Rosa M. Crujeiras Casais
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1 Rosa M. Crujeiras Casais Dpto. de Estatística e Investigación Operativa Universidade de Santiago de Compostela rosa.crujeiras@usc.es
2 ANOVA Análise da Varianza
3 ANOVA Análise da Varianza 1. Introdución 2. Xustificación e contexto 3. ANOVA Motivación e estatístico de contraste Táboa ANOVA Contraste das hipóteses do modelo Comparacións a posteriori 4. Conclusións 5. Algunhas referencias 3
4 4 1. Introdución
5 1. Introdución Para que serve un ANOVA? O ANOVA (ANalysis Of VAriance) é a técnica máis usada para comparar medias en distintos grupos de medidas (xeralmente, tres ou máis). 5
6 1. Introdución Un exemplo (I): xen receptor de estróxeno e cancro de mama Parl et al. (1989) estudaron a relación entre o xenotipo do xen receptor de estróxeno e a idade ao diagnóstico en 59 pacientes de cancro de mama. Xen receptor de estróxeno dous alelos diferentes, que dan lugar a tres xenotipos. 6
7 1. Introdución 7 Figura 1. Diagramas de caixa da idade, para os tres xenotipos.
8 1. Introdución Un exemplo (II): xen receptor de estróxeno e cancro de mama Xenotipo 1 Xenotipo 2 Xenotipo 3 N Media IC (95%) (58.19,71.09) (59.34,69.42) (44.70,56.04) Táboa 1. Resumo dos datos. Número de pacientes en cada grupo (N); media da idade en anos (Media) e intervalo de confianza (IC) ao 95% para a idade media. 8
9 1. Introdución Máis formalmente Mediante o ANOVA, pódese contrastar a hipótese de que as medias en todos os grupos son iguais, pero baixo certas condicións! Pero se rexeitamos a igualdade das medias, será interesante detectar en que grupos se producen as diferenzas, ou ver se hai grupos homoxéneos. 9
10 10 2. Xustificación e contexto
11 2. Xustificación e contexto Que teño e que quero facer: E imos contrastar se as medias son iguais para todos os grupos: 11
12 2. Xustificación e contexto No exemplo E imos contrastar se as medias das idades son as mesmas para os tres xenotipos: 12
13 2. Xustificación e contexto T-Test para comparar dous grupos No contexto de dúas poboacións normais, partindo de mostras independentes, o T-test permite contrastar se as medias son iguais: O estatístico de contraste é e segue unha distribución T-Student. 13
14 2. Xustificación e contexto E por que non facer T-Test repetidos? Exemplo A realización de tests múltiples (pairwise T-test) conleva un incremento do FWER (Familywise Error Rate). É dicir, se a hipótese de partida é certa en todos os casos, incrementáse a probabilidade de rexeitar a hipótese nula nalgún dos tests. Supoñamos que temos 3 grupos coas medias iguais. Se queremos contrastar por pares a igualdade das medias e fixamos un nivel de significación do 5%, a probabilidade de detectar diferenzas entre algún par de medias (rexeitar a hipótese de que son iguais) é de 1-(1-0.05) 3 =0.143=14.3%. 14
15 2. Xustificación e contexto E por que miramos varianzas se estamos interesados nas medias? Figura 2. Densidades coas mesmas medias pero distintas varianzas. 15
16 2. Xustificación e contexto No exemplo Figura 3. Densidades estimadas para a idade nos tres xenotipos. 16
17 17 3. ANOVA
18 3. ANOVA Motivación e estatístico de contraste Motivación Variabilidade intra grupos Variabilidade entre grupos Se a variabilidade dentro dos grupos é grande comparada coa variabilidade entre os grupos, será difícil detectar as diferenzas entre eles. 18
19 3. ANOVA Motivación e estatístico de contraste Motivación Variabilidade intra grupos Variabilidade entre grupos 19
20 3. ANOVA Motivación e estatístico de contraste Fontes de variabilidade (entre/intra) Variabilidade entre (between) grupos Variabilidade intra (within) grupos Variabilidade total 20
21 3. ANOVA Motivación e estatístico de contraste Estatístico de contraste O estatístico de contraste é un cociente entre dous estimadores da varianza: Distribución F de Snedecor. Graos de liberdade (gl). Equivalencia en dúas mostras co T-test. 21
22 3. ANOVA Motivación e estatístico de contraste Significación (p-valor) O p-valor proporciona a probabilidade de obter un valor do estatístico de contraste máis alto do que temos coa mostra observada. Valor do estatístico Área = p-valor 22
