Plotting confidence intervals on graphs

So: you’ve got some data, you’ve read up on confidence intervals and you’re convinced. Your data is a small sample from a large/infinite population (all of contemporary US English, say), and therefore you need to estimate the error in every observation. You’d like to plot a pretty graph like the one below, but you don’t know where to start.

An example graph plot showing the changing proportions of meanings of the verb think over time in the US TIME Magazine Corpus, with Wilson score intervals, after Levin (2013). Many thanks to Magnus for the data!

Of course this graph is not just pretty.

It depicts a pattern whereby two synchronically distinct uses become four over time. Note that for any pair of points across a diachronic contrast we can immediately identify the following:

  • non-overlapping intervals must be statistically distinct (a significant difference);
  • if any point falls within the interval of another, it cannot be significantly distinct (a non-significant difference);
  • in all other cases we need to carry out a 2 × 2 test (either Yates’ χ² test or the Newcombe-Wilson continuity-corrected test) to check.

Probabilities drawn from the same sample sum to 1. To compare points in competition synchronically you should use a single sample z test instead of a 2 × 2 test.

In this case, the 1960s data does not significantly distinguish the rates for quotative and interpretive uses of think because the p value for the quotative use is within the interval for the interpretive. A quick check with the 2 × 2 spreadsheet finds that

  • the initial fall (from 1920s to 1960s) for ‘cogitate’ uses is significant but
  • intention does not change significantly over time (note how the curved line can be misleading), and
  • quotative uses significantly increase their share from the 1960s to 2000s.

Other changes can be easily identified in the graph. Note that this graph expresses semasiological (meaning distribution) change, not onomasiological (choice of alternates) change, and results are therefore indicative. We can’t conclude that speakers in later texts increasingly preferred to employ think in a quotative way, without considering this question relative to the opportunity to employ quotative constructions. See Choice vs. use.

Let us discuss how we arrived at this graph.

Step by step

We want to plot the observed probability p with Wilson score interval error bars. We can’t use the Gaussian interval (some values are zero) and anyway, as other posts clarify, it is wrong to do so!

  1. First we gather the raw data. We need to identify the raw frequencies, F, and the relationship between the different data series. Does it make sense to take proportions out of the total frequency (N)? What should the baseline be for any change?
  2. If we use the total number of cases of think, N, as a meaningful baseline, we can obtain a set of semasiological probabilities for each frequency, p = F/N.
  3. Next we calculate basic Wilson score interval terms. This is the most complicated step and can be broken down into two components for simple calculation.
    • Wilson centre adjusted probability
      p’ = [p + z²/2N] / [1 + z²/N].
    • Wilson adjusted standard deviation
      s’ = √p(1 – p)/N + z²/4 / [1 + z²/N].

    Note that we could simplify each expression further and pre-calculate the Wilson denominator [1 + z²/N] for every cell.

  4. We can now calculate the upper and lower bound of the interval in absolute terms:
    • Wilson score interval
      [w⁻, w⁺] = [p’ – z.s’, p’ + z.s’].
  5. Finally we can work out the upper and lower bounds relative to the probability p. Excel likes these both to be positive, so we have the following:
    • Wilson relative error bars
      [Y⁻, Y⁺] = [pw⁻, w⁺ – p].

Instead of steps 3 and 4 above you can also use the continuity corrected formula for the Wilson score interval. The formula is equation (7) in (Wallis 2013) and is implemented in the 2 × 2 spreadsheet. It is also implemented in the spreadsheet for this example.

  • For an efficient calculation the continuity-corrected Wilson standard deviation can be rewritten as
    s’ = (√a ± b + 1)/d,
    where a = z² – 1/N + 4Np(1 – p), b = 2–4p and d = 2[N+z²]. The sign of b is negative for the lower bound and positive for the upper bound, and the interval is limited to [-1, +1].

The continuity-corrected interval is slightly more conservative and corresponds to Yates’ 2 × 1 χ² test. For most plotting purposes, however, the standard Wilson interval is usually perfectly adequate.

See also

References

Levin, M. 2013. The progressive in modern American English. In Aarts, B., J. Close, G. Leech and S.A. Wallis (eds). The Verb Phrase in English: Investigating recent language change with corpora. Cambridge: CUP. » Table of contents and ordering info

Wallis, S.A. 2013. Binomial confidence intervals and contingency tests: mathematical fundamentals and the evaluation of alternative methods. Journal of Quantitative Linguistics 20:3, 178-208 » Post

» Part 2: Change and certainty

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