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Letter
Endogeneity in Logistic Regression
Models
To the Editor: Ethelberg et al. (1) report on
a study of the determinants of hemolytic uremic syndrome resulting from
Shiga toxin–producing Escherichia coli. The dataset is relatively
small, and the authors use stepwise logistic regression models to detect
small differences. This indicates that the authors were aware of the limitations
of the statistical power of the study. Despite this, the study has an
analytic flaw that seriously reduces the statistical power of the study.
An often overlooked problem in building statistical models is that of
endogeneity, a term arising from econometric analysis, in which the value
of one independent variable is dependent on the value of other predictor
variables. Because of this endogeneity, significant correlation can exist
between the unobserved factors contributing to both the endogenous independent
variable and the dependent variable, which results in biased estimators
(incorrect regression coefficients) (2). Additionally,
the correlation between the dependent variables can create significant
multicollinearity, which violates the assumptions of standard regression
models and results in inefficient estimators. This problem is shown by
model-generated coefficient standard errors that are larger than true
standard errors, which biases the interpretation towards the null hypothesis
and increases the likelihood of a type II error. As a result, the power
of the test of significance for an independent variable X1
is reduced by a factor of (1-r2(1|2,3,….)), where
r(1|2,3,….) is defined as the multiple correlation coefficient
for the model X1 = f(X2,X3,…), and all
Xi are independent variables in the larger model (3,4).
The results of this study clearly show that the presence of bloody diarrhea
is an endogenous variable in the model showing predictors of hemolytic
uremic syndrome, in that the diarrhea is shown to be predicted by, and
therefore strongly correlated with, several other variables used to predict
hemolytic uremic syndrome. Similarly, Shiga toxin 1 and 2 (stx1,
stx2) genes are expected to be key predictors of the presence of
bloody diarrhea, independent of strain, due to the known biochemical effects
of that toxin (5,6). Because the strain is in part determined
by the presence of these toxins, including both strain and genotype in
the model means that the standard errors for variables for the Shiga-containing
strains and bloody diarrhea symptom are likely to be too high, and hence
the significance levels (p values) obtained from the regression models
are higher than the true probability because of a type I error.
This flaw is a particular problem with studies that use a conditional
stepwise technique for including or excluding variables. The authors note
that they excluded variables from the final model if the significance
in initial models for those variables was less than an α level (p
value) of 0.05. Given the inefficiencies due to the endogeneity of bloody
diarrhea, as well as those that may result from other collinearities significant
predictors were likely excluded from the study, although this cannot be
confirmed from the data presented.
The problems associated with the endogeneity of bloody diarrhea can be
overcome by a number of approaches. For example, the simultaneous equations
approach, such as that outlined by Greene (7), would
have used predicted values of bloody diarrhea from the first stage of
the model as instrumental variables for the actual value in the model
for hemolytic uremic syndrome. Structural equations approaches, such as
those suggested by Greenland (8), would also be appropriate.
However, bloody diarrhea is not the only endogenous variable in their
models, and extensive modeling would be necessary to isolate the independent
effects of the various predictor variables. Given the small sample size,
this may not be possible.
The underlying problem in the study is the theoretical specifications
for the model, in which genotypes, strains, and symptoms are mixed, despite
reasonable expectations that differences in 1 level may predict differences
in another. For example, the authors' data demonstrate that all O157 strains
contain the stx2 gene and have higher rates of causing
hemolytic uremic syndrome and bloody diarrhea. This calls into question
the decision to build an analytic model combining 3 distinct levels of
analysis. Such a model depends on the independence of the variables to
gain unbiased, efficient estimators. The model of the relationships one
would develop from a theoretical perspective would predict the opposite
(Figure). We expect that the genotypes (by
definition) will predict the strain, and that strains have a differential
effect on symptoms. The high level of intervariable correlation due to
these relationships, coupled with the decision to exclude variables based
on likely inefficient p values, raises questions concerning the reliability
of the results and conclusions. In particular, the conclusions that strains
O157 and O111 are not predictors of hemolytic uremic syndrome deserve
to be revisited; other excluded variables may also be significant predictors
when considered under an appropriate model. These problems point to the
need to ensure proper specification of analytic models and to demonstrate
due regard for the underlying assumptions of statistical models used.
George Avery*
*University of Minnesota, Duluth, Minnesota, USA
Suggested citation
for this article:
Avery G. Endogeneity
in logistic regression models [letter]. Emerg Infect Dis [serial on the
Internet]. 2005 Mar [date cited]. Available from http://www.cdc.gov/ncidod/EID/vol11no03/04-0462_05-0071.htm
References
- Ethelberg S, Olson KEP, Schuetz F, Jensen C, Schiellerup
P, Engberg J, et al. Virulence
factors for hemolytic uremic syndrome, Denmark. Emerg Infect Dis.
2004;10: 842–7.
- Dowd B, Town R. Does X really cause Y? Washington: Academy Health;
2002.
- Hsieh F, Bloch D, Larsen M. A
simple method of sample size calculation for linear and logistic regression.
