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Previous Article | Next Article 
Applied and Environmental Microbiology, May 2001, p. 2129-2135, Vol. 67, No. 5
0099-2240/01/$04.00+0 DOI: 10.1128/AEM.67.5.2129-2135.2001
Copyright © 2001, American Society for Microbiology. All rights reserved.
Comparison of Logistic Regression and Linear
Regression in Modeling Percentage Data
Lihui
Zhao,
Yuhuan
Chen,
and
Donald W.
Schaffner*
Department of Food Science, Cook College, the
New Jersey Agricultural Experiment Station, Rutgers, The State
University of New Jersey, New Brunswick, New Jersey 08901-8520
Received 3 November 2000/Accepted 27 February 2001
 |
ABSTRACT |
Percentage is widely used to describe different results in food
microbiology, e.g., probability of microbial growth, percent inactivated, and percent of positive samples. Four sets of percentage data, percent-growth-positive, germination extent, probability for one
cell to grow, and maximum fraction of positive tubes, were obtained
from our own experiments and the literature. These data were modeled
using linear and logistic regression. Five methods were used to compare
the goodness of fit of the two models: percentage of predictions closer
to observations, range of the differences (predicted value minus
observed value), deviation of the model, linear regression between the
observed and predicted values, and bias and accuracy factors. Logistic
regression was a better predictor of at least 78% of the observations
in all four data sets. In all cases, the deviation of logistic models
was much smaller. The linear correlation between observations and
logistic predictions was always stronger. Validation (accomplished
using part of one data set) also demonstrated that the logistic model
was more accurate in predicting new data points. Bias and accuracy
factors were found to be less informative when evaluating models
developed for percentage data, since neither of these indices can
compare predictions at zero. Model simplification for the logistic
model was demonstrated with one data set. The simplified model was as powerful in making predictions as the full linear model, and it also
gave clearer insight in determining the key experimental factors.
| |
INTRODUCTION |
Microbial data expressed as
percentages have been modeled for many years. Percentage data may have
very different biological meanings and expressions. In 1971, Genigeorgis et al. initiated the concept of probability for one cell to
grow and produce toxin, presented as the ratio of RG over
RI, where RG is the number of cells initiating
growth, and RI is the number of cells in the inoculum
(14). In a time-to-turbidity model, Whiting and Oriente (32) described the maximum probability of growth with the
parameter Pmax, this value being obtained from fitting the
growth curve with the logistic equation. Chea et al. modeled the extent
of spore germination using the plateau value of the germination curve (6). The percent-growth-positive parameter describes the
maximum proportion of wells that exhibited growth under various
environmental conditions in a study using microplates inoculated with
Clostridium botulinum spores (33).
A conventional approach applied to modeling percentage data is to use
linear regression with polynomial terms. This method usually results in
moderate (R2 < 0.9) (6, 9, 10, 17, 26, 31,
33) to poor (R2 < 0.5) (32)
goodness of fit. Generally, the accuracy of linear models for modeling
bounded variables (e.g., percentage data) is not as good as for other
unbounded variables obtained in the same experiment, and the resulting
linear model also predicts poorly at values close to 0 and 1 (6,
32, 33). An insurmountable limitation of the linear approach is
that the model can predict percentages outside the probability range,
i.e., values of <0 or >1 (6, 26, 31, 33). Generally, all
predicted negative values are forced to 0, and those >1 are forced to
1. Even without this modification, it is not meaningful to compare
these conditions. For example, 120% cannot be interpreted as a higher
percent germination than 101%.
Logistic regression has been widely used in medical research (1,
5, 18, 19, 22, 30). In the field of predictive food
microbiology, logistic models have been developed to describe the
bacterial growth/no growth interface (4, 21, 24, 25). In
these models, the data were presented in the 0-1 format, as in a
typical binomial data set. Genigeorgis et al. first presented the
concept of the probability that one cell could grow in a specific environment (14). Later, this probability was modeled in
various systems using logistic regression combined with a linear
regression of the lag period (3, 11, 12, 15, 16, 20).
