生物統(tǒng)計學(xué)答案 第十章 一元回歸及簡單相關(guān)分析

第十章一元回歸及簡單相關(guān)分析10.1對尿毒癥患者采用低蛋白并補加基本氨基酸的食物進行治療，分析該療法對患者體內(nèi)一些成分的影響。以下數(shù)據(jù)是在治療前患者的基本數(shù)據(jù)[64]：體重（BW）/kg體內(nèi)總鉀（TBK）/mmol血清尿素（UREA）/（mmol·L-1）73314719703647367232662553265025973738347739823663290049543194386639301653341934703978346327472665418146883678418235403969391219914138356228964374341050903679237438553871275050593583318032684766284645115480465111529038642960457136102469290531計算三者之間的相關(guān)系數(shù)，并檢驗相關(guān)的顯著性。答：所用程序及計算結(jié)果如下：optionslinesize=76nodate;datauremia;infile'e:\data\er10-1e.dat';inputbwtbkurea@@;run;proccorrnosimple;varbwtbkurea;run;TheSASSystemCorrelationAnalysis3'VAR'Variables:BWTBKUREAPearsonCorrelationCoefficients/Prob>|R|underHo:Rho=0/N=30BWTBKUREABW1.000000.705940.285820.00.00010.1257TBK0.705941.000000.096610.00010.00.6116UREA0.285820.096611.000000.12570.61160.0三個變量間，只有體重(BW)和體內(nèi)總鉀(TBK)間相關(guān)顯著，r＝0.70594。相關(guān)系數(shù)的顯著性概率P＝0.0001。10.2還是上例，經(jīng)過一年的飲食治療后，體內(nèi)總鉀量與治療前的總鉀量，如下表[64]：病人號治療后/mmol治療前/mmol163246314722327236472531103266282006265039287929004736203930513597397853308027475634203678382280240054236021055824902530以治療前為自變量，治療后為因變量，計算回歸方程，并檢驗回歸的顯著性。答：計算結(jié)果如下：TheSASSystemTheREGProcedureModel:MODEL1DependentVariable:afterAnalysisofVarianceSumofMeanSourceDFSquaresSquareFValuePr>FModel12573589257358939.40<.0001Error1065326465326CorrectedTotal113226853RootMSE255.59029R-Square0.7976DependentMean2946.66667AdjR-Sq0.7773CoeffVar8.67388ParameterEstimatesParameterStandardVariableDFEstimateErrortValuePr>|t|Intercept1560.15163387.316121.450.1787before10.774470.123396.28<.0001回歸方程為：t檢驗的顯著性概率P<0.0001。故回歸系數(shù)極顯著。10.3調(diào)查河流中懸浮物每月沉淀的量與水流速度的關(guān)系，得到以下結(jié)果[65]：流量/（m3·min-1）每月上層沉積物/t流量/（m3·min-1）每月主流沉積物/t流量/（m3·min-1）每月下層沉積物/t1651.342546812181.6199030023640.12010730848.62094559902.3162678611269.9671.326832.61834123592.44885993405.2148.755621.41477993325.34715491398.639156598.61080251763.81124041144.624843574.32005371429.7892011126.432939228.4503861404.479615675.49913204.8576081337.684191285.61189188.1309471128.662034174.026416.31826823.187925104.2881655.95239597.4259595.16637947.1367569.12991345.970273.92049741.3136265.82246932.670236.7227043.413236.2275661.24145.87463142.71128197.2925770.0369963.7395532.8263627.2123218.0106817.058415.640010.24567.91956.6114以流量為自變量，月沉積物為因變量，計算回歸方程。答：首先對自變量和因變量做雙對數(shù)變換，獲得經(jīng)對數(shù)變換后的回歸方程，再通過反對數(shù)得到原始單位的回歸方程。