Conceitos

Conceitos

• preditoras contínuas e categóricas
• interação entre preditoras
• matriz do modelo (algebra linear)
• simplificação do modelo
• colinearidade

Modelo Linear Simples

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

Modelo Linear Múltiplo

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

Retomando o Modelo Linear

Davis 1990

Davis (1990)

peso ~ weight

Modelo Linear

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

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

Partição da Variância

lm(peso ~ altura)

sum((Davis$weight - mean(Davis$weight))^2)

## [1] 32367.75

sum((Davis$weight - predict(lmdavis))^2)

## [1] 13272.71

lm(peso ~ altura)

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

Modelo Linear:

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

Resumo do Modelo

sexo: dummy (mulher = 0, homem = 1)

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

Interpretando o modelo

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

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

weight ~ height + sex

Interação: height:sex

weight ~ height + sex + height:sex

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

## 
## 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

lm(weight ~ height + sex + sex:height)

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

Predição do modelo

Uma mulher de 161 cm de altura

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

\[sex = 0\]

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

\[w = \hat{\alpha} + \hat{\beta_h} height \]

 -45.8 + 0.625 * 161  

## [1] 54.85893

Predito pelo modelo

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

Predito do Modelo

Homem com 182 cm

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

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

\[w = \hat{\alpha}+ \hat{\beta}_s + \hat{\beta}_h * height + \hat{\beta}_{s:h} * height\]

\[w = \hat{\alpha} + \hat{\beta}_s + (\hat{\beta}_h + \hat{\beta}_{s:h}) * height \]

-45.8 -57.4 + 0.625 * 182 + 0.381 * 182  

## [1] 79.892

Predito pelo modelo

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

Matrix do Modelo

Primeiros registros nos dados

##   sex weight height
## 1   M     77    182
## 2   F     58    161

Matrix do Modelo (linhas 1 e 2)

##   (Intercept) height sexM height:sexM
## 1           1    182    1         182
## 2           1    161    0           0

Coeficientes do Modelo

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

Multiplicação Matricial

##       [,1]
## 1 79.99018
## 2 54.85893

Qual o melhor modelo?

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

1. ajuste o modelo máximo (cheio)
2. simplifique o modelo:
   • inspecione os coeficientes (summary)
   • remova termos não significativos
3. 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

Modelo cheio

## 
## 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

Modelo Mínimo Adequado

Diagnóstico do Modelo:

Atividade

Modelo Linear Múltiplo:

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

Simulando dados

y ~ x + z + w …

Quais variáveis estão relacionadas à resposta?

Análise Exploratória

Curvatura da relação: polinômios

Correlação entre preditoras

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;

VIF

vif(lm(y ~ x + z + w ))

##         x         z         w 
## 12.155388 12.183386  1.008662

vif(lm(y ~ x + w))

##       x       w 
## 1.00163 1.00163

vif(lm(y ~ z + w))

##        z        w 
## 1.003937 1.003937

Colinearidade: soluções

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

Definir os termos do modelo cheio

• \(x\)
• \(y\)
• \(w\)
• \(x^2\)
• \(z^2\)
• \(x:z\)
• \(x:w\)
• \(z:w\)
• \(x:z:w\)

Modelo Cheio

lmfull <- lm(y ~ x + w + z +
                       I(x^2) + I(z^2) +
                       x:w + x:z + z:w + z:w:x, data = yxzw)

summary(lmfull)

## 
## Call:
## lm(formula = y ~ x + w + z + I(x^2) + I(z^2) + x:w + x:z + z:w + 
##     z:w:x, data = yxzw)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -16.8495  -5.7956  -0.3322   4.2633  29.6627 
## 
## Coefficients:
##              Estimate Std. Error t value Pr(>|t|)    
## (Intercept) 21.193915  11.442878   1.852   0.0714 .  
## x           -2.572884   0.536259  -4.798 2.25e-05 ***
## w           -4.332059   1.963077  -2.207   0.0331 *  
## z            5.068630   5.895282   0.860   0.3950    
## I(x^2)       0.518748   0.012522  41.426  < 2e-16 ***
## I(z^2)       0.516183   1.069157   0.483   0.6319    
## x:w         -3.022575   0.070387 -42.942  < 2e-16 ***
## x:z         -0.199390   0.224934  -0.886   0.3807    
## w:z          0.279907   0.726247   0.385   0.7020    
## x:w:z       -0.000639   0.006779  -0.094   0.9254    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 9.262 on 40 degrees of freedom
## Multiple R-squared:      1,  Adjusted R-squared:      1 
## F-statistic: 1.243e+05 on 9 and 40 DF,  p-value: < 2.2e-16