Roberts et al. used a similar concept and the regression approach to
model toxin production by C. botulinum in pasteurized pork
slurry (27). Cole et al. modeled the probability of growth
of spoilage yeast in a model fruit drink by directly relating the logit
of probability with the environmental factors (7). In
these studies, probability (a continuous number between 0 and 1)
instead of a dichotomous variable (i.e., 0, 1) was modeled. As pointed
out by Ratkowsky and Ross (25), the response modeled by
logistic regression at a given combination of limiting factors can
either have a value of 0 or 1 or be a probability. Probability,
generally expressed by dividing the number of successes by the total
number of trials, is simply a summarization of binomial data and thus
can be approximated by a logistic general linear model
(8).
In this study, we compared the goodness of fit of linear regression to
logistic regression for modeling percentages. We modeled data from our
own research and from the literature (including publications from our
group) and developed models using both the logistic and linear
approaches in exactly the same manner. Five different approaches were
used to compare the goodness of fit of the two models. In almost all
cases, the logistic models displayed greater accuracy and resulted in
less biased predictions.
| |
MATERIALS AND METHODS |
Data collection.
Four different sets of percentage data were
collected from previous experiments (6, 26, 32, 33). Each
set had its own unique biological meaning and was collected with a
different method.
Weight is the degree of emphasis a model puts on an observation. The
weight for a percentage datum point is the total number of observations
associated with this percentage (2). For example, when 10 of 40 tubes turn turbid, the percentage is 25% (10/40) and the weight
for this percentage is 40. The assignment of weights was determined
differently for each data set, as described below.
Data set I: data for percent-growth-positive were collected by Zhao et
al. (33). This data set contained the exact numbers of
wells that showed growth and no growth. The total number of wells in
each condition is the same, so the weight assigned for each condition
is the same. Environmental factors studied were pH, sodium chloride
concentration, and inoculum size in a complete 3 by 3 factorial design
with a total of 27 different conditions.
Data set II: extent of germination data were collected by Chea et al.
(6). The total number of spores studied for each condition
was between 200 and 300. The small difference in the total number in
each condition is negligible, and equal weight for all the data points
was assumed in logistic regression. Environmental factors studied were
pH, sodium chloride concentration, and temperature in a complete 3 by 3 factorial design with a total of 27 different conditions.
Data set III: Razavilar and Genigeorgis studied the probability of one
cell of Listeria monocytogenes to grow, as affected by
sodium chloride concentration, time, and temperature (26). Weights were not obtainable, so this parameter was assumed to be the
same in each case.
Data set IV: Pmax was the parameter used to indicate the
maximum fraction of positive tubes inoculated with C. botulinum (32). It was obtained by fitting the
experimental data with a logistic equation. The total number of tubes
varied by condition and was used as the weight in logistic regression.
Four environmental factors, pH, sodium chloride concentration,
temperature, and inoculum size, were studied in a total of 103 different conditions. A subset, containing 22 data points at 19°C,
was not used to develop models; instead, these data points were used
later to validate the models developed from the remaining 81 points.
Modeling with linear and logistic regression.
Both linear
and logistic models were developed in S-plus (MathSoft, Inc., Seattle,
Wash.) for an objective comparison. The generalized linear modeling
("glm") function was used for both methods. The link function for
logistic regression is "binomial" and for linear regression is
"gaussian." The full models generated by each approach, with the
same number of terms in the same format, were used to ensure the
validity of the comparison.
The linear model with three predictor variables has the following
general format:
|
(1)
|
where Percentage is the observed percentage,
0 is the intercept, X, Y, and
Z are the predictor variables, and
Cis are the coefficients.
The logistic model with three predictor variables has the following
general format:
|
(2)
|
where P is the probability that the event
would occur according to the model and the remaining symbols have the
same meaning as in equation 1.
Model comparison. (i) Adjustment with predictions from linear
models.
Predictions from linear models can be greater than 1 or
less than 0. In practice, these predictions are generally forced to be
1 and 0, respectively (6, 26, 32, 33). To make the comparison of the two models fairer, predictions from linear models were forced into the range of 0 to 1 in this manner. For all
comparisons, the modified predictions from linear models were used,
except as noted below.
