i Pohl Vyska Hmotnost Vi - mV Hi - mH (Vi-mV)(Hi-mH) H.stř.i ei 1 0 172 86 -2.5 11.04545455 -27.61363636 81.22439642 4.775603576 2 1 169 58 -5.5 -16.95454545 93.25 64.92978486 -6.929784858 3 0 183 80 8.5 5.045454545 42.88636364 85.9713055 -5.971305497 korelace průměr 4 1 170 69 -4.5 -5.954545455 26.79545455 65.36132205 3.638677953 1 (xi - mX) (yi - mY) mX = (1/N) Suma(1…N) xi 5 0 180 85 5.5 10.04545455 55.25 84.67669393 0.323306068 rXY = -------- SUMA (i = 1 … N) {------------------------} 6 1 173 76 -1.5 1.045454545 -1.568181818 66.65593361 9.344066388 (N - 1) sX sY rozptyl (variance) 7 0 190 89 15.5 14.04545455 217.7045455 88.99206582 0.007934183 úprava vzorce korelace s2X = [1/(N-1)] Suma(1…N) (xi - mX)2 8 1 174 62 -0.5 -12.95454545 6.477272727 67.0874708 -5.087470801 = 1/(N - 1) SUMA(i = 1 … N) [ zXi zYi ] 9 1 160 55 -14.5 -19.95454545 289.3409091 61.04595016 -6.045950162 směrodatná odchylka 10 0 182 75 7.5 0.045454545 0.340909091 85.53976831 -10.53976831 s = odmocnina(s2) 11 0 198 101 23.5 26.04545455 612.0681818 92.44436332 8.555636675 kovariance 12 1 153 48 -21.5 -26.95454545 579.5227273 58.02518984 -10.02518984 cXY = sX rXY sY z toho rXY = cXY/(sX sY) 13 1 174 65 -0.5 -9.954545455 4.977272727 67.0874708 -2.087470801 14 1 162 76 -12.5 1.045454545 -13.06818182 61.90902454 14.09097546 15 0 171 69 -3.5 -5.954545455 20.84090909 80.79285924 -11.79285924 Vícenásobná regrese (v tomto případě tří proměnných) 16 1 159 48 -15.5 -26.95454545 417.7954545 60.61441297 -12.61441297 17 0 192 78 17.5 3.045454545 53.29545455 89.85514019 -11.85514019 Cíl: předpovědět hodnoty závislé proměnné (zde Y) pomocí hodnot nezávislých proměnných (zde X1 a X2) 18 1 170 59 -4.5 -15.95454545 71.79545455 65.36132205 -6.361322047 19 1 181 76 6.5 1.045454545 6.795454545 70.10823112 5.89176888 Regresní rovnice: 20 0 179 95 4.5 20.04545455 90.20454545 84.24515674 10.75484326 21 0 165 101 -9.5 26.04545455 -247.4318182 78.2036361 22.7963639 nebo pro každou individuální hodnotu 22 0 182 98 7.5 23.04545455 172.8409091 85.53976831 12.46023169 "Y se stříškou: Hodnota proměnné Y předpovězená pomocí hodnot X1 a X2 a regresních koeficientů a, b1 a b2" m 0.5 174.5 74.95454545 vzorcem 0.651652535 "a: Tzv. regresní konstanta, anglicky intercept, v podstatě hodnota proměnné Y pro X = 0" sd - vzorec 11.18566259 16.15247961 funkcí 0.651652535 "b1 a b2: Tzv. regresní koeficienty, které vyjadřují, o kolik vzroste hodnota závislé proměnné (zde Y), pokud se hodnota nezávislých proměnných (zde X1 a X2) změní o 1" sd - funkcí 0.511766316 11.18566259 16.15247961 a 7 R2: tzv. mnohonásobná korelace se rovná korelaci hodnot Y a hodnot Y předpovězených pomocí regresní rovnice (Y se stříškou) rPV -0.619731492 b1 0.431537189 rPH -0.763283167 b2 -15 Postup rVH 0.651652535 R 0.795440263 "V případě, že máme v regresi více než dvě proměnné, NELZE koeficienty najít analyticky na základě vzorců (tzn. neexistují žádné vzorce pro výpočet a, b1 a b2)" R2 0.632725212 "Nastupuje řešení pomocí metod