ERIC Identifier: ED410231
Publication Date: 1996-11-00
Author: Helberg, Clay
Source: ERIC Clearinghouse on
Assessment and Evaluation Washington DC.
Pitfalls of Data Analysis. ERIC/AE Digest.
It can be surprising to naive observers that statistics from different
research studies concerning the same issue can produce very dissimilar or
contradictory results. In order to resolve this paradox, many people conclude
that statistics are not actually reliable indicators of reality. Those versed in
statistics, however, understand that statistics rely on assumptions. Proceeding
from different assumptions, or claiming assumptions which don't apply to the
research situation in question can lead to divergent results. This digest
attempts to warn against the frequent misuses and abuses of statistics. Although
these issues are familiar to most statisticians, they can be easily overlooked.
These problems can be considered in three broad classes of statistical pitfalls:
sources of bias, errors in methodology, and misinterpretation of results.
SOURCES OF BIAS
Statistical methodology assists researchers
in making inferences about a large group (a population) based on observations of
a smaller subset of that group (a sample). Sources of bias are conditions or
circumstances which affect the external validity of statistical results. Thus,
in order for a researcher to make legitimate conclusions about the specified
population, two characteristics must be present within the sample:
representative sampling and valid statistical assumptions.
Representative sampling is one of the most fundamental principles of
inferential statistics. This type of sampling implies that the observed group
has similar characteristics to the target population in all areas that are
relevant to the study. Representative sampling is necessary to make valid
inferences made about the target population. Unfortunately, representative
sampling can be difficult to achieve. The ideal sample is chosen by selecting
members of the population at random, with each member having an equal
probability of being selected for the sample. When randomization is not
possible, researchers usually try to choose a sample in which their group of
subjects "parallels" the population with respect to the characteristics that are
thought to be important to the particular investigation.
Statistical assumptions made about various aspects of the problem determine
the statistical procedure's validity. This means that certain aspects of the
measured variables must conform to assumptions which underlie the statistical
procedures to be applied. For example, well-known linear methods such as
analysis of variance (ANOVA) depend on the assumptions of normality and
independence. While assumption of normality implies that the scores in each
treatment group are distributed in a way that corresponds to the so-called
"normal" (or Gaussian) distribution, the assumption of independence indicates
that each of the subject's scores are uninfluenced by the scores of anyone else
who was tested.
ERRORS IN METHODOLOGY
There are a number of ways that
statistical techniques can be misapplied to problems in the real world. These
types of errors can lead to invalid or inaccurate results. Three of the most
common hazards are designing experiments with insufficient statistical power,
ignoring measurement error, and performing multiple comparisons.
Two types of errors can occur when making inferences based on a statistical
hypothesis test: a Type I error happens if the null hypothesis is rejected when
it should not be (the probability of this is called "alpha"); and a Type II
error results from the failure to reject a null hypothesis when you should (the
probability of this is called "beta"). Statistical power refers to the
probability of avoiding a Type II error and depends on the ability of one's
statistical test to detect true differences of a particular size. The power of
the test generally depends on four things: the sample size, the desired
detectable effect size, the specified Type I error rate, and the variability of
the sample. Based on these parameters, the power level of the experiment can be
calculated. Nevertheless, the researcher can also specify the desired power
level (e.g. .80), the Type I error level, and the minimum effect size which
would be considered "interesting." (See Cohen, 1988, for more details on power
If there is little statistical power, a researcher risks overlooking the
effect which he/she is attempting to discover. This is especially important if
one intends to make inferences based on a finding of no difference. However, it
should be noted that it is possible to have too much statistical power. If the
sample is too large, nearly any difference, no matter how small or meaningless
from a practical standpoint, will be "statistically significant." This
occurrence can be particularly problematic in applied settings, where important
decisions are determined by statistical results.
Studying the relationship of multiple variables is especially troublesome
because the desired knowledge is complex in nature and many different
combinations of factors need to be examined. The best strategy to check these
different combinations of factors is to rerun the experiment and see which
comparisons show differences in both groups (also known as replication).
Although this method is not irrefutable, it should provide a good notion of
which effects are real and which are not. If replication is not a possibility,
cross-validation--a technique which involves setting aside part of the sample as
a validation sample--can also be helpful. In this system, the statistics of
interest are computed on the main sample and are checked against the validation
sample to verify that the effects are real. Using this technique, results that
are spurious will usually be revealed by the validation sample.