(ii) Methods to compare model predictions.
Out-of-range
predictions from linear models were counted in Method 1. The number of
predictions from logistic regression that were closer to the observed
values was also calculated. For this calculation, the absolute value of
the difference (predicted minus observed) was used. We excluded some
observations whose linear regression predictions were out of range in
the calculation of the percentage of closer predictions. This is
required because logistic regression predicts strictly between 0 and 1. By forcing out-of-range linear predictions to be 0 or 1, we may
inappropriately make some linear predictions seem better. For example,
if the observation is 1, the logistic prediction is 0.999, and the
linear prediction is 1.235, if we force the linear prediction to be 1, it will falsely be judged better.
In Method 2, we compared the ranges of the differences between the
predicted and the observed values. Point summaries of the differences
(predicted minus observed), i.e., minimum, first quarter, median, mean,
third quarter, and maximum, were obtained, and the range and
interquarter range (IQR) were calculated.
|
(3)
|
|
(4)
|
The smaller the values of the range and IQR, the closer
the predictions are to the observations. The range is sensitive to outlying points whose predicted and observed values are very different, while the IQR is not affected as much.
For Method 3, the deviation of the model from observations was
calculated as follows:
|
(5)
|
The smaller the deviation, the closer the model
predictions were to the observations. Method 1 cannot detect
predictions that are far from the observations. Method 2 allows for
detection of these wide deviations by measuring the range of the
differences between the predicted and observed values, but it is unable
to indicate which model results in a greater number of predictions closer to the observed. Method 3 takes both considerations into account.
Method 4 used graphs of the observed values (x axis) versus
predicted values (y axis) from both models. A simple linear
regression was fitted to the points, and the intercept, the slope, and
R2 were obtained. If the predictions are in perfect
agreement with the observed values, the intercept should be 0, the
slope should be 1, and R2 should be 1. The closer the
intercept is to 0, the slope is to 1, and R2 is to 1, the
better is the general predictive power of the model. A slope of less
than 1 indicates that the model underpredicts the observation.
Method 5 used bias and accuracy values as a quantitative way to measure
the goodness of fit of the models (6, 28). The bias factor
indicates by how much, on average, a model overpredicts (bias
factor > 1) or underpredicts (bias factor < 1) the observed data.
|
(6)
|
The accuracy factor indicates by how much the
predictions differ from the observed data.
|
(7)
|
In both equations, n is the number of observations
used in the calculation. In a perfect model, both the bias and accuracy factors are equal to 1.
Simplification of the logistic model.
Data from Chea et al.
(6) were used to demonstrate the procedure for reducing
the number of parameters in the logistic model and to show how better
physiological insight into the experiment might be derived from the
reduced model.
| |
RESULTS |
Data set I: percent-growth-positive.
Thirty percent of the
predictions from linear regression are out of the 0 to 1 range (Table
1). Fifteen predictions from the logistic
model are closer to the observed. Seven linear predictions were
inaccurately made better by forcing predictions over 1 to 1, and one
condition was made falsely better by forcing the prediction lower than
0 to 0. The percentage better predicted by logistic regression is
calculated by excluding these data points:
|
(8)
|
The range of the differences (predicted minus observed)
from logistic regression is more than 2.5 times smaller than that from
linear regression. The IQR from logistic regression is about one-third
of that from linear regression. The deviation value of the logistic
model is more than 8 times smaller.
View this table:
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|
TABLE 1.
Comparison of results for linear and logistic regressions
with five different methods in four different data sets
|
|
Predictions from logistic regression are much better than those from
linear regression over the entire range and especially at points closer
to 1 and 0 (Fig. 1). Three predictions by
the linear model, each with an observation of 1, are 0.761, 0.773, and
0.848, while the logistic predictions are much better: 0.941, 0.990, and 0.999. The two 0 observations were predicted by the linear model to
be 0.195 and <0, while the logistic predictions were 0.036 and 0.013. Another observation at the lower range, 0.136, was predicted to be
0.341 and 0.148 by linear regression and logistic regression,
respectively. The fitted line for the predicted values from logistic
regression was very close to a perfect fit (Table 1). The fitted line
for the linear model predictions versus the observations was
considerably worse, with a slope of about 0.8, suggesting systematic
underprediction. The bias and accuracy factors for logistic regression
are slightly closer to 1 than those for linear regression (Table 1).