Simplificando o modelo

lm01 <-  lm(y ~ x + w + z +
                       I( x^2) + I(z^2) +
                       x:w + x:z + z:w, data = yxzw)

anova(lmfull, lm01)

## Analysis of Variance Table
## 
## Model 1: y ~ x + w + z + I(x^2) + I(z^2) + x:w + x:z + z:w + z:w:x
## Model 2: y ~ x + w + z + I(x^2) + I(z^2) + x:w + x:z + z:w
##   Res.Df    RSS Df Sum of Sq      F Pr(>F)
## 1     40 3431.1                           
## 2     41 3431.9 -1  -0.76203 0.0089 0.9254

summary(lm01)

## 
## Call:
## lm(formula = y ~ x + w + z + I(x^2) + I(z^2) + x:w + x:z + z:w, 
##     data = yxzw)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -16.8821  -5.8719  -0.4094   4.2821  29.5060 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept) 20.39839    7.63268   2.673  0.01075 *  
## x           -2.56245    0.51833  -4.944 1.34e-05 ***
## w           -4.19093    1.25405  -3.342  0.00178 ** 
## z            5.34953    5.02464   1.065  0.29326    
## I(x^2)       0.51873    0.01237  41.938  < 2e-16 ***
## I(z^2)       0.51146    1.05499   0.485  0.63040    
## x:w         -3.02560    0.06186 -48.911  < 2e-16 ***
## x:z         -0.20205    0.22044  -0.917  0.36471    
## w:z          0.23783    0.56583   0.420  0.67645    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 9.149 on 41 degrees of freedom
## Multiple R-squared:      1,  Adjusted R-squared:      1 
## F-statistic: 1.433e+05 on 8 and 41 DF,  p-value: < 2.2e-16

Simplificando o modelo

lm02 <- lm(y ~ x + w + z +
                      I( x^2) + I(z^2) +
                      x:w + z:w, data = yxzw)
anova(lm01, lm02)

## Analysis of Variance Table
## 
## Model 1: y ~ x + w + z + I(x^2) + I(z^2) + x:w + x:z + z:w
## Model 2: y ~ x + w + z + I(x^2) + I(z^2) + x:w + z:w
##   Res.Df    RSS Df Sum of Sq      F Pr(>F)
## 1     41 3431.9                           
## 2     42 3502.2 -1   -70.327 0.8402 0.3647

summary(lm02)

## 
## Call:
## lm(formula = y ~ x + w + z + I(x^2) + I(z^2) + x:w + z:w, data = yxzw)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -14.4363  -6.1137  -0.4808   4.3176  29.9606 
## 
## Coefficients:
##              Estimate Std. Error t value Pr(>|t|)    
## (Intercept) 20.818074   7.604429   2.738  0.00904 ** 
## x           -2.324930   0.448053  -5.189 5.75e-06 ***
## w           -4.240332   1.250502  -3.391  0.00153 ** 
## z            3.209428   4.440771   0.723  0.47386    
## I(x^2)       0.507757   0.003094 164.089  < 2e-16 ***
## I(z^2)      -0.422661   0.272336  -1.552  0.12817    
## x:w         -3.035963   0.060703 -50.013  < 2e-16 ***
## w:z          0.329621   0.555841   0.593  0.55635    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 9.132 on 42 degrees of freedom
## Multiple R-squared:      1,  Adjusted R-squared:      1 
## F-statistic: 1.644e+05 on 7 and 42 DF,  p-value: < 2.2e-16