tzv. numerické matematiky. Abychom si prakticky vyzkoušeli, jak tyto metody mohou postupovat, vyzkoušíme si to formou hry." Zkusíme odhadnout regresní model H = a + b1 V + b2 P (tzn. odhadneme hmotnost na základě výšky a pohlaví). SS 2061.796889 "1. Předstíráme, že známe hodnoty a, b1 a b2 - vlevo v příslušných políčkách jsem napsal nějaké odhady hodnot, které jsem určil zcela ""od oka""" rHe 0.721439076 "2. Vypočítáme H se stříškou pro hodnoty pohlaví a výšky každého respondenta a tyto ""od oka"" stanovené hodnoty a, b1 a b2" rVe 0.09502071 3. Musíme si stanovit nějaká kritéria kvality toho modelu "1. bude to jednak již známý koeficient R (potažmo R2), tzn. korelace H a H se stříškou" "čím vyšší, tím lepší model (odhad hmotnosti na základě výšky a pohlaví" "2. další bude tzv. suma čtverců reziduí, neboli chyb odhadu, neboli hodnot ei (značí se často jako SS - z angl. Sum of Squares)" "čím nižší, tím lepší model" "4. Měníme, přepisujeme hodnoty a, b1 a b2 tak, abychom dosáhli co nejvyšší hodnoty R (nebo R2) a nejnižší hodnoty SS" ##### Sheet/List 2 ##### i Pohlaví (P) Výška (V) Hmotnost (H) H.stř.i.zV ei.HzV P.stř.i.zV ei.PzV H.stř.i.zP ei.HzP V.stř.i.zP ei.VzP V.stř.i.zH ei.VzH P.stř.i.zH ei.PzH 1 0 172 86 72.60202405 13.39797595 0.570884872 -0.570884872 87 -1 181.2727273 -9.272727273 179.4845068 -7.484506832 0.232882849 -0.232882849 2 1 169 58 69.77899836 -11.77899836 0.655946717 0.344053283 62.90909091 -4.909090909 167.7272727 1.272727273 166.8488846 2.151115425 0.91001933 0.08998067 3 0 183 80 82.95311824 -2.953118242 0.258991437 -0.258991437 87 -7 181.2727273 1.727272727 176.7768735 6.223126509 0.377983524 -0.377983524 4 1 170 69 70.72000692 -1.72000692 0.627592769 0.372407231 62.90909091 6.090909091 167.7272727 2.272727273 171.812879 -1.812879033 0.644001427 0.355998573 5 0 180 85 80.13009255 4.869907447 0.344053283 -0.344053283 87 -2 181.2727273 -1.272727273 179.0332346 0.966765392 0.257066295 -0.257066295 6 1 173 76 73.54303261 2.45696739 0.542530923 0.457469077 62.90909091 13.09090909 167.7272727 5.272727273 174.9717846 -1.971784597 0.474717307 0.525282693 7 0 190 89 89.54017819 -0.540178185 0.060513796 -0.060513796 87 2 181.2727273 8.727272727 180.8383235 9.161676498 0.160332512 -0.160332512 8 1 174 62 74.48404117 -12.48404117 0.514176974 0.485823026 62.90909091 -0.909090909 167.7272727 6.272727273 168.6539735 5.346026531 0.813285547 0.186714453 9 1 160 55 61.30992129 -6.309921287 0.911132255 0.088867745 62.90909091 -7.909090909 167.7272727 -7.727272727 165.4950679 -5.495067904 0.982569667 0.017430333 10 0 182 75 82.01210968 -7.012109679 0.287345385 -0.287345385 87 -12 181.2727273 0.727272727 174.5205124 7.479487626 0.498900752 -0.498900752 11 0 198 101 97.06824669 3.931753309 -0.166317793 0.166317793 87 14 181.2727273 16.72727273 186.2535902 11.74640982 -0.129868837 0.129868837 12 1 153 48 54.72286134 -6.722861344 1.109609895 -0.109609895 62.90909091 -14.90909091 167.7272727 -14.72727273 162.3361623 -9.33616234 1.151853788 -0.151853788 13 1 174 65 74.48404117 -9.484041173 0.514176974 0.485823026 62.90909091 2.090909091 167.7272727 6.272727273 170.0077901 3.992209861 0.74073521 0.25926479 14 1 162 76 63.19193841 12.80806159 0.854424358 0.145575642 62.90909091 13.09090909 167.7272727 -5.727272727 174.9717846 -12.9717846 0.474717307 0.525282693 15 0 171 69 71.66101548 -2.661015483 0.59923882 -0.59923882 87 -18 181.2727273 -10.27272727 171.812879 -0.812879033 0.644001427 -0.644001427 16 1 159 48 60.36891272 -12.36891272 0.939486204 0.060513796 62.90909091 -14.90909091 167.7272727 -8.727272727 162.3361623 -3.33616234 1.151853788 -0.151853788 17 0 192 78 91.42219531 -13.42219531 0.003805899 -0.003805899 87 -9 181.2727273 10.72727273 175.874329 16.12567096 0.426350415 -0.426350415 18 1 170 59 70.72000692 -11.72000692 0.627592769 0.372407231 62.90909091 -3.909090909 167.7272727 2.272727273 167.3001568 2.699843202 0.885835884 0.114164116 19 1 181 76 81.07110112 -5.071101116 0.315699334 0.684300666 62.90909091 13.09090909 167.7272727 13.27272727 174.9717846 6.028215403 0.474717307 0.525282693 20 0 179 95 79.18908399 15.81091601 0.372407231 -0.372407231 87 8 181.2727273 -2.272727273 183.5459568 -4.545956843 0.015231838 -0.015231838 21 0 165 101 66.0149641 34.9850359 0.769362512 -0.769362512 87 14 181.2727273 -16.27272727 186.2535902 -21.25359018 -0.129868837 0.129868837 22 0 182 98 82.01210968 15.98789032 0.287345385 -0.287345385 87 11 181.2727273 0.727272727 184.8997735 -2.899773514 -0.0573185 0.0573185 m 0.5 174.5 74.95454545 H predikovaná z V P predikované z V H predikovaná z P V predikovaná z P V predikovaná z H P predikované z H sd 0.511766316 11.18566259 16.15247961 r 0.651652535 r -0.619731492 r -0.763283167 r -0.619731492 r 0.651652535 r -0.763283167 rPV -0.619731492 a -89.25144884 a 5.447764034 a 87 a 181.2727273 a 140.6750956 a 2.312659183 rPH -0.763283167 b 0.941008563 b -0.028353949 b -24.09090909 b -13.54545455 b 0.451272223 b -0.024183446 rVH 0.651652535 Parciální korelace PH.V Parciální korelace VH.P Parciální korelace PV.H vzorcem -0.603790493 vzorcem 0.352283625 vzorcem -0.249639493 korelace e -0.603790493 korelace e 0.352283625 korelace e -0.249639493 ##### Sheet/List 3 ##### Model s 1 faktorem B b1 BT b1 b2 b3 b4 BBT b1*b1 b1*b2 b1*b3 b1*b4 b2 b2*b1 b2*b2 b2*b3 b2*b4 b3 b4 b4*b1 b4*b2 b4*b3 b4*b4 Model s 2 faktory B b11 b21 BT b11 b12 b13 b14 BBT b11*b11+b21*b21 b11*b12+b21*b22 b12 b22 b21 b22 b23 b24 b13 b23 b13*b14+b23*b24 b14 b24 Výpočet v Excelu B 1 10 BT 1 2 3 4 BBT 101 202 303 404 2 20 10 20 30 40 202 404 606 808 3 30 303 606 909 1212 4 40 404 808 1212 1616 ##### Sheet/List 4 ##### n nx ny nz X Y Z oXY oXZ oYZ sXY sXZ sYZ uXY uXZ uYZ uXZY 1 10 1 4 11 1 10 24 42 22 11 110 10 11.04536102 14.86606875 10.04987562 14.89966443 R oXY oXZ oYZ sXY sXZ sYZ uXY uXZ uYZ uXZY 2 14 7 7 5 3 12 16 34 30 15 60 36 5.830951895 13 12.36931688 13.34166406 oXY 1 0.701657296 0.285891699 0.851968987 0.594725687 0.327277036 0.966497504 0.729143246 0.218044068 