程序和結(jié)果如下：optionslinesize=76nodate;datariver;infile'E:\data\er10-3e.dat';inputupflowupsedimmidflowmidsedimlowflowlowsedim@@;x1=log10(upflow);y1=log10(upsedim);x2=log10(midflow);y2=log10(midsedim);x3=log10(lowflow);y3=log10(lowsedim);procreg;modely1=x1;procreg;modely2=x2;procreg;modely3=x3;run;（1）上層沉積物：TheSASSystemModel:MODEL1DependentVariable:Y1AnalysisofVarianceSumofMeanSourceDFSquaresSquareFValueProb>FModel13.921283.92128382.2950.0001Error80.082060.01026CTotal94.00334RootMSE0.10128R-square0.9795DepMean4.89337AdjR-sq0.9769C.V.2.06970ParameterEstimatesParameterStandardTforH0:VariableDFEstimateErrorParameter=0Prob>|T|INTERCEP11.8908410.1568676012.0540.0001X111.1750100.0600955419.5520.0001從參數(shù)估計列，得到如下回歸方程：

變換為原單位后的方程為：由t檢驗的顯著性概率可知，回歸系數(shù)和常數(shù)項都是顯著的。（2）主流沉積物：TheSASSystemModel:MODEL1DependentVariable:Y2AnalysisofVarianceSumofMeanSourceDFSquaresSquareFValueProb>FModel135.5858435.585841438.7270.0001Error280.692560.02473CTotal2936.27840RootMSE0.15727R-square0.9809DepMean4.19618AdjR-sq0.9802C.V.3.74797ParameterEstimatesParameterStandardTforH0:VariableDFEstimateErrorParameter=0Prob>|T|INTERCEP11.3669660.0799251017.1030.0001X211.1942880.0314861637.9310.0001從參數(shù)估計列得到回歸方程如下：變換為原單位后的方程為：由t檢驗的顯著性概率可知，回歸系數(shù)和常數(shù)項都是顯著的。（3）底層沉積物：TheSASSystemModel:MODEL1DependentVariable:Y3AnalysisofVarianceSumofMeanSourceDFSquaresSquareFValueProb>FModel120.9958820.9958826.4140.0001Error1511.923280.79489CTotal1632.91916RootMSE0.89156R-square0.6378DepMean2.92730AdjR-sq0.6137C.V.30.45683ParameterEstimatesParameterStandardTforH0:VariableDFEstimateErrorParameter=0Prob>|T|INTERCEP10.5931560.503014461.1790.2567X310.9964790.193889075.1390.0001從參數(shù)估計列得到回歸方程如下：變換為原單位后的方程為：由t檢驗的顯著性概率可知，回歸系數(shù)是顯著的。10.4一種治療肺動脈高血壓的藥物treprostinilsodium，研究給藥劑量與血漿濃度之間的關(guān)系，當(dāng)用靜脈給藥時得到以下結(jié)果[66]（近似值）：劑量/（ng·kg-1·min-1）血漿藥物濃度/（pg·mL-1）204750242500498000535500709000781250084800090132509618250102145001221750012617000以劑量為自變量，血漿藥物濃度為因變量，計算的回歸方程，檢驗回歸的顯著性并繪出回歸線。答：計算結(jié)果如下：TheSASSystemModel:MODEL1DependentVariable:CONCENAnalysisofVarianceSumofMeanSourceDFSquaresSquareFValueProb>FModel1263507305.51263507305.5152.3870.0001Error1050299986.1535029998.6153CTotal11313807291.67RootMSE2242.76584R-square0.8397DepMean10895.83333AdjR-sq0.8237C.V.20.58370ParameterEstimatesParameterStandardTforH0:VariableDFEstimateErrorParameter=0Prob>|T|INTERCEP189.0365171627.41203990.0550.9574DOSAGE1141.88354719.602869077.2380.0001回歸方程為：從回歸系數(shù)和常數(shù)項的顯著性概率可知，回歸系數(shù)是顯著的，常數(shù)項是不顯著的。