clase2-2009

Estadstica Descriptiva para dos Variables.- Regresin.IntroduccinEstudio conjunto de dos variablesRelacin entre variablesRegresinModelo de regresin lineal simpleBondad de ajuste del modeloEjemplo con SPSS

IntroduccinEn este captulo vamos a tratar diferentes formas de describir la relacin entre dos variables cuando estas son numricas.Estudiar si hay relacin entre la altura y el peso.

Haremos mencin de pasada a otros casos:Alguna de las variables es ordinal.Estudiar la relacin entre el sobrepeso y el dolor de espalda (ordinal) Hay ms de dos variables relacionadas.Conocer el peso de una persona conociendo su altura y contorno de cintura?

El estudio conjunto de dos variables cualitativas no lo trataremos, ya que no veremos los contrastes de la chi-cuadrado (X2).Hay relacin entre fumar y padecer enfermedad de pulmn?

Tema 3: Estadstica bivariante

IntroduccinEl trmino regresin fue introducido por Galton en su libro Natural inheritance (1889) refirindose a la ley de la regresin universal:Cada peculiaridad en un hombre es compartida por sus descendientes, pero en media, en un grado menor.Regresin a la mediaSu trabajo se centraba en la descripcin de los rasgos fsicos de los descendientes (una variable) a partir de los de sus padres (otra variable).Pearson (un amigo suyo) realiz un estudio con ms de 1000 registros de grupos familiares observando una relacin del tipo:Altura del hijo = 85cm + 0,5 altura del padre (aprox.) Conclusin: los padres muy altos tienen tendencia a tener hijos que heredan parte de esta altura, aunque tienen tendencia a acercarse (regresar) a la media. Lo mismo puede decirse de los padres muy bajos.

Hoy en da el sentido de regresin es el de prediccin de una medida basndonos en el conocimiento de otra.

Francis GaltonPrimo de DarwinEstadstico y aventureroFundador (con otros) dela estadstica modernapara explicar las teorasde Darwin.


Estudio conjunto de dos variablesA la derecha tenemos una posible manera de recoger los datos obtenidos observando dos variables en varios individuos de una muestra (tabla de doble entrada).

En cada fila tenemos los datos de un individuo

Cada columna representa los valores que toma una variable sobre los mismos.

Las individuos no se muestran en ningn orden particular.

Dichas observaciones pueden ser representadas en un diagrama de dispersin (scatterplot). En ellos, cada individuo es un punto cuyas coordenadas son los valores de las variables.

Nuestro objetivo ser intentar reconocer a partir del mismo si hay relacin entre las variables, de qu tipo, y si es posible predecir el valor de una de ellas en funcin de la otra.

Altura en cm.Peso en Kg.162611546018078158621716616960166541768416368......


Relacin entre variablesTenemos las alturas y los pesos de 30 individuos representados en un diagrama de dispersin o nube de puntos.Mide 187 cm.Mide 161 cm.Pesa 76 kg.Pesa 50 kg.


Grfico3

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Hoja1

U1UF1UF2Z1Z2med1dt1N1med2dt2N2rabCerrorY

0.32107709790.43902892560.5950575838-0.15343165490.240574730317010168605630.82-10517.216560887365

0.5095050130.21452462770.7282081777-0.79081939810.607402612171.068965517210.33884884716262.99655172411.548410254563.60.8036134967r real68.827586206912.801047816861

0.06005948070.05769547650.9438398223-1.57441758131.587849599815464.760

0.05548652570.83860121620.662221620.98872529190.41853406418063.378

0.14449624920.11387091140.8898571326-1.20619659021.225769024915864.462

0.30899055740.53754795820.49822607190.0942580074-0.004446566317162.566

0.80759120930.4698249680.2895794741-0.0757096811-0.554613554216961.460

0.56459358020.3623377610.1713417684-0.3522171708-0.94887622916660.954

0.08487780230.73047812450.95883435740.61425949151.737316244417664.884

0.79622382230.23698741340.9235514845-0.71602651771.429373400616364.668

0.92548599490.66471351180.41045670910.425361854-0.22637041217462.167

0.40928312970.36272675970.3066671094-0.351179888-0.505319802116661.557

0.78094424680.55083439150.98900851930.12776961942.290661080417164.983

0.86477479320.95588578090.26399508291.7048173074-0.631076828718761.377

0.16457131620.96558168580.90703611441.81948976781.322722601118864.593

0.46622302180.18906278910.2130066129-0.8813550865-0.796032136316161.150

0.98762363060.99697854840.14927259042.7454502187-1.039558342319760.784

0.66300341160.53765111760.21743730330.0945177443-0.780877165217161.160

0.56783209570.48295272630.6630831527-0.04274403060.420892443517063.368

0.89033912970.29005181210.3509696513-0.5532332659-0.382704066516461.856

0.23191944980.81017356710.08427214430.8785359286-1.376896737117960.464

0.892243320.92592929540.84805634671.44612792141.028132991718464.286

0.93297289510.76293712660.97270838780.71578211881.922178021417764.986

0.93813002260.01684722430.1804037537-2.1237079846-0.913827366714960.937

0.58627896790.66791730920.95659177340.43416954991.712435064817464.881

0.05707053280.82722099210.8991330750.94324021231.276627697317964.583

0.70102967010.34869066750.5972566326-0.38885794170.24625268491666363

0.0237166550.70237231980.86871606940.53123577421.120342661417564.378

0.13259568530.20987961530.8718443876-0.80683880891.13515322216264.465

Hoja1

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0

r=0,8

Hoja2

Hoja3

Tenemos las alturas y los pesos de 30 individuos representados en un diagrama de dispersin.Relacin entre variablesParece que el peso aumenta con la altura


Grfico3

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Hoja1


0.32107709790.43902892560.5950575838-0.15343165490.240574730317010168605630.82-10517.216560887365

0.5095050130.21452462770.7282081777-0.79081939810.607402612171.068965517210.33884884716262.99655172411.548410254563.60.8036134967r real68.827586206912.801047816861

