To make sense of their data and effects, scientists might want to standardize (Z-score) their variables. This makes the data unitless, expressed only in terms of deviation from an index of centrality (e.g., the mean or the median). However, aside from some benefits, standardization also comes with challenges and issues, that the scientist should be aware of.
effectsize package offers two methods of standardization via the
Normal standardization: center around the mean, with SD units (default).
Robust standardization: center around the median, with MAD units (
robust = TRUE).
Let’s look at the following example:
library(effectsize) data("hardlyworking") head(hardlyworking)
> salary xtra_hours n_comps age seniority > 1 19745 4.2 1 32 3 > 2 11302 1.6 0 34 3 > 3 20636 1.2 3 33 5 > 4 23047 7.2 1 35 3 > 5 27342 11.3 0 33 4 > 6 25657 3.6 2 30 5
$xtra_hours_z <- standardize(hardlyworking$xtra_hours) hardlyworking $xtra_hours_zr <- standardize(hardlyworking$xtra_hours, robust = TRUE)hardlyworking
We can see that different methods give different central and variation values:
standardize() can also be used to standardize a full data frame - where each numeric variable is standardized separately:
Weighted standardization is also supported via the
weights argument, and factors can also be standardized (if you’re into that kind of thing) by setting
force = TRUE, which converts factors to treatment-coded dummy variables before standardizing.
Standardization is an important step and extra caution is required in repeated-measures designs, in which there are three ways of standardizing data:
Variable-wise: The most common method. A simple scaling of each column.
Participant-wise: Variables are standardized “within” each participant, i.e., for each participant, by the participant’s mean and SD.
Full: Participant-wise first and then re-standardizing variable-wise.
Unfortunately, the method used is often not explicitly stated. This is an issue as these methods can generate important discrepancies (that can in turn contribute to the reproducibility crisis). Let’s investigate these 3 methods.
We will take the
emotion dataset in which participants were exposed to negative pictures and had to rate their emotions (valence) and the amount of memories associated with the picture (autobiographical link). One could make the hypothesis that for young participants with no context of war or violence, the most negative pictures (mutilations) are less related to memories than less negative pictures (involving for example car crashes or sick people). In other words, we expect a positive relationship between valence (with high values corresponding to less negativity) and autobiographical link.
Let’s have a look at the data, averaged by participants:
library(dplyr) library(tidyr) # Download the 'emotion' dataset load(url("https://raw.github.com/neuropsychology/psycho.R/master/data/emotion.rda")) # Discard neutral pictures (keep only negative) <- emotion %>% emotion filter(Emotion_Condition == "Negative") # Summary %>% emotion drop_na(Subjective_Valence, Autobiographical_Link) %>% group_by(Participant_ID) %>% summarise( n_Trials = n(), Valence_Mean = mean(Subjective_Valence), Valence_SD = sd(Subjective_Valence) )
> # A tibble: 19 x 4 > Participant_ID n_Trials Valence_Mean Valence_SD > * <fct> <int> <dbl> <dbl> > 1 10S 24 -58.1 42.6 > 2 11S 24 -73.2 37.0 > 3 12S 24 -57.5 26.6 > 4 13S 24 -63.2 23.7 > 5 14S 24 -56.6 26.5 > 6 15S 24 -60.6 33.7 > 7 16S 24 -46.1 24.9 > 8 17S 24 -1.54 4.98 > 9 18S 24 -67.2 35.0 > 10 19S 24 -59.6 33.2 > 11 1S 24 -53.0 42.9 > 12 2S 23 -43.0 39.2 > 13 3S 24 -64.3 34.4 > 14 4S 24 -81.6 27.6 > 15 5S 24 -58.1 25.3 > 16 6S 24 -74.7 29.2 > 17 7S 24 -62.3 39.7 > 18 8S 24 -56.9 32.7 > 19 9S 24 -31.5 52.7
As we can see from the means and SDs, there is a lot of variability between participants both in their means and their individual within-participant SD.
We will create three data frames standardized with each of the three techniques.
<- emotion %>% Z_VariableWise standardize() <- emotion %>% Z_ParticipantWise group_by(Participant_ID) %>% standardize() <- emotion %>% Z_Full group_by(Participant_ID) %>% standardize() %>% ungroup() %>% standardize()
Let’s see how these three standardization techniques affected the Valence variable.
We can calculate the mean and SD of Valence across all participants:
The means and the SD appear as fairly similar (0 and 1)…
and so do the marginal distributions…
However we can also look at what happens in the participant level. Let’s look at the first 5 participants:
Seems like full and participant-wise standardization give similar results, but different ones than variable-wise standardization.
Let’s do a correlation between the variable-wise and participant-wise methods.
While the three standardization methods roughly present the same characteristics at a general level (mean 0 and SD 1) and a similar distribution, their values are not exactly the same!
Let’s now answer to the original question by investigating the linear relationship between valence and autobiographical link. We can do this by running a mixed model with participants entered as random effects.
library(lme4) <- lmer(Subjective_Valence ~ Autobiographical_Link + (1 | Participant_ID), m_raw data = emotion ) <- update(m_raw, data = Z_VariableWise) m_VariableWise <- update(m_raw, data = Z_ParticipantWise) m_ParticipantWise <- update(m_raw, data = Z_Full)m_Full
We can extract the parameters of interest from each model, and find:
> Model | Coefficient | SE | 95% CI | t(451) | p > ----------------------------------------------------------------------- > m_raw | 0.09 | 0.07 | [-0.04, 0.22] | 1.36 | 0.173 > m_VariableWise | 0.07 | 0.05 | [-0.03, 0.16] | 1.36 | 0.173 > m_ParticipantWise | 0.08 | 0.05 | [-0.01, 0.17] | 1.75 | 0.080 > m_Full | 0.08 | 0.05 | [-0.01, 0.17] | 1.75 | 0.080
As we can see, variable-wise standardization only affects the coefficient (which is expected, as it changes the unit), but not the test statistics or significance. However, using participant-wise standardization does affect the coefficient and the significance.. No method is better or more justified, and its choice depends on the specific case, context, data and goal.
Standardization can be useful in some cases and should be justified
Variable and Participant-wise standardization methods produce “in appearance” similar data
Variable and Participant-wise standardization can lead to different results
The choice of the method can strongly influence the results and thus, should be explicitly stated
We showed here yet another way of sneakily tweaking the data that can change the results. To prevent its use for bad practices (e.g., p-hacking), we can only support the generalization of open-data, open-analysis and preregistration**.
See also the
parameters::demean() function, and the
standardize_parameters(method = "pseudo") for more mixed-model tools.