O que aprendemos?

Parte 1: Transformações dos Dados

Aplicamos transformações como Box-Cox, logaritmo e raiz quadrada.

Entendemos quando as transformações são necessárias para atender premissas da ANOVA.

Parte 2: Regressão Linear

Exploramos modelos de regressão linear simples.

Analisamos o estande de plantas em função da concentração de inóculo.

Ajustamos modelos lineares mistos com o pacote lme4.

Parte 3: Modelos Log-Logísticos

Aplicamos regressão log-logística com os pacotes drc e ec50estimator.

Estimamos valores de EC50 para diferentes isolados fúngicos.

Box-Cox: transformação de variáveis

library(MASS)
library(DHARMa)
This is DHARMa 0.4.7. For overview type '?DHARMa'. For recent changes, type news(package = 'DHARMa')
library(tidyverse)
── Attaching core tidyverse packages ──────────────────────── tidyverse 2.0.0 ──
✔ dplyr     1.1.4     ✔ readr     2.1.5
✔ forcats   1.0.0     ✔ stringr   1.5.1
✔ ggplot2   3.5.2     ✔ tibble    3.2.1
✔ lubridate 1.9.4     ✔ tidyr     1.3.1
✔ purrr     1.0.4     
── Conflicts ────────────────────────────────────────── tidyverse_conflicts() ──
✖ dplyr::filter() masks stats::filter()
✖ dplyr::lag()    masks stats::lag()
✖ dplyr::select() masks MASS::select()
ℹ Use the conflicted package (<http://conflicted.r-lib.org/>) to force all conflicts to become errors
library(agricolae)

insects <- InsectSprays

m1 <- lm(sqrt(count) ~ spray, data = insects)
plot(m1)

b <- boxcox(lm(count + 0.1 ~ 1, data = insects))

lambda <- b$x[which.max(b$y)]
lambda
[1] 0.4242424
insects <- insects |>
  mutate(count2 = (count^lambda - 1) / lambda,
         count3 = sqrt(count))

hist(insects$count)

m2 <- lm(count2 ~ spray, data = insects)
plot(m2)

Regressão Linear

library(gsheet)
library(ggplot2)

estande <- gsheet2tbl("https://docs.google.com/spreadsheets/d/1bq2N19DcZdtax2fQW9OHSGMR0X2__Z9T/edit?gid=401662555#gid=401662555")

ggplot(estande, aes(x = trat, y = nplants)) +
  geom_point() +
  geom_smooth(method = "lm", se = FALSE, color = "black") +
  facet_wrap(~exp) +
  theme_minimal() +
  labs(x = "% de inóculo na semente", y = "Número de Plantas")
`geom_smooth()` using formula = 'y ~ x'

Modelos Individuais por Experimento

exp1 <- filter(estande, exp == 1)
m_exp1 <- lm(nplants ~ trat, data = exp1)
summary(m_exp1)

Call:
lm(formula = nplants ~ trat, data = exp1)

Residuals:
    Min      1Q  Median      3Q     Max 
-25.500  -6.532   1.758   8.573  27.226 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept)  52.5000     4.2044  12.487 1.84e-11 ***
trat         -0.2419     0.1859  -1.301    0.207    
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 15 on 22 degrees of freedom
Multiple R-squared:  0.07148,   Adjusted R-squared:  0.02928 
F-statistic: 1.694 on 1 and 22 DF,  p-value: 0.2066
exp2 <- filter(estande, exp == 2)
m_exp2 <- lm(nplants ~ trat, data = exp2)
summary(m_exp2)

Call:
lm(formula = nplants ~ trat, data = exp2)

Residuals:
     Min       1Q   Median       3Q      Max 
-25.7816  -7.7150   0.5653   8.1929  19.2184 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept)  60.9857     3.6304  16.798 4.93e-14 ***
trat         -0.7007     0.1605  -4.365 0.000247 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 12.95 on 22 degrees of freedom
Multiple R-squared:  0.4641,    Adjusted R-squared:  0.4398 
F-statistic: 19.05 on 1 and 22 DF,  p-value: 0.0002473
exp3 <- filter(estande, exp == 3)
m_exp3 <- lm(nplants ~ trat, data = exp3)
summary(m_exp3)

Call:
lm(formula = nplants ~ trat, data = exp3)

Residuals:
     Min       1Q   Median       3Q      Max 
-26.5887  -3.9597   0.7177   5.5806  19.8952 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept)  95.7500     2.9529  32.425  < 2e-16 ***
trat         -0.7634     0.1306  -5.847 6.97e-06 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 10.53 on 22 degrees of freedom
Multiple R-squared:  0.6085,    Adjusted R-squared:  0.5907 
F-statistic: 34.19 on 1 and 22 DF,  p-value: 6.968e-06

Modelo Linear Misto

library(lme4)
Carregando pacotes exigidos: Matrix

Anexando pacote: 'Matrix'
Os seguintes objetos são mascarados por 'package:tidyr':

    expand, pack, unpack
library(car)
Carregando pacotes exigidos: carData

Anexando pacote: 'car'
O seguinte objeto é mascarado por 'package:dplyr':

    recode
O seguinte objeto é mascarado por 'package:purrr':

    some
m_misto <- lmer(nplants ~ trat + (1 | exp/bloco), data = estande)
summary(m_misto)
Linear mixed model fit by REML ['lmerMod']
Formula: nplants ~ trat + (1 | exp/bloco)
   Data: estande

REML criterion at convergence: 575.8

Scaled residuals: 
     Min       1Q   Median       3Q      Max 
-2.21697 -0.63351  0.04292  0.67094  1.92907 

Random effects:
 Groups    Name        Variance Std.Dev.
 bloco:exp (Intercept)  54.76    7.40   
 exp       (Intercept) 377.43   19.43   
 Residual              134.99   11.62   
Number of obs: 72, groups:  bloco:exp, 12; exp, 3

Fixed effects:
            Estimate Std. Error t value
(Intercept) 69.74524   11.57191   6.027
trat        -0.56869    0.08314  -6.840

Correlation of Fixed Effects:
     (Intr)
trat -0.111
confint(m_misto)
Computing profile confidence intervals ...
                 2.5 %     97.5 %
.sig01       3.3332097 14.4218422
.sig02       7.2377419 47.8269818
.sigma       9.7314178 13.9359486
(Intercept) 43.4631239 96.0274587
trat        -0.7328972 -0.4044812
car::Anova(m_misto)
Analysis of Deviance Table (Type II Wald chisquare tests)

Response: nplants
      Chisq Df Pr(>Chisq)    
trat 46.788  1  7.909e-12 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Visualização do Modelo Misto

ggplot(estande, aes(trat, nplants, color = factor(exp))) +
  geom_point() +
  geom_abline(intercept = 69.74, slope = -0.568, linewidth = 2) +
  geom_abline(intercept = 43, slope = -0.73, linetype = "dashed") +
  geom_abline(intercept = 96, slope = -0.40, linetype = "dashed")

Regressão Log-Logística e Estimativa de EC50

Conclusões

  • Transformações corrigem problemas de normalidade e homocedasticidade.
  • Regressão linear e mista ajudam a explicar variações em estande de plantas.
  • Modelos log-logísticos são essenciais para determinar sensibilidade (EC50).