Modelos Lineares

múltiplas preditoras

Alexandre Adalardo de Oliveira

R IBUSP 2019

Modelos Lineares: múltiplas preditoras

Múltiplas preditoras

preditoras: contínuas e categóricas
interação entre preditoras
matriz do modelo
simplificação do modelo
colinearidade
diagnóstico do modelo

Classes de Modelos Básicos

Modelos	Relação	Resíduos	Observações
Linear	Linear	Normal	Independente
Generalizados	Linearizável	Outras	Independente
Mistos	Linear	Normal	Dependência
Generalizado Mistos	Linearizável	Outras	Dependência

Modelos Lineares

Modelo Linear Simples

\[ y = {\alpha} + {\beta} x + \epsilon\] \[ \epsilon = N(0, \sigma) \]

Modelo Linear Múltiplo

\[ y = {\alpha} + \sum{\beta_i x_i} + \epsilon\] \[ \epsilon = N(0, \sigma) \]

ou

\[ y = \hat{\alpha} + \hat{\beta_1} x_1 + ... + \hat{\beta_n} x_n + \epsilon\] \[ \epsilon = N(0, \sigma) \]

Pink

Modelos lineares: exemplos

Peso ~ altura

library(car)
data(Davis)
str(Davis)

## 'data.frame':    200 obs. of  5 variables:
##  $ sex   : Factor w/ 2 levels "F","M": 2 1 1 2 1 2 2 2 2 2 ...
##  $ weight: int  77 58 53 68 59 76 76 69 71 65 ...
##  $ height: int  182 161 161 177 157 170 167 186 178 171 ...
##  $ repwt : int  77 51 54 70 59 76 77 73 71 64 ...
##  $ repht : int  180 159 158 175 155 165 165 180 175 170 ...

Gráfico dos dados

par(mar=c(4,4,2,2),   cex.lab=1.5, cex.axis=1.2, las=1,
    bg = "gray70", bty="n")
plot(NA, type="n", axes = FALSE, ann = FALSE,
     xlim =range(Davis$height), ylim = range(Davis$weight)) 
rect(par()$usr[1],par()$usr[3], par()$usr[2],par()$usr[4],
     col="gray90")
points(weight ~ height, data = Davis, pch = 19,cex= 1.5,
       col = rgb(0,0,1, 0.4))
axis(1)
axis(2)
mtext("Altura (cm)", 1,  line=2.5, cex = 1.7)
mtext("Peso (kg)", 2, line=2.5, cex = 1.7, las = 0)

Gráfico peso ~ altura

Resumo do lm

lmdavis <- lm(weight~height, data = Davis)
summary(lmdavis)

## 
## Call:
## lm(formula = weight ~ height, data = Davis)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -19.928  -5.406  -0.651   4.891  42.641 
## 
## Coefficients:
##               Estimate Std. Error t value Pr(>|t|)    
## (Intercept) -130.84185   12.30184  -10.64   <2e-16 ***
## height         1.15112    0.07193   16.00   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 8.635 on 178 degrees of freedom
## Multiple R-squared:  0.5899, Adjusted R-squared:  0.5876 
## F-statistic: 256.1 on 1 and 178 DF,  p-value: < 2.2e-16

Predito pelo modelo

newx <- seq(min(Davis$height) -10, max(Davis$height) + 10, length.out=100)
preds <- predict(lmdavis, newdata = data.frame(height=newx), 
                 interval = 'confidence')
kable(rbind(head(preds),tail(preds)))

	fit	lwr	upr
1	28.01250	23.18864	32.83636
2	28.81479	24.08631	33.54328
3	29.61709	24.98382	34.25035
4	30.41938	25.88118	34.95758
5	31.22168	26.77837	35.66498
6	32.02397	27.67539	36.37256
95	103.42820	98.61139	108.24501
96	104.23049	99.31818	109.14281
97	105.03279	100.02484	110.04074
98	105.83508	100.73137	110.93880
99	106.63738	101.43779	111.83697
100	107.43967	102.14409	112.73526

Gráfico: modelo linear

polygon(c(rev(newx), newx), c(rev(preds[ ,3]), preds[ ,2]),
        col = rgb(0,0,0,0.2), border = NA)
abline(lmdavis, lwd = 2.5, col = "black")

Anova do lm

lmdavis <- lm(weight~height, data = Davis)

anova(lmdavis)

## Analysis of Variance Table
## 
## Response: weight
##            Df Sum Sq Mean Sq F value    Pr(>F)    
## height      1  19095 19095.0  256.08 < 2.2e-16 ***
## Residuals 178  13273    74.6                      
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Modelo Linear:

peso ~ altura

ANOVA: comparando modelos

davisNull <- lm(weight ~ 1, data= Davis)
anova(davisNull,lmdavis)

## Analysis of Variance Table
## 
## Model 1: weight ~ 1
## Model 2: weight ~ height
##   Res.Df   RSS Df Sum of Sq      F    Pr(>F)    
## 1    179 32368                                  
## 2    178 13273  1     19095 256.08 < 2.2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

\(p_{valor} = 2.2e{-16} = 2.2 * 10^{-16}\)

\(r^2 = 0.587\)

Anova: comparando modelos

Particionando Variância do Modelo

anova(lmdavis)

## Analysis of Variance Table
## 
## Response: weight
##            Df Sum Sq Mean Sq F value    Pr(>F)    
## height      1  19095 19095.0  256.08 < 2.2e-16 ***
## Residuals 178  13273    74.6                      
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Confrontando com o modelo mínimo

anova(davisNull, lmdavis)

## Analysis of Variance Table
## 
## Model 1: weight ~ 1
## Model 2: weight ~ height
##   Res.Df   RSS Df Sum of Sq      F    Pr(>F)    
## 1    179 32368                                  
## 2    178 13273  1     19095 256.08 < 2.2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Comparando modelos