23 3. ANOVA A táboa ANOVA A táboa ANOVA 23
24 3. ANOVA A táboa ANOVA A táboa ANOVA no exemplo Variabilidade Suma de cadrados Graos de liberdade Promedio de cadrados F-test P-valor Entre grupos Intra grupos Total Rexeitamos a hipótese de que as medias son iguais 24
25 3. ANOVA A táboa ANOVA No exemplo: táboa ANOVA en R > anova(lm(age~genotype)) #>summary(aov(age~genotype)) Analysis of Variance Table Response: age Df Sum Sq Mean Sq F value Pr(>F) genotype *** Residuals Signif. codes: 0 *** ** 0.01 *
26 3. ANOVA A táboa ANOVA No exemplo: táboa ANOVA en SPSS 26
27 3. ANOVA A táboa ANOVA No exemplo: táboa ANOVA en SPSS 27
28 3. ANOVA A táboa ANOVA No exemplo: táboa ANOVA en SPSS idade ANOVA Suma de cuadrados gl Media cuadrática F Sig. Inter-grupos 2315, ,867 7,863,001 Intra-grupos 8245, ,246 Total 10561,
29 3. ANOVA Contraste das hipóteses Condicións para aplicar un ANOVA Independencia (entre/intra) O investigador debe garantizala mediante unha toma de datos adecuada. Normalidade Pode contrastarse a través dalgún test como o de Kolmogorov-Smirnov ou o de Shapiro-Wilk. Homocedasticidade Pode contrastarse utilizando o test de Levene para a igualdade de varianzas. 29
30 3. ANOVA Contraste das hipóteses No exemplo: test KS para os grupos, en SPSS 30
31 3. ANOVA Contraste das hipóteses No exemplo: test de SW, para os grupos, en R > shapiro.test(age[genotype==1]) Shapiro-Wilk normality test data: age[genotype == 1] W = , p-value = > shapiro.test(age[genotype==2]) Shapiro-Wilk normality test data: age[genotype == 2] W = , p-value = > shapiro.test(age[genotype==3]) Shapiro-Wilk normality test data: age[genotype == 3] W = , p-value =
32 3. ANOVA Contraste das hipóteses No exemplo: test de Levene, en SPSS idade Prueba de homogeneidad de varianzas Estadístico de Levene gl1 gl2 Sig., ,441 32
33 3. ANOVA Comparacións a posteriori E se o ANOVA resulta significativo? Se a hipótese se rexeita, só podemos dicir que polo menos hai un par de medias diferentes Pero entre que grupos hai diferenzas? Podemos establecer grupos homoxéneos de medias? Comparacións múltiples, pero con correccións! Bonferroni Tukey (HSD) Scheffé 33
34 3. ANOVA Comparacións a posteriori No exemplo: Tukey HSD en SPSS Variable dependiente:age Comparaciones múltiples Diferencia de Intervalo de confianza al 95% (I) genotype (J) genotype medias (I-J) Error típico Sig. Límite inferior Límite superior HSD de Tukey 1 2,264 3,949,998-9,24 9, ,268 * 4,441,006 3,58 24, ,264 3,949,998-9,77 9, ,004 * 3,779,001 4,91 23, ,268 * 4,441,006-24,96-3, ,004 * 3,779,001-23,10-4,91 Scheffé 1 2,264 3,949,998-9,67 10, ,268 * 4,441,009 3,10 25, ,264 3,949,998-10,19 9, ,004 * 3,779,002 4,50 23, ,268 * 4,441,009-25,44-3, ,004 * 3,779,002-23,51-4,50 Bonferroni 1 2,264 3,949 1,000-9,48 10, ,268 * 4,441,007 3,31 25, ,264 3,949 1,000-10,01 9,48
35 3. ANOVA Comparacións a posteriori No exemplo: Tukey HSD en R > TukeyHSD(aov(age~genotype)) Tukey multiple comparisons of means 95% family-wise confidence level Fit: aov(formula = age ~ genotype) $genotype diff lwr upr p adj
36 3. ANOVA Comparacións a posteriori No exemplo: subconxuntos homoxéneos en SPSS age Subconjunto para alfa = 0.05 genotype N 1 2 HSD de Tukey a,b , , ,64 Sig. 1,000,998 Scheffé a,b , , ,64 Sig. 1,000, Se muestran las medias para los grupos en los subconjuntos homogéneos. a. Usa el tamaño muestral de la media armónica = 17,814. b. Los tamaños de los grupos no son iguales. Se utilizará la media armónica de
37 37 4. CONCLUSIÓNS
38 4. CONCLUSIÓNS Unha píldora de ANOVA! Temos varios grupos de observacións. (*) Queremos comparar os grupos en base ás medias (hipótese: as medias son iguais ). E se son diferentes comparar por pares, detectar grupos homoxéneos. (**) Coidadiño con As condicións do modelo (*) Os contrastes múltiples (**) 38
39 39 5. Algunhas referencias
40 5. Algunhas referencias Referencias WD Dupont (2009) Statistical Modeling for Biomedical Researchers. Cambridge Medicine. P Armitage, G Berry, JNS Matthews (2008) Statistical Methods in Medical Research. Blackwell Science. 40
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