Simplificando o modelo

lm03 <- lm(y ~ x + w + z +
                      I( x^2) + I(z^2) +
                      x:w, data = yxzw)
anova(lm02, lm03)

## Analysis of Variance Table
## 
## Model 1: y ~ x + w + z + I(x^2) + I(z^2) + x:w + z:w
## Model 2: y ~ x + w + z + I(x^2) + I(z^2) + x:w
##   Res.Df    RSS Df Sum of Sq      F Pr(>F)
## 1     42 3502.2                           
## 2     43 3531.6 -1   -29.324 0.3517 0.5564

summary(lm03)

## 
## Call:
## lm(formula = y ~ x + w + z + I(x^2) + I(z^2) + x:w, data = yxzw)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -13.6707  -6.3689  -0.2563   4.6686  29.5533 
## 
## Coefficients:
##              Estimate Std. Error  t value Pr(>|t|)    
## (Intercept) 20.717359   7.545000    2.746 0.008776 ** 
## x           -2.474234   0.367821   -6.727  3.2e-08 ***
## w           -4.355709   1.225927   -3.553 0.000939 ***
## z            4.906277   3.370301    1.456 0.152728    
## I(x^2)       0.507920   0.003059  166.052  < 2e-16 ***
## I(z^2)      -0.461620   0.262293   -1.760 0.085531 .  
## x:w         -3.001515   0.017489 -171.624  < 2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 9.063 on 43 degrees of freedom
## Multiple R-squared:      1,  Adjusted R-squared:      1 
## F-statistic: 1.948e+05 on 6 and 43 DF,  p-value: < 2.2e-16

Simplificando o modelo

lm04 <- lm(y ~ x + w + z +
                      I( x^2) + x:w, data = yxzw)
anova(lm03, lm04)

## Analysis of Variance Table
## 
## Model 1: y ~ x + w + z + I(x^2) + I(z^2) + x:w
## Model 2: y ~ x + w + z + I(x^2) + x:w
##   Res.Df    RSS Df Sum of Sq      F  Pr(>F)  
## 1     43 3531.6                              
## 2     44 3785.9 -1   -254.38 3.0974 0.08553 .
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

summary(lm04)

## 
## Call:
## lm(formula = y ~ x + w + z + I(x^2) + x:w, data = yxzw)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -15.0896  -5.7077  -0.5103   4.5774  30.7566 
## 
## Coefficients:
##              Estimate Std. Error  t value Pr(>|t|)    
## (Intercept) 23.531249   7.547344    3.118  0.00321 ** 
## x           -1.992187   0.251285   -7.928 5.12e-10 ***
## w           -5.205080   1.153480   -4.513 4.73e-05 ***
## z           -0.515414   1.399219   -0.368  0.71437    
## I(x^2)       0.503364   0.001668  301.786  < 2e-16 ***
## x:w         -2.988520   0.016227 -184.169  < 2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 9.276 on 44 degrees of freedom
## Multiple R-squared:      1,  Adjusted R-squared:      1 
## F-statistic: 2.231e+05 on 5 and 44 DF,  p-value: < 2.2e-16

Simplificando o modelo

 lm05 <- lm(y ~ x + w + I( x^2) + x:w, data = yxzw)
anova(lm04, lm05)

## Analysis of Variance Table
## 
## Model 1: y ~ x + w + z + I(x^2) + x:w
## Model 2: y ~ x + w + I(x^2) + x:w
##   Res.Df    RSS Df Sum of Sq      F Pr(>F)
## 1     44 3785.9                           
## 2     45 3797.6 -1   -11.675 0.1357 0.7144

summary(lm05)

## 
## Call:
## lm(formula = y ~ x + w + I(x^2) + x:w, data = yxzw)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -15.9424  -5.5396  -0.7859   4.6774  30.3060 
## 
## Coefficients:
##              Estimate Std. Error  t value Pr(>|t|)    
## (Intercept) 23.437267   7.470240    3.137    0.003 ** 
## x           -2.040476   0.212311   -9.611 1.78e-12 ***
## w           -5.156007   1.134704   -4.544 4.13e-05 ***
## I(x^2)       0.503330   0.001649  305.172  < 2e-16 ***
## x:w         -2.989057   0.016005 -186.753  < 2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 9.186 on 45 degrees of freedom
## Multiple R-squared:      1,  Adjusted R-squared:      1 
## F-statistic: 2.843e+05 on 4 and 45 DF,  p-value: < 2.2e-16