散點圖和回歸線如下：10.5繼續(xù)上題，這次是皮下給藥，結(jié)果如下表[66]（近似值）：劑量/（ng·kg-1·min-1）血漿藥物濃度/（pg·mL-1）劑量/（ng·kg-1·min-1）血漿藥物濃度/（pg·mL-1）5075001210005277501317506414250152500173750661025028625067130006710000293250675750302500701000032525036425073875038625075100008016250387000801025038675080850044350044975087110004750009515250951575049575010011250506000問：（1）計算血漿藥物濃度對劑量的回歸方程，檢驗回歸的顯著性并繪出回歸線。（2）比較10.5和10.4兩種給藥方式的回歸系數(shù)差異是否顯著？答：計算結(jié)果如下：TheSASSystemModel:MODEL1DependentVariable:CONCENAnalysisofVarianceSumofMeanSourceDFSquaresSquareFValueProb>FModel1401262581.37401262581.3784.9350.0001Error32151178595.14724331.0969CTotal33552441176.47RootMSE2173.55264R-square0.7263DepMean7823.52941AdjR-sq0.7178C.V.27.78225ParameterEstimatesParameterStandardTforH0:VariableDFEstimateErrorParameter=0Prob>|T|INTERCEP1301.527681897.279705170.3360.7390DOSAGE1139.90594015.180706609.2160.0001回歸方程為：從回歸系數(shù)和常數(shù)項的顯著性概率可知，回歸系數(shù)是顯著的，常數(shù)項是不顯著的。散點圖和回歸線如下：比較兩個回歸系數(shù)：令10.4的回歸系數(shù)為b1，10.5的回歸系數(shù)為b2。統(tǒng)計假設(shè)為：H0：β1－β2=0HA：β1－β2≠0顯著性概率P＝0.9366，P>0.05，尚無足夠理由拒絕H0。結(jié)論：兩個回歸系數(shù)的差異不顯著。10.6粵東近海漁場雄性條尾鯡鯉4月份和9月份的體重和體長的測定結(jié)果如下表[67]：序號4月份9月份體重/g體長/cm體重/g體長/cm159.714.038.912.7250.113.031.911.9337.112.021.210.3436.211.617.29.9541.211.211.79.6626.610.614.69.1726.510.210.28.6824.19.99.18.2920.19.18.48.11016.58.99.08.01111.77.68.38.0125.06.66.27.2一般來說，魚的體重(Y)在體長(X)上的回歸符合以下關(guān)系：Y=aXb。計算回歸方程，繪出對數(shù)尺度下的回歸線，檢驗回歸的顯著性，并比較4月份和9月份兩個回歸系數(shù)的差異是否顯著。答：記4月份的樣本為樣本1，9月份的樣本為樣本2。程序和結(jié)果如下：optionslinesize=76nodate;datariver;infile'E:\data\er10-6e.dat';inputfwflnwnl@@;y1=log10(fw);x1=log10(fl);y2=log10(nw);x2=log10(nl);procreg;modely1=x1; modely2=x2;symbolv=stari=rll=1w=2cprocgplot;ploty1*x1; ploty2*x2;run;（1）4月份的回歸分析和回歸線：TheSASSystemTheREGProcedureModel:MODEL1DependentVariable:y1AnalysisofVarianceSumofMeanSourceDFSquaresSquareFValuePr>FModel10.940590.94059260.55<.0001Error100.036100.00361CorrectedTotal110.97669RootMSE0.06008R-Square0.9630DependentMean1.39473AdjR-Sq0.9593CoeffVar4.30790ParameterEstimatesParameterStandardVariableDFEstimateErrortValuePr>|t|Intercept1-1.723190.19394-8.89<.0001x113.094390.1917016.14<.0001對數(shù)尺度下的回歸方程和回歸線為：從t的顯著性概率可以得知，常數(shù)項和回歸系數(shù)都是顯著的。