Converts an ANCOVA F to an effect size of d (mean difference), g (unbiased estimate of d), r (correlation coefficient), z' (Fisher's z), and log odds ratio. The variances, confidence intervals and pvalues of these estimates are also computed, along with NNT (number needed to treat), U3 (Cohen's U_(3) overlapping proportions of distributions), CLES (Common Language Effect Size) and Cliff's Delta.
1 
f 
F value from ANCOVA. 
n.1 
Treatment group sample size. 
n.2 
Comparison group sample size. 
R 
Covariate outcome correlation or multiple correlation. 
q 
number of covariates. 
level 
Confidence level. Default is 
cer 
Control group Event Rate (e.g., proportion of cases showing recovery). Default is 
dig 
Number of digits to display. Default is 
verbose 
Print output from scalar values? If yes, then verbose=TRUE; otherwise, verbose=FALSE. Default is TRUE. 
id 
Study identifier. Default is 
data 
name of 
d 
Standardized mean difference (d). 
var.d 
Variance of d. 
l.d 
lower confidence limits for d. 
u.d 
upper confidence limits for d. 
U3.d 
Cohen's U_(3), for d. 
cl.d 
Common Language Effect Size for d. 
cliffs.d 
Cliff's Delta for d. 
p.d 
pvalue for d. 
g 
Unbiased estimate of d. 
var.g 
Variance of g. 
l.g 
lower confidence limits for g. 
u.g 
upper confidence limits for g. 
U3.g 
Cohen's U_(3), for g. 
cl.g 
Common Language Effect Size for g. 
p.g 
pvalue for g. 
r 
Correlation coefficient. 
var.r 
Variance of r. 
l.r 
lower confidence limits for r. 
u.r 
upper confidence limits for r. 
p.r 
pvalue for r. 
z 
Fisher's z (z'). 
var.z 
Variance of z'. 
l.z 
lower confidence limits for z'. 
u.z 
upper confidence limits for z'. 
p.z 
pvalue for z'. 
OR 
Odds ratio. 
l.or 
lower confidence limits for OR. 
u.or 
upper confidence limits for OR. 
p.or 
pvalue for OR. 
lOR 
Log odds ratio. 
var.lor 
Variance of log odds ratio. 
l.lor 
lower confidence limits for lOR. 
u.lor 
upper confidence limits for lOR. 
p.lor 
pvalue for lOR. 
N.total 
Total sample size. 
NNT 
Number needed to treat. 
Detailed information regarding output values of:
(1) Cohen's d, Hedges' g (unbiased estimate of d) and variance
(2) Correlation coefficient (r), Fisher's z', and variance
(3) Log odds and variance
is provided below (followed by general information about NNT, U3, Common Language Effect Size, and Cliff's Delta):
Cohen's d, Hedges' g and Variance of g:
In this particular formula Cohen's d is calculated from the ANCOVA F with independent groups
d= (F(n_(1)+n_(2)))/ (n_(1)n_(2)) sqrt(1R^2)
The variance of d is derived from
v_(d)= ((n_(1)+n_(2))(1R^2))/ (n_(1)n_(2))+ (d^2)/ (2(n_(1)+n_(2)))
The effect size estimate d has a small upward bias (overestimates the population parameter effect size) which can be removed using a correction formula to derive the unbiased estimate of Hedges' g. The correction factor, j, is defined as
J= 1 (3)/ (4df1)
where df= degrees of freedom, which is n_(1)+n_(2)2 for two independent groups. Then, to calculate g
g= Jd
and the variance of g
v_(g)= J^2v_(d)
Correlation Coefficient r, Fisher's z, and Variances:
In this particular formula r is calculated as follows
r= (d)/ (sqrt(d^2+a))
where a corrects for inbalance in n_(1) & n_(2) and is defined as
a= ((n_(1)+n_(2))^2)/ (n_(1)n_(2))
The variance of r is then defined as
v_(r)= (a^2v_(d))/ ((d^2+a)^3)
Often researchers are interested in transforming r to z' (Fisher's z) because r is not normally distributed, particularly at large values of r. Therefore, converting to z' will help to normally distribute the estimate. Converting from r to z' is defined as
z= .5^*log((1+r)/ (1r)
and the variance of z
v_(z)= (1)/ (n3)
where n is the total sample size for groups 1 and 2.
Log Odds Ratio & Variance of Log Odds:
In this particular formula, log odds is calculated as follows
log(o)= (pi d)/ (sqrt(3)
where pi = 3.1459. The variance of log odds is defined as
v_(log(o))= (pi^2v_(d))/ (3)
General information about NNT, U3, Common Language Effect Size, and Cliff's Delta:
Number needed to treat (NNT). NNT is interpreted as the number of participants that would need to be treated in one group (e.g., intervention group) in order to have one additional positive outcome over that of the outcome of a randomly selected participant in the other group (e.g., control group). In the compute.es
package, NNT is calculated directly from d (Furukawa & Leucht, 2011), assuming relative normality of distribution and equal variances across groups, as follows:
NNT= 1/(Phi(dPsi(CER))CER)
U3. Cohen (1988) proposed a method for characterizing effect sizes by expressing them in terms of (normal) distribution overlap, called U3. This statistic describes the percentage of scores in one group that are exceeded by the mean score in another group. If the population means are equal then half of the scores in the treatment group exceed half the scores in the comparison group, and U3 = 50%. As the population mean difference increases, U3 approaches 100% (Valentine & Cooper, 2003).
Common Language Effect Size (CLES). CLES (McGraw & Wong, 1992) expresses the probability that a randomly selected score from one population will be greater than a randomly sampled score from another population. CLES is computed as the percentage of the normal curve that falls between negative infinity and the effect size (Valentine & Cooper, 2003).
Cliff's Delta/success rate difference. Cliff's delta (or success rate difference; Furukawa & Leucht (2011)) is a robust alternative to Cohen's d, when data are either nonnormal or ordinal (with truncated/reduced variance). Cliff's Delta is a nonparametric procedure that provides the probability that individual observations in one group are likely to be greater than the observations in another group. It is the probability that a randomly selected participant of one population has a better outcome than a randomly selected participant of the second population (minus the reverse probability). Cliff's Delta of negative 1 or positive 1 indicates no overlap between the two groups, whereas a value of 0 indicates complete overlap and equal group distributions.
Cliff's Delta= 2*Phi(d/sqrt(2))1
AC Del Re
Much appreciation to Dr. Jeffrey C. Valentine for his contributions in implementing U3 and CLES procedures and related documentation.
Maintainer: AC Del Re acdelre@gmail.com
Borenstein (2009). Effect sizes for continuous data. In H. Cooper, L. V. Hedges, & J. C. Valentine (Eds.), The handbook of research synthesis and meta analysis (pp. 279293). New York: Russell Sage Foundation.
Cohen, J. (1988). Statistical power for the behavioral sciences (2nd ed.). Hillsdale, NJ: Erlbaum.
Furukawa, T. A., & Leucht, S. (2011). How to obtain NNT from Cohen's d: comparison of two methods. PloS one, 6(4), e19070.
McGraw, K. O. & Wong, S. P. (1992). A common language effect size statistic. Psychological Bulletin, 111, 361365.
Valentine, J. C. & Cooper, H. (2003). Effect size substantive interpretation guidelines: Issues in the interpretation of effect sizes. Washington, DC: What Works Clearinghouse.
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# CALCULATE SEVERAL EFFECT SIZES BASED ON FSTATISTIC FROM ANCOVA:
a.fes(3, 30, 30, .4, 2)

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