0.06005948070.05769547650.9438398223-1.57441758131.587849599815464.760

0.05548652570.83860121620.662221620.98872529190.41853406418063.378

0.14449624920.11387091140.8898571326-1.20619659021.225769024915864.462

0.30899055740.53754795820.49822607190.0942580074-0.004446566317162.566

0.80759120930.4698249680.2895794741-0.0757096811-0.554613554216961.460

0.56459358020.3623377610.1713417684-0.3522171708-0.94887622916660.954

0.08487780230.73047812450.95883435740.61425949151.737316244417664.884

0.79622382230.23698741340.9235514845-0.71602651771.429373400616364.668

0.92548599490.66471351180.41045670910.425361854-0.22637041217462.167

0.40928312970.36272675970.3066671094-0.351179888-0.505319802116661.557

0.78094424680.55083439150.98900851930.12776961942.290661080417164.983

0.86477479320.95588578090.26399508291.7048173074-0.631076828718761.377

0.16457131620.96558168580.90703611441.81948976781.322722601118864.593

0.46622302180.18906278910.2130066129-0.8813550865-0.796032136316161.150

0.98762363060.99697854840.14927259042.7454502187-1.039558342319760.784

0.66300341160.53765111760.21743730330.0945177443-0.780877165217161.160

0.56783209570.48295272630.6630831527-0.04274403060.420892443517063.368

0.89033912970.29005181210.3509696513-0.5532332659-0.382704066516461.856

0.23191944980.81017356710.08427214430.8785359286-1.376896737117960.464

0.892243320.92592929540.84805634671.44612792141.028132991718464.286

0.93297289510.76293712660.97270838780.71578211881.922178021417764.986

0.93813002260.01684722430.1804037537-2.1237079846-0.913827366714960.937

0.58627896790.66791730920.95659177340.43416954991.712435064817464.881

0.05707053280.82722099210.8991330750.94324021231.276627697317964.583

0.70102967010.34869066750.5972566326-0.38885794170.24625268491666363

0.0237166550.70237231980.86871606940.53123577421.120342661417564.378

0.13259568530.20987961530.8718443876-0.80683880891.13515322216264.465

Hoja1

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0

r=0,8

Hoja2

Hoja3

Aparentemente el peso aumenta 10Kg por cada 10 cm de altura... o sea, el peso aumenta en una unidad por cada unidad de altura.10 cm.10 kg.Relacin entre variables


Grfico3

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Hoja1


0.09869029090.43902892560.5950575838-0.15343165490.240574730317010168605630.82-10517.216560887365

0.44867371570.21452462770.7282081777-0.79081939810.607402612171.068965517210.33884884716262.99655172411.548410254563.60.8036134967r real68.827586206912.801047816861

0.63588908070.05769547650.9438398223-1.57441758131.587849599815464.760

0.27798826210.83860121620.662221620.98872529190.41853406418063.378

0.33404890920.11387091140.8898571326-1.20619659021.225769024915864.462

0.9056646350.53754795820.49822607190.0942580074-0.004446566317162.566

0.74370709850.4698249680.2895794741-0.0757096811-0.554613554216961.460

0.15874117880.3623377610.1713417684-0.3522171708-0.94887622916660.954

0.20844385880.73047812450.95883435740.61425949151.737316244417664.884

0.33846401810.23698741340.9235514845-0.71602651771.429373400616364.668

0.26644882880.66471351180.41045670910.425361854-0.22637041217462.167

0.00527476350.36272675970.3066671094-0.351179888-0.505319802116661.557

0.01477920190.55083439150.98900851930.12776961942.290661080417164.983

0.35786894770.95588578090.26399508291.7048173074-0.631076828718761.377

0.84226238580.96558168580.90703611441.81948976781.322722601118864.593

0.0702175880.18906278910.2130066129-0.8813550865-0.796032136316161.150

0.81825903790.99697854840.14927259042.7454502187-1.039558342319760.784

0.3333380880.53765111760.21743730330.0945177443-0.780877165217161.160

0.92468923360.48295272630.6630831527-0.04274403060.420892443517063.368

0.58428988110.29005181210.3509696513-0.5532332659-0.382704066516461.856

0.52224880180.81017356710.08427214430.8785359286-1.376896737117960.464

0.21680985320.92592929540.84805634671.44612792141.028132991718464.286

0.50089487940.76293712660.97270838780.71578211881.922178021417764.986

0.49993764190.01684722430.1804037537-2.1237079846-0.913827366714960.937

0.09890800340.66791730920.95659177340.43416954991.712435064817464.881

0.58682803540.82722099210.8991330750.94324021231.276627697317964.583

0.44946246160.34869066750.5972566326-0.38885794170.24625268491666363

0.38216211350.70237231980.86871606940.53123577421.120342661417564.378

0.4254435070.20987961530.8718443876-0.80683880891.13515322216264.465

Hoja1

r=0,8

Hoja2

Hoja3

Para valores de X por encima de la media tenemos valores de Y por encima y por debajo en proporciones similares. Incorrelacin.Para los valores de X mayores que la media le corresponden valores de Y menores. Esto es relacin inversa o decreciente.Para los valores de X mayores que la media le corresponden valores de Y mayores tambin.Para los valores de X menores que la media le corresponden valores de Y menores tambin.Esto se llama relacin directa.Relacin entre variables


Grfico2

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52

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63

Fuerte relacindirecta.

Hoja1


0.28209279550.43902892560.5950575838-0.15343165490.240574730317010168605630.9-10515.007333035264

0.64467139160.21452462770.7282081777-0.79081939810.607402612171.068965517210.33884884716262.99655172411.548410254563.60.8913679189r real68.034482758611.651499115160

0.52906420850.05769547650.9438398223-1.57441758131.587849599815464.757

0.15091860160.83860121620.662221620.98872529190.41853406418063.377

0.38291326010.11387091140.8898571326-1.20619659021.225769024915864.459

0.80245455060.53754795820.49822607190.0942580074-0.004446566317162.566

0.11746811110.4698249680.2895794741-0.0757096811-0.554613554216961.461

0.86564616830.3623377610.1713417684-0.3522171708-0.94887622916660.956

0.72234555310.73047812450.95883435740.61425949151.737316244417664.880

0.36700411250.23698741340.9235514845-0.71602651771.429373400616364.665

0.55871606050.66471351180.41045670910.425361854-0.22637041217462.168

0.36175725350.36272675970.3066671094-0.351179888-0.505319802116661.558

0.02931380190.55083439150.98900851930.12776961942.290661080417164.977

0.10217523180.95588578090.26399508291.7048173074-0.631076828718761.379

0.674278420.96558168580.90703611441.81948976781.322722601118864.590

0.29968969140.18906278910.2130066129-0.8813550865-0.796032136316161.152

0.4637865090.99697854840.14927259042.7454502187-1.039558342319760.787

0.05863233950.53765111760.21743730330.0945177443-0.780877165217161.162

0.03953363160.48295272630.6630831527-0.04274403060.420892443517063.367

0.47112297050.29005181210.3509696513-0.5532332659-0.382704066516461.857

0.11002076870.81017356710.08427214430.8785359286-1.376896737117960.467

0.72529657960.92592929540.84805634671.44612792141.028132991718464.284

0.34903498890.76293712660.97270838780.71578211881.922178021417764.982

0.08395255020.01684722430.1804037537-2.1237079846-0.913827366714960.939

0.7093228480.66791730920.95659177340.43416954991.712435064817464.878

0.47101695160.82722099210.8991330750.94324021231.276627697317964.580

0.06880909080.34869066750.5972566326-0.38885794170.24625268491666362

0.98540780370.70237231980.86871606940.53123577421.120342661417564.376

0.90136719850.20987961530.8718443876-0.80683880891.13515322216264.463

Hoja1

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0

r=0,9

Hoja2

Hoja3

Grfico8

35

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24

55

29

25

24

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29

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31

-8

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Cierta relacininversa

Hoja1


0.84371190120.43902892560.5950575838-0.15343165490.240574730317010168605630.7200-110.547735582135

0.30590917570.21452462770.7282081777-0.79081939810.607402612171.068965517210.33884884716262.99655172411.548410254563.6-0.6931993739r real33.103448275915.166534911244