（2）9月份的回歸分析和回歸線：TheSASSystemTheREGProcedureModel:MODEL1DependentVariable:y2AnalysisofVarianceSumofMeanSourceDFSquaresSquareFValuePr>FModel10.659940.65994320.79<.0001Error100.020570.00206CorrectedTotal110.68051RootMSE0.04536R-Square0.9698DependentMean1.12048AdjR-Sq0.9667CoeffVar4.04795ParameterEstimatesParameterStandardVariableDFEstimateErrortValuePr>|t|Intercept1-2.045920.17727-11.54<.0001x213.289930.1836817.91<.0001對數(shù)尺度下的回歸方程和回歸線為：從t的顯著性概率可以得知，常數(shù)項和回歸系數(shù)都是顯著的。（3）回歸系數(shù)的比較：統(tǒng)計假設(shè)為：H0：β1－β2=0HA：β1－β2≠0顯著性概率P＝0.47，P>0.05，尚無足夠理由拒絕H0。結(jié)論：兩個回歸系數(shù)的差異不顯著。10.7新疆維吾爾族和哈薩克族男生各100名，他們的立定跳遠平均成績與年齡之間的關(guān)系如下表所示[10]：年齡/a789101112維吾爾族/cm124.51132.65138.59143.39151.74160.91哈薩克族/cm135.80146.52153.34162.88171.10174.29年齡/a131415161718維吾爾族/cm169.31184.22195.57200.51207.84217.24哈薩克族/cm185.88190.24211.21228.63235.07233.65分別計算兩個民族的成績與年齡之間的相關(guān)系數(shù)，并檢驗兩個相關(guān)系數(shù)的顯著性。答：程序和結(jié)果如下：optionslinesize=76nodate;datajump;infile'e:\data\er10-7e.dat';inputageweiha@@;run;proccorrnosimple;varageweiha;run;TheSASSystemTheCORRProcedure3Variables:ageweihaPearsonCorrelationCoefficients,N=12Prob>|r|underH0:Rho=0ageweihaage1.000000.994940.98708<.0001<.0001wei0.994941.000000.98651<.0001<.0001ha0.987080.986511.00000<.0001<.0001維吾爾族男生年齡與成績間的相關(guān)系數(shù)r維＝0.99494；哈薩克族男生年齡與成績間的相關(guān)系數(shù)r哈＝0.98708。這兩個相關(guān)系數(shù)都是極顯著的。10.8心臟的冠狀竇口直徑（d）與冠狀竇瓣寬（w）和竇瓣高（h）存在一定關(guān)聯(lián)，下面測量了從新生兒到兒童末期的6個年齡組的竇口直徑、竇瓣寬和竇瓣高，結(jié)果見下表[68]：組別ⅠⅡⅢⅣⅤⅥ竇口直徑/mm3.194.434.965.816.307.98竇瓣寬/mm4.646.427.327.688.9910.30竇瓣高/mm1.683.934.084.414.945.02分別計算竇瓣寬和竇瓣高與竇口直徑間的相關(guān)系數(shù)，并檢驗相關(guān)系數(shù)的顯著性。答：所用程序與第7題一樣，這里僅給出結(jié)果。TheSASSystemTheCORRProcedure3Variables:diameterwidthheightPearsonCorrelationCoefficients,N=6Prob>|r|underH0:Rho=0diameterwidthheightdiameter1.000000.986600.871170.00030.0238width0.986601.000000.913580.00030.0109height0.871170.913581.000000.02380.0109從程序運行的結(jié)果可以得出：rd-w＝0.98660，P＝0.0003，相關(guān)極顯著；rd-h＝0.87117，P＝0.0238，相關(guān)顯著。10.9Cu2+和Zn2+對尾草履蟲的急性毒性試驗結(jié)果如下[69]：Cu2+Zn2+濃度/(mg·L-1)死亡率/%濃度/(mg·L-1)死亡率/%02.504.20.145.11.85.20.1815.43.221.70.2440.25.633.00.3250.410.046.10.4263.018.062.60.5679.532.073.00.7593.256.089.6分別計算Cu2+和Zn2+對尾草履蟲的半致死劑量。答：利用SAS軟件包中正態(tài)分布的分位數(shù)函數(shù)，對死亡率做概率變換，對濃度做常用對數(shù)變換。以正態(tài)尺度的死亡率為自變量，以對數(shù)尺度的濃度為因變量，計算回歸方程。