Sample Size Calculation for Cohen's Kappa Statistic

Description

This function is a sample size estimator for the Cohen's Kappa statistic for a binary outcome. Note that any value of "kappa under null" in the interval [0,1] is acceptable (i.e. k0=0 is a valid null hypothesis).

Usage

 N.cohen.kappa(rate1, rate2, k1, k0, alpha=0.05, 
               power=0.8, twosided=FALSE)

Arguments

Value

returns required sample size

Author(s)

Ian Fellows

References

Cantor, A. B. (1996) Sample-size calculation for Cohen's kappa. Psychological Methods, 1, 150-153.

See Also

kappa2

Examples

  # Testing H0: kappa = 0.7 vs. HA: kappa > 0.7 given that
  # kappa = 0.85 and both raters classify 50% of subjects as positive.
  N.cohen.kappa(0.5, 0.5, 0.7, 0.85)

Sample Size Calculation for Cohen's Kappa Statistic with more than one category

Description

This function calculates the required sample size for the Cohen's Kappa statistic when two raters have the same marginal. Note that any value of "kappa under null" in the interval [-1,1] is acceptable (i.e. k0=0 is a valid null hypothesis).

Usage

 N2.cohen.kappa(mrg, k1, k0, alpha=0.05, power=0.8, twosided=FALSE)

Arguments

Value

Returns required sample size.

Author(s)

Puspendra Singh and Jim Lemon

References

Flack, V.F., Afifi, A.A., Lachenbruch, P.A., & Schouten, H.J.A. (1988). Sample size determinations for the two rater kappa statistic. Psychometrika, 53, 321-325.

See Also

N.cohen.kappa, kappa2

Examples

  require(lpSolve)
  # Testing H0: kappa = 0.4 vs. HA: kappa > 0.4 (=0.6) given that
  # Marginal Probabilities by two raters are (0.2, 0.25, 0.55).
  #
  # one sided test with 80% power:
  N2.cohen.kappa(c(0.2, 0.25, 0.55), k1=0.6, k0=0.4)								
  # one sided test with 90% power:
  N2.cohen.kappa(c(0.2, 0.25, 0.55), k1=0.6, k0=0.4, power=0.9)	  

  # Marginal Probabilities by two raters are (0.2, 0.05, 0.2, 0.05, 0.2, 0.3)
  # Testing H0: kappa = 0.1 vs. HA: kappa > 0.1 (=0.5) given that
  #
  # one sided test with 80% power:
  N2.cohen.kappa(c(0.2, 0.05, 0.2, 0.05, 0.2, 0.3), k1=0.5, k0=0.1)				

Simple and extended percentage agreement

Description

Computes simple and extended percentage agreement among raters.

Usage

agree(ratings, tolerance=0)

Arguments

Details

Missing data are omitted in a listwise way.
Using extended percentage agreement (tolerance!=0) is only possible for numerical values. If tolerance equals 1, for example, raters differing by one scale degree are interpreted as agreeing.

Value

A list with class '"irrlist"' containing the following components:

Author(s)

Matthias Gamer

See Also

kappa2, kappam.fleiss, kappam.light

Examples

data(video)
agree(video)     # Simple percentage agreement
agree(video, 1)  # Extended percentage agreement

Anxiety ratings by different raters

Description

The data frame contains the anxiety ratings of 20 subjects, rated by 3 raters. Values are ranging from 1 (not anxious at all) to 6 (extremely anxious).

Usage

data(anxiety)

Format

A data frame with 20 observations on the following 3 variables.

rater1

ratings of the first rater

rater2

ratings of the second rater

rater3

ratings of the third rater

Source

artificial data

Examples

data(anxiety)
apply(anxiety,2,table)

Bhapkar coefficient of concordance between raters

Description

Calculates the Bhapkar coefficient of concordance for two raters.

Usage

bhapkar(ratings)

Arguments

Details

Missing data are omitted in a listwise way. The Bhapkar (1966) test is a more powerful alternative to the Stuart-Maxwell test. Both tests are asymptotically equivalent and will produce comparable chi-squared values when applied a large sample of rated objects.

Value

A list with class "irrlist" containing the following components:

Author(s)

Matthias Gamer

References

Bhapkar, V.P. (1966). A note on the equivalence of two test criteria for hypotheses in categorical data. Journal of the American Statistical Association, 61, 228-235.