0.54534119620.05769547650.9438398223-1.57441758131.587849599815464.763

0.0072148870.83860121620.662221620.98872529190.41853406418063.324

0.06873210250.11387091140.8898571326-1.20619659021.225769024915864.455

0.22930547090.53754795820.49822607190.0942580074-0.004446566317162.529

0.22035700920.4698249680.2895794741-0.0757096811-0.554613554216961.425

0.33751465980.3623377610.1713417684-0.3522171708-0.94887622916660.924

0.94280061720.73047812450.95883435740.61425949151.737316244417664.842

0.45618808030.23698741340.9235514845-0.71602651771.429373400616364.652

0.43446345790.66471351180.41045670910.425361854-0.22637041217462.124

0.0769471430.36272675970.3066671094-0.351179888-0.505319802116661.529

0.90921836350.55083439150.98900851930.12776961942.290661080417164.953

0.64487470180.95588578090.26399508291.7048173074-0.631076828718761.36

0.52577301590.96558168580.90703611441.81948976781.322722601118864.526

0.82811615720.18906278910.2130066129-0.8813550865-0.796032136316161.131

0.14010696520.99697854840.14927259042.7454502187-1.039558342319760.7-8

0.20183218220.53765111760.21743730330.0945177443-0.780877165217161.121

0.40518825040.48295272630.6630831527-0.04274403060.420892443517063.334

0.5651346180.29005181210.3509696513-0.5532332659-0.382704066516461.832

0.39841003820.81017356710.08427214430.8785359286-1.376896737117960.46

0.99631192490.92592929540.84805634671.44612792141.028132991718464.227

0.9907412970.76293712660.97270838780.71578211881.922178021417764.943

0.28160519090.01684722430.1804037537-2.1237079846-0.913827366714960.941

0.8559068880.66791730920.95659177340.43416954991.712435064817464.844

0.07287398850.82722099210.8991330750.94324021231.276627697317964.534

0.90676773640.34869066750.5972566326-0.38885794170.24625268491666337

0.57726642110.70237231980.86871606940.53123577421.120342661417564.337

0.54484823190.20987961530.8718443876-0.80683880891.13515322216264.450

Hoja1

0

0

0

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0

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0

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0

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0

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0

r=-0,7

Hoja2

Hoja3

Covarianza de dos variables X e YLa covarianza entre dos variables, Sxy, nos indica si la posible relacin entre dos variables es directa o inversa.Directa: Sxy >0 Inversa: Sxy

Propiedades de rEs adimensionalSlo toma valores en [-1,1]Las variables son incorreladas r=0Relacin lineal perfecta entre dos variables r=+1 o r=-1Excluimos los casos de puntos alineados horiz. o verticalmente.Cuanto ms cerca est r de +1 o -1 mejor ser el grado de relacin lineal.Siempre que no existan observaciones anmalas.

-1+10Relacin inversa perfectaRelacin directa casi perfectaVariables incorreladas


Entrenando el ojo: correlaciones positivas


Grfico9

85

114

197

114

168

66

12

-28

233

192

48

14

280

23

207

-18

-5

-7

104

23

-55

175

252

-41

229

193

84

175

163

r=0,1

Hoja1


0.62157410810.43902892560.5950575838-0.15343165490.240574730317010168605630.11-105193.419167674285

0.69064233210.21452462770.7282081777-0.79081939810.607402612171.068965517210.33884884716262.99655172411.548410254563.60.0954028081r real103.344827586297.513668666114

0.00967073250.05769547650.9438398223-1.57441758131.587849599815464.7197

0.18759116220.83860121620.662221620.98872529190.41853406418063.3114

0.54413110950.11387091140.8898571326-1.20619659021.225769024915864.4168

0.12295343450.53754795820.49822607190.0942580074-0.004446566317162.566

0.73700693650.4698249680.2895794741-0.0757096811-0.554613554216961.412

0.35645030210.3623377610.1713417684-0.3522171708-0.94887622916660.9-28

0.90974405250.73047812450.95883435740.61425949151.737316244417664.8233

0.80766615480.23698741340.9235514845-0.71602651771.429373400616364.6192

0.30063374260.66471351180.41045670910.425361854-0.22637041217462.148

0.73531267810.36272675970.3066671094-0.351179888-0.505319802116661.514

0.71713863590.55083439150.98900851930.12776961942.290661080417164.9280

0.22986979150.95588578090.26399508291.7048173074-0.631076828718761.323

0.76254946560.96558168580.90703611441.81948976781.322722601118864.5207

0.20962820130.18906278910.2130066129-0.8813550865-0.796032136316161.1-18

0.96989194320.99697854840.14927259042.7454502187-1.039558342319760.7-5

0.52010091390.53765111760.21743730330.0945177443-0.780877165217161.1-7

0.12240219960.48295272630.6630831527-0.04274403060.420892443517063.3104

0.32332934940.29005181210.3509696513-0.5532332659-0.382704066516461.823

0.62886776380.81017356710.08427214430.8785359286-1.376896737117960.4-55

0.32163576720.92592929540.84805634671.44612792141.028132991718464.2175

0.99619655770.76293712660.97270838780.71578211881.922178021417764.9252

0.85771995550.01684722430.1804037537-2.1237079846-0.913827366714960.9-41

0.87901027940.66791730920.95659177340.43416954991.712435064817464.8229

0.97128092290.82722099210.8991330750.94324021231.276627697317964.5193

0.16127075980.34869066750.5972566326-0.38885794170.24625268491666384

0.05145907510.70237231980.86871606940.53123577421.120342661417564.3175

0.5909037020.20987961530.8718443876-0.80683880891.13515322216264.4163

Hoja1

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

r=0,1

Hoja2

Hoja3

Grfico1

68

71

84

84

80

66

52

40

110

90

64

50

117

68

113

38

69

49

74

50

43

102

115

24

107

103

67

95

82

r=0,4

Hoja1


0.96278561480.43902892560.5950575838-0.15343165490.240574730317010168605630.42-105122.33989176268

0.72884958310.21452462770.7282081777-0.79081939810.607402612171.068965517210.33884884716262.99655172411.548410254563.60.4019895599r real7525.272310048271

