U.S. Department of the Interior

U.S. Geological Survey

Scientific Investigations Report 2014–5145

Prepared in cooperation with the Virginia Department of Environmental Quality

Methods for Estimating Drought Streamflow Probabilities 

for Virginia Streams 

Atlantic

Ocean

WASHINGTON, D.C.

KENTUCKY

MARYLAND WEST

VIRGINIA

TENNESSEE

NORTH CAROLINA

Chesapeake Bay
Cover. Map showing location of 339 selected streamflow study sites used to develop maximum likelihood 

logistic regression models for estimating drought streamflow probabilities for 259 hydrologic basins in Virginia. 

Taken from figure 1 (p. 3).
Methods for Estimating Drought Streamflow Probabilities 

for Virginia Streams

By Samuel H. Austin

Prepared in cooperation with the Virginia Department of Environmental Quality

Scientific Investigations Report 2014–5145

U.S. Department of the Interior

U.S. Geological Survey
U.S. Department of the Interior

SALLY JEWELL, Secretary

U.S. Geological Survey

Suzette M. Kimball, Acting Director

U.S. Geological Survey, Reston, Virginia: 2014

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Suggested citation:

Austin, S.H., 2014, Methods for estimating drought streamflow probabilities for Virginia streams: U.S. Geological 

Survey Scientific Investigations Report 2014–5145, 20 p., http://dx.doi.org/10.3133/sir20145145.

ISSN 2328-0328 (online)

ISSN 2328-031X (print)

ISBN 978-1-4113-3855-5
iii

Acknowledgments

The author thanks Rob Burgholzer and Scott Kudlas of the Virginia Department of Environmental Quality, 

whose continued collaboration and assistance made this report possible. 

George E. Harlow, Jr., of the U.S. Geological Survey is thanked for his advice and review of this report, 

and the author appreciates Jennifer Krstolic and Brian A. Hasty of the U.S. Geological Survey for 

developing a map for this report.
iv

Contents

Acknowledgments........................................................................................................................................iii

Abstract ..........................................................................................................................................................1

Introduction ....................................................................................................................................................1

Purpose and Scope ..............................................................................................................................1

Streamflow Requirements During Dry Times...................................................................................2

Why Predict Drought Streamflow Probabilities? ............................................................................2

Previous Studies ...................................................................................................................................2

Methods of Study...........................................................................................................................................2

Site Selection.........................................................................................................................................2

Data Evaluation .....................................................................................................................................2

Maximum Likelihood Logistic Regression ........................................................................................3

Reading Maximum Likelihood Logistic Regression Plots ..............................................................6

Model Development .............................................................................................................................7

Model Verification and Diagnostics ..................................................................................................9

Drought Streamflow Probability Estimates .............................................................................................10

Selection of an Equation.............................................................................................................................10

Application of the Equations with Examples...........................................................................................11

Example 1. Predicting the probability of exceeding a 5-percent July drought streamflow 

quantile threshold as a function of mean monthly streamflow from the 

previous December...............................................................................................................11

Example 2. Predicting the probability of exceeding a 10-percent August drought 

streamflow quantile threshold as a function of mean monthly streamflow 

from the previous January...................................................................................................12

Example 3. Predicting the probability of exceeding a 25-percent September drought 

streamflow quantile threshold as a function of mean monthly streamflow from 

the previous February...........................................................................................................14

Example 4. Predicting the probability of exceeding a 50-percent July drought 

streamflow quantile threshold as a function of averaged mean monthly 

streamflow from the previous November, December, January, and February...........16

Summary........................................................................................................................................................18

Selected References...................................................................................................................................19
v

Figures

1. Map showing location of 339 selected streamflow study sites used to develop 

maximum likelihood logistic regression models for estimating drought streamflow 

probabilities for 259 hydrologic basins in Virginia ..................................................................3

2. Three categories of maximum likelihood logistic regression curve shape 

illustrating the degree of relation strength...............................................................................5

3. Three “zones of predictive power” for each maximum likelihood logistic 

regression plot, illustrating areas of strong predictive power A and C, and 

relatively weak predictive power B...........................................................................................6

4. Example MLLR drought threshold probability plot showing the chances 

of exceeding, and not exceeding, an historical July 25-percent daily 

streamflow threshold as a function of the average of mean daily streamflow 

values from November, December, January, and February ..................................................7

5. Diagram identifying MLLR probability models developed for evaluation 

and testing .....................................................................................................................................8

6. MLLR drought threshold probability plot showing the chances of exceeding, 

and not exceeding, an historical July 5-percent daily streamflow threshold 

as a function of mean daily streamflow values from December........................................13

7. MLLR drought threshold probability plot showing the chances of exceeding, 

and not exceeding, an historical August 10-percent daily streamflow threshold 

as a function of mean daily streamflow values from January ............................................14

8. MLLR drought threshold probability plot showing the chances of exceeding, 

and not exceeding, an historical September 25-percent daily streamflow 

threshold as a function of mean daily streamflow values from February.........................16

9. MLLR drought threshold probability plot showing the chances of exceeding, 

and not exceeding, an historical July 50-percent daily streamflow threshold 

as a function of the average of mean daily streamflow values from November, 

December, January, and February...........................................................................................18

Tables

[Tables downloadable from http://dx.doi.org/10.3133/sir20145145, or from CD-ROM in back of printed report]

1. Maximum likelihood logistic regression equations for estimating drought 

streamflow threshold probabilities for Virginia streams

2. Range of maximum likelihood logistic regression explanatory variable values 

(x-values) and daily value quantiles for the period of record used for estimating 

drought streamflow threshold probabilities for Virginia streams
vi

Conversion Factors 

 Inch/Pound to SI

Multiply By To obtain

Length

foot (ft) 0.3048 meter (m)

mile (mi) 1.609 kilometer (km)

Area

acre 0.4047 hectare (ha)

square foot (ft2) 0.09290 square meter (m2)

square mile (mi2) 259.0 hectare (ha)

Volume

gallon (gal) 0.003785 cubic meter (m3) 

cubic foot (ft3) 0.02832 cubic meter (m3) 

acre-foot (acre-ft) 1,233 cubic meter (m3)

Flow rate

acre-foot per day (acre-ft/d) 0.01427 cubic meter per second (m3/s)

foot per second (ft/s) 0.3048 meter per second (m/s)

cubic foot per second (ft3/s) 0.02832 cubic meter per second (m3/s)

Hydraulic gradient

foot per mile (ft/mi) 0.1894 meter per kilometer (m/km)

Vertical coordinate information is referenced to the North American Vertical Datum of 1988 (NAVD 88).

Horizontal coordinate information is referenced to the North American Datum of 1983 (NAD 83).

Altitude, as used in this report, refers to distance above the vertical datum.
Methods for Estimating Drought Streamflow Probabilities 

for Virginia Streams

By Samuel H. Austin

Abstract 

Maximum likelihood logistic regression model equations 

used to estimate drought flow probabilities for Virginia 

streams are presented for 259 hydrologic basins in Virginia. 

Winter streamflows were used to estimate the likelihood of 

streamflows during the subsequent drought-prone summer 

months. The maximum likelihood logistic regression models 

identify probable streamflows from 5 to 8 months in advance. 

More than 5 million streamflow daily values collected over 

the period of record (January 1, 1900 through May 16, 2012) 

were compiled and analyzed over a minimum 10-year (maximum 112-year) period of record. The analysis yielded the 

46,704 equations with statistically significant fit statistics and 

parameter ranges published in two tables in this report. These 

model equations produce summer month (July, August, and 

September) drought flow threshold probabilities as a function 

of streamflows during the previous winter months (November, 

December, January, and February). Example calculations are 

provided, demonstrating how to use the equations to estimate 

probable streamflows as much as 8 months in advance.

Introduction 

Planning for drought conditions in Virginia streams is 

essential to the sound management of water resources and 

associated riparian and watershed ecosystems. Reliable 

estimations of the likelihood that streamflows during droughtprone months will exceed specific low-flow thresholds can 

provide advance warning of drought conditions, allowing 

extended lead times for improved drought awareness and 

effective management response. Improved knowledge and 

estimation of low flows in drought-prone months provide 

extended lead time for drought response even as precipitation, 

water withdrawals, land uses, and climate variables change 

over time. Drought streamflow probability estimates can be 

used for the management of riparian systems and the ecology 

associated with forested, agricultural, and urban landscapes. 

Drought streamflow probability estimates provide a basis for 

analysis of future streamflow response to changes in ecosystem and climate.

This study was conducted through a cooperative partnership between the Virginia Department of Environmental Quality (DEQ) and the U.S. Geological Survey (USGS). The DEQ 

and the USGS maintain a statewide network of more than 

290 continuous streamgaging stations with many additional 

partial-record streamgaging stations. Together USGS and 

DEQ develop computations, analyses, and tools to support the 

understanding and effective management of streamflows. For 

this study the probabilities that streamflows in July, August, 

and September would exceed specific drought low-flow 

thresholds, as a function of streamflows during the previous 

November, December, January, and February, were estimated. 

The low-flow characteristics for each station were described 

previously by Austin and others (2011a). Maximum likelihood 

logistic regression (MLLR) was used to determine the probability of exceeding a set of flow thresholds.

Empirically determined MLLR equations were used to 

estimate drought streamflow probabilities for Virginia streams. 

Winter streamflows which correspond to the critical time for 

the majority of effective recharge to the groundwater system 

(Nelms and Moberg, 2010a, b) were used to estimate the 

likelihood of streamflows, composed mostly of base flow from 

groundwater discharge (Harlow and others, 2005; Nelms and 

Moberg, 2010b), during drought-prone summer months. This 

information aids in the understanding of water availability and 

the environmental health of watersheds and their associated 

ecosystems, and extends the lead-time of drought response by 

as much as 8 months.