See Also

mcnemar.test, stuart.maxwell.mh, rater.bias

Examples

data(vision)
bhapkar(vision) # Original example used from Bhapkar (1966)

Psychiatric diagnoses provided by different raters

Description

Psychiatric diagnoses of n=30 patients provided by different sets of m=6 raters. Data were used by Fleiss (1971) to illustrate the computation of Kappa for m raters.

Usage

data(diagnoses)

Format

A data frame with 30 observations (psychiatric diagnoses with levels 1. Depression, 2. Personality Disorder, 3. Schizophrenia, 4. Neurosis, 5. Other) on 6 variables representing different raters.

rater1

a factor including the diagnoses of rater 1 (levels see above)

rater2

a factor including the diagnoses of rater 2 (levels see above)

rater3

a factor including the diagnoses of rater 3 (levels see above)

rater4

a factor including the diagnoses of rater 4 (levels see above)

rater5

a factor including the diagnoses of rater 5 (levels see above)

rater6

a factor including the diagnoses of rater 6 (levels see above)

Source

Fleiss, J.L. (1971). Measuring nominal scale agreement among many raters. Psychological Bulletin, 76, 378-382.

References

Fleiss, J.L. (1971). Measuring nominal scale agreement among many raters. Psychological Bulletin, 76, 378-382.

Examples

data(diagnoses)
table(diagnoses[,1])

Finn coefficient for oneway and twoway models

Description

Computes the Finn coefficient as an index of the interrater reliability of quantitative data. Additionally, F-test and confidence interval are computed.

Usage

finn(ratings, s.levels, model = c("oneway", "twoway"))

Arguments

Details

Missing data are omitted in a listwise way.
The Finn coefficient is especially useful, when variance between raters is low (i.e. agreement is high).
For the computation it could be specified if only the subjects are considered as random effects ('"oneway"' model) or if subjects and raters are randomly chosen from a bigger pool of persons ('"twoway"' model).

Value

A list with class '"irrlist"' containing the following components:

Author(s)

Matthias Gamer

References

Finn, R.H. (1970). A note on estimating the reliability of categorical data. Educational and Psychological Measurement, 30, 71-76.

See Also

icc, meancor, robinson

Examples

data(video)
finn(video, 6, model="twoway")

Intraclass correlation coefficient (ICC) for oneway and twoway models

Description

Computes single score or average score ICCs as an index of interrater reliability of quantitative data. Additionally, F-test and confidence interval are computed.

Usage

icc(ratings, model = c("oneway", "twoway"), 
    type = c("consistency", "agreement"), 
    unit = c("single", "average"), r0 = 0, conf.level = 0.95)

Arguments

Details

Missing data are omitted in a listwise way.
When considering which form of ICC is appropriate for an actual set of data, one has take several decisions (Shrout & Fleiss, 1979):

1. Should only the subjects be considered as random effects ('"oneway"' model) or are subjects and raters randomly chosen from a bigger pool of persons ('"twoway"' model).

2. If differences in judges' mean ratings are of interest, interrater '"agreement"' instead of '"consistency"' should be computed.

3. If the unit of analysis is a mean of several ratings, unit should be changed to '"average"'. In most cases, however, single values (unit='"single"') are regarded.

Value

A list with class '"icclist"' containing the following components:

Author(s)

Matthias Gamer

References

Bartko, J.J. (1966). The intraclass correlation coefficient as a measure of reliability. Psychological Reports, 19, 3-11.

McGraw, K.O., & Wong, S.P. (1996), Forming inferences about some intraclass correlation coefficients. Psychological Methods, 1, 30-46.

Shrout, P.E., & Fleiss, J.L. (1979), Intraclass correlation: uses in assessing rater reliability. Psychological Bulletin, 86, 420-428.

See Also

finn, meancor, robinson

Examples

data(anxiety)
icc(anxiety, model="twoway", type="agreement")

r1 <- round(rnorm(20, 10, 4))
r2 <- round(r1 + 10 + rnorm(20, 0, 2))
r3 <- round(r1 + 20 + rnorm(20, 0, 2))
icc(cbind(r1, r2, r3), "twoway")              # High consistency
icc(cbind(r1, r2, r3), "twoway", "agreement") # Low agreement

iota coefficient for the interrater agreement of multivariate observations

Description

Computes iota as an index of interrater agreement of quantitative or nominal multivariate observations.

