# Negative value in CI after bootstrap

I performed bootstrap (10 000 samples) in boot package for R (multivariate model, binary logistic regression with Firth´s correction). One variable in the model has after bootstrap negative value in confidence interval, i.e. CI 95% = (-0,403, 1,103). Can someone explain please what does it mean and how to interpret it? The variable is birth weighth and the model should estimate predictors for mortality.

Thanks!

• That means that the true value is contained in the interval (-0.403, 1.103) with 95% probability. This means some of the likely true values of your parameter are zero and possibly even negative. In your case, this is likely the result of model misspecification or an inefficient estimator. Aug 14, 2016 at 14:52
• Thank you! But how can be birth weighth negative? Would it be not better to report, as logic dictates, CI in this case (0 - 1,203)? Aug 14, 2016 at 14:56
• That CI does not mean that the birth weight is negative. It means that the impact of birth weight is negative in your model. That is still counterintuitive, and while possible you have to ensure that your model is correctly specified to rule out misspecification biases. Aug 14, 2016 at 15:30
• @tchakravarty: the estimate itself seems to be positive. A negative lower bound could just reflect small sample size and/or collinearity with other predictors. Aug 14, 2016 at 16:24
• @MichaelM Yes, also called an inefficient estimator. Note that OP wants an interpretation of the CI, not the estimate. Aug 14, 2016 at 16:34

As far as I can tell from your description, this question isn't at all specific to the details of your analysis (bootstrap confidence intervals, Firth correction) but is a general issue of interpreting the coefficients from logistic regression.

Let's suppose that you're using the defaults, i.e. you have taken the coefficients as given by R which are by default on the logit or log-odds scale. If the 95% CI on the effect of birth weight is (-0.403, 1.103) that means that the effect of a one-unit increase in birth weight (let's say it's measured in kg for the moment) on the log-odds of mortality is in this range. You could also state this as a proportional, or multiplicative, change in the odds of mortality of exp(c(-0.403,1.103)) = c(0.668,3.013) - this means the odds could be anywhere between decreased by 1/3 to increased threefold by a 1-kg increase in birth weight.

As the comments above point out, the fact that the 95% confidence intervals encompass zero implies that this is a non-significant result at the usual cutoff value of $p=0.05$. In most contexts, this would be considered a fairly large confidence interval (a decrease of 1/3 or a tripling of the odds is a large change in the outcome); however, the interpretation depends entirely on the size of your unit of birth weight.

• If, for example, the birth weight is measured in kg, but the entire range of birthweights in your sample is only $\pm 0.1 \textrm{kg}$, then you should think about the CI per 0.1 kg (i.e., divide the values by 10); this is one reason people sometimes prefer to scale continuous predictors by their standard deviation.
• If on the other hand 1 kg is a typical change of birth weight in your data set, then this result says you have little information about the effect of birth weight - this could be because you have a small data set, or an inefficient study design, or other covariates that are strongly correlated with birth weight, or ...

The actual effect on the probability depends on the baseline probability (i.e., the values of all the other predictors and the intercept). For more on interpreting log-odds, see this FAQ on the UCLA statistics site.

• Thank you very much! For completion, birth weighth was measured in kg, range is 1.8 - 4.6 (kg), p = 0.011, at cut-off p = 0.05. Sample size, N = 176. Aug 14, 2016 at 17:34
• What I wonder is, why was CI before bootstrap with the same data in positive range (0.157, 0.791), p = 0.011 and after bootstrap (N = 10 000 samples) shifted CI-L into negative values. Aug 14, 2016 at 17:41
• I agree that it's quite surprising that you have a p-value of 0.011 and such a large bootstrap confidence interval. (The (0.157, 0.791) range you give there is the Wald confidence interval?) It suggests that the p-value (which will probably be based on Wald statistics) might not be reliable, although one would have to see the whole analysis to be sure ... this might end up being more of a StackOverflow ("is my code working?") than a StackExchange ("what does this mean statistically?") question ... Aug 14, 2016 at 17:42
• Yes, it´s Wald confidence interval. But I think bootstrap has to play some role, because before bootstrap, CI was in range exactly as biological plausibility predicts (it´s known, that a low birth weighth is a risk factor for mortality after Norwood operation in children with hypoplastic left heart syndrome). p-value has changed minimal from 0,01061 before bootstrap to 0,01125..... it doesn´t correlate with such dramatic change in confidence interval.... could I just report the results as you stated above, with exp, i.e. CI = (0.668, 3.013)? Aug 15, 2016 at 10:25