Purpose and Scope

This report presents a set of 46,704 predictive equations 

that describe the likelihood of streamflow during summer 

months exceeding each low-flow threshold, based on previous streamflows at 259 streamgaging stations for a minimum 

of 10 years and a maximum of 112 years. The development 

of the equations is described, and examples are provided to 

demonstrate how to use the equations to estimate probable 

streamflow as much as 8 months in advance. The predictive 

equations are listed in a table. MLLR drought threshold probabilities are shown in illustrations. 
2 Methods for Estimating Drought Streamflow Probabilities for Virginia Streams

Streamflow Requirements During Dry Times

Streamflows support fluvial processes and ecosystem 

functions that are particularly important during dry times and 

include maintaining channel shape (morphology). The dimension, pattern, and profile of a stream contribute to the physical 

properties associated with channel function, such as the movement of sediments and maintenance of streamwater width and 

depth. Ecosystems and habitat for aquatic organisms require 

these and other channel functions to maintain transportation 

corridors and sources of food and water for aquatic and terrestrial animals and plants.

Humans withdraw streamwater for industrial, agricultural,

and domestic uses. Streamflows are an integrated response to 

the conditions upstream created by natural and anthropogenic 

sources, including precipitation received in previous months. 

When streamflows are low and available water is scarce, 

concern for maintaining water uses is heightened. Estimates 

of water availability allow water withdrawals to be adjusted, 

helping to ensure that fluvial processes, ecosystem functions, 

and human uses are maintained.

Why Predict Drought Streamflow Probabilities?

Estimating drought streamflow probabilities allows for 

the anticipation of water availability during critical periods. 

Estimates facilitate improved planning for, and management of, water resources, particularly during intervals when 

streamflows are low. The benefits of accurate drought streamflow probabilities include (1) extended lead-times for drought 

response, as much as 8 months, (2) improved understanding of 

low flows in drought-prone months, and (3) accurate anticipation of future low flows as functions of change in precipitation, water withdrawals, land-use management, and climate. 

These benefits enhance management of water withdrawals and 

land uses, so that water availability may be sustained during 

dry months.

Previous Studies

Previous studies have reported low-flow characteristics 

or analyzed low-flow data for Virginia streams. A study by 

Nuckels (1970) analyzed data from long-term, continuousrecord streamgaging stations throughout Virginia. Hayes 

(1991) provides an analysis of low-flow characteristics in Virginia streams. Austin and others (2011a) provide a comprehensive analysis of low-flow characteristics in Virginia streams. 

Other studies, including those by Cushing and others (1973), 

Trainer and Watkins (1975), Mohler and Hagan (1981), Smith 

(1981), Wetzel and Bettandorff (1986), Lynch (1987), and 

Lynch and others (1987), focused on analysis of streamflow 

characteristics in portions of the State.

Methods of Study

Analysis was conducted for five areas of focus: site 

selection, data evaluation, MLLR, model development, and 

model verification and diagnostics. Each of these focus areas 

is described below.

Site Selection

A total of 361 Virginia streamgaging stations registered in 

the USGS National Water Information System (NWIS) were 

considered for analysis. Of these, 22 streamgaging stations 

were identified as not useful for estimating drought streamflow 

probabilities, including 20 streamgaging stations for which no 

daily value (DV) data are available in NWIS and 2 streamgaging stations with insufficient DV data in NWIS (less than 

1 year of recorded DV data), when streamgaging station data 

were accessed on May 17, 2012. More than 5.12 million 

streamflow daily values (DV) collected over the period of 

record and compiled for the remaining 339 streamgaging stations were obtained from NWIS. These data and their supporting information were used to develop MLLR models to predict 

drought streamflow probabilities. Of these 339 streamgaging 

stations, 259 produced models with accompanying p-value statistics that are less than or equal to 0.05 based on streamflow 

data spanning a minimum 10-year (maximum 112-year) period 

of record. Models predicting drought streamflow probabilities 

for these 259 basins are presented in this report.

Data Evaluation

Daily values of streamflow were grouped by streamgaging station for analysis, then organized into categories corresponding to the 11 major river basins in Virginia—Big Sandy,

Chowan, James, Kanawha, Piankatank, Potomac, Rappahannock,

Roanoke, Tennessee, Ware, and York (fig. 1). Within each 

river basin category, data for each streamgaging station were 

placed in one of two groups—(1) streamgaging stations for 

which flow duration statistics had been previously developed 

and (2) streamgaging stations without previously developed 

flow duration statistics. This helped to speed data evaluation 

by facilitating analysis of data for streamgaging stations with 

previously developed flow duration statistics.

Data were inspected and evaluated for each streamgaging station in an iterative fashion. Each list of daily values was 

visually scanned for errors, then after tabulation, daily values 

were scrutinized a second time. Each dataset was reviewed 

again as models were developed, and for selected models, 

plots of data in probability space produced as part of modeling 

output were evaluated for consistency with a previous tabulation of the same data. Finally, datapoints were plotted on each 

modeled MLLR plot to facilitate continued evaluation.
Methods of Study 3

79°

80°

77°

78°

81° 82°

83°

39°

38°

37°

Study sites 

Drainage area, in square miles

0–75

76–150

151–250

251–1,500

1,501–2,500

2,501–7,500

0 25 50 MILES 12.5

0 40 80 KILOMETERS 20

Water

Salt Marsh

Rooftop

Residential/Industrial

Pavement

Barren

Forest

Forest Harvest

Grassland

Crop

Streams

EXPLANATION

Land-cover type

76° Atlantic

Ocean

WASHINGTON, D.C.

KENTUCKY

MARYLAND WEST

VIRGINIA

TENNESSEE

NORTH CAROLINA

Projection: Albers Equal Area, NAD 83. Base from U.S. Geological Survey 

Digital Line Graph, 1:2,000,000 (1987). Land cover: Fry and others, 2011, Completion of the 2006 National Land Cover Database for the 

Conterminous United States Photogrammetric Engineering and Remote Sensing, Vol. 77 (9): 858-864.

Drainage area: Hayes, D.C. and Wiegand, U. 2006, Drainage Areas of Selected Streams of Virginia:

U.S. Geological Survey Open-File Report 2006-1308, p. 51

Figure 1. 

Chesapeake Bay

James River Basin

Potomac

River

Basin

Kanawha

River

Basin

York River

Basin

Chowan River

Basin

Roanoke River Basin Tennessee River Basin

Big Sandy River

Basin

Rappahannock River Basin

Piankatank River Basin

Ware River Basin

Locations of river basins in this study, in Virginia and North Carolina

Figure 1. Location of 339 selected streamflow study sites used to develop maximum likelihood logistic regression 

models for estimating drought streamflow probabilities for 259 hydrologic basins in Virginia.

Maximum Likelihood Logistic Regression

MLLR is used to predict the chance of exceeding, or not 

exceeding, a particular streamflow threshold. MLLR describes 

the odds of a particular outcome on the basis of the value of 

a predictive variable and provides a method to estimate the 

probability of a response level as a smooth function of a factor. For this study, the probability of streamflow exceeding a 

predefined drought flow threshold during dry months of the 

year was estimated as a function of mean streamflow during 

the previous wet months of the year. The regression is called 

a logistic regression because an S-shaped logistic curve is 

used to fit the probabilities. This type of fitted curve is called a 

logistic function (Sall and others, 2007).

In its simplest form, the response (y-value) of a MLLR 

has two levels, and is, therefore, a binary response. In this 

study, the binary response is an indication of whether a 

drought streamflow threshold will, or will not, be exceeded. A 

“yes” response indicates that a particular drought streamflow 

threshold will be exceeded, and a “no” response indicates 

that a particular drought streamflow threshold will not be 

exceeded. The logistic curve is fitted on the basis of the difference in the logs of the two probabilities of the yes and no 

binary response, expressed as a linear function of the factor 
4 Methods for Estimating Drought Streamflow Probabilities for Virginia Streams

variable. In this case, the factor variable is the mean streamflow during the wet months of the year.

If p denotes the probability of the first response level, 

“yes,” then 1–p denotes the probability of the second response 

level, “no.” The linear model may then be written as shown in 

equation 1, an example of a logistic model in linear form. 

Equation 1: Example of a logistic model in linear form.

 (1)

where 

is called the logit of p, also known as the log 

odds-ratio,

 

p is the probability of exceeding a particular 

streamflow threshold,

β0 is an intercept parameter, 

β1 is a slope parameter, and 

X is an explanatory variable (x-axis value).

There is no error term associated with equation 1

because the predicted value (y-value) is not a response level 

as usually is expected in a linear regression model but rather 

a probability distribution of a response level. Because a 

probability distribution describes a chance, or likelihood, of 

either of two outcomes (yes or no) relative to an explanatory 

variable (x-value), an error cannot be associated with the fitted 

curve in the manner commonly identified with, and expressed 

in, regression models. Just as a weather forecast predicting 

a 90-percent chance of rain is not incorrect if rain does not 

occur, so too the MLLR models are correct even when the less 

likely of this study’s two possible outcomes occurs (Sall and 

others, 2007).

There is, however, a probability value (p-value) associated with each MLLR model, indicating the strength of 

each probability distribution relative to the likelihood that 

the distribution could simply be a product of chance. Each 

p-value is identified with the help of a Likelihood-ratio ChiSquare (Chi-Square) test for the hypothesis that all regression 

parameters are zero. The Chi-Square is computed by taking 

twice the difference in negative log-likelihoods (–2 Log L) 

between the results of the fitted MLLR model and the results 

of the reduced version of the model that has no explanatory (x) 

variable, only intercepts. This score statistic is a function of 

the first and second derivatives of the log-likelihood function 

(Allison, 2012). The p-value is determined as the probability 

of obtaining a greater Chi-Square value by chance alone, if the 

fit of the specified model is no better than the fit of the reduced 

model that includes only intercepts. We can think of this as 

asking, “Is the MLLR probability model better than nothing?” 

Low p-values indicate that, yes, the MLLR probability model 

is highly significant and better than nothing (Allison, 2012).