Usage

iota(ratings, scaledata = c("quantitative","nominal"),
     standardize = FALSE)

Arguments

Details

Each list element must contain observations for each rater and subject without missing values.
In case of one categorical variable (only one list element), iota reduces to the Fleiss exact kappa coefficient, which was proposed by Conger (1980).

Value

A list with class '"irrlist"' containing the following components:

Author(s)

Matthias Gamer

References

Conger, A.J. (1980). Integration and generalisation of Kappas for multiple raters. Psychological Bulletin, 88, 322-328.

Janson, H., & Olsson, U. (2001). A measure of agreement for interval or nominal multivariate observations. Educational and Psychological Measurement, 61, 277-289.

See Also

icc, kappam.fleiss

Examples

data(diagnoses)
iota(list(diagnoses))  # produces the same result as...
kappam.fleiss(diagnoses, exact=TRUE)

# Example from Janson & Olsson (2001), Table 1
photo <- list()
photo[[1]] <- cbind(c( 71, 73, 86, 59, 71),  # weight ratings
                    c( 74, 80,101, 62, 83),
                    c( 76, 80, 93, 66, 77))
photo[[2]] <- cbind(c(166,160,187,161,172),  # height rating
                    c(171,170,174,163,182),
                    c(171,165,185,162,181))
iota(photo)
iota(photo, standardize=TRUE) # iota over standardized values

Cohen's Kappa and weighted Kappa for two raters

Description

Calculates Cohen's Kappa and weighted Kappa as an index of interrater agreement between 2 raters on categorical (or ordinal) data. Own weights for the various degrees of disagreement could be specified.

Usage

kappa2(ratings, weight = c("unweighted", "equal", "squared"), sort.levels = FALSE)

Arguments

Details

Missing data are omitted in a listwise way.
During computation, ratings are converted to factors. Therefore, the categories are ordered accordingly. When ratings are numeric, a sorting of factor levels occurs automatically. Otherwise, levels are sorted when the function is called with sort.levels=TRUE.
kappa2 allows for calculating weighted Kappa coefficients. Beneath '"unweighted"' (default), predifined sets of weights are '"equal"' (all levels disagreement between raters are weighted equally) and '"squared"' (disagreements are weighted according to their squared distance from perfect agreement). The weighted Kappa coefficient with '"squared"' weights equals the product moment correlation under certain conditions. Own weights could be specified by supplying the function with a numeric vector of weights, starting from perfect agreement to worst disagreement. The length of this vector must equal the number of rating categories.

Value

A list with class '"irrlist"' containing the following components:

Author(s)

Matthias Gamer

References

Cohen, J. (1960). A coefficient of agreement for nominal scales. Educational and Psychological Measurement, 20, 37-46.

Cohen, J. (1968). Weighted kappa: Nominal scale agreement with provision for scaled disagreement or partial credit. Psychological Bulletin, 70, 213-220.

Fleiss, J.L., Cohen, J., & Everitt, B.S. (1969). Large sample standard errors of kappa and weighted kappa. Psychological Bulletin, 72, 323-327.

See Also

cor, kappa2, kappam.light

Examples

data(anxiety)
kappa2(anxiety[,1:2], "squared") # predefined set of squared weights
kappa2(anxiety[,1:2], (0:5)^2)   # same result with own set of squared weights

# own weights increasing gradually with larger distance from perfect agreement
kappa2(anxiety[,1:2], c(0,1,2,4,7,11))

data(diagnoses)
# Unweighted Kappa for categorical data without a logical order
kappa2(diagnoses[,2:3])

Fleiss' Kappa for m raters

Description

Computes Fleiss' Kappa as an index of interrater agreement between m raters on categorical data. Additionally, category-wise Kappas could be computed.

Usage

kappam.fleiss(ratings, exact = FALSE, detail = FALSE)

Arguments

Details

Missing data are omitted in a listwise way.
The coefficient described by Fleiss (1971) does not reduce to Cohen's Kappa (unweighted) for m=2 raters. Therefore, the exact Kappa coefficient, which is slightly higher in most cases, was proposed by Conger (1980).
The null hypothesis Kappa=0 could only be tested using Fleiss' formulation of Kappa.