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FIG. 1.
Goodness of fit of linear regression and logistic
regression for C. botulinum percent-growth-positive (Data
set I) from Zhao et al. (33).
|
|
Data set II: germination extent.
Approximately 26% of the
linear predictions are out of range (Table 1). Approximately 87% of
the predictions from logistic regression are closer to the observed
values. The range of the differences (predicted minus observed) from
the logistic model is less than one-third that from the linear model,
and the IQR is almost one-sixth that from the linear model. The
deviation value of the logistic model is 17 times smaller.
The line fitted to the predicted values from the logistic model
compared to observed values is very close to the perfect fitting line
(Fig. 2, Table 1). The fitted line for
predictions from the linear model had a slope of only 0.812, suggesting
underprediction (Table 1). Three of seven observed values of 0 had
higher linear predictions, at 0.106, 0.159, and 0.368, while the
remaining four predictions were negative. All seven logistic
predictions are very close to 0, with the largest being 0.025. Three
higher observations, 0.927, 0.933, and 0.985, were predicted to be
0.805, 0.727, and 0.742 by the linear model, while logistic regression
produced much more accurate predictions at 0.946, 0.862, and 0.949, respectively. The bias factors for the two models are almost the same,
and the logistic model is slightly more accurate as judged by the
accuracy factor (Table 1).
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FIG. 2.
Goodness of fit of linear regression and logistic
regression for C. botulinum germination extent (Data set II)
from Chea et al. (6).
|
|
Data set III: probability of one cell of Listeria
monocytogenes to grow.
This is a very special data set since
21 of 28 data points are either 0 or 1. Results demonstrated that
logistic regression is a much more powerful tool when modeling this
type of data set.
Approximately 32% of the linear predicted values are out of range. All
observations are predicted better by logistic regression. The range of
the differences (predicted minus observed) from the logistic model is
more than 39-fold smaller. The IQR and deviation value from the
logistic model are also much smaller than those from the linear model
(Table 1).
The fitting parameters for the predicted versus the observed values
from logistic regression are very good (Table 1). Predictions from the
linear model are worse. The majority of the linear predictions fall far
from the perfect fit line, indicating substantial deviation of
predicted values from observed values (Fig.
3). The bias and accuracy factors from
logistic regression are better, but not substantially so, as indicated
by the other comparison approaches.
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FIG. 3.
Goodness of fit of linear regression and logistic
regression for probability of one Listeria monocytogenes
cell to grow (Data set III) from Razavilar and Genigeorgis
(26).
|
|
Data set IV: maximum fraction of positive tubes.
There are 103 observations in total (32). Eighty-one data points were
used to develop the models, and the remaining 22 (all at 19°C) were
used to test the predictive power of the models for new data.
Approximately 21% of the predictions from the linear model are out of
range. Seventy-eight percent of the predictions from logistic
regression are closer to the observed values. The minimum of the
differences (predicted minus observed) from the logistic model is
slightly lower. The range from the two models is approximately the
same, while the IQR from the logistic model is less than half of that
from the linear model. This suggests that both models have predictions
far away from the observed values, but if these outlying values are
excluded, the logistic model has much better predictive power. The
deviation value from the logistic model is less than half of that from
the linear model (Table 1).
Although it is not immediately clear from a visual inspection of Fig.
4, the logistic model is better, as
evidenced by an intercept closer to 0, a slope closer to 1, and a much
higher R2 (Table 1). A closer examination of Fig. 4 shows
that at low (0.0 to 0.2) and high (0.8 to 1.0) values, predictions from
the logistic model are closer to observations in general. For 0 observations, most predictions (22 of 25) from the logistic model were
<0.1, while 11 predictions from the linear model were >0.1, and 9 predictions were <0. Of the 22 observations that resulted in a value
of 1, 17 logistic predictions were >0.8, while 11 from the linear
model were <0.8 and 8 were >1. In the middle range (0.2 to 0.8),
predictions from the two models were comparable, with linear
predictions occasionally closer to the observed. In only 3 (labeled 1, 2, 3) of the 19 conditions in the middle range were the linear
predictions substantially better than the corresponding logistic
predictions (Fig. 4). Bias and accuracy factors from logistic
regression are closer to 1 than those from the linear model (Table 1).
