UND STATISTIK INFORMATIONS- UND DOKUMENTATIONSZENTRUM R. KRAFT
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1 TECHNISCHE UNIVERSITÄT MÜNCHEN-WEIHENSTEPHAN MATHEMATIK UND STATISTIK INFORMATIONS- UND DOKUMENTATIONSZENTRUM R. Bioetrische und Ökonoetrische Methoden I WS 99/ Polynoe Quadratische Funktionen (Parabeln) Kubische Funktionen Polynoe -ten Grades Exponentialfunktionen Potenzfunktionen Residuenanalyse Linearisierbarkeit von Funktionen Linearisierbare Funktionen Nicht linearisierbare Funktionen
2 y' j b i x k i i' y' j i' Polynoe Lineare Funktionen (Geraden) y = b + b x Quadratische Funktionen (Parabeln) y = b + bx + bx y = b + b X + b X b i X i (i',,ÿ,; k i ú) y'b %b x'b %b x.5 X = x, X = x Kubische Funktionen y = b + bx + bx + bx y = b + b X + b X + b X X = x, X = x, X = x Polynoe -ten Grades y = b + bx + bx + ÿ + bx y = b + b X + b X + ÿ + b X i Allgeeiner: z.b. i X = x (i =,,ÿ,) y'b %b X X' x'x.5 Körneraisertrag - Phosphordüngung MTB > Print 'Ertrag' 'P' 'P^' 'P^'. Data Display Row Ertrag P P^ P^
3 Körneraisertrag - Phosphordüngung Lineare Regression MTB > Regress 'Ertrag' 'P'. Regression Analysis The regression equation is Ertrag = P Predictor Coef StDev T P Constant P S = 8.6 R-Sq = 88.% R-Sq(adj) = 87.9% Analysis of Variance Source DF SS MS F P Regression Error Total Regression Plot Y = X R-Sq = 88. % Körneraisertrag - Phosphordüngung Quadratische Regression MTB > Let 'P^' = 'P'** MTB > Regress 'Ertrag' 'P' 'P^'. Regression Analysis The regression equation is Ertrag = P -.5 P^ Predictor Coef StDev T P Constant P P^ S = 6.8 R-Sq = 9.8% R-Sq(adj) = 9.4% Analysis of Variance Source DF SS MS F P Regression Error Total Source DF Seq SS P 67 P^ 86 Regression Plot Ertrag Y = X - 5.E-X** R-Sq = 9.8 % P Ertrag P
4 Körneraisertrag - Phosphordüngung Kubische Regression MTB > Let 'P^' = 'P'** MTB > Regress 'Ertrag' 'P' 'P^' 'P^'. Regression Analysis The regression equation is Ertrag = P +.75 P^ -.5 P^ Predictor Coef StDev T P Constant P P^ P^ S = 5.8 R-Sq = 94.% R-Sq(adj) = 9.8% Analysis of Variance Source DF SS MS F P Regression Error Total Source DF Seq SS P 67. P^ 86. P^ Körneraisertrag - Phosphordüngung Schrittweise Regression MTB > Stepwise 'Ertrag' 'P' 'P^' 'P^'; SUBC> FEnter 4.; SUBC> FReove 4.; SUBC> Constant. Stepwise Regression F-to-Enter: 4. F-to-Reove: 4. Response is Ertrag on predictors, with N = 44 Step 4 Constant P T-Value P^ T-Value P^.75.5 T-Value. 5.5 S R-Sq Regression Plot Y = X +.75E-X** -.5E-4X** R-Sq = 94. % 9 Ertrag Ertrag P 5 P
5 Exponentialfunktionen Exponentialfunktion zur Basis e lny'lny %k@x@lne'lny %k@x Y'b Y'lny, b 'lny, b 'k Exponentialfunktion zur Basis lgy'lgy %k@x@lge'lgy %k@x Y'b Y'lgy, b 'lgy, b 'k Potenzfunktionen k logy'logy %k@logx Y'b Y'logy, X'logx, b 'logy, b 'k Herbizidabbau i Boden MTB > Print 't_days' 'c_pp'. Data Display Row t_days c_pp
6 Herbizidabbau - Lineares Modell MTB > Regress 'c_pp' 't_days'; SUBC> Fits 'Fits'; SUBC> Ss 'St.Res.'; SUBC> DW. Regression Analysis The regression equation is c_pp = t_days Predictor Coef StDev T P Constant t_days S =. R-Sq = 78.% R-Sq(adj) = 77.% Analysis of Variance Source DF SS MS F P Regression Error Total 58 Durbin-Watson statistic =. c'c %k@t'65.6pp&.7 pp Herbizidabbau - Lineares Modell Grafische Residuenanalyse Herbizidabbau - Residuenplot für lineares Modell Noral Plot of s I Chart of s - - Observation Nuber Frequency.SL=.664 X=.46 -.SL= Noral Score Histogra of s s vs. Fits Fit
7 Herbizidabbau - Lineares Modell Anderson-Darling-Test auf Noralverteilung der Residuen Herbizidabbau - Anderson-Darling-Test für lineares Modell Probability Average:.466 StDev:.94 N: - St.Res. Anderson-Darling Norality Test A-Squared:.89 P-Value:. Herbizidabbau - Lineares Modell Durbin-Watson-Test auf Autokorrelation k =, n = : dw u =.6, dw o =.5 dw =. <.6 = dw u, also positive Autokorrelation Herbizidabbau - Lineares Modell Runs-Test auf Zufälligkeit MTB > Runs 'St.Res.'. Runs Test St.Res. K =.46 The observed nuber of runs = The expected nuber of runs = Observations above K 7 below The test is significant at.
