

28、10.25342031390.8154416298.939400.013516713.114.1227821246.313400.023616412.918.6299319736.394.5100.02371671527205626064.623700.003815814.437102510144.672.500.473913322.831163340118089900.02401561353226747109022881000.00411698308106899.15328900.004224717.38.65255424177.9373:00.42431668.162.8123325213

29、46490l0.25442096.4386.921572887421900.00451826.4961.73870:432143367：。10.004623515.623.4180616668.818800.344717319.117249729565.828700.004815119.764.22031r40318287400.004919165.4355361392137688:00.005022324.486360335397.747900.005122120.1155317236815073900.00522172528.2234337311049400.005316422.235.5

30、221228115354900.03541738.9936162421610325700.025520218.617.737852253167.3100.005618217.324.8307324650.710900.00572112417383642873.535100.005824621.593.2211235471.719500.005916416.138213515264.324000.38601792135156022647.933000.026185.51.73.9950362.3238762.61.00621440.715.154779.771218.51.006385.71.0

31、94.279017045.8257.90.99641760.5727.331813399.4318.81.00651927.0632.919693431035530.00661888.2822.61208231131413721.00671535.8734.8328163264672.51.006817110.530.567214547330.50.986916213.219.8152116636.21330.25108154416298.9394.50.017116420.128.9106216147.3134.50.067216713.114.1227821236

32、.596.50.017316412.918.6299319765.5237.80.297416715272056r26044.87210.007515814.4371025101180899.5:1.00由上表可以看出，該模型得到的結果中，13號就診人員被誤判，正確率為98.3333%。而61-75號就診人員在此模型下的診斷結果是8人患病，7人健康。2、去除X3,X5,小三個變量，由SPS欹件得到的結果見附錄三回歸方程為：ex)(4.7690.029X10.128x20.011x40.006x7)1exo(4.7690.029x1""0.128x2""0

33、.011x40.006x7)x1x2x3x4x5x6x7是否健康判別結果116615.824.570011217951311.001218515.731.570112518442711.0031939.825.954116312864211.00:415914.239.789699.223972611.001522616.223.860615270.321811.0061719.299.2930718745.525711.00:720113.326.655110149.414111.001814714.53065910215468011.0091728.857.8655175.798.4318

34、11.0011015611.532.563910710355211.0011113215.917.757892.41314137211.00,1218211.311.376711126467211.001131869.2637.19582337334710.95141628.2327.162510862.446511.001151506.632162714017963911.00.1615910.711.761219098.539011.0011711716.17.0498895.513657210.9411818110.14.04143718410154210.221914620.723.8

35、1232128150109210.9912042.310.39.762993.743988811.0012128.212.453.137044.145485211.002215413.853.362110516072311.00:2317912.217.91113915045.221810.5712413.53.3616.813532.651.618210.99251755.8424.980712355.612610.92I2611315.847.362653.616862711.002750.511.66.360858.958.913910.8711/132878.614.69.742170

36、.813346411.0029903.278.1762252.3770852:11.003017828.832.499211270.216910.9813121319.136.222202494016800.003217013.929.8128522647.933000.3413316213.219.8：152116636.213300.01342031390.8154416298.939400.093516713.114.1227821246.313400.003616412.918.6：299319736.394.5100.001371671527205626064.623700.0013

37、815814.437102510144.672.500.593913322.831p63340118089900.261401561353226747109022881000.001411698308106899.15328900.664224717.38.65255424177.9373:00.00：431668.162.81233252134649100.721442096.4386.921572887421900.00451826.4961.7：3870432143367：00.0014623515.623.4180616668.818800.0114717319.117249729565.828700.004815119.764.2203140318287400.00:4919165.435536139213768