Value

A list with class '"irrlist"' containing the following components:

Author(s)

Matthias Gamer

References

Conger, A.J. (1980). Integration and generalisation of Kappas for multiple raters. Psychological Bulletin, 88, 322-328.

Fleiss, J.L. (1971). Measuring nominal scale agreement among many raters. Psychological Bulletin, 76, 378-382.

Fleiss, J.L., Levin, B., & Paik, M.C. (2003). Statistical Methods for Rates and Proportions, 3rd Edition. New York: John Wiley & Sons.

See Also

kappa2, kappam.light

Examples

data(diagnoses)
kappam.fleiss(diagnoses)               # Fleiss' Kappa
kappam.fleiss(diagnoses, exact=TRUE)   # Exact Kappa
kappam.fleiss(diagnoses, detail=TRUE)  # Fleiss' and category-wise Kappa

kappam.fleiss(diagnoses[,1:4])         # Fleiss' Kappa of raters 1 to 4

Light's Kappa for m raters

Description

Computes Light's Kappa as an index of interrater agreement between m raters on categorical data.

Usage

kappam.light(ratings)

Arguments

Details

Missing data are omitted in a listwise way.
Light's Kappa equals the average of all possible combinations of bivariate Kappas between raters.

Value

A list with class '"irrlist"' containing the following components:

Author(s)

Matthias Gamer

References

Conger, A.J. (1980). Integration and generalisation of Kappas for multiple raters. Psychological Bulletin, 88, 322-328.

Light, R.J. (1971). Measures of response agreement for qualitative data: Some generalizations and alternatives. Psychological Bulletin, 76, 365-377.

See Also

kappa2, kappam.fleiss

Examples

data(diagnoses)
kappam.light(diagnoses)   # Light's Kappa

Kendall's coefficient of concordance W

Description

Computes Kendall's coefficient of concordance as an index of interrater reliability of ordinal data. The coefficient could be corrected for ties within raters.

Usage

kendall(ratings, correct = FALSE)

Arguments

Details

Missing data are omitted in a listwise way.
Kendall's W should be corrected for ties if raters did not use a true ranking order for the subjects.
A test for the significance of Kendall's W is only valid for large samples.

Value

A list with class '"irrlist"' containing the following components:

Author(s)

Matthias Gamer

References

Kendall, M.G. (1948). Rank correlation methods. London: Griffin.

See Also

cor, meanrho

Examples

data(anxiety)
kendall(anxiety, TRUE)

calculate Krippendorff's alpha reliability coefficient

Description

calculates the alpha coefficient of reliability proposed by Krippendorff

Usage

 kripp.alpha(x, method=c("nominal","ordinal","interval","ratio"))

Arguments

Value

A list with class '"irrlist"' containing the following components:

Note

Krippendorff's alpha coefficient is particularly useful where the level of measurement of classification data is higher than nominal or ordinal.

Author(s)

Jim Lemon

References

Krippendorff, K. (1980). Content analysis: An introduction to its methodology. Beverly Hills, CA: Sage.

Examples

 # the "C" data from Krippendorff
 nmm<-matrix(c(1,1,NA,1,2,2,3,2,3,3,3,3,3,3,3,3,2,2,2,2,1,2,3,4,4,4,4,4,
 1,1,2,1,2,2,2,2,NA,5,5,5,NA,NA,1,1,NA,NA,3,NA),nrow=4)
 # first assume the default nominal classification
 kripp.alpha(nmm)
 # now use the same data with the other three methods
 kripp.alpha(nmm,"ordinal")
 kripp.alpha(nmm,"interval")
 kripp.alpha(nmm,"ratio") 

Maxwell's RE coefficient for binary data

Description

Computes Maxwell's RE as an index of the interrater agreement of binary data.

Usage

maxwell(ratings)

Arguments

Details

Missing data are omitted in a listwise way.

Value

A list with class '"irrlist"' containing the following components:

Author(s)

Matthias Gamer

References

Maxwell, A.E. (1977). Coefficients of agreement between observers and their interpretation. British Journal of Psychiatry, 130, 79-83.

See Also

kappa2

Examples

data(anxiety)
# Median-split to generate binary data
r1 <- ifelse(anxiety$rater1<median(anxiety$rater1),0,1)
r2 <- ifelse(anxiety$rater2<median(anxiety$rater2),0,1)
maxwell(cbind(r1,r2))

Mean of bivariate correlations between raters

Description

Computes the mean of bivariate Pearson's product moment correlations between raters as an index of the interrater reliability of quantitative data.