Simplificando o modelo

lm06 <- lm(y ~ x + w + I( x^2), data = yxzw)
anova(lm05, lm06)

## Analysis of Variance Table
## 
## Model 1: y ~ x + w + I(x^2) + x:w
## Model 2: y ~ x + w + I(x^2)
##   Res.Df     RSS Df Sum of Sq     F    Pr(>F)    
## 1     45    3798                                 
## 2     46 2947081 -1  -2943284 34877 < 2.2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

lm07 <- lm(y ~ x + w + x:w, data = yxzw)
anova(lm05, lm07)

## Analysis of Variance Table
## 
## Model 1: y ~ x + w + I(x^2) + x:w
## Model 2: y ~ x + w + x:w
##   Res.Df     RSS Df Sum of Sq     F    Pr(>F)    
## 1     45    3798                                 
## 2     46 7863171 -1  -7859373 93130 < 2.2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Modelo Mínimo Adequado

\[ y \sim x + x^2 + x:w \]

## 
## Call:
## lm(formula = y ~ x + w + I(x^2) + x:w, data = yxzw)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -15.9424  -5.5396  -0.7859   4.6774  30.3060 
## 
## Coefficients:
##              Estimate Std. Error  t value Pr(>|t|)    
## (Intercept) 23.437267   7.470240    3.137    0.003 ** 
## x           -2.040476   0.212311   -9.611 1.78e-12 ***
## w           -5.156007   1.134704   -4.544 4.13e-05 ***
## I(x^2)       0.503330   0.001649  305.172  < 2e-16 ***
## x:w         -2.989057   0.016005 -186.753  < 2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 9.186 on 45 degrees of freedom
## Multiple R-squared:      1,  Adjusted R-squared:      1 
## F-statistic: 2.843e+05 on 4 and 45 DF,  p-value: < 2.2e-16

DIAGNÓSTICO DO MODELO

Estimativa do Modelo

coef(lm05)

## (Intercept)           x           w      I(x^2)         x:w 
##  23.4372672  -2.0404761  -5.1560065   0.5033303  -2.9890572

confint(lm05)

##                  2.5 %     97.5 %
## (Intercept)  8.3914314 38.4831031
## x           -2.4680920 -1.6128602
## w           -7.4414185 -2.8705945
## I(x^2)       0.5000084  0.5066522
## x:w         -3.0212937 -2.9568206

\[ y = 23.44 - 2.04 x + 0.50 x^2 - 5.16 w -2.99 xw \]

O que gerou os dados:

x = sort(runif(50,1,100)) 
z = x/10 + rnorm(50,0,1)
w = runif(50,0,10)
y = 12 - 1.6 * x + 0.50 * x^2 -  4.2 * w  - 3 *x * w + rnorm(50, 0, 10)

coef(lm05)

## (Intercept)           x           w      I(x^2)         x:w 
##  23.4372672  -2.0404761  -5.1560065   0.5033303  -2.9890572

Problema da colinearidade

lm10 <- lm(y ~  w + z +  I(z^2) + z:w, data = yxzw)
summary(lm10)

## 
## Call:
## lm(formula = y ~ w + z + I(z^2) + z:w, data = yxzw)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -784.03 -252.24  -22.12  103.84 1594.39 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  169.306    367.407   0.461   0.6472    
## w            -60.680     54.945  -1.104   0.2753    
## z             91.107    102.293   0.891   0.3779    
## I(z^2)        31.160      7.215   4.319 8.53e-05 ***
## w:z          -20.429      7.769  -2.630   0.0117 *  
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 455.7 on 45 degrees of freedom
## Multiple R-squared:  0.9026, Adjusted R-squared:  0.894 
## F-statistic: 104.3 on 4 and 45 DF,  p-value: < 2.2e-16

Atividade