Pink

Multiplas Preditoras

Múltiplas Preditoras

rbind(head(Davis[, 1:3]), tail(Davis[, 1:3]))

##     sex weight height
## 1     M     77    182
## 2     F     58    161
## 3     F     53    161
## 4     M     68    177
## 5     F     59    157
## 6     M     76    170
## 194   F     51    156
## 195   F     62    164
## 196   M     74    175
## 197   M     83    180
## 199   M     90    181
## 200   M     79    177

Gráfico com sexo

Preditora: contínua + fator

lmdavis01 <- lm(weight~ height + sex, data = Davis)
summary(lmdavis01)

## 
## Call:
## lm(formula = weight ~ height + sex, data = Davis)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -20.302  -4.808  -0.335   5.239  41.366 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept) -80.2107    16.8415  -4.763 3.96e-06 ***
## height        0.8341     0.1021   8.169 5.71e-14 ***
## sexM          7.7070     1.8345   4.201 4.20e-05 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 8.258 on 177 degrees of freedom
## Multiple R-squared:  0.6271, Adjusted R-squared:  0.6229 
## F-statistic: 148.8 on 2 and 177 DF,  p-value: < 2.2e-16

lm(weight ~ height + sex, data = Davis)

coef(lmdavis01)

## (Intercept)      height        sexM 
## -80.2107328   0.8340964   7.7070166

\[ peso = -80.21 + 0.83 * altura + 7.71 * sexo \]

lm(weight ~ height + sex)

\[ peso = -80.21 + 0.83 * altura + 7.71 * sexo \]

lm(weight ~ height + sex, data = Davis)

anova(lmdavis01)

## Analysis of Variance Table
## 
## Response: weight
##            Df  Sum Sq Mean Sq F value    Pr(>F)    
## height      1 19095.0 19095.0  280.04 < 2.2e-16 ***
## sex         1  1203.5  1203.5   17.65 4.204e-05 ***
## Residuals 177 12069.2    68.2                      
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

lm(weight ~ sex + height, data = Davis)

lmdavis01a <- lm(weight~sex + height, data=Davis)
anova(lmdavis01a)

## Analysis of Variance Table
## 
## Response: weight
##            Df  Sum Sq Mean Sq F value    Pr(>F)    
## sex         1 15748.5 15748.5 230.958 < 2.2e-16 ***
## height      1  4550.1  4550.1  66.728 5.713e-14 ***
## Residuals 177 12069.2    68.2                      
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Anova do modelo

anova(lmdavis01a, lmdavis01)

## Analysis of Variance Table
## 
## Model 1: weight ~ sex + height
## Model 2: weight ~ height + sex
##   Res.Df   RSS Df  Sum of Sq F Pr(>F)
## 1    177 12069                       
## 2    177 12069  0 -1.819e-12

Anova: múltiplas preditoras

anova(lmdavis01)

## Analysis of Variance Table
## 
## Response: weight
##            Df  Sum Sq Mean Sq F value    Pr(>F)    
## height      1 19095.0 19095.0  280.04 < 2.2e-16 ***
## sex         1  1203.5  1203.5   17.65 4.204e-05 ***
## Residuals 177 12069.2    68.2                      
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

anova(lmdavis01a)

## Analysis of Variance Table
## 
## Response: weight
##            Df  Sum Sq Mean Sq F value    Pr(>F)    
## sex         1 15748.5 15748.5 230.958 < 2.2e-16 ***
## height      1  4550.1  4550.1  66.728 5.713e-14 ***
## Residuals 177 12069.2    68.2                      
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Anova do Modelo

anova(lmdavis01)

## Analysis of Variance Table
## 
## Response: weight
##            Df  Sum Sq Mean Sq F value    Pr(>F)    
## height      1 19095.0 19095.0  280.04 < 2.2e-16 ***
## sex         1  1203.5  1203.5   17.65 4.204e-05 ***
## Residuals 177 12069.2    68.2                      
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Confrontando com o modelo mínimo

anova(davisNull, lmdavis01)

## Analysis of Variance Table
## 
## Model 1: weight ~ 1
## Model 2: weight ~ height + sex
##   Res.Df   RSS Df Sum of Sq      F    Pr(>F)    
## 1    179 32368                                  
## 2    177 12069  2     20298 148.84 < 2.2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Comparação de Modelos

Estimativas do lm

coeflm01 <- coef(lmdavis01)
coeflm01

## (Intercept)      height        sexM 
## -80.2107328   0.8340964   7.7070166

Feminino (\(sex = 0\))

\[w_f = \hat{\alpha}+ \hat{\beta_s} sex + \hat{\beta_h} *height\] \[w_f = \hat{\alpha} + \hat{\beta_h} * height\]

Estimativas do lm

## (Intercept)      height        sexM 
## -80.2107328   0.8340964   7.7070166

Masculino (\(sex=1\))

\[w_h = \hat{\alpha} + \hat{\beta_s}* sex + \hat{\beta} * height\] \[w_h = \hat{\alpha}+ \hat{\beta_s} + \hat{\beta_h} * height\]

Pink

Interação

lmdavisfull <- lm(weight ~ height + sex + sex:height,
                  data = Davis)

Interação

summary(lmdavisfull)

## 
## Call:
## lm(formula = weight ~ height + sex + sex:height, data = Davis)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -20.990  -4.548  -0.926   4.821  41.023 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept) -45.7988    24.8453  -1.843   0.0670 .  
## height        0.6252     0.1507   4.148 5.22e-05 ***
## sexM        -57.4326    34.8293  -1.649   0.1009    
## height:sexM   0.3815     0.2037   1.873   0.0628 .  
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 8.2 on 176 degrees of freedom
## Multiple R-squared:  0.6344, Adjusted R-squared:  0.6282 
## F-statistic: 101.8 on 3 and 176 DF,  p-value: < 2.2e-16

