# Model building demonstration with plaice CPUE in the Baltic

## Research question:

What factors explain best the pattern in age-specific Catch-Per-Unit-Effort (CPUE) of Pleuronectes platessa (plaice) in the first quarter of 2015?

Since we have longitude, latitude, and age in our dataset we can test whether these variables have any effect. Lets start with longitude as it is commonly known that plaice is more frequent in the Western Baltic Sea.

Data import and transformation

library(tidyverse)
library(lubridate)
library(forcats)
library(modelr)

cpue <- read_csv("data/CPUE per age per area_2017-11-20 06_48_16.csv")

# Transform the data (which is already tidy) -----------------

# Subset data and convert to long format --
# (so that CPUE is grouped for all age classes)
plaice <- cpue %>%
mutate(Area = factor(Area, levels = c(21,22,
23,24,25,26,27,28,29,30,31,32))) %>%
filter(Year == 2015, Quarter == 1, Species == "Pleuronectes platessa") %>%
select(Area, Quarter, Species, contains("Age")) %>%
gather(key = "Age", value = "CPUE", Age_0:Age_10) %>%
mutate(Age = factor(Age, levels = paste0("Age_", 0:10),
labels = c("0","1","2","3","4","5","6","7","8","9","10")))

# Create a tibble wiht central coordinates for each area (ICES subdivision) --
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)
)

# Merge this table into the cpue data --
plaice <- left_join(plaice, sd_coord, by = "Area")

## Round 1

### Step 1: Specification of model family

plaice %>%
ggplot(aes(x = Long, y = CPUE)) +
geom_jitter(aes(colour = Age)) +
scale_colour_brewer(
palette = "Paired") +
guides(colour = "none") +
ggtitle("Plaice CPUE ~ Longitude (1.Quarter in 2015)")

Since a straight line seems less appropriate we will compare different model families including polynomials and transformation of Y.

### Step 2: Model fitting

plaice <- plaice %>%
mutate(CPUE_log = log(CPUE + 0.001))

m_lm <- lm(CPUE ~ Long, data = plaice)
m_poly2 <- lm(CPUE ~ poly(Long,2), data = plaice)
m_poly3 <- lm(CPUE ~ poly(Long,3), data = plaice)
m_log_y <- lm(CPUE_log ~ Long, data = plaice)

### Step 3: Model partitioning - predicted values → model fit

pred_dat <- plaice %>%
data_grid(Long = seq_range(Long, 20)) %>%
# generates an evenly spaced grid of data points with
gather(m_lm:m_log_y, key = "Model", value = "Pred") %>%
mutate(Model = factor(Model))

p1 <- pred_dat %>%
filter(Model !=  "m_log_y") %>%
ggplot(aes(x = Long, y = Pred)) +
geom_line(aes(linetype = Model)) +
geom_jitter(data = plaice, aes(y = CPUE, colour = Age)) +
scale_colour_brewer(palette = "Paired") +
ylab("CPUE") +
ggtitle("Models with untransformed data")

p2 <- pred_dat %>%
filter(Model ==  "m_log_y") %>%
ggplot(aes(x = Long, y = Pred)) +
geom_line(aes(linetype = Model)) +
geom_jitter(data = plaice, aes(y = CPUE_log, colour = Age)) +
scale_colour_brewer(palette = "Paired") +
ylab("log(CPUE)") +
ggtitle("Model with log-transformed CPUE data")

gridExtra::grid.arrange(p1, p2, ncol = 2)

Which model do you prefer? Which models are misfitting the data? Which problem becomes more apparent when log-transforming CPUE?

• There are too many zeros in the CPUE data!

### Step 3: Residuals → model assumptions

par(mfrow = c(4,4))
plot(m_lm, main = "m_lm")
plot(m_poly2, main = "m_poly2")
plot(m_poly3, main = "m_poly3")
plot(m_log_y, main = "m_log_y")

### Zero-inflation

Too many zeros in the data can inflate the parameterization of coefficients.

