Conceitos
- razão das variâncias
- dummy variables (indicadoras)
- matriz do modelo
- interação entre preditoras
- ANOVA é similar a uma regressão!
Testes Clássicos
Modelo Linear Simples
\[ y = {\alpha} + {\beta} x + \epsilon\] \[ \epsilon = N(0, \sigma) \]
Simulando dados
\[ y = {\alpha} + {\beta} x + \epsilon\] \[ y_i = 1.2 + 3.5 x_i + \epsilon_i \] \[ \epsilon_i = N(0, 5) \]
| 1.0 |
4.70 |
1.08 |
5.78 |
| 1.5 |
6.45 |
-2.71 |
3.74 |
| 2.0 |
8.20 |
4.46 |
12.66 |
| 2.5 |
9.95 |
2.98 |
12.93 |
| 3.0 |
11.70 |
8.18 |
19.88 |
| 3.5 |
13.45 |
3.45 |
16.90 |
| 4.0 |
15.20 |
-6.41 |
8.79 |
| 4.5 |
16.95 |
-1.07 |
15.88 |
| 5.0 |
18.70 |
9.48 |
28.18 |
lm: dados simulados
MÃnimos Quadrados
Ponto de Fulcro
MÃnimos Quadrados
Estima Parâmetros
Modelo Linear
\[ y_i = 1.2 + 3.5 x_i + \epsilon_i \] \[ \epsilon_i = N(0, 5) \]
##
## Call:
## lm(formula = y1 ~ x1, data = xy)
##
## Residuals:
## Min 1Q Median 3Q Max
## -9.1424 -4.0088 0.9982 2.8714 6.1706
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 1.632 4.520 0.361 0.7287
## x1 4.076 1.384 2.945 0.0216 *
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 5.361 on 7 degrees of freedom
## Multiple R-squared: 0.5534, Adjusted R-squared: 0.4895
## F-statistic: 8.672 on 1 and 7 DF, p-value: 0.02156
Modelo Linear
Incerteza na estimativa
\[ y_i = 1.2 + 3.5 x_i + \epsilon_i \]
Coeficientes estimados
## (Intercept) x1
## 1.632390 4.075949
Intervalo de confiança
## 2.5 % 97.5 %
## (Intercept) -9.0566136 12.321394
## x1 0.8031237 7.348775
Resumo do Modelo Linear
\[ y_i = 1.2 + 3.5 x_i + \epsilon_i \] \[ \epsilon_i = N(0, 5) \]
##
## Call:
## lm(formula = y1 ~ x1, data = xy)
##
## Residuals:
## Min 1Q Median 3Q Max
## -9.1424 -4.0088 0.9982 2.8714 6.1706
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 1.632 4.520 0.361 0.7287
## x1 4.076 1.384 2.945 0.0216 *
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 5.361 on 7 degrees of freedom
## Multiple R-squared: 0.5534, Adjusted R-squared: 0.4895
## F-statistic: 8.672 on 1 and 7 DF, p-value: 0.02156
Resumo do Modelo Linear
Atividade: simulando dados
Exemplo: dieta de lagarta
Modelo Linear: lagartas
lm(growth~tannin, data =lag)
##
## Call:
## lm(formula = growth ~ tannin, data = lag)
##
## Residuals:
## Min 1Q Median 3Q Max
## -2.4556 -0.8889 -0.2389 0.9778 2.8944
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 11.7556 1.0408 11.295 9.54e-06 ***
## tannin -1.2167 0.2186 -5.565 0.000846 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 1.693 on 7 degrees of freedom
## Multiple R-squared: 0.8157, Adjusted R-squared: 0.7893
## F-statistic: 30.97 on 1 and 7 DF, p-value: 0.0008461
Partição da Variância: lagarta
Partição da Variância
Anova
\[ SQ_{total} = SQ_{entre} + SQ_{intra} \]
Modelo Linear
\[ SQ_{total} = SQ_{mod} + SQ_{res} \]
Anova x Modelo Linear
Partição da Variância: lm
Desvios Quadráticos: total
\[ SQ_{total} = \sum_{i=1}^n (y_{i} - \bar{y})^2\]
| 0 |
12 |
6.89 |
5.11 |
26.12 |
| 1 |
10 |
6.89 |
3.11 |
9.68 |
| 2 |
8 |
6.89 |
1.11 |
1.23 |
| 3 |
11 |
6.89 |
4.11 |
16.90 |
| 4 |
6 |
6.89 |
-0.89 |
0.79 |
| 5 |
7 |
6.89 |
0.11 |
0.01 |
| 6 |
2 |
6.89 |
-4.89 |
23.90 |
| 7 |
3 |
