 There are no “groups” in simple linear regression; thus, you do not assess “within” and “amonggroup” independence here.
 You should begin to use the ANOVA table pvalue rather than the summary table slope pvalue to assess the relationship as this will generalize to the next module.
 When describing the relationship, use the confidence interval for the slope. Don’t just use the slope.
 I am fine with interpreting the slope on the transformed scale as long as you remember to include language about the logs. However, the slope can be backtransformed to describe the relationship on the original scale. When backtransforming the slope, however, note that “a change in log(x) of 1” should be backtransformed to “a change in x of e^{1}” which is where the 2.72 in answer 4 below came from. Also, note that the backtransformed slope indicates a “multiplicative” change such that the value of Y is multiplied by the backtransformed slope rather than added to. This is shown below.

The predictions should use confidence intervals as the original authors were interested in predicting the MEAN peak frequency. Also make sure that you backtransform the result from
predict()
as it returns the LOG mean peak frequency.
Rattlesnake Rattling
It is difficult to ultimately assess independence with the amount of information given. However, under the assumption that all Rattlesnakes were unique (no snake was measured twice) and they did not come from a single group (i.e., “nest”) of snakes, then it seems that the data are at least roughly independent. There is evidence for a strong nonlinearity and heteroscedasticity (Figure 1Right), the residuals do not appear to be normal (Anderson Darling p=0.0029) and are rightskewed (Figure 1Left), and significant outliers are evident (outlier test p=0.0012). A transformation will be explored to see if the assumptions can be met.
The authors suggested that the relationship would be exponential but logging just the peak frequency of the rattle variable did not result in meeting all of the assumptions. However, when both variables were logtransformed there is no visual evidence for nonlinearity (Figure 1Right), the residuals appear roughly homoscedastic (Figure 1Right) and approximately normal (AndersonDarling p=0.7765; Figure 1Left), and there is no evidence for significant outliers (outlier test p=0.0844). Thus, the assumptions appear to be adequately met on the loglog scale.
There is a signficant relationship between log peak frequency of the rattle and log weight of Rattlesnakes (p<0.00005; Table 1; Figure 3).
Specifically, as the log weight of the Rattlesnakes increases by one unit, the average log peak frequency of the rattle decreases between 0.099 and 0.188 units.
On the original scale, as the weight of the Rattlesnake increases by a multiple of 2.72 cm, the average peak frequency of the rattle changes (decreases) by a multiple of between 0.829 and 0.906.The predicted mean peak frequency for all 454 g Rattlesnake is between 5.93 and 6.59 kg.
Figure 1: Histogram of residuals (left) and residual plot (right) for regression of peak frequency of rattles on weight of Rattlesnakes.
Figure 2: Histogram of residuals (left) and residual plot (right) for regression of logtransformed peak frequency of rattles on logtransformed weight of Rattlesnakes.
Figure 3: ANOVA table for the linear regression of logtransformed peak frequency of rattles on logtransformed weight of Rattlesnakes.
Table 1: ANOVA table for the linear regression of logtransformed peak frequency of rattles on logtransformed weight of Rattlesnakes.
Df Sum Sq Mean Sq F value Pr(>F)
weight 1 13.706 13.7061 8.9647 0.007785
Residuals 18 27.520 1.5289
R Appendix.
rdf < read.csv("Rattlesnakes.csv")
lm1 < lm(freq~weight,data=rdf)
transChooser(lm1)
rdf$logw < log(rdf$weight)
rdf$logf < log(rdf$freq)
lm2 < lm(logf~logw,data=rdf)
anova(lm2)
summary(lm2)
confint(lm2)
p.logf < predict(lm2,data.frame(logw=log(454)),interval="confidence")
exp(p.logf)*exp(anova(lm2)[2,3]/2)