0.819059851 3 9 1 4 6 1 10 14 32 22 6 60 10 6.08276253 11.66190379 10.04987562 11.70469991 oXZ 0.701657296 1 0.492845985 0.488884093 0.927489291 0.372444583 0.743732247 0.978274268 0.529260086 0.943436494 4 2 2 7 11 4 9 30 40 26 44 99 36 11.70469991 14.2126704 9.848857802 14.76482306 oYZ 0.285891699 0.492845985 1 0.414969594 0.533016464 0.909983404 0.176093749 0.428510102 0.974046227 0.560514606 5 2 3 3 1 1 5 4 12 12 1 5 5 1.414213562 5.099019514 5.099019514 5.196152423 sXY 0.851968987 0.488884093 0.414969594 1 0.43662129 0.484492805 0.708174616 0.495852966 0.318369123 0.616889328 6 3 3 7 14 4 12 36 52 32 56 168 48 14.56021978 18.43908891 12.64911064 18.86796226 sXZ 0.594725687 0.927489291 0.533016464 0.43662129 1 0.392492534 0.622403663 0.84581962 0.577720625 0.826148045 7 11 3 2 10 2 9 24 38 22 20 90 18 10.19803903 13.45362405 9.219544457 13.60147051 sYZ 0.327277036 0.372444583 0.909983404 0.484492805 0.392492534 1 0.203322996 0.331309051 0.810841999 0.476441576 8 11 7 2 3 0 1 6 8 2 0 3 0 3 3.16227766 1 3.16227766 uXY 0.966497504 0.743732247 0.176093749 0.708174616 0.622403663 0.203322996 1 0.785450425 0.134713786 0.843461305 9 1 4 7 5 0 9 10 28 18 0 45 0 5 10.29563014 9 10.29563014 uXZ 0.729143246 0.978274268 0.428510102 0.495852966 0.84581962 0.331309051 0.785450425 1 0.461992661 0.962885975 10 7 3 6 0 4 4 8 8 16 0 0 16 4 4 5.656854249 5.656854249 uYZ 0.218044068 0.529260086 0.974046227 0.318369123 0.577720625 0.810841999 0.134713786 0.461992661 1 0.570353085 11 12 3 6 4 1 4 10 16 10 4 16 4 4.123105626 5.656854249 4.123105626 5.744562647 uXZY 0.819059851 0.943436494 0.560514606 0.616889328 0.826148045 0.476441576 0.843461305 0.962885975 0.570353085 1 12 8 1 11 6 3 7 18 26 20 18 42 21 6.708203932 9.219544457 7.615773106 9.695359715 13 12 0 3 12 2 1 28 26 6 24 12 2 12.16552506 12.04159458 2.236067977 12.20655562 14 2 3 5 14 3 4 34 36 14 42 56 12 14.31782106 14.56021978 5 14.86606875 B F1 F2 F3 BT 15 2 7 12 6 5 1 22 14 12 30 6 5 7.810249676 6.08276253 5.099019514 7.874007874 oXY 0.121 0.477 0.863 0.121039935 0.566956825 0.937319104 0.176612046 0.625651872 0.796068287 0.071989238 0.49004095 0.96544233 0.548203665 16 10 2 3 9 4 12 26 42 32 36 108 48 9.848857802 15 12.64911064 15.5241747 oXZ 0.567 -0.119 0.807 0.476641041 -0.11886755 0.33530231 0.749631152 -0.148726288 0.535402219 0.284588126 -0.095658347 0.171589702 0.1100858 17 8 5 11 6 2 5 16 22 14 12 30 10 6.32455532 7.810249676 5.385164807 8.062257748 oYZ 0.937 0.335 0.007 0.86282761 0.807033082 0.006830901 0.572696131 0.684807428 -0.025804438 0.933804878 0.843339708 0.015483076 0.810145582 18 12 8 5 8 7 4 30 24 22 56 32 28 10.63014581 8.94427191 8.062257748 11.35781669 sXY 0.177 0.750 0.573 19 5 3 3 6 8 3 28 18 22 48 18 24 10 6.708203932 8.544003745 10.44030651 sXZ 0.626 -0.149 0.685 20 3 7 9 8 6 6 28 28 24 48 48 36 10 10 8.485281374 11.66190379 sYZ 0.796 0.535 -0.026 21 4 3 7 9 3 6 24 30 18 27 54 18 9.486832981 10.81665383 6.708203932 11.22497216 uXY 0.072 0.285 0.934 22 1 7 7 8 