6.89 |
-3.89 |
15.12 |
| 8 |
3 |
6.89 |
-3.89 |
15.12 |
## [1] 108.89
Desvios Quadráticos: resÃduo
\[ SQ_{error} = \sum_{i=1}^n (y_{i} - \hat{y}_i)^2\]
| 0 |
12 |
0.24 |
11.76 |
5.11 |
| 1 |
10 |
-0.54 |
10.54 |
3.11 |
| 2 |
8 |
-1.32 |
9.32 |
1.11 |
| 3 |
11 |
2.89 |
8.11 |
4.11 |
| 4 |
6 |
-0.89 |
6.89 |
-0.89 |
| 5 |
7 |
1.33 |
5.67 |
0.11 |
| 6 |
2 |
-2.46 |
4.46 |
-4.89 |
| 7 |
3 |
-0.24 |
3.24 |
-3.89 |
| 8 |
3 |
0.98 |
2.02 |
-3.89 |
## [1] 20.07
Desvios Quadraticos do Modelo
Lógica do Modelo Linear
\[ SQ_{total} = SQ_{mod} + SQ_{erro} \]
\[ SQ_{mod} = SQ_{total} - SQ_{res} \]
Desvios Quadraticos do Modelo
## [1] 88.81667
Tabela de Anova
| Modelo Linear |
88.82 |
1 |
88.82 |
| Erro |
20.07 |
7 |
2.87 |
| Total |
108.89 |
8 |
|
Modelo Linear
Teste de hipótese: F
| Modelo Linear |
88.82 |
1 |
88.82 |
30.97 |
0.00085 |
| Erro |
20.07 |
7 |
2.87 |
|
|
| Total |
108.89 |
8 |
|
|
|
Modelo Linear: lagarta
Tabela de Anova no R
## Analysis of Variance Table
##
## Response: growth
## Df Sum Sq Mean Sq F value Pr(>F)
## tannin 1 88.817 88.817 30.974 0.0008461 ***
## Residuals 7 20.072 2.867
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
| Modelo Linear |
88.82 |
1 |
88.82 |
30.97 |
0.00085 |
| Erro |
20.07 |
7 |
2.87 |
|
|
| Total |
108.89 |
8 |
|
|
|
Coeficiente de Determinação
\[ R^2 = \frac{SQ_{mod}}{SQ_{total}} \]
| Modelo Linear |
88.82 |
1 |
88.82 |
30.97 |
0.00085 |
| Erro |
20.07 |
7 |
2.87 |
|
|
| Total |
108.89 |
8 |
|
|
|
\(R^2\)
## [1] 0.816
Resumo do Modelo
##
## Call:
## lm(formula = growth ~ tannin, data = lag)
##
## Residuals:
## Min 1Q Median 3Q Max
## -2.4556 -0.8889 -0.2389 0.9778 2.8944
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 11.7556 1.0408 11.295 9.54e-06 ***
## tannin -1.2167 0.2186 -5.565 0.000846 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 1.693 on 7 degrees of freedom
## Multiple R-squared: 0.8157, Adjusted R-squared: 0.7893
## F-statistic: 30.97 on 1 and 7 DF, p-value: 0.0008461
Comparando Modelos
Modelo MÃnimo (nulo)
##
## Call:
## lm(formula = growth ~ 1, data = lag)
##
## Residuals:
## Min 1Q Median 3Q Max
## -4.8889 -3.8889 0.1111 3.1111 5.1111
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 6.889 1.230 5.602 0.000509 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 3.689 on 8 degrees of freedom
Comparando Modelos
## Analysis of Variance Table
##
## Model 1: growth ~ 1
## Model 2: growth ~ tannin
## Res.Df RSS Df Sum of Sq F Pr(>F)
## 1 8 108.889
## 2 7 20.072 1 88.817 30.974 0.0008461 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## Analysis of Variance Table
##
## Response: growth
## Df Sum Sq Mean Sq F value Pr(>F)
## tannin 1 88.817 88.817 30.974 0.0008461 ***
## Residuals 7 20.072 2.867
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Comparando Modelos
Anova do modelo: anova(lmlag)
| tannin |
1 |
88.81667 |
88.81667 |
30.97398 |
0.0008461 |
| Residuals |
7 |
20.07222 |
2.86746 |
|
|
Anova da comparação de modelos: anova(nullag, lmlag)
| 8 |
108.88889 |
|
|
|
|
| 7 |
20.07222 |
1 |
88.81667 |
30.97398 |
0.0008461 |
NÃO DESESPERE, ESPERE! KEEP CALM!!
Diagnóstico do Modelos
Modelo linear: variável categórica?!