( ) ( ) 0 1

0 1

log log 1 • 

log • , 1

p X or

p X

p

ρββ

β β

− − =+

    = +

  −

log p

1− p





 





The accounting to determine each modeled probability is 

done by summing the negative logarithms of the probabilities 

attributed by the model to the events that actually did occur. 

For example, if p represents a drought streamflow threshold 

probability, then the MLLR model assigns a value of –log(p)

to represent a chance of “yes”—the drought streamflow 

threshold will be exceeded—and a value of –log(1–p) to 

represent a chance of “no”—the drought streamflow threshold 

will not be exceeded. A model that predicts perfectly yields 

p = 1 when the drought streamflow threshold is exceeded 

since –log(p) = 0 when p = 1. This same perfectly predicting 

model yields p = 0 when the drought streamflow threshold is 

not exceeded because –log(1–p) = 0 when p = 0. A value for 

the negative logarithm of the probability that is equal to zero 

(0) indicates that a perfect prediction has been made. That is 

to say, if –log(p) = 0, then a perfect prediction has been made. 

Why does –log(p) equal to 0 mean a perfect prediction? We 

can think about it this way: If we tried reversing the assignments of these negative logarithms of the probabilities such 

that a probability of zero (0) is attributed to an event that had 

actually occurred, rather than attributing a probability of one 

(1) to that event, then the –log-likelihood of the event would 

be infinity because when p = 0 then –log(p) = infinity (∞). 

Because the accounting mechanism to determine modeled 

probability minimizes the sum of the negative logarithms of 

each modeled probability, that is, minimizes the sum of each –

log(p), the sum of these negative logarithms (–log-likelihood) 

would be infinity in this reversed assignment case, yielding a 

very high minimum value and therefore a very bad forecasting score. However, because the assignments of probability 

attributes are actually not reversed, attributing a probability of 

one (1) to an event that has actually occurred yields a very low 

–log-likelihood value because when p = 1, –log(p) = zero (0).

This yields a low minimum value, which is a good forecasting 

score (Sall and others, 2007).

As described in equation 1, the logit of the logistic model 

is , which is the inverse logit of the model β0 + β1 ̇ X.

The logit expresses the probability of each response level as a 

function of an explanatory variable (x-value), and these 

estimates are in the MLLR so as to maximize the likelihood of 

their occurrence. This is equivalent to minimizing the negative 

log-likelihood (–log-likelihood) of their occurrence, which is 

the negative sum of logs of the probabilities attributed to the 

response level (“yes” or “no”) for each streamflow observation. Therefore, plots of each MLLR trace a curve that solves 

for p in equation 1, which for a “yes” response is equivalent to

Equation 2: Logistic model of a “yes” response

 , and (2)

and for a “no” response is equivalent to

Equation 3: Logistic model of a “no” response

, (3)

log p

1− p





 





p

e 1 X

1

1 0 1

= + ( ) β β+ •

p e 2 X

1

1 0 1

= + −( ) β β+ •

,
Methods of Study 5

where

e is the base of the natural logarithm,

p1 is the probability of exceeding a particular 

streamflow threshold (y-axis value is 

“yes”),

p2 is the probability of not exceeding a particular 

streamflow threshold (y-axis value is 

“no”),

β0 is an intercept parameter, 

β1 is a slope parameter, and 

X is an explanatory variable (x-axis value).

Equation 2 evaluates the probability of a “yes” response. 

Equation 3 evaluates the probability of a “no” response. The 

probability of a “yes” response is equivalent to 1–p2, and the 

probability of a “no” response is equivalent to 1–p1 because 

the “yes” response and the “no” response must sum to 1 (Sall 

and others, 2007).

The strength of the x–y relation in a maximum likelihood logistic regression may be identified using three 

categories of curve shape, each illustrating a typical degree 

of relation strength. The phrase “strength of x–y relation” is 

deliberately used rather than “goodness of fit” because we 

are predicting probabilities, and all equation fits are equally 

“good.” However, predictions of the probability of a “yes” or a 

“no” response relative to an explanatory (x-axis) variable may 

be more distinct and, therefore, potentially more useful for 

decision making as a consequence of the shape of the MLLR 

response curve.

Three useful categories of MLLR response curve shape 

are illustrated in figure 2.

When the fit strength of an x–y relation is weak, the 

absolute value of the slope parameter in the MLLR model is 

relatively small. This results in a gentle slope of the MLLR 

line over the range of the explanatory data, as illustrated by 

curve A in figure 2. When the fit strength of the x–y relation 

is strong, the absolute value of the slope parameter in the 

MLLR model is relatively large. This results in a steeper 

slope of the MLLR line over the range of the explanatory 

data, as illustrated by curve B in figure 2. In curve B, the 

logistic (s-shaped) nature of the regression line also becomes 

more pronounced. When the fit strength of the x–y relation is 

approaching a “perfectly definitive fit,” the absolute value of 

the slope parameter in the MLLR model becomes quite large, 

approaching infinity as the middle line segment of the curve 

approaches an orientation perpendicular to the x-axis of the 

graph. This results in an s-shaped MLLR line with sharper 

breaks in inclination and a section of near vertical slope, as 

illustrated by curve C in figure 2. With a highly definitive 

gradient such as this, distinctions between the y-axis categorical probabilities of “yes” and “no” become quite clear relative 

to the x-axis explanatory variable. With a perfectly definitive fit, the probability of “yes” changes from 0 to 1, and the 

probability of “no” changes from 1 to 0, at the same position 

on the x-axis. A perfectly definitive fit means that, before a 

certain value of X, all the responses are at one level (either a 

level of “yes” or “no”), and after that same value of X, all the 

responses are at another level (either a level of “yes” or “no”) 

(Sall and others, 2007).

In addition to the nuances of curve shape as a useful 

indicator of x–y relation fit strength, curve position within 

the probability space of the MLLR plot helps identify the 

predictive power of each relation. Three zones of predictive 

power may be associated with each MLLR curve—two zones 

of strong predictive power and one zone of weak predictive 

power. These are illustrated in figure 3.

Probability responseProbability responseProbability response

A B C

Explanatory variable Explanatory variable Explanatory variable

Figure 2. Three categories of maximum likelihood logistic regression curve shape illustrating the degree of relation strength.

 Curve (A) illustrates a weak relation between the explanatory variable (x-value) and the probablilty response (y-value), (B)

illustrates a strong relation, and (C) illustrates an even stronger relation, a perfectly definitive fit (after Sall and others, 2007).

Figure 2. Three categories of maximum likelihood logistic regression curve shape illustrating the degree 

of relation strength. Curve A illustrates a weak relation between the explanatory variable (x-value) and the 

probability response (y-value). Curve B illustrates a strong relation. Curve C illustrates an even stronger 

relation, a perfectly definitive fit (after Sall and others, 2007).
6 Methods for Estimating Drought Streamflow Probabilities for Virginia Streams

Probability response

A B C

1.0

0.5

0

Probability response1.0

0.5

0

Probability response1.0

0.5

0

Figure 3. Three "zones of predictive power" for each maximum likelihood logistic regression plot, illustrating areas of

strong predictive power (A) and (C), and (B) relatively weak predictive power.

Explanatory variable

Strong predictive power

Weak predictive power

Line of equal probability

Explanatory variable Explanatory variable

Strong predictive power

Figure 3. Three “zones of predictive power” for each maximum likelihood logistic regression plot, illustrating areas of strong 

predictive power A and C, and relatively weak predictive power B.

Thus, relation fit strength identified in figure 2 and predictive power identified in figure 3 may be used together to 

further refine the understanding of MLLR plots. A curve that 

moves through the three zones identified in figure 3 exhibits 

a strong x–y relation with strong predictive power. Figure 2B

and figure 2C are examples of such a curve. A curve confined 

to the upper one-third or the lower one-third of an MLLR plot 

exhibits strong predictive power, as identified in figure 3A and 

figure 3C; a weak x–y relation is shown in figure 2A. Finally, 

a curve confined to the middle one-third of an MLLR plot 

exhibits weak predictive power, as identified in figure 3B, a 

line of equal probability defines a 0.5 probability of either 

response, and a weak x–y relation is shown in figure 2A.

Reading Maximum Likelihood Logistic 

Regression Plots

MLLR plots may be thought of as cumulative probability plots, where the line of fit partitions the entire probability 

space into response categories. The two response categories 

in the drought threshold probability plots—the probability 

of exceeding a particular drought threshold labeled as “yes” 

on the right vertical axis of the plot and the probability of 

not exceeding the same particular drought threshold labeled 

as “no” on the right vertical axis of the plot—may each be 

identified using the left vertical axis of the plot, as shown in 

figure 4. The probability of “no” is read directly from the left 

vertical axis of the plot. The probability of “yes” is equivalent 

to 1 minus the value read directly from the left vertical axis of 

the plot and represents the distance from the regression line to 

the top of the graph box. The probability of “yes,” therefore, 

equals 1 minus the probability of “no” or, using the variables 

identified in equation 2 and equation 3, p1

 = (1–p2 ). 

Figure 4 illustrates how to read MLLR plots. It is an 

example MLLR drought threshold probability plot showing 

the chance of exceeding and not exceeding a July 25-percent 

daily streamflow threshold as a function of the average of 

mean daily values from November, December, January, and 

February. On the graph, the line of fit partitions the whole 

probability space into two response categories. A “no” 

response category occupies the area below the blue line, and 

a “yes” response category occupies the area above the blue 

line. The division into two areas is identified with the labels 

“Yes” and “No” on the right vertical axis. The probability of 

not exceeding a drought flow threshold (labeled “No”) may 

be read directly from the left vertical axis. For example, the 

probability of not exceeding a July daily streamflow threshold 

equal to 25-percent of historic July daily streamflow, when 

the average daily streamflow during the previous November, 

December, January, and February was 120 cubic feet per second (ft3

/s), may be read from the graph as follows: (1) Enter 

the explanatory variable axis (x-axis) at a value of approximately 120 ft3

/s, then (2) move vertically to the blue regression line, and (3) proceed horizontally to the left vertical axis 

(y-axis), then (4) read the probability value on the left vertical axis. This sequence of steps yields a probability value of 

approximately 0.26, as illustrated by the red and blue arrows 

in figure 4.