0.24308268250.05769547650.9438398223-1.57441758131.587849599815464.784

0.52635142170.83860121620.662221620.98872529190.41853406418063.384

0.50863950040.11387091140.8898571326-1.20619659021.225769024915864.480

0.55985996350.53754795820.49822607190.0942580074-0.004446566317162.566

0.59602864420.4698249680.2895794741-0.0757096811-0.554613554216961.452

0.80864153810.3623377610.1713417684-0.3522171708-0.94887622916660.940

0.87987313340.73047812450.95883435740.61425949151.737316244417664.8110

0.44536999780.23698741340.9235514845-0.71602651771.429373400616364.690

0.19007548650.66471351180.41045670910.425361854-0.22637041217462.164

0.94267983280.36272675970.3066671094-0.351179888-0.505319802116661.550

0.26996493790.55083439150.98900851930.12776961942.290661080417164.9117

0.53698251330.95588578090.26399508291.7048173074-0.631076828718761.368

0.91658998760.96558168580.90703611441.81948976781.322722601118864.5113

0.04159517760.18906278910.2130066129-0.8813550865-0.796032136316161.138

0.71756587380.99697854840.14927259042.7454502187-1.039558342319760.769

0.4257736250.53765111760.21743730330.0945177443-0.780877165217161.149

0.7340984470.48295272630.6630831527-0.04274403060.420892443517063.374

0.79217515180.29005181210.3509696513-0.5532332659-0.382704066516461.850

0.16349300190.81017356710.08427214430.8785359286-1.376896737117960.443

0.87609902720.92592929540.84805634671.44612792141.028132991718464.2102

0.37987287540.76293712660.97270838780.71578211881.922178021417764.9115

0.94283623360.01684722430.1804037537-2.1237079846-0.913827366714960.924

0.80597712590.66791730920.95659177340.43416954991.712435064817464.8107

0.71268062690.82722099210.8991330750.94324021231.276627697317964.5103

0.44776339980.34869066750.5972566326-0.38885794170.24625268491666367

0.69567684650.70237231980.86871606940.53123577421.120342661417564.395

0.45363680980.20987961530.8718443876-0.80683880891.13515322216264.482

Hoja1

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

r=0,4

Hoja2

Hoja3

Grfico3

65

61

60

78

62

66

60

54

84

68

67

57

83

77

93

50

84

60

68

56

64

86

86

37

81

83

63

78

65

r=0,8

Hoja1


0.82764210480.43902892560.5950575838-0.15343165490.240574730317010168605630.82-10517.216560887365

0.48443779890.21452462770.7282081777-0.79081939810.607402612171.068965517210.33884884716262.99655172411.548410254563.60.8036134967r real68.827586206912.801047816861

0.87494951180.05769547650.9438398223-1.57441758131.587849599815464.760

0.09048222120.83860121620.662221620.98872529190.41853406418063.378

0.83722174980.11387091140.8898571326-1.20619659021.225769024915864.462

0.56613209760.53754795820.49822607190.0942580074-0.004446566317162.566

0.19465737450.4698249680.2895794741-0.0757096811-0.554613554216961.460

0.94140210750.3623377610.1713417684-0.3522171708-0.94887622916660.954

0.72142453560.73047812450.95883435740.61425949151.737316244417664.884

0.32169090690.23698741340.9235514845-0.71602651771.429373400616364.668

0.53772385560.66471351180.41045670910.425361854-0.22637041217462.167

0.19731314950.36272675970.3066671094-0.351179888-0.505319802116661.557

0.02376851410.55083439150.98900851930.12776961942.290661080417164.983

0.64190367980.95588578090.26399508291.7048173074-0.631076828718761.377

0.34736649610.96558168580.90703611441.81948976781.322722601118864.593

0.69768502570.18906278910.2130066129-0.8813550865-0.796032136316161.150

0.17596406410.99697854840.14927259042.7454502187-1.039558342319760.784

0.35440034640.53765111760.21743730330.0945177443-0.780877165217161.160

0.77712879430.48295272630.6630831527-0.04274403060.420892443517063.368

0.3932159450.29005181210.3509696513-0.5532332659-0.382704066516461.856

0.98512241010.81017356710.08427214430.8785359286-1.376896737117960.464

0.0985164190.92592929540.84805634671.44612792141.028132991718464.286

0.74107843230.76293712660.97270838780.71578211881.922178021417764.986

0.34261101850.01684722430.1804037537-2.1237079846-0.913827366714960.937

0.99414000130.66791730920.95659177340.43416954991.712435064817464.881

0.66027963270.82722099210.8991330750.94324021231.276627697317964.583

0.8175993160.34869066750.5972566326-0.38885794170.24625268491666363

0.8542092950.70237231980.86871606940.53123577421.120342661417564.378

0.4008132320.20987961530.8718443876-0.80683880891.13515322216264.465

Hoja1

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

r=0,8

Hoja2

Hoja3

Grfico5

63

58

51

76

55

66

63

60

74

60

69

60

69

81

85

55

90

65

66

58

72

81

75

43

72

76

61

72

59

r=0,99

Hoja1


0.44659047840.43902892560.5950575838-0.15343165490.240574730317010168605630.99-10511.473206171963

0.17641518270.21452462770.7282081777-0.79081939810.607402612171.068965517210.33884884716262.99655172411.548410254563.60.9881633659r real66.72413793110.451518786658

0.78483671350.05769547650.9438398223-1.57441758131.587849599815464.751

0.09269008530.83860121620.662221620.98872529190.41853406418063.376

0.52065465670.11387091140.8898571326-1.20619659021.225769024915864.455

0.56071040160.53754795820.49822607190.0942580074-0.004446566317162.566

0.94818143880.4698249680.2895794741-0.0757096811-0.554613554216961.463

0.3012378920.3623377610.1713417684-0.3522171708-0.94887622916660.960

0.66866410.73047812450.95883435740.61425949151.737316244417664.874

0.16145271740.23698741340.9235514845-0.71602651771.429373400616364.660

0.83846469610.66471351180.41045670910.425361854-0.22637041217462.169

0.77310779410.36272675970.3066671094-0.351179888-0.505319802116661.560

0.90297267980.55083439150.98900851930.12776961942.290661080417164.969

0.30601530410.95588578090.26399508291.7048173074-0.631076828718761.381

0.58762811740.96558168580.90703611441.81948976781.322722601118864.585

0.30706795530.18906278910.2130066129-0.8813550865-0.796032136316161.155

0.92571621180.99697854840.14927259042.7454502187-1.039558342319760.790

0.67024332860.53765111760.21743730330.0945177443-0.780877165217161.165

0.67105706480.48295272630.6630831527-0.04274403060.420892443517063.366

0.66276025750.29005181210.3509696513-0.5532332659-0.382704066516461.858

0.17981586580.81017356710.08427214430.8785359286-1.376896737117960.472

0.18294537140.92592929540.84805634671.44612792141.028132991718464.281

0.91781929230.76293712660.97270838780.71578211881.922178021417764.975

0.11459626870.01684722430.1804037537-2.1237079846-0.913827366714960.943

0.66128154380.66791730920.95659177340.43416954991.712435064817464.872

0.65736863710.82722099210.8991330750.94324021231.276627697317964.576

0.22871230540.34869066750.5972566326-0.38885794170.24625268491666361

0.39487873970.70237231980.86871606940.53123577421.120342661417564.372

0.31542994930.20987961530.8718443876-0.80683880891.13515322216264.459

Hoja1

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

r=0,95

Hoja2

Hoja3

Entrenando el ojo: correlaciones negativas


Grfico7

36

49

74

27

64

29

21

17

55

63

22

25

70

2

36

25

-16

15

38

29

-4

34

57

35

57

44

38

45

58

r=-0,5

Hoja1


0.54579486640.43902892560.5950575838-0.15343165490.240574730317010168605630.5200-117.907411494836

0.46270990120.21452462770.7282081777-0.79081939810.607402612171.068965517210.33884884716262.99655172411.548410254563.6-0.494717264r real36.034482758621.392042927149

