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A New Rule of Thumb for 2 × 2 Tables with Low Expected Counts

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A New Rule of Thumb for 2×2 Tables with Low Expected Counts

Bruce Weaver

Northern Health Research Conference

June 4-5, 2010

- I have no affiliations, sponsorships, honoraria, monetary support or conflict of interest from any commercial source.
- However…it is only fair to caution you that this talk has not undergone ethical review of any sort.
- Therefore, you listen at your own peril.

“One of the commonest problems in statistics is the analysis of a 2×2 contingency table.”

Ian Campbell

(Statist. Med. 2007; 26:3661–3675)

What’s a contingency table?

See the example on the next slide.

What the heck is malocclusion?

Counts in the cells

Class IOcclusion. Normal occlusion. The upper teeth bite slightly ahead of the lowers.

Class II Malocclusion. Upper teeth bite greatly ahead of the lower teeth—i.e., overbite.

Class IIIMalocclusion. Upper front teeth bite behind the lower teeth—i.e., under-bite.

What statistical test can I use to analyze the data in my contingency table?

It depends.

- The most common statistical test for contingency tables is Pearson’s chi-squared test of association.

Karl Pearson

Greek letter chi

Observed count

Sum

Expected count

- It is well known (to those who know it well)* that Pearson’s chi-square is an approximate test

A typical chi-square distribution

- The sampling distribution of the test statistic (under a true null hypothesis) is approximated by a chi-square distribution with df = (r-1)(c-1)

- The approximation becomes poor when the expected counts (assuming H0 is true) are too low

* Robert Rankin, author of The Hollow Chocolate Bunnies of the Apocalypse.

How low is too low for expected counts?

It depends.

Again, it depends! This guy is starting to get on my nerves.

- A common rule of thumb for when it’s OK to analyze a 2×2 table with Pearson’s chi-squared test of association says:
- All expected counts should be 5 or greater
- If any expected counts are < 5, another test should be used

- The most frequently recommended alternative test under point 2 above is Fisher’s exact test (aka the Fisher-Irwin test)

- The standard rule of thumb for 2×2 tables dates back to Cochran (1952, 1954), or even earlier
- But, the minimum expected count of 5 appears to have been an arbitrary choice (probably by Fisher)
- Cochran (1952) suggested that it may need to be modified when new evidence became available.
- Computations by Ian Campbell (2007) have provided some new & relevant evidence.

- Three distinct research designs can give rise to 2×2 tables
- Barnard (1947) classified them as follows:

G.A. Barnard

- Model I:
- Model II:
- Model III:

Both row & column totals fixed in advance

Row totals fixed, column totals free to vary

Both row & column totals free to vary

“Here, there is no dispute that the Fisher–Irwin test … should be used.”

“This last research design is rarely used and will not be discussed in detail.”

Ian Campbell

(Statist. Med. 2007; 26:3661–3675, emphasis added)

- Model II
- Sometimes called the 2×2 comparative trial
- Row totals fixed, column totals free to vary
- E.g., researcher fixes group sizes for Treatment & Control groups, or for Males & Females

- Model III
- Also called a cross-sectional study
- Both row & column totals are free to vary
- Only the total N is fixed

“Computer-intensive techniques were used … to compare seven two-sided tests of two-by-two tables in terms of their Type I errors.”

Ian Campbell

(Statist. Med. 2007; 26:3661–3675

- Null hypothesis was always true – i.e., there was no association between the row & column variables
- Therefore, statistically significant results were Type I errors
- For values of N ranging from 4-80, Campbell computed the maximum probability of Type I error(with alpha set to .05)
- He also examined all possible values of π

The proportion of subjects (in the population) having the binary characteristic(s) of interest—e.g., the proportion of males, or the proportion of smokers, etc

- Campbell examined 7 different statistical tests
- I will focus on only 2 of those tests today:
- Pearson’s chi-square
- The ‘N-1’ chi-square

Yoo-hoo! What’s the ‘N-1’ chi-square?