程序和結(jié)果如下：optionslinesize=76nodate;dataparameci;infile'e:\data\er10-9e.dat';inputcuconcencudearatznconcenzndearat@@;xcu=probit(cudearat/100);ycu=log10(cuconcen);xzn=probit(zndearat/100);yzn=log10(znconcen);run;procreg;modelycu=xcu;modelyzn=xzn;run;（1）Cu2+：TheSASSystemTheREGProcedureModel:MODEL1DependentVariable:ycuAnalysisofVarianceSumofMeanSourceDFSquaresSquareFValuePr>FModel10.408350.40835241.67<.0001Error50.008450.00169CorrectedTotal60.41679RootMSE0.04111R-Square0.9797DependentMean-0.49525AdjR-Sq0.9757CoeffVar-8.30000ParameterEstimatesParameterStandardVariableDFEstimateErrortValuePr>|t|Intercept1-0.486620.01555-31.30<.0001xcu10.245450.0157915.55<.0001當(dāng)死亡率為50％時，故Cu2+對草履蟲的半致死劑量為0.32612mg/L。（2）Zn2+：TheSASSystemTheREGProcedureModel:MODEL2DependentVariable:yznAnalysisofVarianceSumofMeanSourceDFSquaresSquareFValuePr>FModel11.703011.70301213.00<.0001Error50.039980.00800CorrectedTotal61.74299RootMSE0.08942R-Square0.9771DependentMean1.00246AdjR-Sq0.9725CoeffVar8.91979ParameterEstimatesParameterStandardVariableDFEstimateErrortValuePr>|t|Intercept11.062620.0340531.21<.0001xzn10.559440.0383314.59<.0001當(dāng)死亡率為50％時，故Zn2+對草履蟲的半致死劑量為11.55101mg/L。10.10在一項關(guān)于碳酸利多卡因注射液熱穩(wěn)定性的實驗研究中，發(fā)現(xiàn)NaHCO3的比值（自變量）與相變點溫度（因變量）存在以下關(guān)系[70]：序號NaHCO3的比值（自變量）相變點溫度/℃（因變量）11.0054.121.5048.031.8146.642.5041.152.9139.163.8735.575.0032.985.8029.697.5026.8108.3925.11110.0022.4做出散點圖，并求出回歸方程。答：程序不再給出，這里只給出結(jié)果。散點圖和回歸線如下：回歸分析見下表：TheSASSystemTheREGProcedureModel:MODEL1DependentVariable:tempAnalysisofVarianceSumofMeanSourceDFSquaresSquareFValuePr>FModel1966.81811966.8181197.17<.0001Error989.543719.94930CorrectedTotal101056.36182RootMSE3.15425R-Square0.9152DependentMean36.47273AdjR-Sq0.9058CoeffVar8.64825ParameterEstimatesParameterStandardVariableDFEstimateErrortValuePr>|t|Intercept151.316441.7809928.81<.0001rate1-3.247430.32943-9.86<.0001由此得出回歸方程：回歸系數(shù)的t檢驗和回歸模型的方差分析都指出，回歸是極顯著的。10.114到10月齡胎兒的肝重與肝的Ca含量存在以下關(guān)系[71]：肝重/g6.4813.0224.1744.8658.3975.5886.47Ca含量/(g·g-1干重)1271.01440.91016.6663.7516.3535.9492.5求鈣含量在肝重上的回歸方程并檢驗回歸的顯著性。答：結(jié)果如下：optionslinesize=76nodate;datafetus;inputlivercalcium@@;cards;6.481271.013.021440.924.171016.644.86663.758.39516.375.58535.986.47492.5;procreg;modelcalcium=liver;run;TheSASSystemTheREGProcedureModel:MODEL1DependentVariable:calciumAnalysisofVarianceSumofMeanSourceDFSquaresSquareFValuePr>FModel178898478898428.650.0031Error513767927536CorrectedTotal6926663RootMSE165.93934R-Square0.8514DependentMean848.12857AdjR-Sq0.8217CoeffVar19.56535ParameterEstimatesParameterStandardVariableDFEstimateErrortValuePr>|t|Intercept11364.29113115.0304111.86<.0001liver1-11.694142.18466-5.350.0031由此得出回歸方程：對回歸模型的方差分析和回歸系數(shù)的t檢驗都指出，回歸是極顯著的。