The Arcsine Transformation

This article presents a brief description of the arcsine transformation 
and why it has been historically applied to the analysis of binary data 
summarized as proportions (i.e., the ratio of the number of outcomes of 
interest or "successes" observed over the total number of independent 
binary trials).

Assume you obtain y successes (or failures) in a sample of n independent 
binary trials where the probability of the observed outcome (success or 
failure) is equal to p for each trial. The observed count y is distributed 
as a binomial random variable with parameters n and p:

y ~ binomial(n,p)

The population proportion %PI is the percent of outcomes expected over 
many replications of the experiment. Even though the calculation of the 
sample proportion p^hat = y/n will be expressed as a number with either 
decimals or converted to a percentage by multiplying by 100, both a 
proportion or a percent still follow the binomial distribution, since 
there is a one-one relationship between y and n with the computed 
proportion/percent.

The variance for the binomial distribution of the observed proportion, 
p=y/n, is a function of p:

VAR(p) = (p)*(1-p)/(n-1)

The fact that the variance of p depends on its particular value violates 
the homogeneity of variance assumption across subjects required for the 
computation of statistical tests, for example, if p were to be used as the 
dependent variable in an ANOVA or regression. One approach to deal with 
this problem is to derive a mathematical function of p whose variance is 
essentially free of the value of p. It can be shown that for the binomial 
distribution the arcsine transformation serves this purpose (see Hogg and 
Craig).

If most of the computed proportions lie between 0.3 and 0.7, this or any 
other transformation should have only a small difference in the analytical 
results.  However, it is wise to use it, especially when a sizeable number 
of the observed proportions are either relatively small (i.e., 0 < p < 
0.2) or large (i.e., 0.8 < p < 1.0).

When n < 50 and the computed p is 0 the proportion should be computed as 
1/(4n) and a proportion of 1 should be expressed as (n-1/4)/n before 
applying the arcsine transformation. This improves the equality of 
variance in the angles.

Also note that the arsine transformation is not particularly good if a 
substantial number of the proportions are equal to 0 or 1 or for values at 
the extreme ends of the possible range of p (near 0 and near 1). It is 
also not recommended if the number of trials, i.e., n, is small.

The arcsine transformation is a transformation that assists a data analyst 
when working with proportions and percentages.  The proportion p can be 
made nearly "normal" if the square root of p is used with the arcsine (or 
inverse sin or sin^-1) transformation. The arcsine transformation is then 
computed as a function of the proportion, p.

arc_p = arcsin(SQRT(p))

If both y and n are small, a somewhat better arcsine transformation is 
given by

arc_P = arcsin[SQRT{(y + 3/8)/(n + 3/4)}]

Tables have been produced by Mosteller and Youtz for small values of n.  

????What does the arcsine transformations do in terms of equalizing the 
variance in data such as we're dealing with here?


How to Compute the Arcsin Transformation

Assume for each subject you have collected the following data:

Let y = the number of outcomes of interest reported for each person
    n = the total number of trials
    
This process first calculates the actual proportion (p = y/n) and then 
transforms the individual proportions to their angular equivalents:

IF y =0          THEN p = 1 / (4*n)
IF 1 <= y <= n-1 THEN p = y/n
If y =n          THEN p = (n - 1/4) / n

angle = t(p) = (360/(2*3.14159) ) * ( arcsin(SQRT( p )) );

where  angle = t(p) = arcsin transformation of p
     3.14159 = the mathematical constant, PI
         360 = the number of degrees in a circle

You can then compute t-tests or ANOVAs on the transformed data using the 
means
of t(p) to check for group differences. For a two sample t-test this would 
be:

t = ( abar_1 - abar_2 ) / std err [( abar_1 - abar_2 )]

where:

 a-bar_1 is the average of the transformed proportions for group 1
 a-bar_2 is the average of the transformed proportions for group 2

The results of the test should be reported by back-transforming the 
predicted values to proportions.  They can be computed with the formula:

 p_hat= (SIN( ((2*3.14159)/360) * pred_ang) )**2;
 
Since the response is linear, it is possible for a predicated value of 
pred_ang to be less than 0 or greater than 90, especially when 
extrapolating beyond the range of the input data, or if the actual data 
are close to the boundaries of 0 and 1. If this is case, you should 
consider p_hat as "missing" or not computable.

When summarizing ANOVA results which have been computed with the arcsin, 
state that the significance tests were made with an arcsine transformation 
of the response variable, p.  Confidence intervals can also be found by 
computing the upper and lower limits with the transformed data (the arcsin 
values) and then convert them back to proportions that are more easily 
interpreted. For example, 1 standard deviation above and below the 
predicted mean percent is:

  p_hat_Upper = (SIN( ((2*3.14159)/360) * (pred_ang + StdErr) ))**2;
  p_hat_Lower = (SIN( ((2*3.14159)/360) * (pred_ang - StdErr) ))**2;


After this brief explanation of the process, recognize that modern 
computing technology has made the need for the arcsine transformation 
nearly obsolete. When analyzing binary data in the form of counts, 
generalized linear models provide a mathematically sound approach to 
analyzing data from a binomial distribution with the logit link. Known as 
logistic regression, which is available in most statistical programs, this 
method is designed to work with data in the forms of counts (e.g., y and 
n) and NOT the computed proportions, p.


The arcsine transformation may still find relevant applications with users 
who are familiar with ANOVA procedures and their interpretation, yet have 
little experience or are not comfortable interpreting more recent 
computing options for count data. It is also perhaps of value when all you 
have to work with are percents that do not have the source data available. 
It is less clear how to redefine 0 and 100 in those cases, but setting 
these values to .0005 and .9995 seems to work reasonably well.

Its major appeal reached its zenith the pre-computer days, when simple
ANOVA computations were done by hand. It was perhaps the only way one
could adjust for the unequal variances of responses as a function of p.
This is true yet today as a rough approximation, but simulation results
show it is not conservative enough when many of the values of p are close
to 0 or 1.

Further discussion of the arcsine transformation and its derivation can be 
found in the References.

References:

Hogg and Craig, (1995) Mathematical Statistics, 5th ed., Prentice Hall, 
pp. 251-2.

Mosteller, F. and Youtz, C. Biometrika 48 (1961) 433-440.

Snedecor and Cochran (1980), "Statistical Methods", 7th ed., Iowa State 
Press, Sect. 15.12, pp. 290-91.

Zar, J.H. (1996) "Biostatistical Analysis". Prentice Hall. Sect 13.3, pp. 
282-283.