ci.mean.w: Within-Subject Confidence Interval for the Arithmetic Mean

In misty: Miscellaneous Functions 'T. Yanagida'

Within-Subject Confidence Interval for the Arithmetic Mean

Description

This function computes difference-adjusted Cousineau-Morey within-subject confidence interval for the arithmetic mean.

Usage

ci.mean.w(x, adjust = TRUE,
          alternative = c("two.sided", "less", "greater"),
          conf.level = 0.95, na.omit = TRUE, digits = 2,
          as.na = NULL, check = TRUE, output = TRUE)

Arguments

x	a matrix or data frame with numeric variables representing the levels of the within-subject factor, i.e., data are specified in wide-format (i.e., multivariate person level format).
adjust	logical: if TRUE (default), difference-adjustment for the Cousineau-Morey within-subject confidence intervals is applied.
alternative	a character string specifying the alternative hypothesis, must be one of "two.sided" (default), "greater" or "less".
conf.level	a numeric value between 0 and 1 indicating the confidence level of the interval.
na.omit	logical: if TRUE, incomplete cases are removed before conducting the analysis (i.e., listwise deletion).
digits	an integer value indicating the number of decimal places to be used.
as.na	a numeric vector indicating user-defined missing values, i.e. these values are converted to NA before conducting the analysis.
check	logical: if TRUE, argument specification is checked.
output	logical: if TRUE, output is shown on the console.

Details

The Cousineau within-subject confidence interval (CI, Cousineau, 2005) is an alternative to the Loftus-Masson within-subject CI (Loftus & Masson, 1994) that does not assume sphericity or homogeneity of covariances. This approach removes individual differences by normalizing the raw scores using participant-mean centering and adding the grand mean back to every score:

Y_{ij}^{'} = Y_{ij} - \hat{\mu}_{i} + \hat{\mu}_{grand}

where Y_{ij}^{'} is the score of the ith participant in condition j (for i = 1 to n), \hat{\mu}_{i} is the mean of participant i across all J levels (for j = 1 to J), and \hat{\mu}_{grand} is the grand mean.

Morey (2008) pointed out that Cousineau's (2005) approach produces intervals that are consistently too narrow due to inducing a positive covariance between normalized scores within a condition introducing bias into the estimate of the sample variances. The degree of bias is proportional to the number of means and can be removed by rescaling the confidence interval by a factor of \sqrt{J - 1}/J:

\hat{\mu}_j \pm t_{n - 1, 1 - \alpha/2} \sqrt{\frac{J}{J - 1}} \hat{\sigma}^{'}_{{\hat{\mu}}_j}

where \hat{\sigma}^{'}_{{\mu}_j} is the standard error of the mean computed from the normalized scores of he jth factor level.

Baguley (2012) pointed out that the Cousineau-Morey interval is larger than that for a difference in means by a factor of \sqrt{2} leading to a misinterpretation of these intervals that overlap of 95% confidence intervals around individual means is indicates that a 95% confidence interval for the difference in means would include zero. Hence, following adjustment to the Cousineau-Morey interval was proposed:

\hat{\mu}_j \pm \frac{\sqrt{2}}{2} (t_{n - 1, 1 - \alpha/2} \sqrt{\frac{J}{J - 1}} \hat{\sigma}^{'}_{{\hat{\mu}}_j})

The adjusted Cousineau-Morey interval is informative about the pattern of differences between means and is computed by default (i.e., adjust = TRUE).

Value

Returns an object of class misty.object, which is a list with following entries:

call	function call
type	type of analysis
data	data frame with variables used in the current analysis
args	specification of function arguments
result	result table

Author(s)

Takuya Yanagida [e-mail]

References

Baguley, T. (2012). Calculating and graphing within-subject confidence intervals for ANOVA. Behavior Research Methods, 44, 158-175. https://doi.org/10.3758/s13428-011-0123-7

Cousineau, D. (2005) Confidence intervals in within-subject designs: A simpler solution to Loftus and Masson’s Method. Tutorials in Quantitative Methods for Psychology, 1, 42–45. https://doi.org/10.20982/tqmp.01.1.p042

Loftus, G. R., and Masson, M. E. J. (1994). Using confidence intervals in within-subject designs. Psychonomic Bulletin and Review, 1, 476–90. https://doi.org/10.3758/BF03210951

Morey, R. D. (2008). Confidence intervals from normalized data: A correction to Cousineau. Tutorials in Quantitative Methods for Psychology, 4, 61–4. https://doi.org/10.20982/tqmp.01.1.p042

See Also

aov.w, test.z, test.t, ci.mean.diff,' ci.median, ci.prop, ci.var, ci.sd, descript

Examples

dat <- data.frame(time1 = c(3, 2, 1, 4, 5, 2, 3, 5, 6, 7),
                  time2 = c(4, 3, 6, 5, 8, 6, 7, 3, 4, 5),
                  time3 = c(1, 2, 2, 3, 6, 5, 1, 2, 4, 6))

# Difference-adjusted Cousineau-Morey confidence intervals
ci.mean.w(dat)

# Cousineau-Morey confidence intervals
ci.mean.w(dat, adjust = FALSE)

misty documentation built on Nov. 15, 2023, 1:06 a.m.