Categorical data and statistical power

Last updated on 2025-04-01 | Edit this page

Estimated time: 10 minutes

Overview

Questions

  • What determines the power of a chi-square test?
  • Why is it difficult to determine significance for discrete data?
  • What is the Fisher test and when should I use it?

Objectives

  • Introduce Fisher’s exact test.
  • Explain difficulties with significance testing for discrete p-values.

Statistical power

Statistical power is the probability that an effect is detected as significant with a given method, provided that the effect really exists.

In the lesson on hypothesis testing, we learned that the power depends on

  • the size of the effect,
  • the significance level,
  • the method we choose for testing, and
  • the sample size.

In this lesson’s example, the estimated association is not quite small (\(6\%\) difference in proportions, or the risk for getting the disease is more than twice as high for exposed mice), the significance level was the usual \(5\%\), and a sample size of 200 doesn’t seem small, either. Still, the effect is not significant. Does this mean that the \(\chi^2\)-test is useless?

They key here is the low overall probability of mice contracting the disease. Even though there were 200 mice, only a total of 14 out of them actually contracted the disease. And it’s this low number of cases that limits the power of the study.

In fact, if there was a sample of 2000 mice, and only 14 of them caught the disease, then the p-value would hardly change:

R 

chisq.test(rbind(
  c(4,996),
  c(10,990)),
correct=FALSE)

OUTPUT 


	Pearson's Chi-squared test

data:  rbind(c(4, 996), c(10, 990))
X-squared = 2.5896, df = 1, p-value = 0.1076

Intuitively, we still have 14 cases, and the question is how we’d expect them to distribute among the two groups (exposure and no exposure) by chance.

Thus, the power of the study is limited by the lowest counts in the table. We could also say that extreme frequencies (a low probability of one outcome always corresponds to a high probability of the other outcome) are unfavorable in terms of power. To overcome the problem of a low number of cases, clinical studies are often conducted as case-control studies. In those, the same number of cases and controls are sampled by design. One can then compare the occurrence of certain risk factors (e.g. smoking, exposure, …) between the two groups.

The Fisher test

There is another test, called Fisher test, which is also used on contingency tables, and which tests exactly the same hypothesis as the \(\chi^2\) test. It doesn’t sum up the differences between observed and expected counts, but instead calculates the probability of the observed data using a hypergeometric distribution. The hypergeometric distribution is similar to the multinomial distribution (except that it assumes draws without replacement) and gives exact probabilities for counts to fall into different categories (in this case the cells of the table) under the assumption of independence. See here for more details on the Fisher test.

Apply the Fisher test

Consider the table we’ve been looking at throughout this lesson.

diseased healthy
not exposed 4 96
exposed 10 90

Look up the help for fisher.test. Can you apply this test on the above data? How do you interpret the result? How does it compare to the outcome of the \(\chi^2\) test?

R 

fisher.test(rbind(c(4,96), c(10,90)))

OUTPUT 


	Fisher's Exact Test for Count Data

data:  rbind(c(4, 96), c(10, 90))
p-value = 0.164
alternative hypothesis: true odds ratio is not equal to 1
95 percent confidence interval:
 0.08322597 1.36465456
sample estimates:
odds ratio
 0.3767541

The p-value for the Fisher test is higher than for the \(\chi^2\) test (without correction), but almost identical to the \(\chi^2\) test with Yates continuity correction (see below). The conclusion stays the same: Not significant at \(\alpha=0.05\). The fisher.test function also gives an odds ratio \(\approx 0.37\). The part that I don’t like about this output is that we have to guess which odds are being compared. If you look up the measures for association, you’ll notice that the calculated odds ratio for the same table was \(\approx 2.7\). There we looked at the odds of getting the disease in case of exposure, relative to non-exposure, because this seemed natural. In the fisher.test function, the ratio was calculated the other way round. We could change this by switching the rows of the table:

R 

fisher.test(rbind(c(10,90),c(4,96)))

OUTPUT 


	Fisher's Exact Test for Count Data

data:  rbind(c(10, 90), c(4, 96))
p-value = 0.164
alternative hypothesis: true odds ratio is not equal to 1
95 percent confidence interval:
  0.7327862 12.0154803
sample estimates:
odds ratio
  2.654251

Alternatively, we can calculate the inverse of the result:

R 

1/0.3767

OUTPUT 

[1] 2.654632

If you do your research on which test to use (e.g. ask google), you’ll stumble upon statements like “the Fisher test is more conservative”, or “the \(\chi^2\)-test overestimates significance for small counts”. What does this mean? These statements both concern errors in hypothesis testing: If the Fisher test is conservative, this means that it is prone to failing to detect cases where an effect exists, i.e. it has a low power. On the other hand, if the \(\chi^2\)-test overestimates significance, then we have a problem with false positives.
In the next sections, we’ll learn why p-values are a bit tricky to interpret for categorical data, and how this leads to discussions on when to use which test.

Significance for discrete data

Let’s recap what it means to choose a significance level of \(\alpha=5\%\): We decide to reject the null hypothesis if the probability of seeing the observed data under the null is \(\leq5\%\).

For continuous probability distributions like the t-distribution used in the t-test, this means that the probability of getting a p-value \(<0.05\) is \(5\%\), the probability of getting a p-value \(<0.1\) is \(10\%\), and so on. This is the definition of the p-value.