0.4852670590.05769547650.9438398223-1.57441758131.587849599815464.774

0.01911357040.83860121620.662221620.98872529190.41853406418063.327

0.92570201880.11387091140.8898571326-1.20619659021.225769024915864.464

0.53085394810.53754795820.49822607190.0942580074-0.004446566317162.529

0.72795158260.4698249680.2895794741-0.0757096811-0.554613554216961.421

0.42381971760.3623377610.1713417684-0.3522171708-0.94887622916660.917

0.54477323810.73047812450.95883435740.61425949151.737316244417664.855

0.42929836280.23698741340.9235514845-0.71602651771.429373400616364.663

0.35054786040.66471351180.41045670910.425361854-0.22637041217462.122

0.94186433010.36272675970.3066671094-0.351179888-0.505319802116661.525

0.2609131380.55083439150.98900851930.12776961942.290661080417164.970

0.63925548160.95588578090.26399508291.7048173074-0.631076828718761.32

0.33941188040.96558168580.90703611441.81948976781.322722601118864.536

0.57540454350.18906278910.2130066129-0.8813550865-0.796032136316161.125

0.25934828650.99697854840.14927259042.7454502187-1.039558342319760.7-16

0.27084887330.53765111760.21743730330.0945177443-0.780877165217161.115

0.21811201870.48295272630.6630831527-0.04274403060.420892443517063.338

0.28946248620.29005181210.3509696513-0.5532332659-0.382704066516461.829

0.02240351990.81017356710.08427214430.8785359286-1.376896737117960.4-4

0.23629557340.92592929540.84805634671.44612792141.028132991718464.234

0.87015346130.76293712660.97270838780.71578211881.922178021417764.957

0.98847085750.01684722430.1804037537-2.1237079846-0.913827366714960.935

0.98361841310.66791730920.95659177340.43416954991.712435064817464.857

0.32776158820.82722099210.8991330750.94324021231.276627697317964.544

0.15788477550.34869066750.5972566326-0.38885794170.24625268491666338

0.79770698560.70237231980.86871606940.53123577421.120342661417564.345

0.49163245750.20987961530.8718443876-0.80683880891.13515322216264.458

Hoja1

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

r=-0,5

Hoja2

Hoja3

Grfico8

35

44

63

24

55

29

25

24

42

52

24

29

53

6

26

31

-8

21

34

32

6

27

43

41

44

34

37

37

50

r=-0,7

Hoja1


0.61729822170.43902892560.5950575838-0.15343165490.240574730317010168605630.7200-110.547735582135

0.69716267940.21452462770.7282081777-0.79081939810.607402612171.068965517210.33884884716262.99655172411.548410254563.6-0.6931993739r real33.103448275915.166534911244

0.04600106330.05769547650.9438398223-1.57441758131.587849599815464.763

0.98776995690.83860121620.662221620.98872529190.41853406418063.324

0.10007391080.11387091140.8898571326-1.20619659021.225769024915864.455

0.03720506460.53754795820.49822607190.0942580074-0.004446566317162.529

0.60226685980.4698249680.2895794741-0.0757096811-0.554613554216961.425

0.07415733990.3623377610.1713417684-0.3522171708-0.94887622916660.924

0.51056231240.73047812450.95883435740.61425949151.737316244417664.842

0.44379665160.23698741340.9235514845-0.71602651771.429373400616364.652

0.73824211370.66471351180.41045670910.425361854-0.22637041217462.124

0.48712595590.36272675970.3066671094-0.351179888-0.505319802116661.529

0.27533967760.55083439150.98900851930.12776961942.290661080417164.953

0.32230035970.95588578090.26399508291.7048173074-0.631076828718761.36

0.52315988260.96558168580.90703611441.81948976781.322722601118864.526

0.16453416770.18906278910.2130066129-0.8813550865-0.796032136316161.131

0.1013879550.99697854840.14927259042.7454502187-1.039558342319760.7-8

0.94243312090.53765111760.21743730330.0945177443-0.780877165217161.121

0.84700707470.48295272630.6630831527-0.04274403060.420892443517063.334

0.66780761830.29005181210.3509696513-0.5532332659-0.382704066516461.832

0.74994647940.81017356710.08427214430.8785359286-1.376896737117960.46

0.43570087950.92592929540.84805634671.44612792141.028132991718464.227

0.22966503720.76293712660.97270838780.71578211881.922178021417764.943

0.75165687450.01684722430.1804037537-2.1237079846-0.913827366714960.941

0.23349169450.66791730920.95659177340.43416954991.712435064817464.844

0.33325891160.82722099210.8991330750.94324021231.276627697317964.534

0.14709819710.34869066750.5972566326-0.38885794170.24625268491666337

0.86272090460.70237231980.86871606940.53123577421.120342661417564.337

0.99333068680.20987961530.8718443876-0.80683880891.13515322216264.450

Hoja1

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

r=-0,7

Hoja2

Hoja3

Grfico9

33

40

51

21

46

29

29

31

30

42

25

32

37

11

16

36

-1

26

31

35

16

19

30

48

32

25

35

29

42

r=-0,95

Hoja1


0.19371888780.43902892560.5950575838-0.15343165490.240574730317010168605630.95200-13.398215281933

0.72452379970.21452462770.7282081777-0.79081939810.607402612171.068965517210.33884884716262.99655172411.548410254563.6-0.9456008699r real30.206896551711.059024432740

0.75956365750.05769547650.9438398223-1.57441758131.587849599815464.751

0.8860076550.83860121620.662221620.98872529190.41853406418063.321

0.69250676470.11387091140.8898571326-1.20619659021.225769024915864.446

0.32026330950.53754795820.49822607190.0942580074-0.004446566317162.529

0.5172898010.4698249680.2895794741-0.0757096811-0.554613554216961.429

0.51446254920.3623377610.1713417684-0.3522171708-0.94887622916660.931

0.7480229540.73047812450.95883435740.61425949151.737316244417664.830

0.54475818930.23698741340.9235514845-0.71602651771.429373400616364.642

0.28150412250.66471351180.41045670910.425361854-0.22637041217462.125

0.86331364820.36272675970.3066671094-0.351179888-0.505319802116661.532

0.81466563140.55083439150.98900851930.12776961942.290661080417164.937

0.0424927570.95588578090.26399508291.7048173074-0.631076828718761.311

0.5326931490.96558168580.90703611441.81948976781.322722601118864.516

0.7907436020.18906278910.2130066129-0.8813550865-0.796032136316161.136

0.10424189380.99697854840.14927259042.7454502187-1.039558342319760.7-1

0.97096570350.53765111760.21743730330.0945177443-0.780877165217161.126

0.06550125910.48295272630.6630831527-0.04274403060.420892443517063.331

0.49919217590.29005181210.3509696513-0.5532332659-0.382704066516461.835

0.77768660630.81017356710.08427214430.8785359286-1.376896737117960.416

0.87148717120.92592929540.84805634671.44612792141.028132991718464.219

0.08683559720.76293712660.97270838780.71578211881.922178021417764.930

0.02372694720.01684722430.1804037537-2.1237079846-0.913827366714960.948

0.23367547340.66791730920.95659177340.43416954991.712435064817464.832

0.13815154190.82722099210.8991330750.94324021231.276627697317964.525

0.99762046750.34869066750.5972566326-0.38885794170.24625268491666335

0.8419379530.70237231980.86871606940.53123577421.120342661417564.329

0.88396354460.20987961530.8718443876-0.80683880891.13515322216264.442

Hoja1

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

r=-0,95

Hoja2

Hoja3

Grfico10

32

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47

20

43

29

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34

25

38

26

34

30

13

13

39

3

29

30

36

20

16

24

51

27

22

34

26

39

r=-0,999

Hoja1


0.30919189280.43902892560.5950575838-0.15343165490.240574730317010168605630.999200-10.462714484832