• One can either apply specific models that account for it (e.g. zero-truncated or zero-inflated regression models),
• convert the data into binary data (presence/absence) and apply a logistic regression,
• or reduce the number of zeros by
• summing up the CPUE over all age classes (disadvantage: we reduce the sample size)
• or, in this demostration, by removing specific age classes

→ In any case, we need to rephrase our research question!

plaice %>% group_by(Age, CPUE) %>%
filter(CPUE < 0.01) %>%
summarize(counts = n()) %>% print(n=40)
## # A tibble: 8 x 3
## # Groups:   Age [?]
##   Age    CPUE counts
##   <fct> <dbl>  <int>
## 1 0         0      5
## 2 1         0      3
## 3 5         0      1
## 4 6         0      2
## 5 7         0      3
## 6 8         0      2
## 7 9         0      3
## 8 10        0      4

Lets remove age 0, 1 and > 6 and start with step 2 again

plaice <- plaice %>% filter(Age %in% 2:6) %>%
mutate(Age = fct_drop(Age))

## Round 2

### Repeat step 2: Model fitting with new data

m_lm <- lm(CPUE ~ Long, data = plaice)
m_poly2 <- lm(CPUE ~ poly(Long,2), data = plaice)
m_poly3 <- lm(CPUE ~ poly(Long,3), data = plaice)
m_log_y <- lm(CPUE_log ~ Long, data = plaice)

For reasons of convenience I named the subset also ‘plaice’ so I can simply run the previous code again without many modifications.

### Repeat step 3: Model partitioning - predicted values

pred_dat <- plaice %>%
data_grid(Long = seq_range(Long, 20)) %>%
gather(m_lm:m_log_y, key = "Model", value = "Pred") %>%
mutate(Model = factor(Model))

p1 <- pred_dat %>%
filter(Model !=  "m_log_y") %>%
ggplot(aes(x = Long, y = Pred)) +
geom_line(aes(linetype = Model)) +
geom_jitter(data = plaice, aes(y = CPUE, colour = Age)) +
scale_colour_brewer(palette = "Paired") +
ylab("CPUE") +
ggtitle("Models with untransformed data")

p2 <- pred_dat %>%
filter(Model ==  "m_log_y") %>%
ggplot(aes(x = Long, y = Pred)) +
geom_line(aes(linetype = Model)) +
geom_jitter(data = plaice, aes(y = CPUE_log, colour = Age)) +
scale_colour_brewer(palette = "Paired") +
ylab("log(CPUE)") +
ggtitle("Model with log-transformed CPUE data")

gridExtra::grid.arrange(p1, p2, ncol = 2)

How do the models perform now?

### Repeat step 3: Model partitioning - residuals

par(mfrow = c(4,4))
plot(m_lm, main = "m_lm")
plot(m_poly2, main = "m_poly2")
plot(m_poly3, main = "m_poly3")
plot(m_log_y, main = "m_log_y")

## What next?

• m_log_y shows the best residual distribution so we could focus on this model and move on the step 4; however, for some age classes this model greatly underestimates the CPUE.
• The models with untransformed data do not look too bad either (particularly the polynomials) so we could keep them for now and see whether including more variables helps reducing the noise and improving the model fit.

### Step 4: Model residuals ~ another X variable

r <- plaice %>% spread_residuals(m_lm, m_poly2, m_poly3, m_log_y) %>%
ggplot() + geom_hline(yintercept = 0,
colour = "blue", size = 0.5)

