Research Methodology: Fish Clutch Size and Belly Redness Analysis

This assignment analyzes the effects of different treatments on the hatching success of female fish eggs and explores the relationship between clutch size and the redness of the female fish bellies. The analysis includes the use of both raw proportions and arcsine square root proportions, and statistical methods like the Mann-Whitney U test, ANOVA, and linear regression are employed to test various hypotheses. The study investigates whether the treatment affects hatching success, if females from different geographical sites lay equally large clutches, and if clutch size is related to the redness of the female's bellies. Diagnostic plots such as boxplots, residual plots, and Q-Q plots are used to check assumptions. The results reveal significant differences in clutch sizes among different geographical sites and no significant linear relationship between clutch size and belly redness. The research also references relevant studies and statistical methods.

Question I1: Did the treatment affect the hatching success of the female’s eggs? (try both
raw proportions and the arcsine square root proportions)
H0: Treatment has no affect on the hatching success of the female’s eggs (raw proportions
and the arcsine square root proportions).
Method: The first step is to examine the data for normality assumption through boxplot.
Figure 1 and 2 shows the boxplot for raw proportions and arcsine square root proportion
respectively (Forsgren et al., 2004).

The boxes are not symmetrical an indication that the data are not following a normal
distribution. Therefore, the assumption of normality is violated w now use Mann-Whitney U
test instead of independent t-test.
Result: We fail to reject H0 of treatment has no effect on the hatching success of the female’s
eggs (Svensson and Forsgren, 2009). There was insufficient evidence at 95% confidence
level (W = 1628.5, p-value = 0.5392) to support the claim that the treatment affects the
hatching success of the female’s eggs.
Question I2: Did females from the four sites lay equally large clutches?
H0: There is no significant difference in size of clutches between the females from the four
different geographical sites.
Method: The initial approach in testing the hypothesis is to perform one-way analysis of
variance. However, we first check for normality of the individual clutch size for each location
using boxplot. We also need to check for equal variance assumption using residual plot.
Finally, we check for the overall normality of the data using normal Q-Q plot. Figure 3, 4 and
5 shows the boxplots, residual verses fitted plot, and normal Q-Q plot respectively.

The plots in figure 3 shows that the data is approximately normal except for existences of
extremely small clutch sizes in Blabergholmen and Rodberget. Also, there is an extremely
large clutch size in Langegap.
In examining the equal variance assumption, we interpret figure 4. The data does not form a
wedge shape hence the variances are equal

The qnorm plots does not form an S-shape also majority of the points lie on or close to the
red line. Therefore, the residuals are normally distributed. These assumptions are met we can
now use the results of the ANOVA.
Result: Reject the H0 of no difference. There was a significant difference in clutch size of fish
females from four different geographical sites (ANOVA, F3,106 = 11.771, P<0,000). The post-
hoc test (Tukey-test  = 0.05) revealed that fishes from Blabergholmen significantly lay
clutch larger than females from Bryggan (P < 0.001). Similarly, fishes from Bryggan
significantly lay clutch smaller than females from Langegap and Rodberget (P < 0.000).
Question I3: Was the redness of the female’s bellies affected by their clutch size? If so,
present the estimated values of the straight-line equation.
H0: There is no significant difference in size of clutches between the females from the four
different geographical sites.
Method: A residual plot (figure 6), a QQ-plot (figure 7) and a residual vs leverage plot (figure
8) to detect any unequal variances, non-normality and outliers respectively.

The initial approach in testing the hypothesis is to perform one-way analysis of variance.
However, we first check for normality of the individual clutch size for each location using
boxplot. We also need to check for equal variance assumption using residual plot. Finally, we
check for the overall normality of the data using normal Q-Q plot. Figure 3, 4 and 5 shows
the boxplots, residual verses fitted plot, and normal Q-Q plot respectively.
The plot in figure 6 shows randomness without any particular pattern in the residuals and
fitted plot (Crawley, 2007). Therefore, the variances are equal.

Majority of the points are along the red line and the points does not form “S” shape.
Therefore, the residuals follow a normal distribution.
The plot does not show any influential outliers (Crawley, 2007). The assumptions for simple
linear regression are satisfied hence we can interpret the results as follows:
Result: Intercept = -735.34, Slope = 13.69, R2 = 0.0155, F1,108 = 1.7, P < 0.1951.
Equation for the straight line: Clutch Size = -735.34 + 13.69*Abelly

H0 should not be rejected, there is no significant linear relationship between the clutch size
and the redness of the female’s bellies (figure 9) (F1,108 = 1.7; P = 0.1951). When all data had
been checked a linear regression was made (figure 9).

References:
Crawley MJ (2007) The R book. 1. England: John Wiley & Sons Ltd.
Forsgren E, Amundsen T, Borg AA, Bjelvenmark J (2004) Unusually dynamic sex roles in a
fish. Nature 429:551-554
Svensson PA, Blount JD, Forsgren E, Amundsen T (2009) Female ornamentation and egg
carotenoids of six sympatric gobies. J Fish Biol 75:2777-2787