Anova: comparando modelos

Multiplos testes

anova(lmdavisfull)

## Analysis of Variance Table
## 
## Response: weight
##             Df  Sum Sq Mean Sq  F value    Pr(>F)    
## height       1 19095.0 19095.0 284.0037 < 2.2e-16 ***
## sex          1  1203.5  1203.5  17.8997  3.74e-05 ***
## height:sex   1   235.8   235.8   3.5075   0.06275 .  
## Residuals  176 11833.4    67.2                       
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Anova: comparando modelos

anova(lmdavisfull, davisNull)

## Analysis of Variance Table
## 
## Model 1: weight ~ height + sex + sex:height
## Model 2: weight ~ 1
##   Res.Df   RSS Df Sum of Sq     F    Pr(>F)    
## 1    176 11833                                 
## 2    179 32368 -3    -20534 101.8 < 2.2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

lm(weight ~ height + sex*height)

## (Intercept)      height        sexM height:sexM 
## -45.7988220   0.6252035 -57.4326307   0.3815088

Feminino (\(sex = 0\))

\[w = \hat{\alpha}+ \hat{\beta_s} sex + \hat{\beta_h} height + \hat{\beta}_{s:h} sex* height\] \[w_m = \hat{\alpha} + \hat{\beta_h} height\]

Masculino (\(sex=1\))

\[w = \hat{\alpha} + \hat{\beta_s} sex + \hat{\beta_h} height + \hat{\beta}_{h:s} sex * height \] \[w_h = \hat{\alpha}+ \hat{\beta_s} + (\hat{\beta_h} + \hat{\beta}_{h:s}) * height\]

Predição do modelo

Uma mulher de 161 cm de altura

\[w = \hat{\alpha}+ \hat{\beta_s} sex + \hat{\beta_h} height + \hat{\beta}_{s:h} sex* height\] \[sex =0\]

(coefull <- coef(lmdavisfull))

## (Intercept)      height        sexM height:sexM 
## -45.7988220   0.6252035 -57.4326307   0.3815088

predMulher <- coefull[1] + coefull[2] * 161
(predMulher <- as.numeric(predMulher))

## [1] 54.85893

Uma mulher com 161cm de altura tem peso 54.86 kg .

points(x= 161, y= predMulher, cex= 3)
points(x= 161, y= predMulher, pch = 19, cex= 1.5)
segments(x0 = 161, y=0, y1= predMulher, lty=2, lwd= 2)

Predito do Modelo

Homem com 182cm

\[w = \hat{\alpha}+ \hat{\beta_s} sex + \hat{\beta_h} height + \hat{\beta}_{s:h} sex* height\] \[ sex = 1\]

coefull

## (Intercept)      height        sexM height:sexM 
## -45.7988220   0.6252035 -57.4326307   0.3815088

predHomem <- (coefull[1]+ coefull[3]) + (coefull[2]+ coefull[4]) * 182 
(predHomem <- as.numeric(predHomem))

## [1] 79.99018

Um homem com 182cm de altura tem peso 79.99 kg.

points(x= 182, y= predHomem, cex= 3)
points(x= 182, y= predHomem, pch = 19, cex= 1.5)
segments(x0 = 182, y=0, y1= predHomem, lty=2, lwd= 2)

Modelos Concorrentes

Tipo de Seleção

Teste de hipótese

Modelos aninhados: o mais simples está contido no mais complexo.

ANOVA (Resíduos)

Razão da Variância

Deviance (Generalização):

Distância ao modelo saturado.

\[ D = 2*(LL_1 - LL_0)\]

Outros tipos de Seleção

Teoria da Informação (AIC)

Baseado no cálculo da verossimilhança, proporcional à probabilidade da realização dos dados e penalizado pelo número de parâmetros.

Distância de Kullback-Leibler

Distância ao modelo verdadeiro

\[ AIC = -2LL + 2k \]

Inferência Bayesiana (Teorema Bayes)

Atualização da probabilidade posteriori, baseado em uma probabilidade priori

\[P(H|dados) \sim L(dados| \theta) * P(prior)\]

Princípio da parcimônia (Navalha de Occam)

mínimo número de parâmetros
linear é melhor que não-linear
reter menos pressupostos
simplificado ao mínimo adequado
explicações mais simples são preferíveis

Método do modelo cheio ao mínimo adequado

ajuste o modelo máximo (cheio)
simplifique o modelo:
- inspecione os coeficientes (summary)
- remova termos não significativos
ordem de remoção de termos:
- interação não significativos (maior ordem)
- termos quadráticos ou não lineares
- variáveis explicativas não significativas
- agrupe níveis de fatores sem diferença
- ANCOVA: intercepto não significativo -> 0

Simplificação do modelo:

Critério para a tomada de decisão (Variância)

Compare o modelo anterior com o simplificado

A diferença não é significativa:

* retenha o modelo mais simples
* continue simplificando

A difereça é significativa

* retenha o modelo complexo 
* modelo MINÍMO ADEQUADO

Simplificando Modelo: exemplo

anova(lmdavisfull, lmdavis01)

## Analysis of Variance Table
## 
## Model 1: weight ~ height + sex + sex:height
## Model 2: weight ~ height + sex
##   Res.Df   RSS Df Sum of Sq      F  Pr(>F)  
## 1    176 11833                              
## 2    177 12069 -1   -235.82 3.5075 0.06275 .
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Simplificando Modelo: exemplo

anova(lmdavis01, lmdavis)

## Analysis of Variance Table
## 
## Model 1: weight ~ height + sex
## Model 2: weight ~ height
##   Res.Df   RSS Df Sum of Sq     F    Pr(>F)    
## 1    177 12069                                 
## 2    178 13273 -1   -1203.5 17.65 4.204e-05 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Modelo Mínimo Adequado

summary(lmdavis01)