r1_lat <- r + geom_point(aes(x = Lat, y = m_lm),
colour = "red", alpha = 0.5) + ggtitle("m_lm")
r2_lat <- r + geom_point(aes(x = Lat, y = m_poly2),
colour = "red", alpha = 0.5) + ggtitle("m_poly2")
r3_lat <- r + geom_point(aes(x = Lat, y = m_poly3),
colour = "red", alpha = 0.5) + ggtitle("m_poly3")
r4_lat <- r + geom_point(aes(x = Lat, y = m_log_y),
colour = "red", alpha = 0.5) + ggtitle("m_log_y")
r1_age <- r + geom_boxplot(aes(x = Age, y = m_lm))
r2_age <- r + geom_boxplot(aes(x = Age, y = m_poly2))
r3_age <- r + geom_boxplot(aes(x = Age, y = m_poly3))
r4_age <- r + geom_boxplot(aes(x = Age, y = m_log_y))

gridExtra::grid.arrange(grobs = list(
r1_lat,r2_lat,r3_lat,r4_lat,r1_age,r2_age,r3_age,r4_age), ncol = 4, nrow = 2)

Lets include the factor Age in all 4 models!

### Step 5: Re-specify the model and visualize the predictions

m_lm <- update(m_lm, .~. + Age)
m_poly2 <- update(m_poly2, .~. + Age)
m_poly3 <- update(m_poly3, .~. + Age)
m_log_y <- update(m_log_y, .~. + Age)
pred_dat <- plaice %>%
data_grid(Long = seq_range(Long, 20), Age = Age) %>%
gather(m_lm:m_log_y, key = "Model", value = "Pred") %>%
mutate(Model = factor(Model))

p <- plaice %>%
ggplot(aes(x = Long, y = CPUE)) +
geom_jitter(aes(colour = Age)) +
scale_colour_brewer(palette = "Paired") +
ylab("CPUE")

p1 <- p + geom_line(data = filter(pred_dat, Model == "m_lm"),
aes(y = Pred, group = Age, colour = Age)) + ggtitle("m_lm") +
theme(legend.position = c(1,1) ,
legend.justification = c(1.1,1.1))

p2 <- p + geom_line(data = filter(pred_dat, Model == "m_poly2"),
aes(y = Pred, group = Age, colour = Age)) + ggtitle("m_poly2") +
guides(colour = "none")

p3 <- p + geom_line(data = filter(pred_dat, Model == "m_poly3"),
aes(y = Pred, group = Age, colour = Age)) + ggtitle("m_poly3") +
guides(colour = "none")

p4 <- plaice %>%
ggplot(aes(x = Long, y = CPUE_log)) +
geom_jitter(aes(colour = Age)) +
scale_colour_brewer(palette = "Paired") +
ylab("CPUE") +
geom_line(data = filter(pred_dat, Model == "m_log_y"),
aes(y = Pred, group = Age, colour = Age)) + ggtitle("m_log_y") +
guides(colour = "none")

gridExtra::grid.arrange(grobs = list(p1, p2, p3, p4), ncol = 4)

Happy now?

Plotting the residual diagnostics of these models

par(mfrow = c(4,4))
plot(m_lm, main = "m_lm")
plot(m_poly2, main = "m_poly2")
plot(m_poly3, main = "m_poly3")
plot(m_log_y, main = "m_log_y")

### Step 6: Test for an interaction between Long and Age

m_lm_int <- lm(CPUE ~ Long * Age, data = plaice)
m_poly2_int <- lm(CPUE ~ poly(Long,2) * Age, data = plaice)
m_poly3_int <- lm(CPUE ~ poly(Long,3) * Age, data = plaice)
m_log_y_int <- lm(CPUE_log ~ Long * Age, data = plaice)
pred_dat <- plaice %>%
data_grid(Long = seq_range(Long, 20), Age = Age) %>%
gather(m_lm_int:m_log_y_int, key = "Model", value = "Pred") %>%
mutate(Model = factor(Model))

p <- plaice %>%
ggplot(aes(x = Long, y = CPUE)) +
geom_jitter(aes(colour = Age)) +
scale_colour_brewer(palette = "Paired") +
ylab("CPUE")