Similarly, the probability of exceeding a July daily 

streamflow threshold (labeled “Yes”) equal to 25-percent of 

historic July daily streamflow, when the average daily streamflow during the previous November, December, January, and 

February was 120 ft3/s, may be traced in like manner from the 

top of the graph-box downward and determined using simple 

subtraction, as follows: (1) Find the vertical above the blue 

regression line that corresponds to the vertical below the blue 

regression line associated with the explanatory variable (x) 

value of interest, in this case again, a value of approximately 

120 ft3/s, (2) move vertically downward from the top of the 

graph box to the blue regression line, and (3) proceed horizontally to the left vertical axis, (4) read the probability value on 

the left vertical axis, and (5) subtract this value from the left 

vertical axis value of 1.00 at the top of the graph box. This 

sequence of steps yields a probability value of approximately 

1.00 minus 0.26, which equals 0.74, as illustrated by the green 

and blue arrows in figure 4. For each explanatory variable 

value on the x-axis, a “yes” probability value is equal to 1 minus

a “no” probability value, and a “no” probability value is equal 

to 1 minus a “yes” probability value (see fig. 4).
Methods of Study 7

Yes

Figure 4. An example MLLR drought threshold probability plot showing the chances of exceeding,

and not exceeding, an historical July twenty-five percent daily streamflow threshold as a function

of the average of mean daily streamflow values from November, December, January, and February.

Reading the 0.26 non-exceedance probability (”No” -value) of an explanatory variable value (x-axis

value) of 120 cubic feet per second is demonstrated by red and blue arrows. Reading the 0.74 (1.00

minus 0.26) exceedance probability (”Yes” -value) an explanatory variable value (x-axis value)

120 cubic feet per second is demonstrated by orange and blue arrows. (Site number 02030000,

Hardware River at route 637 below Briery Creek (Run), near Scottsville, Virginia).

40 60 80 100 130 160 190 220 250 280

Daily value greater than quantile 25-percent July daily 

value of period of record

1.00

0.75

0.50

0.25

0

Average (N-D-J-F) mean daily value

No

Figure 4. An example MLLR drought threshold probability plot showing the chances 

of exceeding, and not exceeding, an historical July 25-percent daily streamflow 

threshold as a function of the average of mean daily streamflow values from November, 

December, January, and February (N-D-J-F). Reading the 0.26 non-exceedance 

probability (“No”-value) of an explanatory variable value (x-axis value) of 120 ft3

/s is 

demonstrated by red and blue arrows. Reading the 0.74 (1.00 minus 0.26) exceedance 

probability (“Yes”-value) of an explanatory variable value (x-axis value) of 120 ft3

/s is 

demonstrated by green and blue arrows. (Site number 02030000, Hardware River at 

Route 637 below Briery Creek [Run], near Scottsville, Virginia).

Model Development

MLLR models were developed to describe the chance 

of a particular average streamflow exceeding or not exceeding an identified drought streamflow threshold. A total of 

70,120 MLLR models were developed describing a range of 

streamflow probabilities for each of 339 streamgaging stations. 

Of these, 55,428 models had very low p-values, less than or 

equal to a p-value of 0.05, and were identified as candidates for 

publication. Of these, 46,704 models were based on streamflow 

data spanning a period of record of 10 years or more. These 

46,704 models for 259 basins were selected for publication in 

this report. Specifically, the models describe the chance of a 

particular average streamflow exceeding or not exceeding an 

identified streamflow threshold, as a function of streamflows 

from an earlier time period. The models answer the question: 

“What is the chance (likelihood) that a particular flow value 

will exceed a certain stream-flow threshold as a function of 

measured streamflows from an earlier time period?”

These logistic regression models fit nominal Y responses 

to a linear model of X terms. More specifically, the models fit 

probabilities for two response levels using a logistic function, 

as described in the previous section “Maximum Likelihood 

Logistic Regression.”

The fitting principal of maximum likelihood means that 

the variable coefficients (βs) are chosen to maximize the joint 

probability attributed by the model to the responses that did 

occur. This fitting principal essentially minimizes the negative 

log-likelihood (–LogLikelihood), as attributed by the model. 

Each model is fitted iteratively with negative log-likelihood 

values converging to the final estimates (SAS Institute, 2012). 

(See the section “Maximum Likelihood Logistic Regression.”) 

The probability of exceeding nominal monthly and daily 

streamflow threshold response values for July, August, and 

September is predicted using November, December, January, 

and February mean daily streamflow values as explanatory 

variables. These explanatory variables are used as surrogates 

for precipitation.
8 Methods for Estimating Drought Streamflow Probabilities for Virginia Streams

The MLLR models describe the MLLR probability of 

“Yes” (P[Yes]), as shown in equation 2, and the maximum 

likelihood logistic regression probability of “No” (P[No]), as 

shown in equation 3. Drought streamflow threshold values for 

models of drought streamflow probabilities were identified 

using guidance from the Virginia DEQ and indicator streamflow values provided by the Virginia Drought Monitoring 

Task Force (http://www.deq.virginia.gov/Programs/Water/

WaterSupplyWaterQuantity/Drought/DroughtIndicators.aspx, 

accessed May 18, 2012).

The following drought streamflow threshold categories 

were used to test streamflow probabilities in MLLR models:

• the 5-percent streamflow quantile for the month, over 

the period of record;

• the 10-percent streamflow quantile for the month, over 

the period of record;

• the 25-percent streamflow quantile for the month, over 

the period of record; and

• the 50-percent streamflow quantile for the month, over 

the period of record.

The questions addressed using these drought flow threshold categories are 

• What is the chance that streamflow exceeds the 

monthly 5-percent streamflow quantile?

• What is the chance that streamflow exceeds the 

monthly 10-percent streamflow quantile?

• What is the chance that streamflow exceeds the 

monthly 25-percent streamflow quantile? 

• What is the chance that streamflow exceeds the 

monthly 50-percent streamflow quantile?

Data were prepared and models developed to perform 

analyses. As many as 240 individual MLLR probability models

were developed for each streamgaging station, represented 

using 120 probability plots each illustrating 2 probability 

models, (P[Yes] and P[No]). The group of MLLR models 

developed and tested for each streamgaging station may be 

visualized using three matrices. For each station a 12-element 

matrix may be identified containing response (y) variable 

choices (fig. 5A), and a 10-element matrix may be identified 

containing explanatory (x) variable choices (fig. 5B). A third 

matrix may also be identified containing the two potential 

model probability outcomes (P[Yes] and P[No]) (fig. 5C). 

Multiplying the 12 elements in the first matrix, the 10 elements 

in the second matrix, and the 2 elements in the third matrix 

yields 240 individual MLLR probability models for testing 

and evaluation per streamgaging station, a potential total of 

81,360 MLLR probability models for the 339 streamgaging 

stations. The 70,120 MLLR models were developed rather 

than 81,360 because data for some explanatory variables were 

unavailable in some instances.

For each station, the response (y) variable choices combine three response variable months (July [J], August [A], 

and September [S]) with four distinct streamflow thresholds 

(the 5-, 10-, 25-, and 50-percent streamflow quantiles). The 

explanatory (x) variable choices combine two explanatory 

variable flow statistics, mean monthly streamflow (Flow), and 

daily value flow (DV) with four distinct explanatory variable 

months plus the average of these four as a fifth explanatory 

variable (November [N] mean streamflow, December [D] mean 

streamflow, January [J] mean streamflow, February [F] mean 

streamflow, and the average N, D, J, F mean streamflow). 

Individual probability models combine selected response and 

explanatory variables (fig. 5). Ten examples of individual 

probability model types are listed in figure 5.

=

J A S

5

10

25

50

J5 A5 DVN P[Yes]

P[No]

MEANN

DVD MEAND

DVJ MEANJ

DVF MEANF

DVN-DJ-F

MEAN 

N-D-J-F

S5

J10 A10 S10

J25 A25 S25

J50 A50 S50

Response drought flow

quantile threshold

A. Response variable month

N

D

J

F

N, D,

Explanatory variable month

J, F

B. Explanatory variable C. Model

probability

P

D. 240 individual probability models for each

flow statistic streamgaging station

Daily

value

Monthly

mean P[Yes] P[No]

1. J5 as a function of DVN

2. A10 as a function of MEANJ

3. S25 as a function of DV N-D-J-F

4. A25 as a function of DVF

5. J50 as a function of MEAN

 N-D-J-F

6. J5 as a function of (DVN)

7. A10 as a function of MEANJ

8. S25 as a function of DV N-D-J-F

9. A25 as a function of DVF

10. J50 as a function of MEAN

 N-D-J-F

Figure 5. A diagram identifying MLLR probability models developed for evaluation and testing. For each streamgaging station up to

240 individual MLLR probability models were developed for a combination of 3 potential response variable months (matrix

(A) column labels), and 4 potential response drought flow quantile threshold categories (matrix (A) row labels), from 5 potential

explanatory variable months (matrix B row labels), 2 potential explanatory variable flow statistics (matrix B column labels), and 2

potential model probability outcomes (matrix C). Colors show the progression of several example variable selections listed at the

right of the figure.