Usage

meancor(ratings, fisher = TRUE)

Arguments

Details

Missing data are omitted in a listwise way.
The mean of bivariate correlations should not be used as an index of interrater reliability when the variance of ratings differs between raters.
The null hypothesis r=0 could only be tested when Fisher z-standardized values are used for the averaging.
When computing Fisher z-standardized values, perfect correlations are omitted before averaging because z equals +/-Inf in that case.

Value

A list with class '"irrlist"' containing the following components:

Author(s)

Matthias Gamer

See Also

cor

Examples

data(anxiety)
meancor(anxiety)

Mean of bivariate rank correlations between raters

Description

Computes the mean of bivariate Spearman's rho rank correlations between raters as an index of the interrater reliability of ordinal data.

Usage

meanrho(ratings, fisher = TRUE)

Arguments

Details

Missing data are omitted in a listwise way.
The mean of bivariate rank correlations should not be used as an index of interrater reliability when ties within raters occur.
The null hypothesis r=0 could only be tested when Fisher z-standardized values are used for the averaging.
When computing Fisher z-standardized values, perfect correlations are omitted before averaging because z equals +/-Inf in that case.

Value

A list with class '"irrlist"' containing the following components:

Author(s)

Matthias Gamer

See Also

cor, kendall

Examples

data(anxiety)
meanrho(anxiety, TRUE)

Default printing function for ICC results

Description

Prints the results of the ICC computation.

Usage

  ## S3 method for class 'icclist'
print(x, ...)

Arguments

Details

'"print.icclist"' is only a printing function and is usually not called directly.

Author(s)

Matthias Gamer

See Also

icc

Examples

data(anxiety)
# "print.icclist" is the default printing function of "icc"
icc(anxiety, model="twoway", type="agreement")

Default printing function for various coefficients of interrater reliability

Description

Prints the results of various functions computing coefficients of interrater reliability.

Usage

  ## S3 method for class 'irrlist'
print(x, ...)

Arguments

Details

'"print.irrlist"' is only a printing function and is usually not called directly.

Author(s)

Matthias Gamer

See Also

bhapkar, finn, iota, kappa2, kappam.fleiss, kappam.light, kripp.alpha, kendall, maxwell, meancor, meanrho, rater.bias, robinson, stuart.maxwell

Examples

data(anxiety)
# "print.irrlist" is the default printing method of various functions, e.g.
finn(anxiety, 6)
meancor(anxiety)

Coefficient of rater bias

Description

Calculates a coefficient of systematic bias between two raters.

Usage

rater.bias(x)

Arguments

Details

rater.bias calculates a reliability coefficient for two raters classifying n objects into any number of categories. It will accept either a c x c classification matrix of counts of objects falling into c categories or a 2 x n or n x 2 matrix of classification scores.
The function returns the absolute value of the triangular off-diagnonal sum ratio of the cxc classification table and the corresponding test statistic. A systematic bias between raters can be assumed when the ratio substantially deviates from 0.5 while yielding a significant Chi-squared statistic.

Value

Author(s)

Jim Lemon

References

Bishop Y.M.M., Fienberg S.E., & Holland P.W. (1978). Discrete multivariate analysis: theory and practice. Cambridge, Massachusetts: MIT Press.

See Also

mcnemar.test

Examples

 # fake a 2xn matrix of three way classification scores
 ratings <- matrix(sample(1:3,60,TRUE), nrow=2)
 rater.bias(ratings)

 # Example from Bishop, Fienberg & Holland (1978), Table 8.2-1
 data(vision)
 rater.bias(vision)                      

Inter- and intra-rater reliability

Description

‘⁠relInterIntra⁠’ calculates inter- and intra-rater reliability coefficients.

Usage

 relInterIntra(x, nrater=1, raterLabels=NULL, rho0inter=0.6,
               rho0intra=0.8, conf.level=.95)

Arguments

Value

nil

Author(s)

Tore Wentzel-Larsen

References

Eliasziw, M., Young, S.L., Woodbury, M.G., & Fryday-Field, K. (1994). Statistical methodology for the concurrent assessment of interrater and intrarater reliability: Using goniometric measurements as an example. Physical Therapy, 74, 777-788.