0.78490643980.21452462770.7282081777-0.79081939810.607402612171.068965517210.33884884716262.99655172411.548410254563.6-0.9986429307r real29.27586206910.395279272538

0.77747221160.05769547650.9438398223-1.57441758131.587849599815464.747

0.76591717580.83860121620.662221620.98872529190.41853406418063.320

0.28391026010.11387091140.8898571326-1.20619659021.225769024915864.443

0.49399174920.53754795820.49822607190.0942580074-0.004446566317162.529

0.70429608740.4698249680.2895794741-0.0757096811-0.554613554216961.431

0.10320091660.3623377610.1713417684-0.3522171708-0.94887622916660.934

0.74752938910.73047812450.95883435740.61425949151.737316244417664.825

0.69745751490.23698741340.9235514845-0.71602651771.429373400616364.638

0.94158042580.66471351180.41045670910.425361854-0.22637041217462.126

0.47268893190.36272675970.3066671094-0.351179888-0.505319802116661.534

0.48932701720.55083439150.98900851930.12776961942.290661080417164.930

0.89196259980.95588578090.26399508291.7048173074-0.631076828718761.313

0.17601936970.96558168580.90703611441.81948976781.322722601118864.513

0.89755704180.18906278910.2130066129-0.8813550865-0.796032136316161.139

0.11903461790.99697854840.14927259042.7454502187-1.039558342319760.73

0.25030891630.53765111760.21743730330.0945177443-0.780877165217161.129

0.49519415370.48295272630.6630831527-0.04274403060.420892443517063.330

0.51311001040.29005181210.3509696513-0.5532332659-0.382704066516461.836

0.4647394070.81017356710.08427214430.8785359286-1.376896737117960.420

0.41704355860.92592929540.84805634671.44612792141.028132991718464.216

0.99611579630.76293712660.97270838780.71578211881.922178021417764.924

0.06456219630.01684722430.1804037537-2.1237079846-0.913827366714960.951

0.2766573180.66791730920.95659177340.43416954991.712435064817464.827

0.26284736910.82722099210.8991330750.94324021231.276627697317964.522

0.63533924570.34869066750.5972566326-0.38885794170.24625268491666334

0.87805949470.70237231980.86871606940.53123577421.120342661417564.326

0.63362402120.20987961530.8718443876-0.80683880891.13515322216264.439

Hoja1

0

0

0

0

0

0

0

0

0

0

0

0

0

0

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0

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0

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0

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0

0

r=-0,999

Hoja2

Hoja3

Evolucin de r y diagrama de dispersin


Bondad de ajuste del modeloLo adecuado del modelo depende de la relacin entre: la dispersin marginal de Y La dispersin de Y condicionada a X

Es decir, fijando valores de X, vemos cmo se distribuye Y

La distribucin de Y, para valores fijados de X, se denomina distribucin condicionada.

La distribucin de Y, independientemente del valor de X, se denomina distribucin marginal.

Si la dispersin se reduce notablemente, el modelo de regresin ser adecuado.


Preguntas frecuentesSi r = 0 eso quiere decir que no las variables son independientes?En la prctica, casi siempre s, pero no tiene por qu ser cierto en todos los casos.Lo contrario si es cierto: Independencia implica incorrelacin.

Me ha salido r = 12 la relacin es superlineal[sic]?Superqu? Eso es un error de clculo. Siempre debe tomar un valor entre -1 y +1.

A partir de qu valores se considera que hay buena relacin lineal?Imposible dar un valor concreto (mirad los grficos anteriores). Para este curso digamos que si |r| > 0,7 hay buena relacin lineal y que si |r| > 0,4 hay cierta relacin (por decir algo... la cosa es un poco ms complicada observaciones atpicas, homogeneidad de varianzas...)


Otros coeficientes de correlacinCuando las variables en vez de ser numricas son ordinales, es posible preguntarse sobre si hay algn tipo de correlacin entre ellas.

Disponemos para estos casos de dos estadsticos, aunque no los usaremos en clase: (ro) de Spearman (tau) de Kendall

No tenis que estudiar nada sobre ellos en este curso. Recordad slo que son estadsticos anlogos a r y que los encontrareis en publicaciones donde las variables no puedan considerarse numricas.

Maurice George KendallCharles Edward Spearman


RegresinEl anlisis de regresin sirve para predecir una medida en funcin de otra medida (o varias).Y = Variable dependientepredichaExplicada

X = Variable independientepredictora explicativa

Es posible descubrir una relacin? Y = f(X) + errorf es una funcin de un tipo determinadoel error es aleatorio, pequeo, y no depende de X


RegresinSe pueden considerar otros tipos de modelos, en funcin del aspecto que presente el diagrama de dispersin (regresin no lineal)

Incluso se puede considerar el que una variable dependa de varias (regresin mltiple).


Grfico11

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155

224

56

158

60

recta o parbola?

Hoja1

U1UF1UF2Z1Z2med1dt1N1med2dt2N2rabCerrorYYNLb2

0.82827150790.43902892560.5950575838-0.15343165490.240574730317010168605630.5200-117.907411494836680.5

0.66580588720.21452462770.7282081777-0.79081939810.607402612171.068965517210.33884884716262.99655172411.548410254563.6-0.494717264r real36.034482758621.39204292714951

0.09094520610.05769547650.9438398223-1.57441758131.587849599815464.77492

0.38419582620.83860121620.662221620.98872529190.41853406418063.327227

0.39853652680.11387091140.8898571326-1.20619659021.225769024915864.46466

0.23855711690.53754795820.49822607190.0942580074-0.004446566317162.52989

0.0807724210.4698249680.2895794741-0.0757096811-0.554613554216961.42162

0.47727346210.3623377610.1713417684-0.3522171708-0.94887622916660.91735

0.51399794250.73047812450.95883435740.61425949151.737316244417664.855183

0.185120210.23698741340.9235514845-0.71602651771.429373400616364.66367

0.27690945840.66471351180.41045670910.425361854-0.22637041217462.122120

0.73911763210.36272675970.3066671094-0.351179888-0.505319802116661.52543

0.08559200750.55083439150.98900851930.12776961942.290661080417164.970131

0.81043304870.95588578090.26399508291.7048173074-0.631076828718761.32366

0.47429788370.96558168580.90703611441.81948976781.322722601118864.536428

0.29084249060.18906278910.2130066129-0.8813550865-0.796032136316161.12525

0.57542750070.99697854840.14927259042.7454502187-1.039558342319760.7-16669

0.4848115980.53765111760.21743730330.0945177443-0.780877165217161.11576

0.54603101930.48295272630.6630831527-0.04274403060.420892443517063.33888

0.78196795340.29005181210.3509696513-0.5532332659-0.382704066516461.82937

0.91859773020.81017356710.08427214430.8785359286-1.376896737117960.4-4177

0.75963493550.92592929540.84805634671.44612792141.028132991718464.234322

0.58602831080.76293712660.97270838780.71578211881.922178021417764.957202

0.59536752150.01684722430.1804037537-2.1237079846-0.913827366714960.93595

0.31575595690.66791730920.95659177340.43416954991.712435064817464.857155

0.25058008890.82722099210.8991330750.94324021231.276627697317964.544224

0.15838036630.34869066750.5972566326-0.38885794170.2462526849166633856

0.66490467320.70237231980.86871606940.53123577421.120342661417564.345158

0.24012203090.20987961530.8718443876-0.80683880891.13515322216264.45860

Hoja1

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

Parablico?