10.12青菜對14CO2的富集系數(shù)（CF值）如下[72]：時間/d菜心葉子624.613.81253.430.91882.041.924100.163.236114.196.848156.4135.6以時間為自變量，菜心和葉子分別為因變量，計算回歸方程，并比較兩者回歸系數(shù)的差異顯著性。答：程序不再給出，這里只給出結(jié)果。（1）菜心：TheSASSystemTheREGProcedureModel:MODEL1DependentVariable:cfheartAnalysisofVarianceSumofMeanSourceDFSquaresSquareFValuePr>FModel1102801028086.050.0008Error4477.84863119.46216CorrectedTotal510758RootMSE10.92987R-Square0.9556DependentMean88.43333AdjR-Sq0.9445CoeffVar12.35945ParameterEstimatesParameterStandardVariableDFEstimateErrortValuePr>|t|Intercept118.880398.725132.160.0965time12.898040.312419.280.0008由上表得出回歸方程：回歸系數(shù)b1是極顯著的。（2）葉子：TheSASSystemTheREGProcedureModel:MODEL2DependentVariable:cfleafAnalysisofVarianceSumofMeanSourceDFSquaresSquareFValuePr>FModel110269102691108.25<.0001Error437.065599.26640CorrectedTotal510307RootMSE3.04408R-Square0.9964DependentMean63.70000AdjR-Sq0.9955CoeffVar4.77877ParameterEstimatesParameterStandardVariableDFEstimateErrortValuePr>|t|Intercept1-5.817652.43003-2.390.0748time12.896570.0870133.29<.0001由上表得出回歸方程：回歸系數(shù)b2是極顯著的。（3）回歸系數(shù)的比較：統(tǒng)計假設(shè)為：H0：β1－β2=0HA：β1－β2≠0顯著性概率P＝0.9965，P>0.05，尚無足夠理由拒絕H0。結(jié)論：兩個回歸系數(shù)的差異不顯著。10.13人工測定蚊密度與氣溫存在以下關(guān)系[73]：序號蚊密度*氣溫/℃序號蚊密度*氣溫/℃序號蚊密度*氣溫/℃152.823.011134.325.321193.727.92104.423.512162.727.222165.127.4374.721.913341.428.32374.928.7479.623.714292.429.324102.126.8543.822.515265.227.825185.024.4647.521.016230.628.326175.825.07191.524.917259.830.127203.526.38157.825.618148.529.428138.523.39204.326.019331.430.22993.126.610232.825.520326.327.53097.024.8注：*單位：只/h。以氣溫作為自變量，蚊密度作為因變量，求回歸方程并對回歸方程做方差分析。答：結(jié)果如下：TheSASSystemTheREGProcedureModel:MODEL1DependentVariable:densityAnalysisofVarianceSumofMeanSourceDFSquaresSquareFValuePr>FModel110356510356525.51<.0001Error281136744059.77886CorrectedTotal29217239RootMSE63.71639R-Square0.4767DependentMean170.35000AdjR-Sq0.4580CoeffVar37.40323ParameterEstimatesParameterStandardVariableDFEstimateErrortValuePr>|t|Intercept1-459.98640125.34184-3.670.0010temp124.175524.786535.05<.0001由以上結(jié)果得到回歸方程：方差分析表：變差來源平方和自由度均方FP回歸103565110356525.51<0.0001剩余113674284059.77886總和2172392910.14馬鹿下臼齒咀嚼面寬度與年齡之間存在以下關(guān)系[48]：序號年齡/a下臼齒咀嚼面寬度/mm12.58.658.908.308.8023.59.608.307.808.408.709.407.507.908.908.358.4034.510.138.6510.0010.909.9210.0010.1410.1210.159.1010.179.809.729.8210.0010.158.8045.510.7511.6810.3010.2210.0011.9011.8511.9011.8510.6856.511.3012.7011.4811.8710.2010.8211.5211.6010.2511.0011.3067.510.4011.0012.5013.509.9878.512.1612.8011.8811.1011.4811.4012.1010.1589.512.7211.6812.8011.3513.33913.512.201017.514.03以年齡為自變量，咀嚼面寬度為因變量，計算回歸方程。