# Wide or narrow confidence interval

I have a project in which I am looking at the correlation betwen fortnightly income and the amount saved each fortnight by low income workers.

I am reporting the effect size and the confidence interval. One of my results is as follows: r (214) = 0.34, p < 001, 95% CI [0.22, 0.45]

I know it is accepted that a narrow confidence interval is better than a wide confidence interval.

However, I am unsure if this is a wide or a narrow confidence interval.

Is there a quick way to determine what is 'wide' or 'narrow' in this context.

• Re: "Is there an easy way to determine what is 'wide' or 'narrow' in this context." - what exactly is "this context"? You've given almost no information. At this point the answer is entirely subjective. – Macro May 10 '12 at 19:24
• I have updated my question with additional details. – Amarald May 10 '12 at 19:42
• So this is a confidence interval for a correlation (which thus has a range of [-1, 1])? If so, you have unfortunately stepped out of one hole and into another: what correlation values are meaningful. It totally depends on the field in question, with some fields requiring high correlations (say, 0.9), while others being happy to find relatively low correlations (say 0.3). In most fields, I'd think that a correlation of less than 0.5 isn't that remarkable. – Wayne May 10 '12 at 20:42
• Thanks for the update - much better question. Now though, I think the issue is that the correlation coefficient is not the best way to describe the relationship between these two variables. A better way would be a regression of some sort - then you could report the slope coefficient and have a confidence interval for that instead. You will also have a problem that income and savings variables are usually quite skewed distributions. The confidence interval for your correlation coefficient probably assumes they (or one of them) is normally distributed and hence will be incorrectly calculated. – Peter Ellis May 10 '12 at 20:46
• I voted this question up because it is clearly stated, even though (a) it is very close to a series of questions asked and answered last year and (b) it may be based on a misconception about what may be feasible, as @Peter Ellis nicely explains in a reply. – whuber May 10 '12 at 22:40

Often, a number is useful only in reference to other numbers. For example, customers rated their satisfaction as 9.1 out of 10 sounds good, but not when you hear that elsewhere in the industry reported satisfaction of 9.5/10 is the norm.

The same applies to the width of confidence intervals. Unless you have a reference of some sort, as @Macro says there is no way of providing an answer to your question. It looks like you have some statistical output that compares your value to zero and says you are unlikely to have gotten the result you have if the real value is zero (this is the p value). This may or may not be of interest.

If your main research question is "can we find evidence that this parameter is not zero", then your confidence interval is plenty narrow enough. If your research question is "can we decide between X who claims this number is 0.3 and Y who claims it is 0.4" then it is too wide.

To explain the last sentence further. If all you want is to establish that there is some relationship between income and savings, you have already done it, because your confidence interval is a long way from zero. So there is no need to look for a smaller confidence interval from more data, etc. If, however, you are interested in the nature of that relationship, eg because there is a theoretical dispute about whether the correlation coefficient is really 0.3 or 0.4, then you do not have enough data to answer the question. Both alternatives are in your confidence interval - you will need to find more data.

In my comments on the original question however, I suggested that the correlation coefficient is not a good way of showing the relationship between two variables like this. Consider the two examples shown below. The first has a very high correlation coefficient and a negligible confidence interval, yet the relationship between savings and income is what I would call weak - an increase in income does not lead to much of an increase in savings. The second plot is the other way around - there is a lot more (realistic) noise in the data, but an increase in income leads to quite a significant increase in savings. For most purposes, you want to say the second plot shows a stronger relationship than the first. To do this you need to fit a regression model of some sort. The R code that produced these plots is below.

income <- exp(rnorm(100,6,.5))
summary(income)
savings1 <- 80 + income*.05
savings2 <- 8 + income*.5 + exp(rnorm(100,4.5,.5)) + rnorm(100,0,150)

win.graph(10,5)
text(200,1500, "r=1.00")
text(200,1300, "Confidence interval for r=[1,1]")
text(200,1100, "Slope confidence interval=[0.05, 0.05]")

library(boot)
test <- data.frame(income, savings2)
test.b <- boot(test, function(x,w){cor(test[w,1], test[w,2])}, R=1000)
ci <- round(as.vector(round(quantile(test.b\$t, prob=c(0.025, 0.975)),2)),2)
ci2 <- round(as.vector(confint(lm(savings2~income))[2,]),2)