This function computes difference-adjusted Cousineau-Morey within-subject confidence interval for the arithmetic mean.
ci.mean.w(..., data = NULL, adjust = TRUE,
alternative = c("two.sided", "less", "greater"),
conf.level = 0.95, na.omit = TRUE, digits = 2,
as.na = NULL, write = NULL, append = TRUE, check = TRUE,
output = TRUE)Returns an object of class misty.object, which is a list with following
entries:
callfunction call
typetype of analysis
datadata frame used for the current analysis
argsspecification of function arguments
resultresult table
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). Alternatively, an expression indicating the variable
names in data e.g., ci.mean.w(x1, x2, x3, data = dat).
Note that the operators
., +, -, ~, :, ::,
and ! can also be used to select variables, see 'Details'
in the df.subset function.
a data frame when specifying one or more variables in the
argument .... Note that the argument is NULL
when specifying a matrix or data frame
for the argument ....
logical: if TRUE (default), difference-adjustment
for the Cousineau-Morey within-subject confidence intervals
is applied.
a character string specifying the alternative hypothesis,
must be one of "two.sided" (default), "greater"
or "less".
a numeric value between 0 and 1 indicating the confidence level of the interval.
logical: if TRUE (default), incomplete cases are removed
before conducting the analysis (i.e., listwise deletion).
an integer value indicating the number of decimal places to be used.
a numeric vector indicating user-defined missing values,
i.e. these values are converted to NA before
conducting the analysis.
a character string naming a text file with file extension
".txt" (e.g., "Output.txt") for writing the
output into a text file.
logical: if TRUE (default), output will be appended
to an existing text file with extension .txt specified
in write, if FALSE existing text file will be
overwritten.
logical: if TRUE (default), argument specification is checked.
logical: if TRUE (default), output is shown on the console.
Takuya Yanagida takuya.yanagida@univie.ac.at
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 \(i\)th 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 \(j\)th 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).
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
aov.w, test.z, test.t,
ci.mean.diff,' ci.median, ci.prop,
ci.var, ci.sd, descript
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))
# Example 1: Difference-adjusted Cousineau-Morey confidence intervals
ci.mean.w(dat)
# Example 1: Alternative specification using the 'data' argument
ci.mean.w(., data = dat)
# Example 2: Cousineau-Morey confidence intervals
ci.mean.w(dat, adjust = FALSE)
if (FALSE) {
# Example 3: Write results into a text file
ci.mean.w(dat, write = "WS_Confidence_Interval.txt")
}
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