Confidence Interval for a Standardized Contrast in a Fixed Effects ANOVA

Description

Function to obtain the confidence interval for a standardized contrast in a fixed effects analysis of variance context.

Usage

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ci.sc(means = NULL, s.anova = NULL, c.weights = NULL, n = NULL, 
N = NULL, Psi = NULL, ncp = NULL, conf.level = 0.95, 
alpha.lower = NULL, alpha.upper = NULL, df.error = NULL, ...)

Arguments

means

a vector of the group means or the means of the particular level of the effect (for fixed effect designs)

s.anova

the standard deviation of the errors from the ANOVA model (i.e., the square root of the mean square error)

c.weights

the contrast weights (chose weights so that the positive c-weights sum to 1 and the negative c-weights sum to -1; i.e., use fractional values not integers).

n

sample sizes per group or sample sizes for the level of the particular factor (if length 1 it is assumed that the sample size per group or for the level of the particular factor are are equal)

N

total sample size

Psi

the (unstandardized) contrast effect, obtained by multiplying the jth mean by the jth contrast weight (this is the unstandardized effect)

ncp

the noncentrality parameter from the t-distribution

conf.level

desired level of confidence for the computed interval (i.e., 1 - the Type I error rate)

alpha.lower

the Type I error rate for the lower confidence interval limit

alpha.upper

the Type I error rate for the upper confidence interval limit

df.error

the degrees of freedom for the error. In one-way designs, this is simply N-length (means) and need not be specified; it must be specified if the design has multiple factors.

...

optional additional specifications for nested functions

Value

Lower.Conf.Limit.Standardized.Contrast

the lower confidence limit for the standardized contrast

Standardized.contrast

standardized contrast

Upper.Conf.Limit.Standardized.Contrast

the upper confidence limit for the standardized contrast

Note

Be sure to use the standard deviation and not the error variance for s.anova, not the square of this value (the error variance) which would come from the source table (i.e., do not use the variance of the error but rather use its square root, the standard deviation).

Be sure to use the error variance and not its square root (i.e., use the variance of the standard deviation of the errors). Be sure to use the standard deviations of errors for s.anova and s.ancova, not the square of these values (i.e., do not use the variance of the errors).

Be sure to use fractional c-weights when doing complex contrasts (not integers) to specify c.weights. For exmaple, in an ANCOVA of four groups, if the user wants to compare the mean of group 1 and 2 with the mean of group 3 and 4, c.weights should be specified as c(0.5, 0.5, -0.5, -0.5) rather than c(1, 1, -1, -1). Make sure the sum of the contrast weights are zero.

Author(s)

Ken Kelley (University of Notre Dame; KKelley@ND.Edu)

References

Kelley, K. (2007). Constructing confidence intervals for standardized effect sizes: Theory, application, and implementation. Journal of Statistical Software, 20 (8), 1–24.

Lai, K., & Kelley, K. (2007). Sample size planning for standardized ANCOVA and ANOVA contrasts: Obtaining narrow confidence intervals. Manuscript submitted for publication.

Steiger, J. H. (2004). Beyond the F Test: Effect size confidence intervals and tests of close fit in the Analysis of Variance and Contrast Analysis. Psychological Methods, 9, 164–182.

See Also

conf.limits.nct, ci.src, ci.smd, ci.smd.c, ci.sm, ci.c

Examples

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# Here is a four group example. Suppose that the means of groups 1--4 are 2, 4, 9, 
# and 13, respectively. Further, let the error variance be .64 and thus the standard
# deviation would be .80 (note we use the standard deviation in the function, not the 
# variance). The standardized contrast of interest here is the average of groups 1 and 4
# versus the average of groups 2 and 3. 

ci.sc(means=c(2, 4, 9, 13), s.anova=.80, c.weights=c(.5, -.5, -.5, .5), 
n=c(3, 3, 3, 3), N=12, conf.level=.95)


# Here is an example with two groups. 
ci.sc(means=c(1.6, 0), s.anova=.80, c.weights=c(1, -1), n=c(10, 10), N=20, conf.level=.95)

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