Rendimento Colheita: dados
| 1 |
are |
6 |
| 2 |
are |
10 |
| 3 |
are |
8 |
| 4 |
are |
6 |
| 5 |
are |
14 |
| 11 |
arg |
17 |
| 12 |
arg |
15 |
| 13 |
arg |
3 |
| 14 |
arg |
11 |
| 15 |
arg |
14 |
| 21 |
hum |
13 |
| 22 |
hum |
16 |
| 23 |
hum |
9 |
| 24 |
hum |
12 |
| 25 |
hum |
15 |
Variavel Dammy ou Indicadora
| 1 |
6 |
are |
0 |
0 |
| 2 |
10 |
are |
0 |
0 |
| 3 |
8 |
are |
0 |
0 |
| 4 |
6 |
are |
0 |
0 |
| 5 |
14 |
are |
0 |
0 |
| 11 |
17 |
arg |
1 |
0 |
| 12 |
15 |
arg |
1 |
0 |
| 13 |
3 |
arg |
1 |
0 |
| 14 |
11 |
arg |
1 |
0 |
| 15 |
14 |
arg |
1 |
0 |
| 21 |
13 |
hum |
0 |
1 |
| 22 |
16 |
hum |
0 |
1 |
| 23 |
9 |
hum |
0 |
1 |
| 24 |
12 |
hum |
0 |
1 |
| 25 |
15 |
hum |
0 |
1 |
Número de nÃveis do fator menos 1 (intercepto)
Modelo linear: dummy
Modelo
\(y = \alpha_{d_1} + \beta_{2} x_{d_2}+ \beta_3 x_{d_3}\)
Intercepto:
\(\alpha_{d_1} = \bar{x}_1\)
Coeficientes:
\(\beta_{2}= \bar{x}_2 - \bar{x}_1\)
\(\beta_{3}= \bar{x}_3 - \bar{x}_1\)
| 1 |
6 |
are |
0 |
0 |
| 2 |
10 |
are |
0 |
0 |
| 3 |
8 |
are |
0 |
0 |
| 4 |
6 |
are |
0 |
0 |
| 5 |
14 |
are |
0 |
0 |
| 11 |
17 |
arg |
1 |
0 |
| 12 |
15 |
arg |
1 |
0 |
| 13 |
3 |
arg |
1 |
0 |
| 14 |
11 |
arg |
1 |
0 |
| 15 |
14 |
arg |
1 |
0 |
| 21 |
13 |
hum |
0 |
1 |
| 22 |
16 |
hum |
0 |
1 |
| 23 |
9 |
hum |
0 |
1 |
| 24 |
12 |
hum |
0 |
1 |
| 25 |
15 |
hum |
0 |
1 |
Variável Dummy
##
## Call:
## lm(formula = colhe ~ arg + hum, data = croplin)
##
## Residuals:
## Min 1Q Median 3Q Max
## -8.5 -1.8 0.3 1.7 7.1
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 9.900 1.081 9.158 9.04e-10 ***
## arg 1.600 1.529 1.047 0.30456
## hum 4.400 1.529 2.878 0.00773 **
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 3.418 on 27 degrees of freedom
## Multiple R-squared: 0.2392, Adjusted R-squared: 0.1829
## F-statistic: 4.245 on 2 and 27 DF, p-value: 0.02495
Modelo Linear Normal
##
## Call:
## lm(formula = colhe ~ solo, data = crop)
##
## Residuals:
## Min 1Q Median 3Q Max
## -8.5 -1.8 0.3 1.7 7.1
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 9.900 1.081 9.158 9.04e-10 ***
## soloarg 1.600 1.529 1.047 0.30456
## solohum 4.400 1.529 2.878 0.00773 **
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 3.418 on 27 degrees of freedom
## Multiple R-squared: 0.2392, Adjusted R-squared: 0.1829
## F-statistic: 4.245 on 2 and 27 DF, p-value: 0.02495
Coeficientes do modelo
## (Intercept) arg hum
## 9.9 1.6 4.4
## are arg hum
## 9.9 11.5 14.3
Modelo Dummy
\[y = \hat{\alpha}_{d_1} + \hat{\beta}_{2} x_{d_2}+ \hat{\beta}_3 x_{d_3}\]
| 1 |
6 |
are |
0 |
0 |
| 2 |
10 |
are |
0 |
0 |
| 3 |
8 |
are |
0 |
0 |
| 11 |
17 |
arg |
1 |
0 |
| 12 |
15 |
arg |
1 |
0 |
| 13 |
3 |
arg |
1 |
0 |
| 21 |
13 |
hum |
0 |
1 |
| 22 |
16 |
hum |
0 |
1 |
| 23 |
9 |
hum |
0 |
1 |
## (Intercept) arg hum
## 9.9 1.6 4.4
Modelo
\[y = \alpha_{d_1} + \beta_{2} x_{d_2}+ \beta_3 x_{d_3}\]
Intercepto:
\(\alpha_{d_1} = \bar{x}_1\)
Coeficientes:
\(\beta_{2}= \bar{x}_2 - \bar{x}_1\)
\(\beta_{3}= \bar{x}_3 - \bar{x}_1\)