Figure 5. A diagram identifying MLLR probability models developed for evaluation and testing. For each streamgaging 

station, up to 240 individual MLLR probability models were developed for a combination of 3 potential response variable 

months (matrix A column labels), and 4 potential response drought streamflow quantile threshold categories (matrix A row 

labels), from 5 potential explanatory variable months (matrix B row labels), 2 potential explanatory variable streamflow 

statistics (matrix B column labels), and 2 potential model probability outcomes (matrix C). Colors show the progression of 

several example variable selections listed at the right of the figure.
Methods of Study 9

Models were prepared using iterative methods in which 

the MLLR model fit converged to minimize the negative loglikelihood of each probable outcome, P[Yes] and P[No], over 

the range of explanatory variable (x) values. Minimizing negative log-likelihood is equivalent to maximizing log-likelihood, 

yielding MLLR probabilities. Model results were compiled 

in a table organized by streamgaging station number, listing 

model parameters and equations, and in graphs, showing a 

logistic regression response in probability space of probable 

outcomes over the range of a selected explanatory variable. 

These tabulated and plotted results were then evaluated.

Model Verification and Diagnostics

Visual inspections of MLLR model response curves and 

evaluations of p-values associated with each MLLR model 

were used for model verification and diagnostics. Since a 

MLLR model describes a chance, or likelihood, of either of 

two possible outcomes, P[Yes] or P[No], relative to an explanatory variable (x-value), an error cannot be associated with the 

fitted model in the manner customarily used with regression 

models. An error term is not associated with a MLLR model 

because the predicted value (y-value) is not a response level 

as usually expected in a linear regression model, but rather a 

probability distribution of a response level. Thus, these MLLR 

models are correct even when the less likely of the two possible outcomes, P[Yes] or P[No], occurs (Sall and others, 2007).

A probability value (p-value) is associated with each 

MLLR model, indicating the strength of each probability distribution relative to the likelihood that the distribution could 

simply be a product of chance. Each p-value is determined 

by a Chi-Square test for the hypothesis that all regression 

parameters are zero. The Chi-Square is computed by calculating twice the difference in negative log-likelihoods (–2 Log L) 

between the fitted MLLR model and a reduced version of 

the fitted MLLR model that has no explanatory (x) variable 

parameters, only intercepts, and then the resulting p-value 

is evaluated. The resulting p-value gives the probability of 

obtaining a greater Chi-Square value by chance alone, if the fit 

of the specified model is no better than the fit of the reduced 

model that includes only intercepts. Low p-values indicate 

that there is very little likelihood that the probability distribution associated with the MLLR model could be produced by 

chance alone. Thus, low p-values indicate it is highly likely 

that the fitted curve explaining the probabilities expressed in 

the MLLR model is an accurate response to the explanatory 

variable in the model. Model equations were ranked by the 

probability of obtaining a greater Chi-Square value by chance 

alone (Prob>ChiSq, or p-value). Smaller p-values were considered better than larger p-values. Only models with very low 

p-values, those less than or equal to a p-value of 0.05, were 

identified as candidates for publication. Of these 55,428 models, only those based on streamflow data spanning a period of 

record of 10 years or more (46,704 models), were selected for 

publication in this report. This set of 46,704 models may be 

considered highly significant, with a high likelihood that the 

fitted model curve explains the probabilities expressed in the 

model and is an accurate response to the explanatory variable.

Visual inspection of each MLLR model response curve 

was used to identify the strength of the x–y relation and 

the predictive power of the x–y relation within each MLLR 

model. The strength of the x–y relation in each MLLR model 

was identified by comparing the shape of the response curve 

plotted for each model with the three categories of curve shape 

identified in figure 2. Each curve shape shown in figure 2

illustrates a distinct degree of relation strength. Note that 

relation strength is not goodness of fit because probabilities 

are predicted and all equation fits are considered “good” at 

predicting the likelihood of each response outcome within the 

given probability distribution. Predictions of P[Yes] or P[No] 

response relative to an explanatory (x-axis) variable may be 

more definitive, however, and therefore potentially more useful for decision making as a consequence of the shape of the 

MLLR response curve. When the fit strength of an x–y relation 

is weak, the absolute value of the slope parameter in the 

MLLR model is relatively small, producing a gentle slope of 

the MLLR line over the range of the explanatory data. When 

the fit strength of the x–y relation is strong, the absolute value 

of the slope parameter in the MLLR model is relatively large, 

producing a steeper slope of the MLLR line over the range of 

the explanatory data. When the fit strength of the x–y relation 

is approaching a “perfectly definitive fit,” the absolute value of 

the slope parameter in the MLLR model becomes quite large, 

approaching infinity as a segment of the curve approaches 

an orientation perpendicular to the x-axis of the graph, that 

is almost vertical. (See figure 2 and the section “Maximum 

Likelihood Logistic Regression” for additional illustration and 

discussion of the strength of the x–y relation.)

The predictive power of the x–y relation in each MLLR 

model was identified by comparing the position of the 

response curve plotted in probability space for each model 

with the three categories of curve position identified in figure 3.

A curve that moves through all three zones of predictive power 

identified in figure 3 exhibits strong predictive power for probabilities P[Yes] and P[No] and strong explanatory power associated with the MLLR explanatory variable. A curve confined 

to the upper one-third or the lower one-third of an MLLR 

plot as identified in figure 3 also exhibits strong predictive 

power for probabilities P[Yes] and P[No] but weak explanatory power associated with the MLLR explanatory variable. 

A curve confined to the middle one-third of an MLLR plot 

exhibits weak predictive power for probabilities P[Yes] and 

P[No] and weak explanatory power associated with the MLLR 

explanatory variable. (See figure 3 and the section “Maximum 

Likelihood Logistic Regression” for additional illustration and 

discussion of the predictive power of the x–y relation.)
10 Methods for Estimating Drought Streamflow Probabilities for Virginia Streams

Drought Streamflow Probability 

Estimates

A total of 70,120 MLLR models were developed, 

describing 70,120 drought streamflow probability estimates 

for 339 streamgaging stations. Of these, 55,428 models had 

an accompanying p-value statistic of 0.05 or less, indicating 

a highly statistically significant model of drought streamflow 

threshold probabilities. Of these, 46,704 models were based 

on streamflow data spanning a period of record of 10 years 

or more. These 46,704 highly significant models based on at 

least 10 years of streamflow data were selected for publication (table 1). Each of these models describes the chance of 

a particular daily-value streamflow or mean-monthly streamflow exceeding or not exceeding a 5-, 10-, 25-, or 50-percent 

streamflow quantile threshold for July, August, or September, as a function of mean streamflow from the previous 

November, December, January, or February, or the average 

of the previous November-December-January-February 

mean streamflows. The models were prepared using iterative 

methods in which the MLLR model fit converged to minimize 

the negative log-likelihood of each probable outcome, P[Yes] 

and P[No], over the range of explanatory variable (x) values. 

Because minimizing negative log-likelihood is equivalent to 

maximizing log-likelihood, this yields MLLR probabilities. 

Drought streamflow probability estimates are compiled 

in a table organized by streamgaging station number, listing 

model parameters and equations, and in graphs, for each 

model showing a logistic regression response in probability 

space of probable outcomes over the range of an explanatory variable. These tabulated drought streamflow probability 

model parameters and equations are listed in table 1. A list of 

the minimum and maximum values of each explanatory (x) 

variable, identifying the range of x-variable values over which 

drought streamflow threshold probabilities are estimated for 

Virginia streams, is presented in table 2.

Selection of an Equation

As several models are available for each response variable (y) at each streamgaging station for predicting the chance 

of July, August, or September streamflow exceeding or not 

exceeding a particular drought streamflow threshold as a 

function of November, December, January, or February mean 

daily-flow values, questions may arise as to which model 

is best or which equation to choose. An analyst potentially 

interested in using the models may wonder “How do I choose 

the best equation?” In this section, guidance is provided for 

selecting an appropriate equation.

The choice of most appropriate equation is often indicated by a mix of at least five factors: (1) the availability of an 

explanatory (x) variable value to use in the equation; (2) the 

time-frame within which the prediction will be useful; (3) the 

degree of accuracy and reliability associated with the selected 

explanatory variable measurement; (4) the strength of the x–y 

relation in the MLLR accompanying each equation choice, 

as described in the section “Maximum Likelihood Logistic 

Regression;” and (5) the predictive power of the MLLR curve 

corresponding to each equation choice, also described in the 

section “Maximum Likelihood Logistic Regression.”

Reliable and accessible explanatory (x) variable values 

are essential and may narrow the field of equation choices. 

If reliable explanatory variable values are available for only 

one particular month of the year, say for example February, 

then equations that use only February as their explanatory (x) 

variable become the viable alternatives. Potential explanatory 

variable data need to be carefully evaluated, ensuring 

that measurements have been made accurately and using 

accepted methods.

The time frame within which the estimate will be useful 

may dictate viable equation options. For a given management 

decision, an analyst may need to predict drought streamflow 

probabilities within a particular interval of time. For example, 

if management policies direct adjustments to July streamflow withdrawals in order to avoid July drought streamflow 

conditions and at least 5 months are required to implement 

the adjustments, then equations that use explanatory variable values for March are best avoided because these values, 

although potentially accurate, provide predictions 4 months 

in advance of July streamflows when a minimum 5-month 

window is needed to successfully make the streamflow adjustments dictated by management policy. In contrast, if the time 

window for successful implementation of a particular drought 

management policy is relatively short, perhaps 2 months rather 

than 5 months, then the decision calculation may change. For 

example, if one set of equations using February values as the 

explanatory variable is available and a second set of equations 

using March values as the explanatory variable is available, 

and both equation sets have comparable x–y relation strength 

and predictive power, then equations that use March values 

as the explanatory variable may be preferred as these provide 

predicted drought streamflow probabilities using the most 

recently acquired data as explanatory variable values, while 

still providing a time-window sufficient to successfully implement the streamflow adjustment management policy.

The degree of accuracy and reliability of an explanatory 

variable, relative to other explanatory variable choices may 

point to an appropriate equation choice. A certain explanatory 

variable streamflow statistic may be more accurate or reliable 

than another. For example, if a streamgaging station has 

few daily value data available for November and many data 

accurately measured and collected during January or February, 

then streamflow values from January or February may prove 

to be more accurate and reliable explanatory variables than 

those for November. Graphs of each MLLR model, plotting 
Application of the Equations with Examples 11

the fitted MLLR line, can be compared to identify explanatory 

variables determined using more complete data coverage. 

