1. Independence (most important!)
2. Homogeneity / homogenous variances (2nd most important)
3. Normality / normal error distribution
4. Linearity

1. Independence (most important!)
2. Homogeneity / homogenous variances (2nd most important)
3. Normality / normal error distribution
4. Linearity

1. Independence (most important!)
2. Homogeneity / homogenous variances (2nd most important)
3. Normality / normal error distribution
4. Linearity

Leverage / Cook‘s distance ⇒ step of model validation

• Leverage: tool that identifies observations that have rather extreme values for the explanatory variables and may potentially bias the regression results
• Cook's distance statistic: measure for influential points → identifies single observations that are influential on all regression parameters: it calculates a test statistic D that measures the change in all regression parameters when omitting an observation.
  • D > 0.5 considered as too influential
  • D > 1: very critical
• It is easier to justify omitting influential points if they have both, large Cook and large leverage values.

Adapted from: Zuur et al. (2007)

p

Lets use the salinity ~ depth example from lecture 12:

(taken from the ICES hydro dataset, station 0076, 2015-02-01)

par(mfrow = c(2,2))
plot(mod) 

Lets use the salinity ~ depth example:

par(mfrow = c(2,2))
plot(mod) 

Lets use the salinity ~ depth example:

par(mfrow = c(2,2))
plot(mod) 

• Ordinary residuals: observed – fitted value
• Standardized residuals = the residual divided by its standard deviation: \[e_{stand.} = \frac{e_{i}}{\sqrt{MS_{Residual}*(1-h_{i})}}\]
  
  • where \(e_{i}\) = observed - fitted value; \(h_{i}\) = leverage for observation i; \(MS_{Residual}\) represents the residual variance → more on this later
• Standardised residuals are assumed to be normally distributed with expectation 0 and variance 1; N(0,1).
• Consequently, large residual values (>2) indicate a poor fit of the regression model.

You can compute both types of residuals using

• residuals(model) (works too: resid()) from the stats package → returns a vector with the ordinary residuals

• rstandard(model) from the stats package → returns a vector with the standardized residuals

• add_residuals(data, model, var = "resid") from the modelr package (in tidyverse) → adds the variable 'resid' containing the ordinary residuals to your data frame; useful when piping operations!

Load the following dataset, which is a subset of the full CPUE dataset ("CPUE per age per area_2017-11-20 06_48_16.csv") we used already in lecture 10:

load("data/cod_2015_q1.R")
ls()

## [1] "cod15"

p

To get the latitude and longitude for each area

1. search the internet for a map that shows the ICES subdivisions (SD) for the Baltic Sea and estimate the central coordinates for each SD (= area),
2. create a tibble that contains the Area variable as well as the respective Lat and Long values and than
3. merge this tibble into the cod15 tibble using a join function from the dplyr package

Here are some rough approximations of the central coordinates of each area:

sd_coord <- tibble(
  Area = factor(c(21,22,23,24,25,26,27,28,29,30,31,32)),
  Lat = c(57,55,55.75,55,55.5,55.5,58,57.5,59.5,62,64.75,60),
  Long = c(11.5,11,12.5,13.5,16,19.5,18,20,21,19.5,22.5,26)
  )

The merging can be done using the left_join() function from dpylr:

cod15 <- left_join(cod15, sd_coord, 
  by = "Area") %>% print(n = 5)

## # A tibble: 88 x 5
##   Area  Age    CPUE   Lat  Long
##   <fct> <fct> <dbl> <dbl> <dbl>
## 1 21    0         0  57    11.5
## 2 22    0         0  55    11  
## 3 23    0         0  55.8  12.5
## 4 24    0         0  55    13.5
## 5 25    0         0  55.5  16  
## # ... with 83 more rows

m_lat <- lm(formula = CPUE ~ Lat, data = cod15)
m_long <- lm(formula = CPUE ~ Long, data = cod15)

par(mfrow = c(2,4))
plot(m_lat)
plot(m_long)

m_lat <- lm(formula = CPUE ~ Lat, data = cod15)
m_long <- lm(formula = CPUE ~ Long, data = cod15)

par(mfrow = c(2,4))
plot(m_lat)
plot(m_long)

m_lat <- lm(formula = CPUE ~ Lat, data = cod15)
m_long <- lm(formula = CPUE ~ Long, data = cod15)

par(mfrow = c(2,4))
plot(m_lat)
plot(m_long)

Compute the residuals and generate histograms for both models

cod15 <- cod15 %>% 
  add_residuals(model = m_lat, var = "Res_lat") %>% 
  add_residuals(model = m_long, var = "Res_long")

p_lat <- cod15 %>% 
  ggplot(aes(x = Res_lat)) +
  geom_histogram(binwidth = 5) +
  geom_vline(xintercept = 0, 
    colour = "blue", size = 0.5)
p_long <- cod15 %>% 
  ggplot(aes(x = Res_lat)) +
  geom_histogram(binwidth = 5) +
  geom_vline(xintercept = 0, 
    colour = "blue", size = 0.5)
grid.arrange(p_lat, p_long, nrow = 2)

One should stop here, in fact, as the model assumptions are violated, and do something such as transforming the data or excluding age groups. This will be discussed in the next lecture.

Lets visualize again the relationship between CPUE and Lat/Long but this time colour the data points by the age groups and plot the predictions manually.

p_lat <- cod15 %>% 
  add_predictions(m_lat, "Pred") %>%
  ggplot(aes(x = Lat)) +
  geom_point(aes(y = CPUE, colour = Age)) +
  geom_line(aes(y = Pred)) 

p_long <- cod15 %>% 
  add_predictions(m_long, "Pred") %>%
  ggplot(aes(x = Long)) +
  geom_point(aes(y = CPUE, colour = Age)) +
  geom_line(aes(y = Pred)) +
  guides(colour = "none")

grid.arrange(p_lat, p_long, nrow = 2)

Try out the exercises and read up on linear regressions in

• chapter 23 on model basics in 'R for Data Science'
• chapter 10 (linear regressions) in "The R book" (Crawley, 2013, 2nd edition) (an online pdf version is freely available here)
• or any other textbook on linear regressions

Then go grab a coffee, lean back and enjoy the rest of the day...!