Hoja2

Hoja3

Grfico12

59804

57801

43490

63827

52952

59833

59817

59561

60719

58491

60078

59569

59874

79454

83164

56909

138516

59819

59838

58965

62712

70810

61229

22791

60113

62760

59582

60345

59874

recta o cbica?

Hoja1

U1UF1UF2Z1Z2med1dt1N1med2dt2N2rabCerrorYYNLb2

0.44521822660.43902892560.5950575838-0.15343165490.240574730317010168605630.560000-117.907411494859836598044

0.69953027470.21452462770.7282081777-0.79081939810.607402612171.068965517210.33884884716262.99655172411.548410254563.6-0.494717264r real59836.034482758621.39204292715984957801

0.29815526350.05769547650.9438398223-1.57441758131.587849599815464.75987443490

0.39416940050.83860121620.662221620.98872529190.41853406418063.35982763827

0.34900926860.11387091140.8898571326-1.20619659021.225769024915864.45986452952

0.83135368960.53754795820.49822607190.0942580074-0.004446566317162.55982959833

0.93591247460.4698249680.2895794741-0.0757096811-0.554613554216961.45982159817

0.80773956360.3623377610.1713417684-0.3522171708-0.94887622916660.95981759561

0.02158149480.73047812450.95883435740.61425949151.737316244417664.85985560719

0.16318699890.23698741340.9235514845-0.71602651771.429373400616364.65986358491

0.87084296920.66471351180.41045670910.425361854-0.22637041217462.15982260078

0.76012732520.36272675970.3066671094-0.351179888-0.505319802116661.55982559569

0.42178730780.55083439150.98900851930.12776961942.290661080417164.95987059874

0.58447688160.95588578090.26399508291.7048173074-0.631076828718761.35980279454

0.35878109760.96558168580.90703611441.81948976781.322722601118864.55983683164

0.80048620940.18906278910.2130066129-0.8813550865-0.796032136316161.15982556909

0.78638977490.99697854840.14927259042.7454502187-1.039558342319760.759784138516

0.34530652460.53765111760.21743730330.0945177443-0.780877165217161.15981559819

0.46670555680.48295272630.6630831527-0.04274403060.420892443517063.35983859838

0.72660037220.29005181210.3509696513-0.5532332659-0.382704066516461.85982958965

0.15358275210.81017356710.08427214430.8785359286-1.376896737117960.45979662712

0.5475353840.92592929540.84805634671.44612792141.028132991718464.25983470810

0.55633311220.76293712660.97270838780.71578211881.922178021417764.95985761229

0.95882179270.01684722430.1804037537-2.1237079846-0.913827366714960.95983522791

0.80015355320.66791730920.95659177340.43416954991.712435064817464.85985760113

0.51937276890.82722099210.8991330750.94324021231.276627697317964.55984462760

0.9712908260.34869066750.5972566326-0.38885794170.2462526849166635983859582

0.25852924330.70237231980.86871606940.53123577421.120342661417564.35984560345

0.227025450.20987961530.8718443876-0.80683880891.13515322216264.45985859874

Hoja1

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

Cbico?

Hoja2

Hoja3

Regresin1 variable explicativa2+ variables explicativasEn clase slo tratamos el modelo de regresin lineal simple.


Regresin

El ejemplo del estudio de la altura en grupos familiares de Pearson es del tipo que desarrollaremos en el resto del tema.

Altura del hijo = 85cm + 0,5 altura del padre (Y = 85 + 0,5 X)

Si el padre mide 200cm cunto mide el hijo?Se espera (predice) 85 + 0,5x200=185 cm.Alto, pero no tanto como el padre. Regresa a la media.

Si el padre mide 120cm cunto mide el hijo?Se espera (predice) 85 + 0,5x120=145 cm.Bajo, pero no tanto como el padre. Regresa a la media.

Es decir, nos interesaremos por modelos de regresin lineal simple.


Modelo de regresin lineal simpleEn el modelo de regresin lineal simple, dado dos variablesY (dependiente)X (independiente, explicativa, predictora)

buscamos encontrar una funcin de X muy simple (lineal) que nos permita aproximar Y mediante = b0 + b1Xb0 (ordenada en el origen, constante)b1 (pendiente de la recta)

Y e rara vez coincidirn por muy bueno que sea el modelo de regresin. A la cantidad e=Y- se le denomina residuo o error residual.


Modelo de regresin lineal simpleEn el ejemplo de Pearson y las alturas, l encontr: = b0 + b1Xb0=85 cm (No interpretar como altura de un hijo cuyo padre mide 0 cm Extrapolacin salvaje!b1=0,5 (En media el hijo gana 0,5 cm por cada cm del padre.)

b0=85 cmb1=0,5


Grfico2

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Hoja1


0.62882127510.43902892560.5950575838-0.15343165490.240574730317010168170101760.7900.55.273867791175

0.86506927050.21452462770.7282081777-0.79081939810.607402612171.068965517210.338848847162176.00344827593.109438663177.30.6808795128r real177.51724137937.5091699787174