## 
## Call:
## lm(formula = weight ~ height + sex, data = Davis)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -20.302  -4.808  -0.335   5.239  41.366 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept) -80.2107    16.8415  -4.763 3.96e-06 ***
## height        0.8341     0.1021   8.169 5.71e-14 ***
## sexM          7.7070     1.8345   4.201 4.20e-05 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 8.258 on 177 degrees of freedom
## Multiple R-squared:  0.6271, Adjusted R-squared:  0.6229 
## F-statistic: 148.8 on 2 and 177 DF,  p-value: < 2.2e-16

Anova

Anova do modelo

kable(anova(lmdavisfull), align= "c", digits = 4)

	Df	Sum Sq	Mean Sq	F value	Pr(>F)
height	1	19095.0407	19095.0407	284.0037	0.0000
sex	1	1203.4919	1203.4919	17.8997	0.0000
height:sex	1	235.8241	235.8241	3.5075	0.0628
Residuals	176	11833.3933	67.2352

Anova entre modelos

kable(anova(lmdavis01, lmdavisfull), align= "c", digits = 4)

Res.Df	RSS	Df	Sum of Sq	F	Pr(>F)
177	12069.22
176	11833.39	1	235.8241	3.5075	0.0628

Anova sequencial

kable(anova(davisNull, lmdavis,lmdavis01,lmdavisfull), align= "c", digits = 4)

Res.Df	RSS	Df	Sum of Sq	F	Pr(>F)
179	32367.75
178	13272.71	1	19095.0407	284.0037	0.0000
177	12069.22	1	1203.4919	17.8997	0.0000
176	11833.39	1	235.8241	3.5075	0.0628

Anova do cheio

	Df	Sum Sq	Mean Sq	F value	Pr(>F)
height	1	19095.0407	19095.0407	284.0037	0.0000
sex	1	1203.4919	1203.4919	17.8997	0.0000
height:sex	1	235.8241	235.8241	3.5075	0.0628
Residuals	176	11833.3933	67.2352

Modelo sem interação!

Modelo Mínimo Adequado

coef(lmdavis01)

## (Intercept)      height        sexM 
## -80.2107328   0.8340964   7.7070166

confint(lmdavis01)

##                  2.5 %     97.5 %
## (Intercept) -113.44661 -46.974852
## height         0.63259   1.035603
## sexM           4.08671  11.327323

Diagnóstico do Modelo: plot(modelo)

par(mfrow = c(2,2), mar=c(4,4,2,2), cex.lab=1.2, cex.axis=1.2, las=1, bg = "gray70", bty="n")
plot(lmdavis01)

Diagnóstico: plot(modelo)

Apresenta o Modelo

## 
## Call:
## lm(formula = weight ~ height + sex, data = Davis)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -20.302  -4.808  -0.335   5.239  41.366 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept) -80.2107    16.8415  -4.763 3.96e-06 ***
## height        0.8341     0.1021   8.169 5.71e-14 ***
## sexM          7.7070     1.8345   4.201 4.20e-05 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 8.258 on 177 degrees of freedom
## Multiple R-squared:  0.6271, Adjusted R-squared:  0.6229 
## F-statistic: 148.8 on 2 and 177 DF,  p-value: < 2.2e-16

Apresenta o Modelo

Poluição

Modelo Linear Múltiplo:

quais variáveis incluir
curvatura em resposta a variável preditora
interações entre variáveis
correlação entre variáveis preditoras (colinearidade)
saturação do modelo (complexidade

Poluição: ozônio

Quais variáveis climáticas estão relacionadas à concentração de ozônio?

rad	temp	wind	ozone
190	67	7.4	41
118	72	8.0	36
149	74	12.6	12
313	62	11.5	18
299	65	8.6	23
99	59	13.8	19
19	61	20.1	8
256	69	9.7	16
290	66	9.2	11
274	68	10.9	14

Ozônio data

var	natureza	tipo	descrição
rad	pred	contínua	radiação
temp	pred	contínua	temperatura
wind	pred	contínua	vento
ozone	resposta	contínua	ozônio

Linearidade

Curvatura da relação: polinômios

Correlação entre preditoras

Indíce de colinearidade (confirmar)

VIF: Variance Inflation Factor

Proporcional a variação compartilhada com outras preditoras

\[ VIF = \frac{1}{1-R_k^2} \]

\(R_k^2\) : coeficiente de determinação da preditora (k) em relação a outras preditoras do modelo

\(VIF = 1\) : não há variação compartilhada;
\(VIF= 4\) : 75% de variação explicada ;
\(VIF = 10\) : 90% de variação explicada;

Colinearidade: soluções

reter apenas uma das variáveis colineares
reduzir as dimensões das variáveis colineares (PCA)

Definir os termos do modelo cheio

Modelo para concentração de ozônio:

Modelo Cheio:

temp
wind
rad
temp^2
wind^2
rad^2
temp : wind
temp : rad
wind : rad
temp : wind : rad

Modelo Cheio: Ozônio

lmozfull <- lm(ozone ~ temp + wind + rad + 
                  I(temp^2)+ I(wind^2) + I(rad^2) + 
                 temp:wind + temp:rad + wind:rad + 
                 temp:wind:rad, data = ozo)
lmozfull <- lm(ozone ~  temp * wind * rad + 
                  I(temp^2) + I(wind^2) + I(rad^2), data = ozo)

summary(lmozfull)

