Confidence Intervals from Normalized Data: A correction to Cousineau (2005)

Abstract

Presenting confidence intervals around means is a common method of expressing uncertainty in data. Loftus and Masson (1994) describe confidence intervals for means in within-subjects designs. These confidence intervals are based on the ANOVA mean squared error. Cousineau (2005) presents an alternative to the Loftus and Masson method, but his method produces confidence intervals that are smaller than those of Loftus and Masson. I show why this is the case and offer a simple correction that makes the expected size of Cousineau confidence intervals the same as that of Loftus and Masson confidence intervals. Confidence intervals (CIs) are a staple in the presentation of psychological data because they allow researchers to quickly gage the amount of uncertainty in data (Rouder & Morey, 2005). For within-subjects designs, there are several approaches to creating confidence intervals. For a given design it may not be clear which to choose. Consider a simple within-subjects design with two conditions, a pre-test and post-test. For this design, there are multiple methods of generating confidence intervals. I will discuss each in turn. Approaches to confidence intervals The standard way to build confidence intervals is to compute the standard error of the mean for each condition, and multiply it by the appropriate t-distribution quantile. In order to make this concrete, Table 1 lists hypothetical data for N = 10 participants. A paired t t-test reveals a significant

Associatively mediated stopping: Training stimulus-specific inhibitory control

“…3 Mean reaction times of no-signal trials (top) and p(respond | signal) (bottom) for the single-signal (left) and multiple-signal (right) groups from Experiment 1. Error bars are normalized 95 % confidence intervals (see Morey, 2008) 25 % and 50 % stop cues did not differ significantly (p = .81, n 2 G ¼ :000). At test, whereas the main effect of cue type was not significant (p = .19,n 2 G ¼ :000), a two-way interaction between cue type and group was observed in the measured RTs (p < .05,n 2 G ¼ :001).…”

“…4 Mean reaction times of no-signal trials (top) and p(respond | signal) (bottom) for the single-signal (left) and multiple-signal (right) groups from Experiment 2. Error bars are normalized 95 % confidence intervals (see Morey, 2008) p < .06,n 2 G ¼ :032) cues. However, the 75 % and 50 % stop cues did not differ (p = .89,n 2 G ¼ :001).…”

Hippocampal volume and functional connectivity changes during the female menstrual cycle

“…1. For visual display of the error bars, between-subject variance was removed before plotting following Morey (2008). Estrogen levels were significantly higher during LFP, OVP and MLP than during EFP, and did not significantly differ between each of the three phases (LFP N EFP T(df = 16) = 2.14, P = 0.049; OVP N EFP T(df = 15) = 3.60, P b 0.001; MLP N EFP T(df = 15) = 2.54, P = 0.023).…”

“…Estrogen, progesterone, LH, FSH and BDNF levels on the four measurement occasions. Error bars represent standard error of the mean (SEM) at each time point after removing between-person variability (Morey, 2008).…”

Congruency effects on recognition memory: A context effect.

“…These d' values were analysed with a 2 x 2 mixed-factors ANOVA, with difficulty (easyselection/hard-selection) as a within-subject factor and order (easy-first/hard-first) as a betweensubjects factor. This analysis also revealed only a significant main effect of difficulty, Error bars represent the standard error of the mean corrected for between-subject variability (Morey, 2008). …”