The strength of the x-y relation in the MLLR associated 

with a potential equation contributes to the usefulness of the 

MLLR equation as an estimator. As described in the section 

“Maximum Likelihood Logistic Regression,” the strength 

of the x-y relation in each MLLR model may be identified 

by comparing the shape of the response curve plotted for 

each model with the three categories of curve shape shown 

in figure 2. Each curve shape shown in figure 2 illustrates a 

distinct degree of x–y relation strength. Predictions of P[Yes] 

or P[No] response relative to an explanatory variable may be 

more definitive or less definitive and, therefore, potentially 

more or less useful for management decision making as a 

consequence of the shape of the MLLR response curve. When 

the fit strength of an x–y relation is weak, the MLLR line has a 

relatively gentle slope over the range of the explanatory data, 

as shown in figure 2A. When the fit strength of the x-y relation is strong, the MLLR line has a varying yet consistently 

relatively steep slope over the range of the explanatory data, 

expressing a more pronounced sigmoidal (s-shaped) curve, as 

shown in figure 2B. When the fit strength of the x-y relation is 

very strong, approaching a perfectly definitive fit, the absolute 

value of the slope of the MLLR line becomes quite large, often 

approaching infinity in a segment of the curve with a nearperpendicular orientation relative to the x-axis of the graph 

(almost vertical), as shown in figure 2C. Equations exhibiting 

highly definitive curve shapes are best chosen over equations 

exhibiting weak curve shapes.

The predictive power of the MLLR curve, corresponding to each equation choice can also help determine usefulness as a predictor. As described in the section “Maximum 

Likelihood Logistic Regression,” the predictive power of each 

MLLR model may be identified by comparing the position of 

the response curve plotted in probability space for each model 

with the three categories of curve position identified in figure 3.

Equations with accompanying model graphs displaying 

MLLR curves that move through all three zones of predictive 

power (fig. 3) exhibit strong predictive power with strong and 

highly definitive curve shapes. Equations and curves of this 

type are preferred over equations and curves exhibiting weak 

predictive power and weak curve shape. In practice, a combination of the three zones of predictive power plus curve shape 

and the relative importance of each, as determined by the 

analyst, will identify the most appropriate equation for useful 

predictions of drought streamflow probability in each 

particular circumstance.

Application of the Equations 

with Examples

MLLR equations are provided that predict the chance of 

exceeding or not exceeding one of a set of identified drought 

streamflow thresholds during the historically dry months of 

the year—July, August, and September—as a function of mean 

streamflow values from the previous historically wet months 

of the year—November, December, January, and February. 

Because equations are provided for each of the two probability 

categories associated with each response variable, these calculations are easy to make and easy to interpret. For a given 

streamgaging station, an equation may be chosen from table 1

that addresses the probability of exceeding a streamflow 

threshold value of interest P[Yes] or not exceeding a streamflow threshold value of interest P[No] on the basis of a mean 

monthly streamflow value for one of the preceding explanatory variable months of November, December, January, or 

February or the averaged mean monthly streamflow of these 

four months.

The following four example exercises provide step-bystep guidance for how to perform these calculations using 

the equations supplied in this report to predict the probability 

of exceeding or not exceeding a streamflow threshold. Each 

example exercise shows how to apply equations to calculate 

the probability of exceeding a drought streamflow threshold, 

P[Yes], and how to apply equations to calculate the probability 

of not exceeding a drought streamflow threshold, P[No]. The 

example exercises describe these calculations for each of the 

four response drought streamflow quantile thresholds identified in figure 5A and 4 of the 5 explanatory variable months 

identified in figure 5B.

Example 1. Predicting the probability of 

exceeding a 5-percent July drought streamflow 

quantile threshold as a function of mean monthly 

streamflow from the previous December

Problem: At the start of the new year, you need to 

predict July drought streamflow probabilities 

for Stave Run, a stream in the Potomac River 

Basin near Reston, Virginia, in order to manage water withdrawals and avoid severe low 

flows. 

Question: What is the chance that July streamflow will 

exceed the monthly 5-percent streamflow 

quantile if the average December daily streamflow is 0.02 ft3

/s?

Given: Mean monthly streamflow from the 

previous December.
12 Methods for Estimating Drought Streamflow Probabilities for Virginia Streams

Solution: A streamgage identified as USGS 01644291 is 

on this stream below Parking Lot Road, near 

Reston, Virginia.

Step 1: In table 1, find station 01644291.

Step 2: Find the streamflow threshold criteria listed 

in table 1 that corresponds to this station number and the information given. In this case the 

flow threshold criteria in table 1 is listed as DV 

> Quantile 5% July DV POR By December 

Mean DV.

Step 3: Read Equation: Probability of “Yes” from 

table 1 corresponding to this station number 

and flow threshold criteria. In this instance, it 

is

P[Yes] = 1/(1 + e(0.46159633 + –11.06522•X)) .

Step 4: Read Equation: Probability of “No” from 

table 1 corresponding to this station number 

and flow threshold criteria. In this instance, it 

is

P[No] = 1/(1 + e–(0.46159633 + –11.06522•X)) .

Step 5: Calculate the probability P[Yes] that July 

streamflow will exceed the monthly 5-percent 

streamflow quantile if the average December 

daily streamflow is 0.02 ft3/s by substituting 

0.02 for the X variable in the equation and then 

simplifying as follows: 

P[Yes]0.02 = 1/(1 + e(0.46159633 + –11.06522•X)) 

P[Yes]0.02 = 1/(1 + e(0.46159633 + –11.06522•0.02))

P[Yes]0.02 = 1/(1 + e(0.46159633 + –0.22130))

P[Yes]0.02 = 1/(1 + e0.24029)

P[Yes]0.02 = 1/(1 + 1.27162)

P[Yes]0.02=1/2.27162

P[Yes]0.02 = 0.44

Step 6: Calculate the probability P[No] that July 

streamflow will not exceed the monthly 

5-percent streamflow quantile if the average 

December daily streamflow is 0.02 ft3/s by substituting 0.02 for the X variable in the equation 

and then simplifying as follows: 

P[No]0.02 = 1/(1 + e–(0.46159633 + –11.06522•X))

P[No]0.02 = 1/(1 + e–(0.46159633 + –11.06522•0.02))

P[No]0.02 = 1/(1 + e–(0.46159633 + –0.22130))

P[No]0.02 = 1/(1 + e–0.24029)

P[No]0.02 = 1/(1 + 0.78640)

P[No]0.02 = 1/1.78640

P[No]0.02 = 0.56

Answer: P[Yes]: There is a 0.44 (44-percent) chance 

that July streamflow will exceed the monthly 

5-percent streamflow quantile if average 

December daily streamflow is 0.02 ft3/s.

P[No]: There is a 0.56 (56-percent) chance that 

July streamflow will not exceed the monthly 

5-percent streamflow quantile if average 

December daily streamflow is 0.02 ft3/s.

A plot of the MLLR line for the Potomac River Basin site 

number 01644291 confirms these values as shown in figure 6.

Example 2. Predicting the probability of 

exceeding a 10-percent August drought 

streamflow quantile threshold as a function of 

mean monthly streamflow from the previous 

January

Problem: You need to predict August drought streamflow 

probabilities for the South Fork of the Roanoke River in the Roanoke River Basin near 

Shawsville, Virginia, in order to determine the 

feasibility of potential large-scale water withdrawals requested by a new business.

Question: What is the chance that August streamflow 

will exceed the monthly 10-percent streamflow 

quantile if the average January daily streamflow is 20 ft3

/s?

Given: Mean monthly streamflow from the previous 

January.

Solution: A streamgage identified as USGS 02053800 

is on the South Fork of the Roanoke River at 

Route 637 near Shawsville, Virginia.

Step 1: In table 1, find station number 02053800.

Step 2: Find the streamflow threshold criteria listed in 

table 1 that corresponds to this station number 

and the information given. In this case the flow 

threshold criteria in table 1 is listed as “DV 

> Quantile 10% August DV POR By January 

Mean DV.”

Step 3: Read Equation: Probability of “Yes” from table 1

corresponding to this station number and flow 

threshold criteria. In this instance, it is

P[Yes] = 1/(1 + e(0.09725373 + –0.0223555•X)) .

Step 4: Read Equation: Probability of “No” from table 1

corresponding to this station number and flow 

threshold criteria. In this instance, it is

P[No] = 1/(1 + e–(0.09725373 + –0.0223555•X)) .

Step 5: Calculate the probability P[Yes] that August 

streamflow will exceed the monthly 10-percent 

streamflow quantile if the average January 

daily streamflow is 20 ft3/s by substituting 20 
Application of the Equations with Examples 13

No

Yes

Figure 6. A MLLR drought threshold probability plot showing the chances of exceeding,

and not exceeding, an historical July five percent daily streamflow threshold as a function

of mean daily streamflow values from December. Red and blue arrows illustrate reading a

P [no] = 0.56 response with an explanatory variable value of 0.02 cubic feet per second.

Orange and blue arrows illustrate reading a P [yes] = 0.44 response with an explanatory

variable value of 0.02 cubic feet per second. (site number 01644291, Stave Run below

Parking Lot Road, near Reston, Virginia). 