## 
## Call:
## lm(formula = ozone ~ temp * wind * rad + I(temp^2) + I(wind^2) + 
##     I(rad^2), data = ozo)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -38.894 -11.205  -2.736   8.809  70.551 
## 
## Coefficients:
##                 Estimate Std. Error t value Pr(>|t|)    
## (Intercept)    5.683e+02  2.073e+02   2.741  0.00725 ** 
## temp          -1.076e+01  4.303e+00  -2.501  0.01401 *  
## wind          -3.237e+01  1.173e+01  -2.760  0.00687 ** 
## rad           -3.117e-01  5.585e-01  -0.558  0.57799    
## I(temp^2)      5.833e-02  2.396e-02   2.435  0.01668 *  
## I(wind^2)      6.106e-01  1.469e-01   4.157 6.81e-05 ***
## I(rad^2)      -3.619e-04  2.573e-04  -1.407  0.16265    
## temp:wind      2.377e-01  1.367e-01   1.739  0.08519 .  
## temp:rad       8.403e-03  7.512e-03   1.119  0.26602    
## wind:rad       2.054e-02  4.892e-02   0.420  0.67552    
## temp:wind:rad -4.324e-04  6.595e-04  -0.656  0.51358    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 17.82 on 100 degrees of freedom
## Multiple R-squared:  0.7394, Adjusted R-squared:  0.7133 
## F-statistic: 28.37 on 10 and 100 DF,  p-value: < 2.2e-16

Simplificando o modelo: Ozônio

lmoz01 <- lm(ozone ~ temp + wind + rad + 
                  I(temp^2)+ I(wind^2) + I(rad^2) + 
                 temp:wind + temp:rad + wind:rad, data = ozo)
anova(lmozfull, lmoz01)

## Analysis of Variance Table
## 
## Model 1: ozone ~ temp * wind * rad + I(temp^2) + I(wind^2) + I(rad^2)
## Model 2: ozone ~ temp + wind + rad + I(temp^2) + I(wind^2) + I(rad^2) + 
##     temp:wind + temp:rad + wind:rad
##   Res.Df   RSS Df Sum of Sq      F Pr(>F)
## 1    100 31742                           
## 2    101 31879 -1   -136.44 0.4298 0.5136

summary(lmoz01)

## 
## Call:
## lm(formula = ozone ~ temp + wind + rad + I(temp^2) + I(wind^2) + 
##     I(rad^2) + temp:wind + temp:rad + wind:rad, data = ozo)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -39.611 -11.455  -2.901   8.548  70.325 
## 
## Coefficients:
##               Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  5.245e+02  1.957e+02   2.680   0.0086 ** 
## temp        -1.021e+01  4.209e+00  -2.427   0.0170 *  
## wind        -2.802e+01  9.645e+00  -2.906   0.0045 ** 
## rad          2.628e-02  2.142e-01   0.123   0.9026    
## I(temp^2)    5.953e-02  2.382e-02   2.499   0.0141 *  
## I(wind^2)    6.173e-01  1.461e-01   4.225 5.25e-05 ***
## I(rad^2)    -3.388e-04  2.541e-04  -1.333   0.1855    
## temp:wind    1.734e-01  9.497e-02   1.825   0.0709 .  
## temp:rad     3.750e-03  2.459e-03   1.525   0.1303    
## wind:rad    -1.127e-02  6.277e-03  -1.795   0.0756 .  
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 17.77 on 101 degrees of freedom
## Multiple R-squared:  0.7383, Adjusted R-squared:  0.715 
## F-statistic: 31.66 on 9 and 101 DF,  p-value: < 2.2e-16

Simplificando o modelo: Ozônio

lmoz02 <- lm(ozone ~ temp + wind + rad + 
                  I(temp^2)+ I(wind^2) + I(rad^2) + 
                 temp:wind +  wind:rad, data = ozo)
anova(lmoz01, lmoz02)

## Analysis of Variance Table
## 
## Model 1: ozone ~ temp + wind + rad + I(temp^2) + I(wind^2) + I(rad^2) + 
##     temp:wind + temp:rad + wind:rad
## Model 2: ozone ~ temp + wind + rad + I(temp^2) + I(wind^2) + I(rad^2) + 
##     temp:wind + wind:rad
##   Res.Df   RSS Df Sum of Sq      F Pr(>F)
## 1    101 31879                           
## 2    102 32613 -1   -734.23 2.3262 0.1303

summary(lmoz02)

## 
## Call:
## lm(formula = ozone ~ temp + wind + rad + I(temp^2) + I(wind^2) + 
##     I(rad^2) + temp:wind + wind:rad, data = ozo)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -42.040 -11.962  -2.863   9.661  70.475 
## 
## Coefficients:
##               Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  5.488e+02  1.963e+02   2.796  0.00619 ** 
## temp        -1.144e+01  4.158e+00  -2.752  0.00702 ** 
## wind        -2.876e+01  9.695e+00  -2.967  0.00375 ** 
## rad          3.061e-01  1.113e-01   2.751  0.00704 ** 
## I(temp^2)    7.145e-02  2.265e-02   3.154  0.00211 ** 
## I(wind^2)    6.363e-01  1.465e-01   4.343 3.33e-05 ***
## I(rad^2)    -2.690e-04  2.516e-04  -1.069  0.28755    
## temp:wind    1.840e-01  9.533e-02   1.930  0.05644 .  
## wind:rad    -1.381e-02  6.090e-03  -2.268  0.02541 *  
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 17.88 on 102 degrees of freedom
## Multiple R-squared:  0.7322, Adjusted R-squared:  0.7112 
## F-statistic: 34.87 on 8 and 102 DF,  p-value: < 2.2e-16

Simplificando o modelo: Ozônio

lmoz03 <- lm(ozone ~ temp + wind + rad + 
                  I(temp^2)+ I(wind^2) + 
                 temp:wind +  wind:rad, data = ozo)
anova( lmoz02, lmoz03)