8 Herbizidabbau - Exponentielles Modell MTB > Let 'lg c' = LOGT('c_pp') MTB > Regress 'lg c' 't_days'; SUBC> Fits 'Fits'; SUBC> Ss 'St.Res.'; SUBC> DW. Regression Analysis The regression equation is lg c = t_days Predictor Coef StDev T P Constant t_days S =. R-Sq = 99.9% R-Sq(adj) = 99.9% Analysis of Variance Source DF SS MS F P Regression Error Total 9.67 Durbin-Watson statistic =.58 Y lgc'lgc %k@t'.!.64@t Y Anfangskonzentration: c 'pp Halbwertszeit: H Y th ' lg.6 days'5days Herbizidabbau - Exponentielles Modell Grafische Residuenanalyse Frequency Histogra of s Herbizidabbau - Residuenplot für exponentielles Modell - - Noral Plot of s - - Noral Score I Chart of s Observation Nuber s vs. Fits Fit.SL=.85 X=.6 -.SL=-.8
9 Herbizidabbau - Exponentielles Modell Anderson-Darling-Test auf Noralverteilung der Residuen Herbizidabbau - Anderson-Darling-Test für exponentielles Modell Probability Average:.65 StDev:.4 N: - - St.Res. Anderson-Darling Norality Test A-Squared:.74 P-Value:.99 Herbizidabbau - Exponentielles Modell Durbin-Watson-Test auf Autokorrelation k =, n = : dw u =.6, dw o =.5 4! dw o =.5 <.58 <.64 = 4! dw u, also keine Aussage über Autokorrelation öglich Herbizidabbau - Exponentielles Modell Runs-Test auf Zufälligkeit MTB > Runs 'St.Res.'. Runs Test St.Res. K =. The observed nuber of runs = 7 The expected nuber of runs = Observations above K 7 below The test is significant at.89 Cannot reject at alpha =.5
10 Herbizidabbau Lineares - Exponentielles Modell Grundusatz MTB > Print 'Maal' '_kg' 'E_kJ/d'. Herbizidabbau - Lineares Modell Y = X R-Sq = 77.8 % Data Display Row Maal _kg E_kJ/d c_pp 5 Regression 95% CI Mouse. Bird. 5 Chicken Dog. 7 5 Man Cow Bull. 5 t_days 95% PI Regression Plot W = Logten(Y), Z = Logten(X) W = Z R-Sq =. % 6 Herbizidabbau - Exponentielles Modell 5 W = Logten(Y) W = E-X R-Sq = 99.9 % E_kJ/d 4 Regression 95% CI c_pp 5 5 _kg 95% PI Regression 95% CI t_days 95% PI k Y lge'lge %k@lg'.5%.76@lg Y E'.46 kj '88 (etabolische Körpergröße) d
11 Linearisierbare Funktionen Nicht linearisierbare Funktionen Polynoe y' j b i x k i i' y' j b i X i (i',,ÿ,; k i ú) i' Sättigungsfunktion (Mitscherlich-Funktion) y')y@ &e %y '(y ax &y )@ &e %y Sigoide Funktion (Logistische Funktion) Exponentialfunktionen y' y ax &y v %a@e %y v it y ' y ax %a@y v %a log a y'log a y %k@x Potenzfunktionen k logy'logy %k@logx Weibull-Funktion y')y@(&e & y')y@e & x&x k x&x k )%y '(y ax &y )@(&e & %y '(y ax &y )@e & x&x k Kreisfunktionen (Periodische Funktionen) %y x&x k )%y Logarithusfunktionen y'y %k@logx y'y ))%y V y'y )%y V y'y %k@x (X'logx)
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