Using Statistical Methods for
Environmental Science and Management
Graham McBride, NIWA, Hamilton
g.mcbride@niwa.co.nz
Statistics Teachers’ Day, 25 November 2008
What do statisticians really do?
THE ROLE OF STATISTICAL
METHODS: MY VIEW
• Separate randomness from pattern
• Make inferences about the world, based on
data from samples
• Help to design sampling programmes (use
resources efficiently)
• Help to establish cause and effect
• Can’t “prove anything with statistics”
“Three kinds of lies”
Insult, or compliment?
There are three kinds of lies
– lies, damned lies, and statistics
Who said that?
– Mark Twain (1835 – 1910)
“Figures often beguile me, particularly when I have
the arranging of them myself”
– Benjamin Disraeli (1804 – 1881)
Sought to discredit true British soldier casualty
figures in the Crimean War (1853 – 1856)
Who came first? (Twain cites Disraeli!)
What you should do
• Establish the context of your work (what do
people want to know, and why do they want to
know that?)
• Consult with others, e.g., to discuss whether a
proposed sampling programme can actually be
done
• Discuss the appropriate burden-of-proof (e.g.,
drinking water standards minimise the consumer’s
risk, not the producer’s risk)
What you should not do
• Confuse association and causation (pp. 267-8 of
Barton, Sigma Mathematics)
• Ignore other lines-of-evidence (Bradford-Hill
criteria), such as
– Can the cause reach the location of the effect?
– Is the finding plausible?
– Can you explain inconsistencies with other evidence?
• Be ignorant of how statistical procedures work
– The computer said so
What you should not do
• Believe that there is only one “statistically
correct” way of analysing data
– There are lots of good ways; many more bad
and wrong ways too
• Not consider bias and imprecision in your
data
Bias and Imprecision
INACCURATE INACCURATE INACCURATE
(a) Biased, imprecise
(b) Unbiased, imprecise
(c) Biased, precise
ACCURATE
(d) Unbiased, precise
What you might have to do
• Use non-standard methods, e.g.,
– non-parametric (rank) methods for highly skewed data
(very common in aquatic studies)
• e.g., linear trend or monotonic trend?
• Read rather widely
– Statistics is not a cut-and-dried subject; there are still
some fundamental debates about statistical inference,
especially the Bayesians versus the frequentists—both
approaches have their place
What you also might have to do
• Answer this question: “What is P”
– Result of a hypothesis test
– Used (over-used!) routinely, so you’ll need to
know
• P = Prob(data at least as extreme if the tested
hypothesis is true)
• Not the probability of the truth of the hypothesis
• Relate results to confidence intervals
EXAMPLE
Increasing pressure on freshwaters
Is there evidence of associated deterioration
(or improvements) in rivers?
600000
4
Cows
3
400000
2.5
300000
2
1.5
200000
1
100000
Data source: 1Fertilizer consumption – UN Food & Agriculture Organisation
2Cows –Livestock Improvement NZ Dairy Statistics
2003
2002
2001
2000
1999
1998
1997
1996
1995
1994
1993
1992
1991
1990
0
1989
0.5
0
Cow numbers (millions)2
3.5
Total Phosphorus
500000
1988
Fertilizer consumption (tonnes)1
Total Nitrogen
A National River Water Quality
Network for New Zealand (1989)
GOAL
To provide scientifically defensible information on the
important physical, chemical, and biological
characteristics of a selection of the nation’s rivers as
a basis for advising the Minister of Science and other
Ministers of the Crown of the trends and status of
these waters
OBJECTIVES
1. Detect significant trends in water quality
2. Develop better understanding of water resources, and
hence to better assist their management
NRWQN
structure
• 77 sites on 35 rivers
• All sites have reliable flow data
• Sites are sampled by regional
Field Teams
• 14 WQ parameters (monthly)
• Data available (search for
WQIS www.niwa.co.nz
WQ state & land
use
Correlations with % Pasture
Temperature
0.50***
Conductivity
0.55***
pH
-0.19
Dissolved oxygen -0.17
Visual clarity
-0.60***
NOx-N
0.71***
NH4-N
0.77***
Total nitrogen
0.84***
DRP
0.67***
Total phosphorus
0.74***
E. coli
0.79***
***
P < 0.001; Spearman rank correlation
WQ Trends 1989-2005
• Calculated annual medians from monthly data at each site
for each parameter
• Took the 77 datapoints for each year and calculated the
5th, 50th, and 95th percentile values
• The 50th percentile gives us a picture of what is happening
in a national “average” river in terms of annual median
water quality data
• The 5th and 95th percentiles tell us about changes over
time in our “best” and “worst” rivers.