Stat Med. 1998;17:1623–34.
- Menard S. Applied logistic regression analysis, 2nd ed. Thousand Oaks
(CA): Sage Publications: 2002. p. 75–8.
- Blackall DP, Marques MB. Hemolytic
uremic syndrome revisited: Shiga toxin, factor H, and fibrin generation.
Am J Clin Pathol. 2004;121 (Suppl):S81–8.
- Harrison LM, van Haaften WC, Tesh VL. Regulation
of proinflammatory cytokine expression by Shiga toxin 1 and/or lipopolysaccharides
in the human monocytic cell line THP-1. Infect Immun. 2004;72:2618–27.
- Greene W. Gender economics courses in liberal arts colleges: further
results. J Econ Educ. 1998;29:291–300.
- Greenland S, Brumback B. An
overview of relations among causal modelling methods. Int J Epidemiol.
2002;31:1030–7.
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In response: We appreciate Avery's interest
(1) in our article (2), although we
believe the critique of the methods is largely based on misunderstandings.
We developed a model for the risk of progression to hemolytic uremic syndrome
(HUS) containing 3 variables: whether the infecting Shiga toxin–producing
Escherichia coli isolate had the stx2 gene, age
of the patient, and occurrence of bloody diarrhea. The critique relates
to the fact that bloody diarrhea and stx2 are not independent,
since we showed that stx2 was strongly associated with
progression to HUS (odds ratio [OR] = 18.9) and also weakly associated
with development of bloody diarrhea (OR = 2.5) (2).
Avery uses the term endogeneity as it is used in econometric analyses;
however, the term "intermediary variable," i.e., a factor in
the causal pathway leading from exposure to disease, is more frequently
used in epidemiology. In this context, we chose to consider bloody diarrhea
as a potential confounder (3). A confounder is a risk
factor but is also independently associated with the exposure variable
of interest and is not regarded as part of the causal pathway (Figure).
Bloody diarrhea may act as a confounder if patients with bloody stools
are treated differently by the examining physicians or if, for instance,
unknown virulence factors contribute to the risk of having bloody stools.
A second line of critique of our methods apparently develops from the
idea that virulence factors determine the serogroup. This idea, however,
is a biological misconception. In fact, virulence genes and serogroup
are independent at the genetic level, and an important point of our article
is that HUS is determined by the virulence gene composition of the strain
rather than the serogroup.
Regardless of the status of the bloody diarrhea variable, excluding it
from the model doesn't change the conclusions of the article. A revised
model contains only the significant variables age and stx2
(Table). Serotype O157 is still not an independent
predictor of HUS, and this result is robust.
Steen Ethelberg*
and Kåre Mølbak*
*Statens Serum Institut, Copenhagen, Denmark
Suggested citation
for this article:
Ethelberg S, Mølbak
K. Endogeneity in logistic regression models [reply to letter]. Emerg
Infect Dis [serial on the Internet]. 2005 Mar [date cited]. Available
from http://www.cdc.gov/ncidod/EID/vol11no03/04-0462_05-0071.htm
References
- Avery G. Endogeneity in logistic regression models.
Emerg Infect Dis. 2005;11[this issue].
- Ethelberg S, Olsen KE, Scheutz F, Jensen C, Schiellerup P, Enberg
J, et al. Virulence
factors for hemolytic uremic syndrome, Denmark. Emerg Infect Dis.
2004;10:842–7.
- Griffin PM, Mead PS, Sivapalasingam S. Escherichia coli O157:H7
and other enterohaemorrhagic E. coli. In: Blaser MJ, Smith PD,
Ravdin JI, Greenberg HB, Guerrant RL, editors. Infections of the gastrointestinal
tract. Philadelphia: Lippincott Williams & Wilkins; 2002. p. 627–42.
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Table. Risk factors for
HUS among 343 STEC patients, Denmark 1997–2003, comparison of models
with and without bloody diarrhea as a variable*
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Determinant
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No. of patients
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No. (%) with HUS
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Original model, OR (95% CI)
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New model, OR (95% CI)
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eae
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Negative
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111
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0 (0.0)
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Positive
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232
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21 (9.1)
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NI
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NI
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stx2
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Negative
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159
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1 (0.6)
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1
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1
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Positive
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184
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20 (10.9)
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18.9 (2.4–146)
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24.6 (3.2–187)
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Age
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>8 y
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178
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3 (1.7)
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1
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1
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<7 y
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165
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18 (10.9)
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11.4 (3.2–41.3)
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9.7 (2.7–34.1)
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Bloody diarrhea
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No
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218
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6 (2.8)
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Yes
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125
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15 (12.0)
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4.5 (1.6–12.7)
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EX
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O157
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No
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262
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10 (3.8)
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Yes
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81
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11 (13.6)
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NS
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NS
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*HUS, hemolytic uremic syndrome; STEC, Shiga toxin–producing
Escherichia coli; OR, odds ratio; CI, confidence interval;
NI, not included (test not appropriate); NS, not significant; EX,
excluded from model.
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