這是一個有重復(fù)數(shù)據(jù)的回歸問題，它的計算與無重復(fù)時相似，只是DATA步略有不同。答：程序如下：optionslinesize=76nodate;datadeer;infile'E:\data\er10-14e.dat';doi=1to10;inputnage@@;doj=1ton;inputwidth@@;output;end;end;procreg;modelwidth=age;run;TheSASSystemTheREGProcedureModel:MODEL1DependentVariable:widthAnalysisofVarianceSumofMeanSourceDFSquaresSquareFValuePr>FModel198.0938198.09381113.00<.0001Error7161.635520.86811CorrectedTotal72159.72934RootMSE0.93172R-Square0.6141DependentMean10.53699AdjR-Sq0.6087CoeffVar8.84240ParameterEstimatesParameterStandardVariableDFEstimateErrortValuePr>|t|Intercept17.817800.2780828.11<.0001age10.455800.0428810.63<.0001由以上數(shù)據(jù)得到回歸方程：經(jīng)t檢驗回歸系數(shù)和常數(shù)項都是顯著的。10.15端粒（telomere）的長度隨著年齡的增長而逐漸縮短，因此有可能根據(jù)端粒的大小推斷出個體的年齡。采用末端限制片段（terminalrestrictionfragment,TRF）長度來確定不同年齡組端粒的大小。年齡組（歲）和各年齡組外周血白細胞TRF平均長度（kb）的測定結(jié)果見下表[74]：序號年齡組中值/aTRF/kb12.014.2929.012.92319.012.16429.011.91539.011.68649.011.25759.010.94869.010.39977.510.31以TRF為自變量，年齡為因變量，求出最佳擬合回歸方程。答：求最佳擬合方程，可以通過繪圖法，也可以通過比較剩余均方來確定。繪圖法比較直觀，在這里我們采用繪圖法。（1）不變換：（2）log10(age)變換：（3）sqrt(age)變換：比較以上三個圖形，顯然對年齡做平方根變換后，直線化的效果最好，則方差分析表為：ParameterEstimatesParameterStandardVariableDFEstimateErrortValuePr>|t|Intercept128.650831.4236720.12<.0001trf1-1.942970.12043-16.13<.0001變換后的回歸方程為：10.16用18種不同水體配制成培養(yǎng)基，培養(yǎng)基中的磷（P）濃度及用該培養(yǎng)基培養(yǎng)的玫瑰擬衣藻（Chloromonasrosae）的生長速率見下表[75]：水體號P濃度/（mg·L-1）生長速率/（）10.10100.244020.08200.198930.06100.238240.02800.246050.03000.171660.03200.216370.21000.413880.15000.332890.16000.2684100.01680.0948110.01200.0993120.01280.1650130.00840.0915140.00600.0067150.00640.0592160.00420.0333170.00300.0198180.0032-0.0147以P濃度為自變量，生長速率為因變量，在直角坐標(biāo)系中畫出散點圖，求出回歸方程，并檢驗回歸的顯著性。答：對自變量（P濃度）做自然對數(shù)變換，用變換后的數(shù)據(jù)進行分析。程序和結(jié)果如下：optionslinesize=76nodate;dataleaves;infile'e:\data\er10-16e.dat';inputprate@@; x=log(p);y=rate;run;procgplot;ploty*x;procreg;modely=x;run;TheSASSystemTheREGProcedureModel:MODEL1DependentVariable:yAnalysisofVarianceSumofMeanSourceDFSquaresSquareFValuePr>FModel10.216850.21685128.01<.0001Error160.027100.00169CorrectedTotal170.24395RootMSE0.04116R-Square0.8889DependentMean0.16027AdjR-Sq0.8820CoeffVar25.67984ParameterE