p1 <- p + geom_line(data = filter(pred_dat, Model == "m_lm_int"),
aes(y = Pred, group = Age, colour = Age)) + ggtitle("m_lm_int") +
theme(legend.position = c(1,1) ,
legend.justification = c(1.1,1.1))

p2 <- p + geom_line(data = filter(pred_dat, Model == "m_poly2_int"),
aes(y = Pred, group = Age, colour = Age)) + ggtitle("m_poly2_int") +
guides(colour = "none")

p3 <- p + geom_line(data = filter(pred_dat, Model == "m_poly3_int"),
aes(y = Pred, group = Age, colour = Age)) + ggtitle("m_poly3_int") +
guides(colour = "none")

p4 <- plaice %>%
ggplot(aes(x = Long, y = CPUE_log)) +
geom_jitter(aes(colour = Age)) +
scale_colour_brewer(palette = "Paired") +
ylab("CPUE") +
geom_line(data = filter(pred_dat, Model == "m_log_y_int"),
aes(y = Pred, group = Age, colour = Age)) + ggtitle("m_log_y_int") +
guides(colour = "none")

gridExtra::grid.arrange(grobs = list(p1, p2, p3, p4), ncol = 4)

Better?

Plotting the residual diagnostics of these models

par(mfrow = c(4,4))
plot(m_lm_int, main = "m_lm_int")
plot(m_poly2_int, main = "m_poly2_int")
plot(m_poly3_int, main = "m_poly3_int")
plot(m_log_y_int, main = "m_log_y_int")

### Go back to step 4: Model residuals ~ latitude

No clear pattern that suggests to include latitude in the model (in the mod_log_y model it seems residuals decrease with latitude but that is mainly caused by the 2 outliers).

## So which one to choose?

• From the residual diagnostics it seems that mod_log_y with Age and Long is the best, but the cubic model is also quite good.
• Should an interaction term be included or not? Difficult to say from the graphics.
• Lets use the AIC to guide our decision!
• It is wise to include a NULL model (with only an intercept) to see whether any of the covariate should be included or not.
• The AIC is only useful for comparison between models fitted to the same observations and same Y variables: you can’t compare models fitted to different data subsets or if Y is partly transformed!

### Overview of models we fitted

# Generate a null-model for the untransformed models --
m0 <- lm(CPUE ~ 1, data = plaice)
m_lm_1 <- lm(CPUE ~ Long, data = plaice)
m_lm_2 <- lm(CPUE ~ Long + Age, data = plaice)
m_lm_3 <- lm(CPUE ~ Long * Age, data = plaice)
m_poly2_1 <- lm(CPUE ~ poly(Long,2), data = plaice)
m_poly2_2 <- lm(CPUE ~ poly(Long,2) + Age, data = plaice)
m_poly2_3 <- lm(CPUE ~ poly(Long,2) * Age, data = plaice)
m_poly3_1 <- lm(CPUE ~ poly(Long,3), data = plaice)
m_poly3_2 <- lm(CPUE ~ poly(Long,3) + Age, data = plaice)
m_poly3_3 <- lm(CPUE ~ poly(Long,3) * Age, data = plaice)

# Generate a null-model for the log-models --
m_log_y_0 <-lm(CPUE_log ~ 1, data = plaice)
m_log_y_1 <-lm(CPUE_log ~ Long, data = plaice)
m_log_y_2 <-lm(CPUE_log ~ Long + Age, data = plaice)
m_log_y_3 <-lm(CPUE_log ~ Long * Age, data = plaice)