## Analysis of Variance Table
## 
## Model 1: ozone ~ temp + wind + rad + I(temp^2) + I(wind^2) + I(rad^2) + 
##     temp:wind + wind:rad
## Model 2: ozone ~ temp + wind + rad + I(temp^2) + I(wind^2) + temp:wind + 
##     wind:rad
##   Res.Df   RSS Df Sum of Sq     F Pr(>F)
## 1    102 32613                          
## 2    103 32978 -1   -365.45 1.143 0.2875

summary(lmoz03)

## 
## Call:
## lm(formula = ozone ~ temp + wind + rad + I(temp^2) + I(wind^2) + 
##     temp:wind + wind:rad, data = ozo)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -41.379 -11.375  -2.217   8.921  71.247 
## 
## Coefficients:
##               Estimate Std. Error t value Pr(>|t|)    
## (Intercept) 514.401470 193.783580   2.655  0.00920 ** 
## temp        -10.654041   4.094889  -2.602  0.01064 *  
## wind        -27.391965   9.616998  -2.848  0.00531 ** 
## rad           0.212945   0.069283   3.074  0.00271 ** 
## I(temp^2)     0.067805   0.022408   3.026  0.00313 ** 
## I(wind^2)     0.619396   0.145773   4.249 4.72e-05 ***
## temp:wind     0.169674   0.094458   1.796  0.07538 .  
## wind:rad     -0.013561   0.006089  -2.227  0.02813 *  
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 17.89 on 103 degrees of freedom
## Multiple R-squared:  0.7292, Adjusted R-squared:  0.7108 
## F-statistic: 39.63 on 7 and 103 DF,  p-value: < 2.2e-16

Simplificando o modelo: Ozônio

lmoz04 <- lm(ozone ~ temp + wind + rad + 
                  I(temp^2)+ I(wind^2) + 
                  wind:rad, data = ozo)
anova( lmoz03, lmoz04)

## Analysis of Variance Table
## 
## Model 1: ozone ~ temp + wind + rad + I(temp^2) + I(wind^2) + temp:wind + 
##     wind:rad
## Model 2: ozone ~ temp + wind + rad + I(temp^2) + I(wind^2) + wind:rad
##   Res.Df   RSS Df Sum of Sq      F  Pr(>F)  
## 1    103 32978                              
## 2    104 34011 -1   -1033.1 3.2267 0.07538 .
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

summary(lmoz04)

## 
## Call:
## lm(formula = ozone ~ temp + wind + rad + I(temp^2) + I(wind^2) + 
##     wind:rad, data = ozo)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -44.478 -10.735  -2.437   9.685  77.543 
## 
## Coefficients:
##               Estimate Std. Error t value Pr(>|t|)    
## (Intercept) 223.573855 107.618223   2.077 0.040221 *  
## temp         -5.197139   2.775039  -1.873 0.063902 .  
## wind        -10.816032   2.736757  -3.952 0.000141 ***
## rad           0.173431   0.066398   2.612 0.010333 *  
## I(temp^2)     0.043640   0.018112   2.410 0.017731 *  
## I(wind^2)     0.430059   0.101767   4.226 5.12e-05 ***
## wind:rad     -0.009819   0.005783  -1.698 0.092507 .  
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 18.08 on 104 degrees of freedom
## Multiple R-squared:  0.7208, Adjusted R-squared:  0.7047 
## F-statistic: 44.74 on 6 and 104 DF,  p-value: < 2.2e-16

Simplificando o modelo: Ozônio

lmoz05 <- lm(ozone ~ temp + wind + rad + 
                  I(temp^2)+ I(wind^2), 
                  data = ozo)
anova( lmoz04, lmoz05)

## Analysis of Variance Table
## 
## Model 1: ozone ~ temp + wind + rad + I(temp^2) + I(wind^2) + wind:rad
## Model 2: ozone ~ temp + wind + rad + I(temp^2) + I(wind^2)
##   Res.Df   RSS Df Sum of Sq     F  Pr(>F)  
## 1    104 34011                             
## 2    105 34954 -1   -942.85 2.883 0.09251 .
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Simplificando o Modelo

summary(lmoz05)

## 
## Call:
## lm(formula = ozone ~ temp + wind + rad + I(temp^2) + I(wind^2), 
##     data = ozo)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -48.044 -10.796  -4.138   8.131  80.098 
## 
## Coefficients:
##              Estimate Std. Error t value Pr(>|t|)    
## (Intercept) 291.16758  100.87723   2.886  0.00473 ** 
## temp         -6.33955    2.71627  -2.334  0.02150 *  
## wind        -13.39674    2.29623  -5.834 6.05e-08 ***
## rad           0.06586    0.02005   3.285  0.00139 ** 
## I(temp^2)     0.05102    0.01774   2.876  0.00488 ** 
## I(wind^2)     0.46464    0.10060   4.619 1.10e-05 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 18.25 on 105 degrees of freedom
## Multiple R-squared:  0.713,  Adjusted R-squared:  0.6994 
## F-statistic: 52.18 on 5 and 105 DF,  p-value: < 2.2e-16

Simplificando o modelo: Ozônio

lmoz06 <- lm(ozone ~ temp + wind + rad + 
                  I(wind^2) 
                  , data = ozo)
anova(lmoz05, lmoz06)

## Analysis of Variance Table
## 
## Model 1: ozone ~ temp + wind + rad + I(temp^2) + I(wind^2)
## Model 2: ozone ~ temp + wind + rad + I(wind^2)
##   Res.Df   RSS Df Sum of Sq      F   Pr(>F)   
## 1    105 34954                                
## 2    106 37708 -1   -2753.7 8.2718 0.004877 **
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Modelo Mínimo Adequado

summary(lmoz05)