So if you have \(1000\) scenarios where nothing is going on and calculate a p-value using a one-sample t-test:

R 

set.seed(30)
pvals <- rep(NA,1000)
for(i in seq_along(pvals)){
  pvals[i] <- t.test(rnorm(100), mu=0)$p.value
}

Then in roughly \(10\%\) of the cases, the p-value is below \(0.1\):

R 

mean(pvals <0.1)

OUTPUT 

[1] 0.111

Then in roughly \(20\%\) of the cases, the p-value is below \(0.2\):

R 

mean(pvals <0.2)

OUTPUT 

[1] 0.218

If you make a histogram of the p-values, it will be roughly uniform:

R 

data.frame(pvals) %>% 
  ggplot(aes(x=pvals))+
  geom_histogram()

OUTPUT 

`stat_bin()` using `bins = 30`. Pick better value with `binwidth`.

Unfortunately, for discrete data, this is not actually the case.

The following simulation is an attempt to demonstrate this. Suppose we test the association between the preferred animal of a person (we give the choice between cat and dog), and their handedness. And let’s say that being left-handed gives absolutely no information on whether a person likes cats more than dogs, i.e. the null hypothesis of independence is true. Further, we assume that the probability of being left-handed is \(P(L)=0.1\) and the probability of preferring cats is \(P(C)=0.6\). The hypothetical experiment will sample 50 left-handed and 50 right-handed persons, and ask them about their preference for cats or dogs. With this information, we can simulate such a study using rbinom for generating random binomial numbers:

R 

set.seed(30)
n <- 25
cats_l <- rbinom(n=1, size=n,prob = 0.6)
dogs_l <- n-cats_l
cats_r <- rbinom(n=1, size=n,prob = 0.6)
dogs_r <- n-cats_r

simtable <- rbind(c(cats_l, dogs_l),
                  c(cats_r, dogs_r))

We can also compute the p-value from this table:

R 

chisq.test(simtable)

OUTPUT 


	Pearson's Chi-squared test with Yates' continuity correction

data:  simtable
X-squared = 0.35651, df = 1, p-value = 0.5505

To see how the p-values behave, let’s run 2000 such simulations and report the p-value. For this, I use a for-loop around the above code. In each iteration of the loop, a contingency table under always the same assumptions is generated, and the p-value is stored in a variable calle p_vals.

R 

set.seed(20)
N=2000
p_vals <- rep(NA,2000)

for(i in seq_along(p_vals)){ # as many iterations as p_vals is long
  cats_l <- rbinom(n=1, size=n,prob = 0.9)
  dogs_l <- n-cats_l
  cats_r <- rbinom(n=1, size=n,prob = 0.9)
  dogs_r <- n-cats_r
  
  p_vals[i] <- chisq.test(rbind(c(cats_l, dogs_l),
                    c(cats_r, dogs_r)))$p.value
}

As a result of the simulation, p_vals contains 2000 p-values under the assumption that the null hypothesis is true.

R 

head(p_vals)

OUTPUT 

[1] 1 1 1 1 1 1

What percentage of the p-values is smaller than \(0.05\)?

R 

mean(p_vals<0.05, na.rm=TRUE)

OUTPUT 

[1] 0.004515805

We can also make a histogram of all the p-values in the simulation:

R 

data.frame(p_vals) %>% 
  ggplot(aes(x=p_vals))+
  geom_histogram()

In theory, the histogram should show a uniform distribution (the probability of getting a p-value \(<0.05\) is \(5\%\), the probability of getting a p-value \(<0.1\) is \(10\%\), and so on…). But here, instead, the p-values are discrete: They can only take certain values, because there’s only a limited number of options how 25 observations can fall into two categories (dogs/cats).

Since for small counts, the chances to get a p-value \(p<0.05\) are actually below 5% – and this holds also for other values of \(p\) which are not exactly \(0.05\) – we say that the Fisher test is conservative. It means that interpreting the discrete hypergeometric probabilities as continuous p-values will lead us to overestimated p-values. This also holds true when the null hypothesis is actually false, so we’re reducing our chances to run a significant test, even if there’s something to detect.

For a more detailed explanation, see An introduction to Categorical Data Analysis by Alan Agresti.

Now what?

The Fisher test is conservative, especially for low counts. The \(\chi^2\)-test doesn’t have this problem, but at the same time should not be applied to tables with low counts, because it might produce false positives (see the rule of thumb). So what should we do? There is not one commonly accepted answer to this. Some suggestions are:

  • Use Fisher’s exact test and accept that it can be conservative. In the end, low counts indicate a small amount of evidence anyways, and you might need to collect more data to detect or confirm an effect.
  • In Agresti’s book, you’ll find mid p-values and similar methods that aim at correcting the p-value for its discreteness.
  • You can use the \(\chi^2\)-test in conjunction with Yates continuity correction. It applies a correction on the \(\chi^2\) statistic for tables with low expected counts to avoid underestimating the p-value. This method is not perfect either, and can lead to over-correcting.

The important part is that you don’t choose the method based on the p-value it produces for your specific data set. The worst thing you can do is try 3 different methods, choose the one with the lowest p-value and report this one. This is a form of p-value hacking called method hacking and increases your chances to produce false positive results. If your test comes out significant in only one of the tests you tried, be honest and report all of them, or otherwise investigate further, for example by collecting more data.