• Trends in these values were assessed using the Spearman
rank correlation coefficient (rS).
NOx-N Trends 1989-2005
1200
5th
95th
3
NOx-N (mg/m )
1000
50th
800
600
400
Year
Concentrations of NOx-N increased dramatically between 1989 & 2005
in our most enriched rivers
2005
2004
2003
2002
2001
2000
1999
1998
1997
1996
1995
1994
1992
1991
1990
1989
1988
0
1993
200
Trends 1989-2005
5th
50th
95th
TEMP
0.70
0.33
0.28
COND
0.30
0.48
0.22
PH
-0.27
-0.11
-0.64
DO
-0.09
-0.25
-0.11
CLAR
0.13
0.39
0.44
NOx-N
-0.81
0.37
0.80
NH4-N
-0.96
-0.94
0.44
• Increasing nitrogen enrichment
TN
0.39
0.59
0.71
• Decreases in BOD5 most rivers
DRP
-0.26
0.48
0.05
TP
-0.10
0.24
-0.37
BOD5
-0.75
-0.88
-0.70
Results indicative of:
• Warming in our coolest rivers
• Drops in pH
Trends 1989-2003
• More formal analysis of trends carried out on
monthly data (1989-2003) at all 77 sites
• Seasonal Kendall test
• Data were flow-adjusted using LOWESS (many
WQ parameters can be strongly influenced by
discharge)
• Used a binomial test to indicate a “national trend”
• Discriminate between “significant” (i.e. P < 0.05)
and “meaningful” trends (i.e., P < 0.05 and slope
> 1% of median value per annum).
Trends in TN
Total nitrogen exhibited a
strong increasing trend at the
national scale during 19892003 (P < 0.001).
Increasing trends in TN
were particularly evident in
the South Island, where 25 of
33 sites showed meaningful
increases.
Trends in DRP
There was a strong national
trend of increasing DRP
concentrations during 19892003 (P < 0.001).
This result contrasts with the
relatively weak trends observed
for 1989-2005.
Summary of trends 1989-2003
15
10
RSKSE
5
Plot 1
0
-5
-10
-15
Temp Cond
No significant trend
pH
DO
Clar NOx-N NH4-N
TN
Significant improving trend
DRP
TP
BOD5
Significant deteriorating trend
Links between land use and trends
10
Trend in Dissolved Reactive Phospohorus (SKSE
as % of median)
y = 0.0406x - 0.0027
8
R 2 = 0.31
6
4
2
0
-2
Lower
Manawatu Rv.
-4
0
10
20
30
40
50
60
70
80
% Pastoral land use
The magnitude of trends in DRP increase with % pastoral land use
90
100
Land use and trends
Parameter
SKSE
Temperature
0.19
0.20
Conductivity
0.47
0.40
pH
-0.28
-0.28
Dissolved oxygen
-0.27
-0.27
Visual clarity
-0.26
-0.11
Oxidised nitrogen
0.30
0.23
Ammoniacal nitrogen
0.29
0.68
Total nitrogen
0.35
-0.01
Dissolved reactive phosphorus
0.59
0.48
Total phosphorus
0.31
0.18
Spearman rank correlation coefficients (bold P < 0.01)
RSKSE
Conclusions
• Strong associations between nutrient concentrations and
%pastoral land cover at the national scale (State)
• Rivers draining large areas of pastoral land have
deteriorated significantly over the last 17 years with
respect to nitrogen concentrations (Trends)
• The magnitude of trends in some parameters is associated
with extent of pastoral land use
• Decreasing trends in NH4-N and BOD5 indicative of
improvements in point source management
• Increasing trends in nutrients indicative of increasing
pressure from agriculture
EXAMPLE:
Water quality-human health risk
assessment, quantitative approach
Christchurch City Wastewater Outfall
Oxidation Ponds
Avon-Heathcote
Estuary
Pump Station
Pipeline
route
Quantitative Microbial Health Risk
Assessment (QMHRA)
• Identify hazards (pathogens)
• Quantify exposure (swimming, shellfish consumption)
• Assess dose-response
• Characterise risk
Hazard vs. Risk
• Hazards can cause harm, after exposure
• Risk cannot occur if no exposure
• Can have hazard without risk
• But not vice versa!