## 
## Call:
## lm(formula = ozone ~ temp + wind + rad + I(temp^2) + I(wind^2), 
##     data = ozo)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -48.044 -10.796  -4.138   8.131  80.098 
## 
## Coefficients:
##              Estimate Std. Error t value Pr(>|t|)    
## (Intercept) 291.16758  100.87723   2.886  0.00473 ** 
## temp         -6.33955    2.71627  -2.334  0.02150 *  
## wind        -13.39674    2.29623  -5.834 6.05e-08 ***
## rad           0.06586    0.02005   3.285  0.00139 ** 
## I(temp^2)     0.05102    0.01774   2.876  0.00488 ** 
## I(wind^2)     0.46464    0.10060   4.619 1.10e-05 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 18.25 on 105 degrees of freedom
## Multiple R-squared:  0.713,  Adjusted R-squared:  0.6994 
## F-statistic: 52.18 on 5 and 105 DF,  p-value: < 2.2e-16

DIAGNÓSTICO DO MODELO

Transformando variável

 lmoz07 <- lm(log(ozone) ~ temp + wind + rad + 
    I(temp^2)+ I(wind^2) + I(rad^2) + temp:wind + 
      temp:rad + wind:rad + temp:wind:rad, data = ozo)
summary(lmoz07)

## 
## Call:
## lm(formula = log(ozone) ~ temp + wind + rad + I(temp^2) + I(wind^2) + 
##     I(rad^2) + temp:wind + temp:rad + wind:rad + temp:wind:rad, 
##     data = ozo)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -1.91943 -0.24169 -0.01742  0.28213  1.11802 
## 
## Coefficients:
##                 Estimate Std. Error t value Pr(>|t|)  
## (Intercept)    2.803e+00  5.676e+00   0.494   0.6225  
## temp          -3.018e-02  1.178e-01  -0.256   0.7983  
## wind          -9.812e-02  3.211e-01  -0.306   0.7605  
## rad            2.771e-02  1.529e-02   1.812   0.0729 .
## I(temp^2)      6.034e-04  6.559e-04   0.920   0.3598  
## I(wind^2)      8.732e-03  4.021e-03   2.172   0.0322 *
## I(rad^2)      -1.489e-05  7.043e-06  -2.114   0.0370 *
## temp:wind     -1.985e-03  3.742e-03  -0.530   0.5971  
## temp:rad      -2.507e-04  2.056e-04  -1.219   0.2256  
## wind:rad      -2.001e-03  1.339e-03  -1.494   0.1382  
## temp:wind:rad  2.535e-05  1.805e-05   1.404   0.1634  
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.4877 on 100 degrees of freedom
## Multiple R-squared:  0.7116, Adjusted R-squared:  0.6827 
## F-statistic: 24.67 on 10 and 100 DF,  p-value: < 2.2e-16

Simplificando o modelo

 lmoz08 <- lm(log(ozone) ~ temp + wind + rad + 
                  I(temp^2)+ I(wind^2) + I(rad^2) + 
                 temp:wind + temp:rad + wind:rad, data = ozo)
anova(lmoz07, lmoz08)

## Analysis of Variance Table
## 
## Model 1: log(ozone) ~ temp + wind + rad + I(temp^2) + I(wind^2) + I(rad^2) + 
##     temp:wind + temp:rad + wind:rad + temp:wind:rad
## Model 2: log(ozone) ~ temp + wind + rad + I(temp^2) + I(wind^2) + I(rad^2) + 
##     temp:wind + temp:rad + wind:rad
##   Res.Df    RSS Df Sum of Sq      F Pr(>F)
## 1    100 23.787                           
## 2    101 24.256 -1  -0.46883 1.9709 0.1634

Simplificando o modelo

 lmoz09 <- lm(log(ozone) ~ temp + wind + rad + 
                  I(temp^2)+ I(wind^2) + I(rad^2) + 
                 temp:wind +  wind:rad, data = ozo)
anova(lmoz08, lmoz09)

## Analysis of Variance Table
## 
## Model 1: log(ozone) ~ temp + wind + rad + I(temp^2) + I(wind^2) + I(rad^2) + 
##     temp:wind + temp:rad + wind:rad
## Model 2: log(ozone) ~ temp + wind + rad + I(temp^2) + I(wind^2) + I(rad^2) + 
##     temp:wind + wind:rad
##   Res.Df    RSS Df Sum of Sq      F Pr(>F)
## 1    101 24.256                           
## 2    102 24.281 -1  -0.02515 0.1047 0.7469

Simplificando o modelo

 lmoz10 <- lm(log(ozone) ~ temp + wind + rad + 
                  I(temp^2)+ I(wind^2) + I(rad^2) + 
                  wind:rad, data = ozo)
anova(lmoz09, lmoz10)

## Analysis of Variance Table
## 
## Model 1: log(ozone) ~ temp + wind + rad + I(temp^2) + I(wind^2) + I(rad^2) + 
##     temp:wind + wind:rad
## Model 2: log(ozone) ~ temp + wind + rad + I(temp^2) + I(wind^2) + I(rad^2) + 
##     wind:rad
##   Res.Df    RSS Df Sum of Sq      F Pr(>F)
## 1    102 24.281                           
## 2    103 24.401 -1  -0.11987 0.5035 0.4796

Simplificando o modelo

 lmoz11 <- lm(log(ozone) ~ temp + wind + rad + 
          I(temp^2)+ I(wind^2) + I(rad^2), data = ozo)
anova(lmoz10, lmoz11)

## Analysis of Variance Table
## 
## Model 1: log(ozone) ~ temp + wind + rad + I(temp^2) + I(wind^2) + I(rad^2) + 
##     wind:rad
## Model 2: log(ozone) ~ temp + wind + rad + I(temp^2) + I(wind^2) + I(rad^2)
##   Res.Df    RSS Df Sum of Sq    F Pr(>F)
## 1    103 24.401                         
## 2    104 24.522 -1  -0.12081 0.51 0.4768