### AIC comparison

# Untransf. models --
AIC(m0, m_lm_1, m_lm_2, m_lm_3, m_poly2_1, m_poly2_2, m_poly2_3,
m_poly3_1, m_poly3_2, m_poly3_3)
##           df      AIC
## m0         2 262.6016
## m_lm_1     3 256.8791
## m_lm_2     7 254.4298
## m_lm_3    11 252.9204
## m_poly2_1  4 257.0557
## m_poly2_2  8 253.6051
## m_poly2_3 16 252.1079
## m_poly3_1  5 258.3897
## m_poly3_2  9 254.5438
## m_poly3_3 21 255.0540
# Log-models --
AIC(m_log_y_0, m_log_y_1, m_log_y_2, m_log_y_3)
##           df      AIC
## m_log_y_0  2 137.3974
## m_log_y_1  3 135.0501
## m_log_y_2  7 129.8373
## m_log_y_3 11 135.0000
• All models perform better than the NULL model but do not differ much from each other; no model sticks out as most adequate but simple model! Given an AIC difference of 2, none of the polynomial models performs better than the simple linear model and, in fact, m_lm_2 (Age + Long) would be the best choice based on the AIC.
• The best log-model contains Age and Long but no interaction!

## Summary of final model

### Numerical output of m_log_y_2

summary(m_log_y_2)
##
## Call:
## lm(formula = CPUE_log ~ Long + Age, data = plaice)
##
## Residuals:
##     Min      1Q  Median      3Q     Max
## -7.7375 -0.6531  0.1546  1.4860  3.1414
##
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)
## (Intercept)   8.5793     2.8565   3.003  0.00731 **
## Long         -0.4436     0.1793  -2.474  0.02295 *
## Age3          1.1864     1.7805   0.666  0.51320
## Age4          0.0158     1.7805   0.009  0.99301
## Age5         -2.6477     1.7805  -1.487  0.15340
## Age6         -4.4044     1.7805  -2.474  0.02298 *
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 2.815 on 19 degrees of freedom
## Multiple R-squared:  0.5046, Adjusted R-squared:  0.3742
## F-statistic: 3.871 on 5 and 19 DF,  p-value: 0.01377

### Graphical output of m_log_y_2

pred_dat <- plaice %>%
data_grid(Long = seq_range(Long, 20), Age = Age)
pred_dat$Pred <- predict(m_log_y_2, se = TRUE, newdata = pred_dat)$fit
pred_dat$Pred_se <- predict(m_log_y_2, se = TRUE, newdata = pred_dat)$se.fit

pred_dat_untrans <- pred_dat %>%
mutate(Pred = exp(Pred - 0.001), Pred_se = exp(Pred_se - 0.001))

p_fin1 <- plaice %>%
ggplot(aes(x = Long)) +
geom_ribbon(data = pred_dat,
aes(ymin = (Pred - Pred_se), ymax = (Pred + Pred_se),
group = Age, fill = Age), alpha = 0.5) +
geom_line(data = pred_dat,
aes(y = Pred, group = Age, colour = Age)) +
geom_jitter(aes(y = CPUE_log, colour = Age)) +
scale_colour_brewer(palette = "Paired") +
scale_fill_brewer(palette = "Paired") +
facet_grid(. ~ Age, labeller = label_both) +
guides(colour = "none") +
ylab("CPUE") + theme_bw() +
theme(panel.grid = element_blank()) +
ggtitle("Obs. and pred. (with s.e.) CPUE on the modelled log-scale")

p_fin2 <- plaice %>%
ggplot(aes(x = Long)) +
geom_ribbon(data = pred_dat_untrans,
aes(ymin = (Pred - Pred_se), ymax = (Pred + Pred_se),
group = Age, fill = Age), alpha = 0.5) +
geom_line(data = pred_dat_untrans,
aes(y = Pred, group = Age, colour = Age)) +
geom_jitter(aes(y = CPUE, colour = Age)) +
scale_colour_brewer(palette = "Paired") +
scale_fill_brewer(palette = "Paired") +
facet_grid(. ~ Age, labeller = label_both) +
guides(colour = "none") +
ylab("CPUE") + theme_bw() +
theme(panel.grid = element_blank()) +
ggtitle("Obs. and pred. (with s.e.) CPUE back transformed")

gridExtra::grid.arrange(p_fin1, p_fin2, nrow = 2)