0.05663284170.05769547650.9438398223-1.57441758131.5878495998154179.4175

0.40246557430.83860121620.662221620.98872529190.418534064180176.6182

0.8257325710.11387091140.8898571326-1.20619659021.2257690249158178.9175

0.73090655110.53754795820.49822607190.0942580074-0.0044465663171175175

0.44456492290.4698249680.2895794741-0.0757096811-0.5546135542169172.9172

0.28343466610.3623377610.1713417684-0.3522171708-0.948876229166171.7168

0.82318254810.73047812450.95883435740.61425949151.7373162444176179.6187

0.68713189180.23698741340.9235514845-0.71602651771.4293734006163179.2179

0.53363632570.66471351180.41045670910.425361854-0.226370412174174.1176

0.05368156590.36272675970.3066671094-0.351179888-0.5053198021166173.1170

0.41798535710.55083439150.98900851930.12776961942.2906610804171179.9188

0.24551871020.95588578090.26399508291.7048173074-0.6310768287187172.6180

0.45057967940.96558168580.90703611441.81948976781.3227226011188179.1191

0.3543584370.18906278910.2130066129-0.8813550865-0.7960321363161172.1166

0.36553687250.99697854840.14927259042.7454502187-1.0395583423197171.5183

0.14895155140.53765111760.21743730330.0945177443-0.7808771652171172.2171

0.06451326450.48295272630.6630831527-0.04274403060.4208924435170176.6177

0.79609741360.29005181210.3509696513-0.5532332659-0.3827040665164173.5170

0.68402626250.81017356710.08427214430.8785359286-1.3768967371179170.8172

0.03325125720.92592929540.84805634671.44612792141.0281329917184178.5187

0.77192411640.76293712660.97270838780.71578211881.9221780214177179.7189

0.27807411450.01684722430.1804037537-2.1237079846-0.9138273667149171.8160

0.17720593670.66791730920.95659177340.43416954991.7124350648174179.6186

0.55083111920.82722099210.8991330750.94324021231.2766276973179179186

0.92374910230.34869066750.5972566326-0.38885794170.2462526849166176174

0.3512604180.70237231980.86871606940.53123577421.1203426614175178.7183

0.9398921530.20987961530.8718443876-0.80683880891.135153222162178.7177

Hoja1

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Hoja2

Hoja3

Modelo de regresin lineal simpleLa relacin entre las variables no es exacta. Es natural preguntarse entonces: Cul es la mejor recta que sirve para predecir los valores de Y en funcin de los de XQu error cometemos con dicha aproximacin (residual).

b0=85 cmb1=0,5


Grfico2

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0.62882127510.43902892560.5950575838-0.15343165490.240574730317010168170101760.7900.55.273867791175

0.86506927050.21452462770.7282081777-0.79081939810.607402612171.068965517210.338848847162176.00344827593.109438663177.30.6808795128r real177.51724137937.5091699787174

0.05663284170.05769547650.9438398223-1.57441758131.5878495998154179.4175

0.40246557430.83860121620.662221620.98872529190.418534064180176.6182

0.8257325710.11387091140.8898571326-1.20619659021.2257690249158178.9175

0.73090655110.53754795820.49822607190.0942580074-0.0044465663171175175

0.44456492290.4698249680.2895794741-0.0757096811-0.5546135542169172.9172

0.28343466610.3623377610.1713417684-0.3522171708-0.948876229166171.7168

0.82318254810.73047812450.95883435740.61425949151.7373162444176179.6187

0.68713189180.23698741340.9235514845-0.71602651771.4293734006163179.2179

0.53363632570.66471351180.41045670910.425361854-0.226370412174174.1176

0.05368156590.36272675970.3066671094-0.351179888-0.5053198021166173.1170

0.41798535710.55083439150.98900851930.12776961942.2906610804171179.9188

0.24551871020.95588578090.26399508291.7048173074-0.6310768287187172.6180

0.45057967940.96558168580.90703611441.81948976781.3227226011188179.1191

0.3543584370.18906278910.2130066129-0.8813550865-0.7960321363161172.1166

0.36553687250.99697854840.14927259042.7454502187-1.0395583423197171.5183

0.14895155140.53765111760.21743730330.0945177443-0.7808771652171172.2171

0.06451326450.48295272630.6630831527-0.04274403060.4208924435170176.6177

0.79609741360.29005181210.3509696513-0.5532332659-0.3827040665164173.5170

0.68402626250.81017356710.08427214430.8785359286-1.3768967371179170.8172

0.03325125720.92592929540.84805634671.44612792141.0281329917184178.5187

0.77192411640.76293712660.97270838780.71578211881.9221780214177179.7189

0.27807411450.01684722430.1804037537-2.1237079846-0.9138273667149171.8160

0.17720593670.66791730920.95659177340.43416954991.7124350648174179.6186

0.55083111920.82722099210.8991330750.94324021231.2766276973179179186

0.92374910230.34869066750.5972566326-0.38885794170.2462526849166176174

0.3512604180.70237231980.86871606940.53123577421.1203426614175178.7183

0.9398921530.20987961530.8718443876-0.80683880891.135153222162178.7177

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Modelo de regresin lineal simpleEl modelo lineal de regresin se construye utilizando la tcnica de estimacin mnimo cuadrtica:Buscar b0, b1 de tal manera que se minimice la cantidadi ei2

Se comprueba que para lograr dicho resultado basta con elegir:

Se obtiene adems unas ventajas de regaloEl error residual medio es nuloLa varianza del error residual es mnima para dicha estimacin.Traducido: En trmino medio no nos equivocamos. Cualquier otra estimacin que no cometa error en trmino medio, si es de tipo lineal, ser peor por presentar mayor variabilidad con respecto al error medio (que es cero).


Bondad de ajuste del modelo


Bondad de ajuste del modeloQue el error medio de las predicciones sea nulo no quiere decir que las predicciones sean buenas.

Hay que encontrar un medio de expresar la bondad del ajuste (bondad de la prediccin)

Cometi un error de -30 en su ltima prediccinNo importa. Con los dos ltimos clientes me equivoqu en +10 y +20. En trmino medio el error es cero.


Bondad de ajuste del modeloImaginemos un diagrama de dispersin, y vamos a tratar de comprender en primer lugar qu es el error residual, su relacin con la varianza de Y, y de ah, cmo medir la bondad de un ajuste.


Bondad de ajuste del modeloYEn primer lugar olvidemos que existe la variable X. Veamos cul es la variabilidad en el eje Y.La franja sombreada indica la zona donde varan los valores de Y.

Proyeccin sobre el eje Y = olvidar X


Bondad de ajuste del modeloYFijmonos ahora en los errores de prediccin (lneas verticales). Los proyectamos sobre el eje Y.Se observa que los errores de prediccin, residuos, estn menos dispersos que la variable Y original.

Cuanto menos dispersos sean los residuos, mejor ser la bondad del ajuste.


Bondad de ajuste del modeloResumiendo:

La dispersin del error residual ser una fraccin de la dispersin original de Y

Cuanto menor sea la dispersin del error residual mejor ser el ajuste de regresin.

Eso hace que definamos como medida de bondad de un ajuste de regresin, o coeficiente de determinacin a:Y


Bondad de ajuste del modeloLa bondad de un ajuste de un modelo de regresin se mide usando el coeficiente de determinacin R2

R2 es una cantidad adimensional que slo puede tomar valores en [0, 1]

Cuando un ajuste es bueno, R2 ser cercano a uno.

Cuando un ajuste es malo R2 ser cercano a cero.

A R2 tambin se le denomina porcentaje de variabilidad explicado

por el modelo de regresin.

R2 puede ser pesado de calcular en modelos de regresin general,

pero en el modelo lineal simple, la expresin es de lo ms sencilla: R2=r2


Ejemplo con SPSSA continuacin vamos a analizar un ejemplo realizado con datos simulados, de lo que podra parecer el estudio sobre alturas de hijos y padres, realizado con SPSS.

Suponemos que hemos recogido la altura de 60 varones, junto a las de su padre.

El estudio descriptivo univariante de ambas variables por separado no revela nada sobre una posible relacin.


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Ejemplo con SPSSEn el diagrama de dispersin se aprecie una clara relacin lineal directa.La tabla de correlaciones nos muestra que r=0,759El modelo de regresin lineal simple esAltura hijo = b0 + b1 Altura del padreb0=89,985b1=0,466La bondad del ajuste es de R2=0,577= 57,7%


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clase2-2009

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