• A model is a simplified version of reality that allows us to make inferences or predictions about a population
• A model is an adequate summary of reality
• A model is a simplification or approximation to reality and therefore will not reflect all of reality (Burnham and Anderson)
• All models are wrong, some are useful (George Box)

• ¿How much \(CO_2\) do plants capture?

some_function(Y ~ X1 + X2 + ... + Xn, data = data.frame)

• Y: Response variable (\(CO_2\) Uptake)
• ~: Explained by
• \(X_n\): Explainatory variable n (Subespecies, treatment, etc.)
• data.frame: data base (CO2)
• some_function: The model to test (our simplification of reality)

Fit1 <- lm(uptake ~ Type, data = CO2)

• For this exercise we will start by using a simple linear model
• Equivalent to an ANOVA

• To see general data of the model

library(broom)
glance(Fit1)

• To see model parameters

tidy(Fit1)

• To view model predictions and residuals

augment(Fit1)

• Based on information criteria
• We will work with AIC
• \(K\) number of parameters
• \(\ln{(\hat{L})}\) fit, more positive better, more negative is bad

\[AIC = 2 K - 2 \ln{(\hat{L})}\]

Fit1 <- lm(uptake ~ Type, data = CO2)
Fit2 <- lm(uptake ~ Treatment, data = CO2)
Fit3 <- lm(uptake ~ conc, data = CO2)
Fit4 <- lm(uptake ~ Type + Treatment + conc, data = CO2)
Fit5 <- lm(uptake ~ Type + conc + I(log(conc)), data = CO2)
Fit6 <- lm(uptake ~ Type:Treatment + conc + I(log(conc)),
    data = CO2)

• uptake ~ Type

\[ \operatorname{\widehat{uptake}} = 33.54 - 12.66(\operatorname{Type}_{\operatorname{Mississippi}}) \]

• uptake ~ Treatment

\[ \operatorname{\widehat{uptake}} = 30.64 - 6.86(\operatorname{Treatment}_{\operatorname{chilled}}) \]

• uptake ~ conc

\[ \operatorname{\widehat{uptake}} = 19.5 + 0.02(\operatorname{conc}) \]

• uptake ~ Type + Treatment + conc

\[ \begin{aligned} \operatorname{\widehat{uptake}} &= 29.26 - 12.66(\operatorname{Type}_{\operatorname{Mississippi}})\ - \\ &\quad 6.86(\operatorname{Treatment}_{\operatorname{chilled}}) + 0.02(\operatorname{conc}) \end{aligned} \]

• uptake ~ Type + conc + I(log(conc))

\[ \operatorname{\widehat{uptake}} = -59.04 - 12.66(\operatorname{Type}_{\operatorname{Mississippi}}) - 0.03(\operatorname{conc}) + 17.79(\operatorname{\log(conc)}) \]

• uptake ~ Type:Treatment + conc + I(log(conc))

\[ \begin{aligned} \operatorname{\widehat{uptake}} &= -76.77 - 0.03(\operatorname{conc}) + 17.79(\operatorname{\log(conc)})\ + \\ &\quad 19.52() + 10.14() + 15.94()\ + \\ &\quad NA() \end{aligned} \]

Model1 <- glance(Fit1) %>%
    dplyr::select(r.squared, AIC) %>%
    mutate(Model = "Fit1")
Model2 <- glance(Fit2) %>%
    dplyr::select(r.squared, AIC) %>%
    mutate(Model = "Fit2")
Model3 <- glance(Fit3) %>%
    dplyr::select(r.squared, AIC) %>%
    mutate(Model = "Fit3")
Model4 <- glance(Fit4) %>%
    dplyr::select(r.squared, AIC) %>%
    mutate(Model = "Fit4")
Model5 <- glance(Fit5) %>%
    dplyr::select(r.squared, AIC) %>%
    mutate(Model = "Fit5")
Model6 <- glance(Fit6) %>%
    dplyr::select(r.squared, AIC) %>%
    mutate(Model = "Fit6")
Models <- bind_rows(Model1, Model2, Model3, Model4,
    Model5, Model6) %>%
    arrange(AIC) %>%
    mutate(DeltaAIC = AIC - min(AIC))

library(lme4)

Fit7 <- lmer(uptake ~ Type:Treatment + conc + I(log(conc)) +
    (1 | Plant), CO2)

y ~ I(abs(YEAR - 1)) + I((YEAR - 1)^2) + YEAR:InitialHabitat +
    YEAR:Treatment