1、化驗(yàn)結(jié)果診斷模型問題重述與分析人們到醫(yī)院就診時(shí)，通常要化驗(yàn)一些指標(biāo)來協(xié)助醫(yī)生的診斷。本題給出了人們是否患某種疾病時(shí)通常要化驗(yàn)的幾種指標(biāo)以及其檢驗(yàn)值。表1是確診病例的化驗(yàn)結(jié)果，其中130號(hào)病例是已經(jīng)確診為患該種疾病的化驗(yàn)結(jié)果；3160號(hào)病例是已經(jīng)確診為健康人的結(jié)果。表2是某些就診人員的化驗(yàn)結(jié)果，但未確診其是否患有該種疾病。根據(jù)已知數(shù)據(jù)，需要解答如下問題：1）問題：根據(jù)表1中的數(shù)據(jù)，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(yàn)?zāi)闾岢龇椒ǖ恼_性。分析：根據(jù)表1當(dāng)中60個(gè)化驗(yàn)結(jié)果，將Zn、CuFe、CaMgK、Na看成是七個(gè)指標(biāo)，則前30個(gè)為該疾病患者的指標(biāo)值，后30個(gè)為健康人的指標(biāo)值

2、，可以將這些數(shù)據(jù)進(jìn)行標(biāo)準(zhǔn)化處理，再采用主成分分析方法，將多個(gè)指標(biāo)轉(zhuǎn)化為幾個(gè)綜合指標(biāo)，當(dāng)給定一個(gè)患者的各指標(biāo)值時(shí)，可以算出各綜合指標(biāo)的得分，當(dāng)這些得分滿足一定條件時(shí)，如根據(jù)正負(fù)值可以判定為健康或疾病。2）問題：按照（1）提出的方法，對(duì)表2中的15名就診人員的化驗(yàn)結(jié)果進(jìn)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(biāo)，根據(jù)給定的15名就診人員的指標(biāo)值計(jì)算出綜合指標(biāo)的得分，以此判斷他們的健康狀況。3）問題：能否根據(jù)表1的數(shù)據(jù)特征，確定哪些指標(biāo)是影響人們患該疾病的關(guān)鍵或主要因素，以便減少化驗(yàn)的指標(biāo)。并根據(jù)你給出的結(jié)果，重復(fù)2的工作。分析：為了確定哪些指標(biāo)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(quán)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(shè)1）假設(shè)醫(yī)院化驗(yàn)設(shè)備先進(jìn)，化驗(yàn)過程科學(xué)可靠，化驗(yàn)結(jié)果真實(shí)可信，確診情況（有病/健康人）符合實(shí)際。2）在解決本題過程中，所有的化驗(yàn)結(jié)果只是針對(duì)該類疾病檢驗(yàn)，并不考慮其他疾病的影響。3）本文所建模型的檢驗(yàn)結(jié)果只是作為醫(yī)生為病人診斷的一個(gè)參考，醫(yī)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(guān)。1/13二、符號(hào)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(shù)據(jù)“標(biāo)準(zhǔn)化”題目已2&出了60為確診病例的化驗(yàn)結(jié)果以及診斷結(jié)果，但是60個(gè)病例中各元素的含量的呈無規(guī)律性。所以我們需要對(duì)原始數(shù)據(jù)進(jìn)行處理，首先對(duì)其進(jìn)行標(biāo)準(zhǔn)化分析：用向量X=（Xi,X2,X3,X4,X5,X6,X7）'表示每個(gè)就診人員的化驗(yàn)結(jié)果,則X=（Xi，X2，X3，X4，X5,X6,X7）表示第a病人的化驗(yàn)結(jié)果。將每個(gè)指“標(biāo)準(zhǔn)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(biāo)準(zhǔn)化的數(shù)據(jù)見附錄2、主成分分析對(duì)標(biāo)準(zhǔn)化的數(shù)據(jù)運(yùn)用SPS球件進(jìn)行主成分分析，結(jié)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(jì)力差的累積合計(jì)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(gè)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(gè)指標(biāo)的信息，這樣得到了2個(gè)新指標(biāo)。SPSSB件得到的這成分系數(shù)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(gè)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(yàn)證3/13將60個(gè)就診人員的化驗(yàn)結(jié)果帶入（4.1）式得到結(jié)果如表3,我們的判別標(biāo)準(zhǔn)為：第一主成分為正值表示健康，為負(fù)值表示患病。表3:病例號(hào)第一主成分第二主成分正誤判標(biāo)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(gè)就診人員的第一主成分均為負(fù)值，判定為患病，后30個(gè)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(gè)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(yàn)結(jié)果帶入上式得:表4:病例號(hào)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(yàn)結(jié)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(jìn)：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(shí)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(gè)二元變量，且只取0與1兩個(gè)值，因變量取1的概率p為要研究的對(duì)象，且lnPbobe3*7是為，X7的線性函1P數(shù)，故考慮采用Logistic線性回歸模型。對(duì)附錄一中白數(shù)據(jù)運(yùn)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(gè)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(jù)以上公式，我們可以將這個(gè)模型計(jì)算出來的p應(yīng)用于實(shí)際病例的判別。只要給出某一個(gè)受檢者的化驗(yàn)結(jié)果，就能應(yīng)用此計(jì)算公式算出其患病幾率，我們以0.5為參照，當(dāng)p>0.5時(shí)表示該受檢者患病，當(dāng)p<0.5時(shí)表示該受檢者健康。具體數(shù)據(jù)如下：6/13病號(hào)x1x2x3x4x5x6x7是否健康判別結(jié)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(jìn)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(jìn)行組合排除，將XX3,X4分別去掉，比較去掉后該回歸的各個(gè)參量的值，以標(biāo)準(zhǔn)誤差和正確率作為評(píng)判假設(shè)是否合理的依據(jù)。再將X1,X2去除，重新建立回歸模型，以相同標(biāo)準(zhǔn)進(jìn)行評(píng)判。同理，分別從Xi,X2,X3,X4,%,X7去除23個(gè)量作為一組，進(jìn)行評(píng)判。最后依照相同的方法每兩個(gè)、每三個(gè)進(jìn)行分組，均依

24、上述標(biāo)準(zhǔn)評(píng)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(jié)果。1、去除為？4,%三個(gè)變量，由SPS欹件得到的結(jié)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(jié)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(jié)果中，13號(hào)就診人員被誤判，正確率為98.3333%。而61-75號(hào)就診人員在此模型下的診斷結(jié)果是8人患病，7人健康。2、去除X3,X5,小三個(gè)變量，由SPS欹件得到的結(jié)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(jié)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