Simplificando o modelo

 lmoz12 <- lm(log(ozone) ~ temp + wind + rad + 
          I(wind^2) + I(rad^2), data = ozo)
anova(lmoz11, lmoz12)

## Analysis of Variance Table
## 
## Model 1: log(ozone) ~ temp + wind + rad + I(temp^2) + I(wind^2) + I(rad^2)
## Model 2: log(ozone) ~ temp + wind + rad + I(wind^2) + I(rad^2)
##   Res.Df    RSS Df Sum of Sq      F Pr(>F)
## 1    104 24.522                           
## 2    105 24.707 -1  -0.18512 0.7851 0.3776

Simplificando o modelo

summary( lmoz12)

## 
## Call:
## lm(formula = log(ozone) ~ temp + wind + rad + I(wind^2) + I(rad^2), 
##     data = ozo)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -1.85551 -0.25578  0.00248  0.31349  1.16251 
## 
## Coefficients:
##               Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  7.724e-01  6.350e-01   1.216 0.226543    
## temp         4.193e-02  6.237e-03   6.723 9.52e-10 ***
## wind        -2.211e-01  5.874e-02  -3.765 0.000275 ***
## rad          7.466e-03  2.323e-03   3.215 0.001736 ** 
## I(wind^2)    7.390e-03  2.585e-03   2.859 0.005126 ** 
## I(rad^2)    -1.470e-05  6.734e-06  -2.183 0.031246 *  
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.4851 on 105 degrees of freedom
## Multiple R-squared:  0.7004, Adjusted R-squared:  0.6861 
## F-statistic:  49.1 on 5 and 105 DF,  p-value: < 2.2e-16

Simplificando o modelo

 lmoz13 <- lm(log(ozone) ~ temp + wind + rad + 
          I(wind^2), data = ozo)
anova(lmoz12, lmoz13)

## Analysis of Variance Table
## 
## Model 1: log(ozone) ~ temp + wind + rad + I(wind^2) + I(rad^2)
## Model 2: log(ozone) ~ temp + wind + rad + I(wind^2)
##   Res.Df    RSS Df Sum of Sq      F  Pr(>F)  
## 1    105 24.707                              
## 2    106 25.828 -1   -1.1216 4.7665 0.03125 *
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Diagnóstico do Modelo

Modelo Mínimo Adequado

summary(lmoz12)

## 
## Call:
## lm(formula = log(ozone) ~ temp + wind + rad + I(wind^2) + I(rad^2), 
##     data = ozo)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -1.85551 -0.25578  0.00248  0.31349  1.16251 
## 
## Coefficients:
##               Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  7.724e-01  6.350e-01   1.216 0.226543    
## temp         4.193e-02  6.237e-03   6.723 9.52e-10 ***
## wind        -2.211e-01  5.874e-02  -3.765 0.000275 ***
## rad          7.466e-03  2.323e-03   3.215 0.001736 ** 
## I(wind^2)    7.390e-03  2.585e-03   2.859 0.005126 ** 
## I(rad^2)    -1.470e-05  6.734e-06  -2.183 0.031246 *  
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.4851 on 105 degrees of freedom
## Multiple R-squared:  0.7004, Adjusted R-squared:  0.6861 
## F-statistic:  49.1 on 5 and 105 DF,  p-value: < 2.2e-16

IMPORTÂNCIA DAS VARIÁVEIS

Escalas diferentes

lmoz12a <- lm(log(ozone) ~ I(temp/100) + wind + rad + 
          I((wind/100)^2) + I(rad^2), data = ozo)
anova(lmoz12, lmoz12a)

## Analysis of Variance Table
## 
## Model 1: log(ozone) ~ temp + wind + rad + I(wind^2) + I(rad^2)
## Model 2: log(ozone) ~ I(temp/100) + wind + rad + I((wind/100)^2) + I(rad^2)
##   Res.Df    RSS Df  Sum of Sq F Pr(>F)
## 1    105 24.707                       
## 2    105 24.707  0 1.0658e-14

IMPORTÂNCIA DAS VARIÁVEIS

Escalas diferentes: problema

coef(lmoz12)

##   (Intercept)          temp          wind           rad     I(wind^2) 
##  7.723892e-01  4.193355e-02 -2.211428e-01  7.465764e-03  7.390204e-03 
##      I(rad^2) 
## -1.470231e-05

coef(lmoz12a)

##     (Intercept)     I(temp/100)            wind             rad 
##    7.723892e-01    4.193355e+00   -2.211428e-01    7.465764e-03 
## I((wind/100)^2)        I(rad^2) 
##    7.390204e+01   -1.470231e-05

Rescalonando os coeficientes:

tempR <- scale(ozo$temp)
windR <- scale(ozo$wind)
radR <- scale (ozo$rad)
## modelo rescalonado
lmoz12R <- lm(log(ozone) ~ tempR + windR + radR + I(radR^2)+ I(windR^2), data = ozo)

Modelo rescalonado

## 
## Call:
## lm(formula = log(ozone) ~ tempR + windR + radR + I(radR^2) + 
##     I(windR^2), data = ozo)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -1.85551 -0.25578  0.00248  0.31349  1.16251 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  3.44421    0.07836  43.951  < 2e-16 ***
## tempR        0.39963    0.05944   6.723 9.52e-10 ***
## windR       -0.26425    0.05688  -4.646 9.86e-06 ***
## radR         0.18520    0.05268   3.516 0.000649 ***
## I(radR^2)   -0.12216    0.05595  -2.183 0.031246 *  
## I(windR^2)   0.09362    0.03274   2.859 0.005126 ** 
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.4851 on 105 degrees of freedom
## Multiple R-squared:  0.7004, Adjusted R-squared:  0.6861 
## F-statistic:  49.1 on 5 and 105 DF,  p-value: < 2.2e-16

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