# Bootstrapping regression slopes - p-value and significance level

I have a data set showing an (exponential) increase in the size of an animal population over time. I can fit an exponential model to these data and obtain an estimate of the population growth rate (under the assumption of unconstrained exponential growth).

However, I would like to get a handle on how variable this estimate is however. To do this, I have written an R script which conducts some Bootstrap resampling of the initial observations and harvests the value of the population growth rate for each Bootstrap resample. I am achieving this by fitting a lm of the form log(abundance)~Year.

While the initial exponential model was highly significant (p<<0.05), this is not necessarily the case in all Bootstrap resamples.

How can I construct reliable confidence intervals for the population growth rate, knowing that sometimes (in some runs) it is found to be non-significant at alpha=0.05?

I see 3 options here: 1) Generate a distribution of growth rate values based on all resamples, regardless of whether the regression was significant or not; 2) Only base the CI on the subset of resamples where the regression was significant; 3) Assign a value of zero to the regression coefficient whenever its associated p-value is larger than the chosen significance threshold, i.e. alpha=0.05.

I have only found one published paper that uses method 3) [Austin (2007). Using the bootstrap to improve estimation and confidence intervals for regression coefficients selected using backwards variable elimination. STATISTICS IN MEDICINE, 27, VOL. 17:3286-3300].

Is this a valid approach?

Thanks!

• My answer here: how-do-i-compare-bootstrapped-regression-slopes, may help you. For a CI on the slope, #1 is the way to go. It is fine that some bootsamples are not significant. "The difference between significant and not significant is not itself statistically significant" -Gelman. – gung Nov 13 '13 at 0:30
• Usually you would count all cases, significant and not - though it can depend on the actual question you're trying to answer. However, I'm concerned that you don't seem to be using weights at all in your model; it would surprise me if the log-counts had nearly constant variance across time. Do they? You might consider whether a GLM is worthwhile, or at least iterating your weights once or twice. – Glen_b Nov 13 '13 at 0:48
• Thanks for your help! The log counts seem to satisfy variance assumptions - all good! – PhB74 Nov 13 '13 at 10:57