Pearson’s chi-square (shortcut for 2×2 tables only)

The ‘N-1’ chi-square (for 2×2 tables only)

- First derived by E.S. Pearson (1947)
- Egon Sharpe Pearson, son of Karl

- Derived again by Kendall & Stuart (1967)
- Richardson (1994) asserted that it is “the appropriate chi-square statistic to use in analysing all 2×2 contingency tables” (p. 116, emphasis added)
- Campbell summarizes the theoretical argument for preferring the N-1 chi-square on his website:
- www.iancampbell.co.uk/twobytwo/n-1_theory.htm

- Campbell computed the maximum Type I error probability for:
- N ranging from 4 to 80
- Over all values of π
- For minimum expected count = 0, 1, 3, and 5

- He did all of that using both:
- Pearson’s chi-squared test of association
- The N-1 chi-squared test

- Compared the actual Type I error rate to the nominal alpha
- All of the above done for Models II and III separately

- For an ideal test, the actual proportion of Type I errors is equal to the nominal alpha level
- E.g., if you set alpha at .05, Type I errors occur 5% of the time (when the null hypothesis is true)

- A test is if the actual Type I error rate is lower than the nominal alpha
- Conservative tests have low power – they don’t reject H0 as often as they should (i.e., too many Type II errors)

- A test is if the actual Type I error rate is higher than the nominal alpha
- Liberal tests reject H0 too easily, or too frequently (i.e., too many Type I errors)

- With discrete data (like counts) and small sample sizes, the actual Type I error rate is generally not exactly equal to the nominal alpha
- Cochran (1942) suggested allowing a 20% error in the actual Type I error rate—e.g., for nominal alpha = .05, an actual Type I error rate between .04 and .06 is acceptable
- Cochran’s criterion is admittedly arbitrary, but other authors have generally followed it (or a similar criterion) – and Campbell (2007) uses it.

Minimum value of E

Maximum over all values of π

.05 ± 20% (from Cochran)

For Model II, Pearson’s chi-squared test meets Cochran’s criterion only if the minimum E≥ 5 (the blue line).

Minimum value of E

For Model II, the N-1 chi-squared test meets Cochran’s criterion quite well for expected counts as low as 1.

Minimum value of E

For Model III, Pearson’s chi-squared test meets Cochran’s criterion fairly well for E as low as 3.

Minimum value of E

For Model III, the N-1 chi-squared test meets Cochran’s criterion very well for expected counts as low as 1.

- For Model I – row & column totals both fixed
- Use the two-sided Fisher Exact Test(as computed by SPSS)
- Aka the Fisher-Irwin Test “by Irwin’s rule”

- For Models II and III – comparative trials & cross-sectional
- If all E≥ 1, use the ‘N − 1’ chi-squared test
- Otherwise, use the Fisher–Irwin Test by Irwin’s rule

- Campbell’s new rule of thumb “extends the use of the chi-squared test to smaller samples … with a resultant increase in the power to detect real differences.” (Campbell, 2007, p. 3674, emphasis added)

And as everyone knows, the more power, the better!

Tim “the Stats-Man” Taylor & Al

http://www.iancampbell.co.uk/twobytwo/calculator.htm

- I have written 2 SPSS syntax files to compute the N-1 chi-square
- Ian Campbell provides a link to them beside his online calculator

A link to my two SPSS syntax files

Yeah, I have a question. Did you have to include that picture?

Severe Malocclusion

Barnard GA. Significance tests for 2×2 tables. Biometrika 1947; 34:123–138.

Campbell I. Chi-squared and Fisher–Irwin tests of two-by-two tables with small sample recommendations. Statist. Med. 2007; 26:3661–3675. [See also: http://www.iancampbell.co.uk/twobytwo/twobytwo.htm]

Cochran WG. The χ2 test of goodness of fit. Annals of Mathematical Statistics 1952; 25:315–345.

Cochran WG. Some methods for strengthening the common χ2 tests. Biometrics 1954; 10:417–451.

Kempthorne O. In dispraise of the exact test: reactions. Journal of Statistical Planning and Inference 1979;3:199–213.

Kendall MG, Stuart A. The advanced theory of statistics, Vol. 2, 2nd Ed. London: Griffin, 1967.

Pearson ES. The choice of statistical tests illustrated on the interpretation of data classed in a 2×2 table. Biometrika 1947; 34:139–167.

Rankin R. The Hollow Chocolate Bunnies of the Apocalypse. Gollancz (August 1, 2003).