0.02 0.06 0.1 0.14 0.18 0.22 0.26

Daily value greater than quantile 5-percent July daily 

value of period of record

1.00

0.75

0.50

0.25

0

December mean daily value

Figure 6. A MLLR drought threshold probability plot showing the chances of exceeding, and not 

exceeding, an historical July 5-percent daily streamflow threshold as a function of mean daily streamflow 

values from December. Red and blue arrows illustrate reading a P[No] = 0.56 response with an 

explanatory variable value of 0.02 ft3

/s. Green and blue arrows illustrate reading a P[Yes] = 0.44 

response with an explanatory variable value of 0.02 ft3

/s. (Site number 01644291, Stave Run Below 

Parking Lot Road, near Reston, Virginia.)

for the X variable in the equation and then 

simplifying as follows:

P[Yes]20 = 1/(1 + e(0.09725373 + –0.0223555•X)) 

P[Yes]20 = 1/(1 + e(0.09725373 + –0.0223555•20))

P[Yes]20 = 1/(1 + e(0.09725373 + –0.44711))

P[Yes]20 = 1/(1 + e–0.34986)

P[Yes]20 = 1/(1 + 0.70479)

P[Yes]20 = 1/1.70479

P[Yes]20 = 0.59

Step 6: Calculate the probability P[No] that August 

streamflow will not exceed the monthly 

10-percent streamflow quantile if the average 

January daily streamflow is 20 ft3

/s by substituting 20 for the X variable in the equation and 

then simplifying as follows:

P[No]20 = 1/(1 + e–(0.09725373 + –0.0223555•X)) 

P[No]20 = 1/(1 + e–(0.09725373 + –0.0223555•20))

P[No]20 = 1/(1 + e–(0. 09725373 + –0.44711))

P[No]20 = 1/(1 + e0.34986)P[No]20 = 1/(1 + 1.41886)

P[No]20 = 1/2.41886

P[No]20 = 0.41

Answer: P[Yes]: There is a 0.59 (59-percent) chance 

that August streamflow will exceed the 

monthly 10-percent streamflow quantile if average January daily streamflow is 20 ft3

/s.

P[No]: There is a 0.41 (41-percent) chance that 

August streamflow will not exceed the monthly 

10-percent streamflow quantile if average 

January daily streamflow is 20 ft3

/s.

A plot of the MLLR line for the Roanoke River Basin site 

number 02053800 confirms these values, as shown in figure 7.
14 Methods for Estimating Drought Streamflow Probabilities for Virginia Streams

No

Yes

Figure 7. A MLLR drought threshold probability plot showing the chances of exceeding, and

not exceeding, an historical August ten percent daily streamflow threshold as a function

of mean daily streamflow values from January. Red and blue arrows illustrate reading a

P[No] = 0.41 response with an explanatory variable value of 20 cubic feet per second

Orange and blue arrows illustrate reading a P[Yes] = 0.59 response with an explanatory variable

value of 20 cubic feet per second.(Site number 02053800, South Fork Roanoke River at route 637

near Shawsville, Virginia).

4020 60 80 120 160 200 240 280 320

Daily value greater than quantile 10-percent August daily 

value of period of record

1.00

0.75

0.50

0.25

0

January mean daily value

Figure 7. A MLLR drought threshold probability plot showing the chances of exceeding, 

and not exceeding, an historical August 10-percent daily streamflow threshold as a 

function of mean daily streamflow values from January. Red and blue arrows illustrate 

reading a P[No] = 0.41 response with an explanatory variable value of 20 ft3

/s. Green 

and blue arrows illustrate reading a P[Yes] = 0.59 response with an explanatory variable 

value of 20 ft3

/s. (Site number 02053800, South Fork Roanoke River at Route 637 near 

Shawsville, Virginia.)

Example 3. Predicting the probability of 

exceeding a 25-percent September drought 

streamflow quantile threshold as a function 

of mean monthly streamflow from the 

previous February

Problem: You have noticed over the past 5 years that 

intervals of drought during the month of 

September have become more frequent and 

of longer duration than they seemed to be 

previously. You have good streamflow data 

from February and need to predict September 

drought streamflow probabilities for the North 

Mayo River in the Roanoke River Basin near 

Spencer, Virginia, in order to anticipate conditions that may warrant implementing voluntary 

water-use reduction programs or mandatory 

water-rationing policies during occasions of 

severe drought.

Question: What is the change in the probability that 

September streamflow will exceed the monthly 

25-percent streamflow quantile if the average 

February daily streamflow drops from 200 ft3

/s 

to 40 ft3

/s?

Given: Mean monthly streamflow from the previous 

February.

Solution: A streamgage identified as USGS 02070000 

is on the North Mayo River below Route 629, 

near Spencer, Virginia.

Step 1: In table 1, find station number 02070000.

Step 2: Find the streamflow threshold criteria listed in 

table 1 that corresponds to this station number 

and the information given. In this case the 

flow threshold criteria in table 1 is listed as 

DV > Quantile 25% September DV POR By 

February Mean DV.
Application of the Equations with Examples 15

Step 3: Read Equation: Probability of “Yes” from table 1

corresponding to this station number and flow 

threshold criteria. In this instance, it is 

P[Yes] = 1/(1 + e(1.4162127 + -0.0184251•X)).

Step 4: Read Equation: Probability of “No” from table 1

corresponding to this station number and flow 

threshold criteria. In this instance, it is

P[No] = 1/(1 + e-(1.4162127 + -0.0184251•X)).

Step 5: Calculate the change in probability P[Yes] 

that September streamflow will exceed the 

monthly 25-percent streamflow quantile if the 

average February daily streamflow drops from 

200 ft3/s to 40 ft3/s by first substituting 200 for 

the X variable in the equation, then simplify 

the equation. Then, for comparison, solve the 

equation again this time substituting 40 for the 

X variable in the equation, and simplifying as 

follows:

Solve the first P[Yes] equation:

P[Yes]200 = 1/(1 + e(1.4162127 + –0.0184251•X)) 

P[Yes]200 = 1/(1 + e(1.4162127 + –0.0184251•200))

P[Yes]200 = 1/(1 + e(1.4162127 + –3.68502))

P[Yes]200 = 1/(1 + e–2.26881)

P[Yes]200 = 1/(1 + 0.10344)

P[Yes]200 = 1/1.10344

P[Yes]200 = 0.91

Solve the comparative P[Yes] equation:

P[Yes]40 = 1/(1 + e(1.4162127 + –0.0184251•X)) 

P[Yes]40 = 1/(1 + e(1.4162127 + –0.0184251•40))

P[Yes]40 = 1/(1 + e(1.4162127 + –0.73700))

P[Yes]40 = 1/(1 + e0.67921)

P[Yes]40 = 1/(1 + 1.97232)

P[Yes]40 = 1/2.97232

P[Yes]40 = 0.34

Step 6: Calculate the change in probability P[No] 

that September streamflow will not exceed 

the monthly 25-percent streamflow quantile if 

the daily streamflow drops from 200 ft3/s to 

40 ft3/s by substituting 200 for the X variable 

in the equation, then simplify the equation. For 

comparison, solve the equation again, this time 

substituting 40 for the X variable in the equation, and simplify as follows:

Solve the first P[No] equation:

P[No]200 = 1/(1 + e-(1.4162127 + -0.0184251•X)) 

P[No]200 = 1/(1 + e-(1.4162127 + -0.0184251•200))

P[No]200 = 1/(1 + e-(1.4162127 + -3.68502))

P[No]200 = 1/(1 + e2.26881)

P[No]200 = 1/(1 + 9.66786)

P[No]200 = 1/10.66786

P[No]200 = 0.09

Solve the comparative P[No] equation:

P[No]40 = 1/(1 + e-(1.4162127 + -0.0184251•X)) 

P[No]40 = 1/(1 + e-(1.4162127 + -0.0184251•40))

P[No]40 = 1/(1 + e-(1.4162127 + -0.73700))

P[No]40 = 1/(1 + e-0.67921)

P[No]40 = 1/(1 + 0.50702)

P[No]40 = 1/1.50702P[No]40 = 0.66

Answer: P[Yes]200: There is a 0.91 (91-percent) chance 

that September streamflow will exceed the 

monthly 25-percent streamflow quantile if 

average February daily streamflow is 200 ft3/s.

P[No]200: There is a 0.09 (9-percent) chance

that September streamflow will not exceed 

the monthly 25-percent streamflow quantile if 

average February daily streamflow is 200 ft3/s.

P[Yes]40: There is a 0.34 (34-percent) chance

that September streamflow will exceed the 

monthly 25-percent streamflow quantile if 

average February daily streamflow is 40 ft3/s.

P[No]40: There is a 0.66 (66-percent) chance

that September streamflow will not exceed 

the monthly 25-percent streamflow quantile if 

average February daily streamflow is 40 ft3/s.

The probability that September streamflow 

will exceed the monthly 25-percent streamflow quantile if the average February daily 

streamflow drops from 200 ft3/s to 40 ft3/s is 

reduced from P[Yes]200 = 0.91 to P[Yes]40 = 0.34,

a decrease of 0.57 (57 percentage points).

The probability that September streamflow will 

not exceed the monthly 25-percent streamflow 

quantile if the average February daily streamflow drops from 200 ft3

/s to 40 ft3/s is increased 

from P[No]200 = 0.09 to P[No]40 = 0.66, an 

increase of 0.57 (57 percentage points).

A plot of the MLLR line for the Roanoke River 

Basin site number 02070000 confirms these 

values, as shown in figure 8.
16 Methods for Estimating Drought Streamflow Probabilities for Virginia Streams

No

Yes

Figure 8. A MLLR drought threshold probability plot showing the chances of exceeding, and

not exceeding, an historical September 25 percent daily streamflow threshold as a function

of mean daily streamflow values from February. Red and blue arrows illustrate reading a

P[No]200 = 0.09 response with an explanatory variable value of 200 cubic feet per second and

a P[No]40 = 0.66 response with an explanatory variable value of 40 cubic feet per second.

Orange and blue arrows illustrate reading a P[Yes]200 = 0.91 response with an explanatory variable

value of 200 cubic feet per second and a P[Yes]40 = 0.34 response with an explanatory variable

value of 40 cubic feet per second.(Site number 0207000, North Mayo River below route 629

near Spencer, Virginia).