Examples

# testing code for the Goniometer data from the article:
 table4<-matrix(c(
  -2,16,5,11,7,-7,18,4,0,0,-3,3,7,-6,1,-13,2,4,-10,8,7,-3,-5,5,0,7,-8,1,-3,
  0,16,6,10,8,-8,19,5,-3,0,-2,-1,9,-7,1,-14,1,4,-9,9,6,-2,-5,5,-1,6,-8,1,-3,
  1,15,6,10,6,-8,19,5,-2,-2,-2,1,9,-6,0,-14,0,3,-10,8,7,-4,-7,5,-1,6,-8,2,-3,
  2,12,4,9,5,-9,17,5,-7,1,-4,-1,4,-8,-2,-12,-1,7,-10,2,8,-5,-6,3,-4,4,-10,1,-5,
  1,14,4,7,6,-10,17,5,-6,2,-3,-2,4,-10,-2,-12,0,6,-11,8,7,-5,-8,4,-3,4,-11,-1,-4,
  1,13,4,8,6,-9,17,5,-5,1,-3,1,2,-9,-3,-12,0,4,-10,8,7,-5,-7,4,-4,4,-10,0,-5
  ),ncol=6)
 relInterIntra(x=table4,nrater=2,raterLabels=c('universal','Lamoreux'))

Robinson's A

Description

Computes Robinson's A as an index of the interrater reliability of quantitative data.

Usage

robinson(ratings)

Arguments

Details

Missing data are omitted in a listwise way.

Value

A list with class '"irrlist"' containing the following components:

Author(s)

Matthias Gamer

References

Robinson, W.S. (1957). The statistical measurement of agreement. American Sociological Review, 22, 17-25.

See Also

finn, icc, meancor

Examples

data(anxiety)
robinson(anxiety)

Stuart-Maxwell coefficient of concordance for two raters

Description

Calculates the Stuart-Maxwell coefficient of concordance for two raters.

Usage

stuart.maxwell.mh(x)

Arguments

Details

stuart.maxwell.mh calculates a reliability coefficient for two raters classifying n objects into any number of categories. It will accept either a c x c classification matrix of counts of objects falling into c categories or a c x n or n x c matrix of classification scores.

Value

A list with class "irrlist" containing the following components:

Author(s)

Jim Lemon

References

Stuart, A.A. (1955). A test for homogeneity of the marginal distributions in a two-way classification. Biometrika, 42, 412-416.

Maxwell, A.E. (1970) Comparing the classification of subjects by two independent judges. British Journal of Psychiatry, 116, 651-655.

See Also

bhapkar, rater.bias

Examples

 # fake a 2xn matrix of three way classification scores
 ratings<-matrix(sample(1:3,60,TRUE), nrow=2)
 stuart.maxwell.mh(ratings)
 
 # Example used from Stuart (1955)
 data(vision)
 stuart.maxwell.mh(vision)

Different raters judging the credibility of videotaped testimonies

Description

The data frame contains the credibility ratings of 20 subjects, rated by 4 raters. Judgements could vary from 1 (not credible) to 6 (highly credible). Variance between and within raters is low.

Usage

data(video)

Format

A data frame with 20 observations on the following 4 variables.

rater1

ratings of rater 1

rater2

ratings of rater 2

rater3

ratings of rater 3

rater4

ratings of rater 4

Source

artificial data

Examples

data(video)
apply(video,2,table)

Eye-testing case records

Description

Case records of the eye-testing of N=7477 female employees in Royal Ordnance factories between 1943 and 1946. Data were primarily used by Stuart (1953) to illustrate the the estimation and comparison of strengths of association in contingency tables.

Usage

data(anxiety)

Format

A data frame with 7477 observations (eye testing results with levels 1st grade, 2nd grade, 3rd grade, 4th Grade) on the following 2 variables.

r.eye

unaided distance vision performance of the right eye

l.eye

unaided distance vision performance of the left eye

Source

Stuart, A. (1953). The Estimation and Comparison of Strengths of Association in Contingency Tables. Biometrika, 40, 105-110.

References

Stuart, A. (1953). The Estimation and Comparison of Strengths of Association in Contingency Tables. Biometrika, 40, 105-110.

Examples

data(vision)
table(vision$r.eye, vision$l.eye)