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FIG. 4.
Goodness of fit of linear regression and logistic
regression for maximum fraction of positive tubes inoculated with
C. botulinum (Data set IV) from Whiting and Oriente
(32). (A) Linear model. (B) Logistic model. Three points
in the 0.2 to 0.8 range, numbered in both panels, were substantially
better predicted by linear model.
|
|
The above analysis shows that logistic models are almost always better
at predicting the values used to develop that model. It is more
important, however, to be able to predict new observations not used to
create the model. Models should not be used for conditions outside the
range studied (23). Based on this principle, we picked one
temperature (19°C) from the seven temperatures studied (5 to 28°C)
and did not use it in the model development. We then used the models to
predict the data at the 19°C temperature.
Two of 22 predictions from the linear model are out of range. More than
80% of the values were better predicted by logistic regression. Both
the range and the IQR from logistic regression were more than 1.5-fold
smaller. The parameter derived from fitting the predicted to the
observed values also indicated that logistic regression was much better
at predicting data not included in developing the model. The deviation
value of the logistic model is less than half that of the linear model
(Fig. 5; Table 1). Bias and accuracy
factors for the two models exhibit little difference.
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FIG. 5.
Validation of linear regression and logistic regression
for maximum fraction of positive tubes inoculated with C. botulinum (Data set IV) from Whiting and Oriente
(32). Both models were validated by 22 of 103 conditions
not used in model development.
|
|
Reducing the logistic model.
The logistic model developed for
Data set II (6) was simplified using stepwise selection in
S-plus 2000. Two parameters, pH and sodium chloride concentration, were
retained. Approximately 58% of the time, this model predicts closer to
the observed value than does the full linear model. The differences
(predicted minus observed) from this reduced model have a slightly
larger range but a smaller IQR. This model has a slightly smaller
R2 but an intercept closer to 0 and a slope closer to 1. In
general, the reduced 2-parameter logistic model has a predictive power very similar to that of the full 9-parameter linear model.
| |
DISCUSSION |
Limitation of bias and accuracy factors in evaluating goodness of
fit.
We used five different methods to compare the goodness of fit
of the models. Our analysis shows good agreement among the first four
methods. Bias and accuracy factors on the other hand sometimes indicate
a close degree of goodness of fit between the two models, while the
other four approaches indicate substantially better fit by logistic
regression (Data set I, Data set II, and Data set IV-validation). Bias
and accuracy factors also indicate that logistic regression is only
moderately good in the case of Data set III (bias factor = 0.89, accuracy factor = 1.14), while the other four methods indicate the
clear superiority of the logistic model.
The discrepancy between bias and accuracy factors with the other four
methods is due to differences in the treatment of observation values of
0 in the data set. The first four methods all take 0 observations into
consideration. However, because the observed value is in the
denominator of both equation 6 and equation 7, neither bias factor nor
accuracy factor can be used to compare predictions when 0 is observed.
Zero was frequently observed in percentage data: 7% in Data set I,
26% in Data set II, 36% in Data set III, 31% in Data set IV-model
development, and 27% in Data set IV-validation (Table 1). One major
disadvantage of the linear model is that it predicts far less
accurately at values closer to 0 or 1. Due to the limitation of their
formulae, bias and accuracy factors cannot reflect this. As a result,
bias and accuracy factors show that logistic regression is only
slightly better or equal to linear regression for the four data sets
studied. Similarly, the bias factor loses its meaning of an overall
measurement of overprediction or underprediction for percentage data.
If the bias factor is 1, a conclusion as to whether or not the model is
good cannot be drawn without first checking predictions at 0.
Predictions at the two extremes.
