Standardized coefficients
Jane E. Miller, PhD
The Chicago Guide to Writing about Multivariate Analysis, 2 nd edition.
Overview
• Review of unstandardized coefficients
• Interpretation of standardized coefficients
• Descriptive statistics for multivariate models
with standardized coefficients
The Chicago Guide to Writing about Multivariate Analysis, 2nd edition.
Unstandardized coefficients
• Unstandardized βs estimate the effect of a 1unit increase in Xi on Y, where the effect size is
measured in the original units of Y.
• A “one-size-fits-all” approach to interpreting
βs can be misleading because variables
– Represent different levels of measurement,
– Have different units of measurement,
– Have varying distributions of values,
– Occur in different real-world circumstances.
Standardized coefficients
• A standardized coefficient estimates the effect of a
one-standard-deviation increase in Xi on Y
– Measured in standard deviation units of Y
• e.g., an effect size of 0.3 would mean 30% of a standard deviation
in the dependent variable
– Similar to standardized scores or z-scores
• Standardized βs provide a consistent metric in which
to compare the relative sizes of the βs on continuous
independent variables with different ranges and
scales.
– Contrast for each IV is its standard deviation
Using standardized coefficients
• Commonly used for psychological or
attitudinal scales for which the units have no
inherent meaning.
• Should not be used for variables for which a
one-standard-deviation increase lacks an
intuitive interpretation. E.g.,
– dummy variables
– interaction terms
Specifying a model with standardized
coefficients
• Easily specified as an option to an OLS model
in most statistical packages.
• Identify the dependent and independent
variables as usual.
– Enter them in the model specification in their
original, untransformed versions.
• Do not create versions in the metric of standard
deviations. The software will do that for you!
• Request “standardized betas”
Descriptive statistics to report if you
use standardized coefficients
• In table of descriptive statistics, report the
mean, minimum and maximum values and
standard deviation in the original units for
– each independent variable (IV)
– the dependent variable (DV)
Describing standardized coefficients in
prose
• In the results section, interpret the effect sizes
for different IVs in terms of multiples or
percentages of the standard deviation in the
DV
– E.g., “A one-standard-deviation increase in the
income-to-poverty ratio (IPR) is associated with an
increase of 19.6% of a standard deviation in birth
weight (about 38 grams), roughly twice the size of
the corresponding standardized coefficient on
mother’s age (9.7%).”
Reporting the effect size
in original units
“A one-standard-deviation increase in the income-topoverty ratio (IPR) is associated with an increase of
19.6% of a standard deviation in birth weight (about
38 grams), roughly twice the size of the
corresponding standardized coefficient on mother’s
age (9.7%).”
• Note that the effect size is also reported back in the original
units of the DV (grams in this case), to facilitate intuitive
understanding in the context of the specific research question
and variables.
Summary
• Consider whether standardized coefficients fit
your topic and data
– Continuous variables
– Field or discipline
• Report descriptive statistics in original units to
facilitate real-world interpretation of effect
sizes.
• Clearly convey the units in which the
standardized coefficients are measured.
Suggested resources
• Miller, J. E., 2013. The Chicago Guide to
Writing about Multivariate Analysis, 2nd
Edition.
– Chapter 8, on standardized scores and z-scores
– Chapter 10, on standardized coefficients
• Kachigan, Sam Kash. 1991. Multivariate
Statistical Analysis: A Conceptual Introduction.
2nd Edition. New York: Radius Press.
Supplemental online resources
• Podcasts on
– Interpreting multivariate coefficients
– Defining the Goldilocks problem
– Resolving the Goldilocks problem using model
specification
Suggested practice exercises
• Study guide to The Chicago Guide to Writing
about Multivariate Analysis, 2nd Edition.
– Questions #5 through 8 from the problem set for
chapter 10
– Suggested course extensions for chapter 10
• “Reviewing” exercise #3.
• “Applying statistics and writing” question #5.
• “Revising” questions #8 and 9.
Contact information
Jane E. Miller, PhD
jmiller@ifh.rutgers.edu
Online materials available at
http://press.uchicago.edu/books/miller/multivariate/index.html
The Chicago Guide to Writing about Multivariate Analysis, 2 nd edition.