Most statistical models assume error free measurement, at least of
independent (predictor) variables. However, measurements are seldom perfect.
Therefore, close attention must be paid to the effects of measurement errors.
This is especially important when dealing with noisy data such as questionnaire
responses or processes which are difficult to measure precisely.
Methods are available for taking measurement error into account in some
statistical models. In particular, structural equation modeling allows one to
specify relationships between "indicators," or measurement tools, and the
underlying latent variables being measured, in the context of a linear path
model. For more information on structural equation modeling and its uses, see
PROBLEMS WITH INTERPRETATION
In addition to difficulties
with bias and methodology, there are a number of problems which can arise in the
context of substantive interpretation as well. These problems usually involve
determining the significance of certain findings, avoiding confusion between
precision and accuracy, and unraveling the causal relationships among variables.
The difference between "significance" in the statistical sense and
"significance" in the practical sense continues to elude many consumers of
statistical results. Significance (in the statistical sense) is really as much a
function of sample size and experimental design as it is a function of strength
of relationship. With low power, a researcher may overlook a useful
relationship; with excessive power, one may find microscopic effects that have
no real practical value. A reasonable way to handle this sort of thing is to
cast results in terms of effect sizes (see Cohen, 1994)--that way the size of
the effect is presented in terms that make quantitative sense.
Precision and Accuracy are two concepts which seem to get confused
frequently. It's a subtle but important distinction: precision refers to how
finely an estimate is specified, whereas accuracy refers to how close an
estimate is to the true value. Estimates can be precise without being accurate,
a fact often glossed over when interpreting computer output containing results
specified to the fourth or sixth or eighth decimal place. Therefore, one should
not report any more decimal places than he/she is fairly confident of reflecting
Assessing causality is the reason of most statistical analysis, yet its
subtleties escape many statistical consumers. For one to determine a causal
inference, he/she must have random assignment. That is, the experimenter must be
the one assigning values of predictor variables to cases. If the values are not
assigned or manipulated, the most one can hope for is to show evidence of a
relationship of some kind. Observational studies are very limited in their
ability to illuminate causal relationships.
Now, of course, many of the things that are of interest to study are not
subject to experimental manipulation (e.g. health problems/risk factors). In
order to understand them in a causal framework, a multifaceted approach to the
research (you might think of it as "conceptual triangulation"), the use of
chronologically structured designs (placing variables in the roles of
antecedents and consequents), and plenty of replication is required to come to
any strong conclusions regarding causality.
In this paper, some of the trickier aspects of
applied data analysis have been discussed. In future research or data analysis,
people should be certain of the following:
The sample is representative of the population of interest.
The right amount of power should be included.
The best available measurement tools should be used. If there are errors in
the measures, that fact must be taken into account when interpreting the
Multiple comparisons need to be watched closely. If many test need to be
done, replication or cross-validation should be used to verify the results.
The objective of the study should remain the focus when interpreting the
data. Therefore, magnitudes rather than p-values should be studied so that one
isn't seduced by "stars in the tables."
Numerical notation should be used in a rational way. This will help to avoid
confusion between precision and accuracy.
The conditions for causal inference should be understood.
If causal inference must be made, random assignment should be use. In the
absence of random assignment, much effort will be needed to uncovering causal
relationships, requiring a variety of approaches to the question.
Although errors and misconceptions about statistical information are
difficult to avoid, one can use the above suggestions to help present the
information in the clearest way possible.
Mr. Helberg can be reached at SPSS, Inc., 444 N. Michigan Avenue, Chicago, IL
60611; or via e-mail at [email protected]
Bollen, K. (1989). Structural Equations
with Latent Variables. New York: John Wiley & Sons.
Cohen, J. (1988). Statistical Power Analysis for the Behavioral Sciences.
Hillsdale, NJ: Lawrence Erlbaum Associates.
Cohen, J. (1990). Things I have learned (so far). American Psychologist, 45,
Huff, D. (1954). How to Lie with Statistics, New York: W.W. Norton & Co.
Paulos, J.A. (1988). Innumeracy: mathematical illiteracy and its
consequences. New York: Hill & Wang.
Tufte, E.R. (1983). The Visual Display of Quantitative Information. Cheshire,
CT: Graphics Press.