In addition to always making
biologically meaningful predictions, logistic regression makes it
possible to compare conditions at very low and very high ranges. Linear
regression often makes out-of-range predictions at these ranges. For
example, a germination probability of 
0.999 cannot be interpreted as
less likely than a probability of 0.001. On the other hand,
out-of-range prediction is not a problem for logistic regression,
which, by definition, predicts strictly between 0 and 1. For linear
models, predictions greater than 1 occasionally occur when observations
are in the middle range as well as when very high. The two >1
predictions for Data set IV-validation had observed values of 0.67 and
0.70.
Reducing the logistic model.
Many times, especially when the
data set is not very large, a reduced model is more desirable than the
full model (13). Moreover, a reduced model can give better
(or at least more straightforward) insight into the physiological
factors influencing the experiment. Logistic models can be reduced in a
fashion similar to that of linear models (29). As an
example, the logistic model developed for Data set II (6)
was simplified using the stepwise method in S-plus 2000.
The reduced model gives clear insight as to which environmental factors
influence the germination extent most under the conditions studied: pH
and salt concentration. As a comparison, the reduced linear model has
terms for pH, NaCl, temperature, pH2, and interaction
between temperature and pH (6), which results in a far
less clear picture.
Why logistic regression is better.
We have demonstrated using
four different data sets that logistic regression is better than linear
regression for modeling percentage data. This approach can be extended
to any data set that is presented as percentages. This is true because
percentage is a simple and convenient way to present binomial data, and
logistic regression, not linear regression, should be used for binomial data.
When linear regression is used for binary data, three problems arise
(23): the variance of the error term is not constant, the
error term is not normally distributed, and there is no restriction requiring the prediction to fall between 0 and 1. The first problem can
be handled by using weighted least-square regression. When the sample
size is very large, the method of least squares provides estimators
that are asymptotically normal under fairly general regulations, even
when the distribution of the error term is far from normal. The third
problem is insurmountable (23).
Logistic regression may seem much more complicated than its linear
counterpart. Yet most statistical software packages can do logistic
regression with no more effort than linear regression. However, it is
not as easy and straightforward to interpret the coefficients and test
for goodness of fit of logistic models. There is no R2
associated with a logistic model, since a residual in the commonly accepted sense does not exist. In linear regression, R2 can
be easily obtained and is often used to evaluate the goodness of fit of
models (6, 23), but R2, despite its
usefulness, is not a "gold standard." Approaches such as the
deviance test and graphical tools such as index plot and half-normal
plot can be readily employed to evaluate the goodness of fit of
logistic models (23).
In conclusion, logistic regression was demonstrated to be a better
approach than linear regression to model percentage. It has the
inherent advantage of always making biologically meaningful predictions, and in most of the cases it also predicts closer to the
observations. We believe that logistic and not linear regression should
be used whenever the observations are presented as percentages, even
though under certain circumstances linear models may give acceptable
goodness of fit.
| |
ACKNOWLEDGMENTS |
We thank Thomas J. Montville and Kristin Jackson in Department of
Food Science, Rutgers University, for help and support with the
manuscript. Thanks also to Minge Xie, Department of Statistics, Rutgers
University, for suggestions and help with the statistical discussion
section of this paper.
This work was supported by funds from the USDA National Research
Initiative (96-35201-3458), the U.S. Hatch Act, and the state of New Jersey.
| |
FOOTNOTES |
*
Corresponding author. Mailing address: Department of
Food Science, Cook College, the New Jersey Agricultural Experiment
Station, Rutgers, The State University of New Jersey, 65 Dudley Rd.,
New Brunswick, NJ 08901-8520. Phone: (732) 932-9611, ext. 214. Fax: (732) 932-6776. E-mail: schaffner{at}aesop.rutgers.edu.
Present address: National Food Processors Association, Washington,
DC 20005.
| |
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Applied and Environmental Microbiology, May 2001, p. 2129-2135, Vol. 67, No. 5
0099-2240/01/$04.00+0 DOI: 10.1128/AEM.67.5.2129-2135.2001
Copyright © 2001, American Society for Microbiology. All rights reserved.
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Abstract
Introduction