40 60 80 120 160 200 240 280 320 360

Daily value greater than quantile 25-percent September daily 

value of period of record

1.00

0.75

0.50

0.25

0

February mean daily value

Figure 8. A MLLR drought threshold probability plot showing the chances of 

exceeding, and not exceeding, an historical September 25-percent daily streamflow 

threshold as a function of mean daily streamflow values from February. Red and blue 

arrows illustrate reading a P[No]200 = 0.09 response with an explanatory variable value 

of 200 ft3

/s, and a P[No]40 = 0.66 response with an explanatory variable value of 40 ft3/s. 

Green and blue arrows illustrate reading a P[Yes]200 = 0.91 response with an explanatory 

variable value of 200 ft3

/s, and a P[Yes]40 = 0.34 response with an explanatory variable 

value of 40 ft3

/s. (Site number 02070000, North Mayo River below Route 629 near 

Spencer, Virginia.)

Example 4. Predicting the probability 

of exceeding a 50-percent July drought 

streamflow quantile threshold as a function of 

averaged mean monthly streamflow from the 

previous November, December, January, and 

February

Problem: A land-use re-classification (development) is 

proposed that may increase the demand for 

potable water. July river flow is frequently the 

lowest flow of the year in your district. County 

supervisors have scheduled a vote on the landuse re-classification in 2 weeks time. You need 

to predict July drought streamflow probabilities 

for the Robinson River in the Rappahannock 

River Basin in order to determine whether 

water flows are sufficient to sustain the water 

demands of the proposed development.

Question 1: What is the chance that July streamflow will 

exceed the monthly 50-percent streamflow 

quantile if the average of November, December, January, and February daily streamflow is 

maintained at 500 ft3

/s?

Question 2: What is the chance that July streamflow will 

exceed the monthly 50-percent streamflow 

quantile if the average of November, December, January, and February daily streamflow is 

reduced to 120 ft3

/s?

Given: The average of mean monthly streamflow from 

the previous November, December, January, 

and February.

Solution: A streamgage identified as USGS 01666500 

is on the Robinson River at Route 614 near 

Locust Dale, Virginia.

Step 1: In table 1, find station number 01666500.
Application of the Equations with Examples 17

Step 2: Find the streamflow threshold criteria listed in 

table 1 that corresponds to this station number 

and the information given. In this case, the 

flow threshold criteria in table 1 is listed as 

DV > Quantile 50% July DV POR By Average 

N-D-J-F Mean DV.

Step 3: Read Equation: Probability of “Yes” from table 1

corresponding to this station number and flow 

threshold criteria. In this instance, it is 

P[Yes] = 1/(1 + e(2.12923861 + -0.0085836•X)).

Step 4: Read Equation: Probability of “No” from table 1

corresponding to this station number and flow 

threshold criteria. In this instance, it is

P[No] = 1/(1 + e-(2.12923861 + -0.0085836•X)).

Step 5: Calculate the probability P[Yes] that July 

streamflow will exceed the monthly 50-percent 

streamflow quantile if the average November, 

December, January, and February daily streamflow is maintained at 500 ft3

/s by substituting 

500 for the X variable in the equation and then 

simplifying as follows: 

P[Yes]500 = 1/(1 + e(2.12923861 + -0.0085836•X)) 

P[Yes]500 = 1/(1 + e(2.12923861 + -0.0085836•500))

P[Yes]500 = 1/(1 + e(2.12923861 + -4.2918))

P[Yes]500 = 1/(1 + e-2.16256)

P[Yes]500 = 1/(1 + 0.11503)

P[Yes]500 = 1/ 1.11503

P[Yes]500 = 0.90

Step 6: Calculate the probability P[No] that July 

streamflow will not exceed the monthly 

50-percent streamflow quantile if the average 

November, December, January, and February 

daily streamflow is maintained at 500 ft3/s by 

substituting 500 for the X variable in the equation and then simplifying as follows:

P[No]500 = 1/(1 + e-(2.12923861 + -0.0085836•X)) 

P[No]500 = 1/(1 + e-(2.12923861 + -0.0085836•500))

P[No]500 = 1/(1 + e-(2.12923861 + -4.2918))

P[No]500 = 1/(1 + e2.16256)

P[No]500 = 1/(1 + 8.69338)

P[No]500 = 1/ 9.69338

P[No]500= 0.10

Answer: P[Yes]500: There is a 0.90 (90-percent) chance 

that July streamflow will exceed the monthly 

50-percent streamflow quantile if the average 

November, December, January, and February 

daily streamflow is maintained at 500 ft3/s.

P[No]500: There is a 0.10 (10-percent) chance 

that July streamflow will not exceed the 

monthly 50-percent streamflow quantile if the 

average November, December, January, and 

February daily streamflow is maintained at 

500 ft3/s.

Step 7: Calculate the probability P[Yes] that July 

streamflow will exceed the monthly 50-percent 

streamflow quantile if the average November, 

December, January, and February daily streamflow is reduced to 120 ft3

/s by substituting 120 

for the X variable in the equation and then 

simplifying as follows: 

P[Yes]120 = 1/(1 + e(2.12923861 + -0.0085836•X)) 

P[Yes]120 = 1/(1 + e(2.12923861 + -0.0085836•120))

P[Yes]120 = 1/(1 + e(2.12923861 + -1.030032))

P[Yes]120 = 1/(1 + e1.09921)

P[Yes]120 = 1/(1 + 3.00178)

P[Yes]120 = 1/4.00178

P[Yes]120 = 0.25

Step 8: Calculate the probability P[No] that July 

streamflow will not exceed the monthly 

50-percent streamflow quantile if the average 

November, December, January, and February 

daily streamflow is reduced to 120 ft3/s by substituting 120 for the X variable in the equation 

and then simplifying as follows: 

P[No]120 = 1/(1 + e-(2.12923861 + -0.0085836•X)) 

P[No]120 = 1/(1 + e-(2.12923861 + -0.0085836•120))

P[No]120 = 1/(1 + e-(2.12923861 + -1.030032))

P[No]120 = 1/(1 + e-1.09921)

P[No]120 = 1/(1 + 0.33314)

P[No]120 = 1/1.33314

P[No]120 = 0.75

Answer: P[Yes]120: There is a 0.25 (25-percent) chance 

that July streamflow will exceed the monthly 

50-percent streamflow quantile if the average 

November, December, January, and February 

daily streamflow is reduced to 120 ft3/s.

P[No]120: There is a 0.75 (75-percent) chance 

that July streamflow will not exceed the 

monthly 50-percent streamflow quantile if 

the average November, December, January, 

and February daily streamflow is reduced to 

120 ft3/s.

A plot of the MLLR line for the Rappahannock 

River Basin site number 01666500 confirms 

these values, as shown in figure 9.
18 Methods for Estimating Drought Streamflow Probabilities for Virginia Streams

No

Yes

80 120 180 220 280 340 380 440 500 540

Daily value greater than quantile 50-percent July daily 

value of period of record

1.00

0.75

0.50

0.25

0

Figure 9. An example MLLR drought threshold probability plot showing the chances of exceeding,

and not exceeding, an historical July fifty percent daily streamflow threshold as a function

of the average of mean daily streamflow values from November, December, January, and February.

Red and blue arrows illustrate reading a P[No]500 = 0.10 response with an explanatory variable value

of 500 cubic feet per second and a P[No]120 = 0.75 response with an explanatory variable value of 

120 cubic feet per second. Orange and blue arrows illustrate reading a P[Yes]500 = 0.90 response with

an explanatory variable value of 500 cubic feet per second and a P[Yes]120 = 0.25 response with an

explanatory variable value of 120 cubic feet per second. (Site number 01666500, Robinson (Robertson)

River below route 614 near Locust Dale, Virginia).

Average (N-D-J-F) mean daily value

Figure 9. A MLLR drought threshold probability plot showing the chances of exceeding, 

and not exceeding, an historical July 50-percent daily streamflow threshold as a 

function of the average of mean daily streamflow values from November, December, 

January, and February (N-D-J-F). Red and blue arrows illustrate reading a P[No]500 = 0.10 

response with an explanatory variable value of 500 ft3

/s, and a P[No]120 = 0.75 response 

with an explanatory variable value of 120 ft3

/s. Green and blue arrows illustrate 

reading a P[Yes]500 = 0.90 response with an explanatory variable value of 500 ft3

/s, and 

a P[Yes]120 = 0.25 response with an explanatory variable value of 120 ft3

/s. (Site number 

01666500, Robinson (Robertson) River at Route 614 near Locust Dale, Virginia.)

Summary

Anticipating drought conditions in Virginia streams is 

essential to sound management of the water resources and 

associated riparian and watershed ecosystems. Reliable estimates of the likelihood that stream low flows during droughtprone months will exceed specific low-flow thresholds can 

provide advance warning of drought conditions, allowing 

extended lead times for improved drought awareness and 

effective management response. This report, a product of a 

cooperative partnership between the Virginia Department 

of Environmental Quality (DEQ) and the U.S. Geological 

Survey, describes empirically determined maximum likelihood logistic regression (MLLR) equations that predict 

drought streamflow probabilities for Virginia streams. Winter 

streamflows are used to estimate the likelihood of probable 

streamflows during drought-prone summer months. Probabilities of streamflows exceeding specific drought streamflow 

thresholds are characterized in a set of 70,120 MLLR equations for 339 basins in Virginia. A 46,704-member subset of 

these equations is published in this report for 259 Virginia 

basins, each with an accompanying p-value statistic that is less 

than or equal to 0.05 and is based on streamflow data spanning 

a minimum 10-year (maximum 112-year) period of record. 

More than 5.12 million streamflow daily values, collected over 

the period of record, were compiled and used in the analysis. 

Example calculations are provided demonstrating how to use 

the equations to estimate probable streamflows up to 8 months 

in advance.
Selected References 19

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Austin—Methods for Estimating Drought Streamflow Probabilities for Virginia Streams—Scientific Investigations Report 2014–5145

Printed on recycled paper

ISSN 2328-0328 (online)

ISSN 2328-031X (print)

http://dx.doi.org/10.3133/sir20145145