7 10 30 36 34 56 80 70 10.63014581 12.80624847 12.20655562 14.59451952 uXZ 0.490 -0.096 0.843 23 3 2 9 7 3 1 20 16 8 21 7 3 7.615773106 7.071067812 3.16227766 7.681145748 uYZ 0.965 0.172 0.015 24 9 5 7 1 0 7 2 16 14 0 7 0 1 7.071067812 7 7.071067812 uXZY 0.548 0.110 0.810 25 15 2 7 11 3 9 28 40 24 33 99 27 11.40175425 14.2126704 9.486832981 14.52583905 26 6 5 7 13 6 7 38 40 26 78 91 42 14.31782106 14.76482306 9.219544457 15.93737745 R.se.stř. 0.986308833 0.70829769 0.279165775 0.872820118 0.595710566 0.329285943 0.950072585 0.741376615 0.212002996 0.817841923 27 1 7 4 7 0 3 14 20 6 0 21 0 7 7.615773106 3 7.615773106 0.70829769 0.986871931 0.497075662 0.473209311 0.925058578 0.366869363 0.760597926 0.969805778 0.539463025 0.951536466 28 11 7 7 9 0 9 18 36 18 0 81 0 9 12.72792206 9 12.72792206 0.279165775 0.497075662 0.991041403 0.420806932 0.541245036 0.925515346 0.169278673 0.433011049 0.962567727 0.556287815 29 11 7 10 6 7 10 26 32 34 42 60 70 9.219544457 11.66190379 12.20655562 13.60147051 0.872820118 0.473209311 0.420806932 0.921119538 0.391194364 0.52717133 0.760836732 0.497816046 0.30800483 0.643310356 30 6 4 2 14 3 7 34 42 20 42 98 21 14.31782106 15.65247584 7.615773106 15.93737745 0.595710566 0.925058578 0.541245036 0.391194364 0.882520987 0.400762159 0.642190983 0.898347245 0.589113828 0.881405709 31 6 5 4 4 5 10 18 28 30 20 40 50 6.403124237 10.77032961 11.18033989 11.87434209 0.329285943 0.366869363 0.925515346 0.52717133 0.400762159 0.921046122 0.185581154 0.317128461 0.860027997 0.474442382 32 2 4 4 4 2 5 12 18 14 8 20 10 4.472135955 6.403124237 5.385164807 6.708203932 0.950072585 0.760597926 0.169278673 0.760836732 0.642190983 0.185581154 0.958164402 0.795569178 0.132792022 0.827311773 33 3 8 5 6 5 4 22 20 18 30 24 20 7.810249676 7.211102551 6.403124237 8.774964387 0.741376615 0.969805778 0.433011049 0.497816046 0.898347245 0.317128461 0.795569178 0.960512515 0.469749782 0.941339558 34 14 8 2 12 6 6 36 36 24 72 72 36 13.41640786 13.41640786 8.485281374 14.69693846 0.212002996 0.539463025 0.962567727 0.30800483 0.589113828 0.860027997 0.132792022 0.469749782 0.961761645 0.560692159 35 13 2 0 13 7 1 40 28 16 91 13 7 14.76482306 13.03840481 7.071067812 14.79864859 0.817841923 0.951536466 0.556287815 0.643310356 0.881405709 0.474442382 0.827311773 0.941339558 0.560692159 0.968982006 36 4 1 6 10 3 6 26 32 18 30 60 18 10.44030651 11.66190379 6.708203932 12.04159458 37 6 5 7 4 1 7 10 22 16 4 28 7 4.123105626 8.062257748 7.071067812 8.124038405 R.rezid 0.013691167 -0.006640394 0.006725924 -0.02085113 -0.000984879 -0.002008907 0.016424919 -0.012233369 0.006041072 0.001217928 38 13 0 10 7 5 9 24 32 28 35 63 45 8.602325267 11.40175425 10.29563014 12.4498996 -0.006640394 0.013128069 -0.004229677 0.015674782 0.002430713 0.00557522 -0.016865678 0.00846849 -0.010202939 -0.008099972 39 13 7 2 12 5 9 34 42 28 60 108 45 13 15 10.29563014 15.8113883 0.006725924 -0.004229677 0.008958597 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