Richardson JTE. The analysis of 2x1 and 2x2 contingency tables: A historical review. Statistical Methods in Medical Research 1994; 3:107-133.

The Cutting Room Floor

- However, there is no solid evidence to support that claim
- http://www.phrases.org.uk/meanings/rule-of-thumb.html
- http://www.canlaw.com/rights/thumbrul.htm
- http://womenshistory.about.com/od/mythsofwomenshistory/a/rule_of_thumb.htm
- http://www.straightdope.com/columns/read/2550/does-rule-of-thumb-refer-to-an-old-law-permitting-wife-beating

- Some have claimed that the expression rule of thumb derives an old legal ruling in England that allowed men to beat their wives with a stick, provided it was no thicker than their thumb

"The importance of the topic cannot be stressed too heavily."

"2×2 contingency tables are the most elemental structures leading to ideas of association.... The comparison of two binomial parameters runs through all sciences."

Dr. Oscar Kempthorne

(J Stat Planning and Inf 1979;3:199–213, emphasis added)

- Farm boy from Cornwall who became a Cambridge-trained statistician
- In 1941, he joined Rothamsted Experiment Station, where he met Ronald Fisher and Frank Yates
- Strongly influenced by Fisher—e.g., areas of interest were experimental design, genetic statistics, and statistical inference

Kempthorne & Fisher

“J. O. Irwin was a soft spoken kind soul who took a tremendous interest in his students and their achievements.... He was a lovable absent-minded kind of professor who smoked more matches than he did tobacco in his ever-present pipe while he was deeply involved in thinking about other important matters.”

Major Greenwood

“His old boss Pearson and his new boss R. A. Fisher were bitter enemies but Irwin's conciliatory nature allowed him to remain on good terms with both men.”

From http://en.wikipedia.org/wiki/Joseph_Oscar_Irwin

- A variation on that rule of thumb says that:
- All expected counts should be 10 or greater.
- If any expected counts are less than 10, but greater than or equal to 5, Yates' Correction for continuity should be applied. (However, the use of Yates' correction is controversial, and is not recommended by all authors).
- If any expected counts are less than 5, then some other test should be used.

- Again, the most frequently recommended alternative test under point 3 has been Fisher’s exact test.

Maximum over all values of π

Far too liberal if we impose no restrictions on minimum value of E

Cochran’s range: ± 20% of .05

Arguably too conservative for smaller values of N

Too liberal if we impose no restrictions on minimum value of E

Again, the FET is too conservative

- O = observed count
- E = expected count (assuming a true null hypothesis)
- Σ = Greek letter sigma & means to sum across all cells

General formula for contingency tables of any size

I don’t remember what expected counts are—can you explain that?

Of course. See the next slide.

E = row total × column total / grand total

How low is too low for expected counts?

It depends.

If I had a dollar for every time I heard a statistician say that, I’d be rich.

- For contingency tables larger than 2×2, the chi-square approximation is pretty good if:
- Many people do not know this, and mistakenly assume that all expected counts must be 5 or more for tables of any size

“…no more than 20% of the expected counts are less than 5 and all individual expected counts are 1 or greater."

(Yates, Moore & McCabe, 1999, p. 734)

- Each person is classified on 2 different categorical variables
- Each person appears in only one cell of the table

Two of 10 cells (20%) have E < 5; but all E >= 1

La-la-la-la-la …

MAJOR

- Fisher’s formula for working out the exact probability of an observed set of counts (and of more extreme sets under H0):

- For Model I, if a is known, b, c, and d can be worked out using the fixed row & column totals
- Kendall & Stuart demonstrated that under a true null hypothesis, a is asymptotically normal with:

i.e., row total × column total divided by grand total

N-1 chi-square

- “It will become clear later that [the N-1 chi-square] rather than [Pearson’s chi-square] is in fact the appropriate chi-square statistic to use in analysing all 2×2 contingency tables regardless of the underlying model.” (Richardson, 1994, p. 116, emphasis added)

J.T.E. Richardson

“The purpose of most research is to discover relations—relations between or among variables or between treatment interventions and outcomes.”

(Can J Psychiatry 2002;47:262–266)

Dr. David Streiner

They test the null hypothesis that in the population from which you have sampled, there is no association between the variables.

So when you reject the null hypothesis, you infer that there is an association between the variables (in the population).

Yours truly