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I apologize if this questions is poorly worded or has a very simple answer but I work in a field very different than statistics. I am looking at diction (word use) in books from two different artistic movements over time. I want to compare words that were first used in each century by each set of authors. For example both sets use words from the 18th century at a rate of 15.5 and 15.2 percent of their total word use. How do I find out if this .3 percent means anything? Would I be looking at effect size? And, if so, what measure should I use?

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  • $\begingroup$ Well 0.3 percent sounds like a trivial difference to me, but I'd not in your field. Does it sound like a meaningful difference to you? $\endgroup$
    – Glen_b
    Apr 11, 2015 at 14:07
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    $\begingroup$ To be honest it does not, but I am worried about assuming triviality. And also, I would like to develop the skills to figure out what threshold would not be trivial. $\endgroup$ Apr 11, 2015 at 14:15
  • $\begingroup$ It would come from subject-area knowledge, numbers don't carry inherent meaningfulness. If I was in high energy physics that 0.3% might be a big change (it might be much larger than effects they look for there), but in some other subject areas it could be so tiny as to be not worth mentioning. $\endgroup$
    – Glen_b
    Apr 11, 2015 at 14:21
  • $\begingroup$ Thanks glen, I appreciate the comments. I think part of what I am trying to figure out is what constitutes a big change. Im on a fact finding mission to more or less see what areas I should be looking at and if statistics can help. $\endgroup$ Apr 11, 2015 at 14:29

3 Answers 3

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Look up Cohen's $h$. First, you calculate $h$, which is pretty straightforward:

$$h = 2\times\arcsin\left(\sqrt{p_1}\right) - 2\times\arcsin\left(\sqrt{p_2}\right)$$

Where $p_1$ and $p_2$ are the two proportions.

Then, you have to decide on a cutoff. The "rule of thumb" cutoff is that if $h \ge 0.2$, then you have something interesting. Though in your particular science, a different cutoff might be more appropriate.

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You can test whether two proportions are equal using a test like the one described here.

This is implemented in R in the prop.test function. For example, your data might look like this:

# count of target words for author 1 and author 2
word_counts <- c(27, 46) 

# total words for author 1 and author 2
total_words <- c(173, 302)


# here are the proportions for the two authors:
> word_counts/total_words
[1] 0.1560694 0.1523179

This tests the null hypothesis that the rate in both groups is the same:

> prop.test(x=word_counts, n=total_words)

    2-sample test for equality of proportions with continuity correction

data:  word_counts out of total_words
X-squared = 2.2201e-30, df = 1, p-value = 1
alternative hypothesis: two.sided
95 percent confidence interval:
 -0.06757982  0.07508279
sample estimates:
   prop 1    prop 2 
0.1560694 0.1523179 

Note that if the sample size is large enough, even this small difference in proportions can be significant:

# count of target words for author 1 and author 2
word_counts <- c(27000, 46000) 

# total words for author 1 and author 2
total_words <- c(173000, 302000)

> prop.test(x=word_counts, n=total_words)

    2-sample test for equality of proportions with continuity correction

data:  word_counts out of total_words
X-squared = 11.873, df = 1, p-value = 0.0005696
alternative hypothesis: two.sided
95 percent confidence interval:
 0.001609876 0.005893091
sample estimates:
   prop 1    prop 2 
0.1560694 0.1523179
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There are a few ways to do this, but I think that the way to go there is making a contingency table with the counting and applying a chi-square test to check if counting of the sets differed on the centuries.

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