Export the data as .CSV

Make an RStudio Project

Geese <- read.csv(file = "my data.csv")

Calculate time fractions and frequencies

To account for the fact that the individual video segments (observation times) vary somewhat, we

• convert durations to time-fractions
  (between 0 and 1, like percent);
• convert head-up counts to frequencies (events per minute).

Geese$feed.tf <- Geese$time.feeding / Geese$clip.length
Geese$vig.tf <- Geese$time.vigilant / Geese$clip.length

Geese$hu.freq <- Geese$n.headups / Geese$clip.length * 60  # per minute

Always plot the data first! (1)

We’ll use package ggplot2. Simples:

library(ggplot2)  # load the "ggplot2" plotting package;

feeding <- ggplot(Geese, aes(x=flock.size, y=feed.tf))  # set up plot
feeding + geom_point()  # make scatter plot

Always plot the data first! (2)

Straight line fit (1)

You may feel the urge to add a regression line, which is easy enough:

feeding + geom_point() + 
  geom_smooth(method = "lm", se = FALSE, linetype = 2)

Straight line fit (2)

Is it sensible?

Look at the data: it is pretty clear that if there is a relationship between feeding time and flock size, it is not a linear one.

There is no reason to assume a priori that the relationship would be linear:

1. Would you expect any net benefits of group living to increase linearly forever as group size increases? – Think ‘diminishing returns curves’ from the lectures!

2. It can’t – \(0 \leq time fraction \leq 1\)!

Nonlinear fit (1)

Fit a smooth regression using "loess" instead of "lm".

The span=1 bit just sets how bendy the line is allowed to be. Lower span means more wobbly (try it!).

feeding + geom_point() + 
  geom_smooth(method = "loess", span = 1, se = FALSE)

Nonlinear fit (2)

Hypohesis test (1)

• Testing for correlation between feed.tf and flock.size.
• By default, cor.test will carry out a Pearson’s correlation test. This assumes linearity!
• Since the relationship is clearly non-linear, you should do Spearman’s rank correlation test.

cor.test( ~ feed.tf + flock.size, Geese, method="spearman")

Hypohesis test (2)

cor.test( ~ feed.tf + flock.size, Geese, method="spearman")

    Spearman's rank correlation rho

data:  feed.tf and flock.size
S = 297.62, p-value = 0.006139
alternative hypothesis: true rho is not equal to 0
sample estimates:
      rho 
0.6352753 

• Strict Frequentist: reject \(H_0\) at \(\alpha = 0.05\) (‘significant’ correlation)

• Strength of Evidence: the data provide good evidence that time investment in feeding increases with group size.

Plot the data (1)

vigil <- ggplot(Geese, aes(x=flock.size, y=vig.tf)) # set up plot
vigil + geom_point()  # make scatter plot

Plot the data (2)

Plot the data (3)

Let’s add both linear and smooth regression lines:

vigil + geom_point() + 
  geom_smooth(method = "lm", se = FALSE, linetype = 2) +
  geom_smooth(method = "loess", span = 2, se = FALSE)

Plot the data (4)

Hypothesis test

cor.test( ~ vig.tf + flock.size, Geese, method="spearman")

    Spearman's rank correlation rho

data:  vig.tf and flock.size
S = 1268.6, p-value = 0.02085
alternative hypothesis: true rho is not equal to 0
sample estimates:
       rho 
-0.5546251 

• Strict Frequentist: reject \(H_0\) at \(\alpha = 0.05\) (‘significant’ correlation).

• Strength of Evidence: the data provide weak evidence that time investment in vigilance decreases with group size.

Plot the data (1)

Let’s add the regression lines straight away

ggplot(Geese, aes(x=flock.size, y=hu.freq)) + 
  geom_point() +
  geom_smooth(method = "lm", se = FALSE, linetype = 2) +
  geom_smooth(method = "loess", span = 1, se = FALSE)

Plot the data (2)

Hypothesis test

cor.test( ~ hu.freq + flock.size, Geese, method="spearman")

    Spearman's rank correlation rho

data:  hu.freq and flock.size
S = 621.37, p-value = 0.3566
alternative hypothesis: true rho is not equal to 0
sample estimates:
      rho 
0.2385211 

• Strict Frequentist: accept \(H_0\) at \(\alpha = 0.05\) (‘no significant’ correlation).

• Strength of Evidence: the data provide no evidence that the frequency of head-up events correlates with group size.

Either way, you have not proven the absence of a correlation!

• You can never prove the absence of a correlation with a \(P\)-value.
• Put formally, accepting \(H_0\) does not reject \(H_1\).
• but with sufficient \(N\) you can have compelling evidence that the correlation is very weak.

Are the vigilance bouts longer in smaller groups?

We can easily calculate the mean bout length, because we know both

• the total time the head was up and
• the number of bouts.

Geese$mean.vbout.length <- with(Geese, time.vigilant / hu.freq)

Plot the data (1)

ggplot(Geese, aes(x=flock.size, y=mean.vbout.length)) + geom_point() +
  geom_smooth(method = "lm", se = FALSE, linetype = 2) +
  geom_smooth(method = "loess", span = 1.5, se = FALSE)

Plot the data (2)

Hypothesis test

cor.test( ~ mean.vbout.length + flock.size, Geese, method="spearman")

    Spearman's rank correlation rho

data:  mean.vbout.length and flock.size
S = 1146.9, p-value = 0.1064
alternative hypothesis: true rho is not equal to 0
sample estimates:
       rho 
-0.4054838 

Curiously, this doesn’t come out significant at \(\alpha = 0.05\) – even though we have good evidence that total time vigilant is higher in small groups, and we found no support for claiming that the frequency of vigilance bouts depends on group size.

Log-transform \(x\) and \(y\) (1)

ggplot(Geese, aes(x=log(flock.size), y=log(mean.vbout.length))) + 
  geom_point() +
  geom_smooth(method = "lm", se = FALSE, linetype = 2) +
  geom_smooth(method = "loess", span = 1.5, se = FALSE)

Log-transform \(x\) and \(y\) (2)

Try again…

cor.test( ~ log(mean.vbout.length) + log(flock.size),
          Geese, method="pearson")

    Pearson's product-moment correlation

data:  log(mean.vbout.length) and log(flock.size)
t = -2.9325, df = 15, p-value = 0.01029
alternative hypothesis: true correlation is not equal to 0
95 percent confidence interval:
 -0.8404458 -0.1732788
sample estimates:
       cor 
-0.6036483 

Suggests that the individual vigilance bouts are indeed shorter in larger groups.

Engagement Marks

• Do the Quiz on Top Hat
• Deadline: Monday 16 March 2026 @ 10:00 AM