Note:
  • On the first question, you must use the p-value from the ANOVA table. That is the p-value that assesses whether all group means are equal or not. Tukey’s multiple comparisons are only used to assess difference in paired means AFTER it has been determined that there is a difference in means. The linear models coefficients table is not appropriate for answering that question either as it does not assess all pairs of means (you should get out of the habit of using summary() on your lm() results).
  • The second question should include a plot of means with appropriate significance letters. Note that when using Tukey’s method that there should be letters on each point. With Dunnett’s method it may be appropriate to leave one point without a letter, but this should be explained in the figure label.
  • In the second to last questions, make sure to clearly indicate which group is greater (or lesser). Don’t just say that you are 95% confident that the difference is between such-and-such.
  • I round my p-values to four decimal places. As such, if the fifth decimal place is less than 5, then the p-value would round to “0.0000”, which does not make much sense. Thus, if the fifth decimal place is less than 5 then I will report my p-values as “<0.00005.” You will see examples of this in the answers below.
  • Note how concise the answer key is. Please work to get your answers this concise.

Temperature and Turtle Hatchlings II

  1. The mean days to hatch is significantly different among the four temperature groups (p<0.00005; Table 1).
  2. It appears that the mean days to hatch for the 30oC group is significantly lower than the mean days to hatch for 15oC (p<0.00005), 20oC (p=0.0008), and 25oC (p=0.0001) groups (Table 2). The mean days to hatch for all other pairs of temperature groups are not significantly different (Table 2). These results are shown visually in Figure 1.
  3. The mean days to hatch for sea turtles at 30oC is between 22.7 and 53.9 days lower than that at 15oC, between 8.9 and 40.1 days lower than that at 20oC, and between 13.5 and 44.7 days lower than that at 25oC (Table 3).
  4. It appears that mean days to hatch is longer at 15, 20, and 25oC than at 30oC. Thus, it appears that the days to hatch will not decrease substantially until temperatures rise above somewhere between 25 and 30oC.

Table 1: Analysis of variance table for the days to hatch by temperature group.

            Df Sum Sq Mean Sq F value    Pr(>F)
Temperature  3 8025.5 2675.16  15.978 9.082e-07
Residuals   36 6027.3  167.42                  

Table 2: Tukey’s multiple comparison results for the days to hatch by sea turles at various temperature groups.

            Estimate Std. Error    t value      p value
20 - 15 = 0    -13.8   5.786623 -2.3848108 9.819862e-02
25 - 15 = 0     -9.2   5.786623 -1.5898739 3.970328e-01
30 - 15 = 0    -38.3   5.786623 -6.6187141 4.147556e-07
25 - 20 = 0      4.6   5.786623  0.7949369 8.562519e-01
30 - 20 = 0    -24.5   5.786623 -4.2339033 7.910900e-04
30 - 25 = 0    -29.1   5.786623 -5.0288402 6.706843e-05

Figure 1: Plot of mean (with 95% CI) days to hatch by sea turtles at various temperature groups. Different letters indicate means that are significantly different.

Table 3: Tukey’s confidence interval results for the difference in mean days to hatch by sea turtles for each pair of temperature groups.

        Estimate       lwr        upr
20 - 15    -13.8 -29.38918   1.789182
25 - 15     -9.2 -24.78918   6.389182
30 - 15    -38.3 -53.88918 -22.710818
25 - 20      4.6 -10.98918  20.189182
30 - 20    -24.5 -40.08918  -8.910818
30 - 25    -29.1 -44.68918 -13.510818

R Appendix.

library(multcomp)
d <- read.csv("turtles.csv")
d$Temperature <- factor(d$Temperature)
lm1 <- lm(Days~Temperature,data=d)
anova(lm1)
mc1 <- glht(lm1,mcp(Temperature="Tukey"))
summary(mc1)
confint(mc1)
fitPlot(lm1,xlab="Temperature (C)",ylab="Days to Hatch")
addSigLetters(lm1,c("b","b","b","a"),pos=c(2,2,4,4))