Christchurch hazards—viruses only
From an extensive list (next slide):
• Swimming
– adenovirus (respiratory)
– rotavirus
– enterovirus (Echovirus 12)
• Shellfish consumption (raw)
– enteroviruses
– rotavirus
– hepatitis A
Pathogen
Main disease caused
Comments
Include?
Campylobacter spp.
Gastroenteritis
Poor survival in seawater
No
Pathogenic E. coli
Gastroenteritis
Low concentration expected in sewage
No
Legionella pneumophila
Legionnaires' disease
No evidence of environmental infection
route
No
Leptospira sp.
Leptospirosis
Low concentration expected in sewage
No
Salmonella sp.
Gastroenteritis
Low concentration expected in sewage
No
Salmonella typhi
Typhoid fever
Rare in New Zealand
No
Shigella sp.
Dysentery
Low concentration expected in sewage
No
Vibrio cholerae
Cholera
Rare in New Zealand
No
Yersinia enterolitica
Gastroenteritis
Low concentration expected in sewage
No
Ascaris lumbricoides
Roundworm
Rare in New Zealand
No
Enterobius vernicularis
Pinworm
Low concentration expected in sewage
No
Fasciola hepatica
Liver fluke
Rare in New Zealand
No
Hymnolepis nana
Dwarf tapeworm
Rare in New Zealand
No
Taenia sp.
Tapeworm
Rare in New Zealand
No
Trichuris trichiura
Whipworm
Rare in New Zealand
No
Balantidium coli
Dysentery
Low concentration expected in sewage
No
Cryptosporidium oocysts
Gastroenteritis
Can accumulate in shellfish, but virus
groups of more concern
No
Entamoeba histolytica
Amoebic dysentery
Rare in New Zealand
No
Giardia cysts
Gastroenteritis
Poor survival in seawater
No
adenoviruses
Respiratory disease2
Very infective, present in substantial
concentrations in raw sewage
Yes (SW
only)1
enteroviruses
Gastroenteritis
Less infective, but health consequences
can be more severe than adenovirus
Yes (SW
and SF)
hepatitis A virus
Infectious hepatitis
Low sewage concentration; very infective
Can affect surfers in contaminated waters 4
Yes (SF)
noroviruses3
Gastroenteritis
No reliable method for viability
enumeration; limited data on occurrence in
water and infectivity.
No
rotaviruses
Gastroenteritis
Limited evidence of waterborne infection in
NZ; infection in children would be of
concern.5
Yes (SF
and SW)
Bacteria
Helminths
Protozoa
Viruses
Dose-response curves
Probability of infection
1
adenovirus
rotavirus
Constant susceptibility
Variable susceptibility
0.8
0.6
0.4
0.2
0
0
2
4
6
Dose
8
10 0
20
40
60
Dose
80
100
Accounting for variability and
uncertainty
• Exposure is variable
– e.g., individuals’ swim duration
• Dose-response is uncertain
– only some pathogen strains in clinical trials
– trials limited to healthy adults
• Describe using statistical distributions in a Monte
Carlo analysis
Scenariosis!
• 1,000 people; 1,000 occasions
–
–
–
–
–
–
–
8 beaches
2 influent virus conditions (normal & outbreak)
2 seasons summer/winter
3 viruses for 2 activities
2 outfall lengths
2 virus inactivation regimes
2 UV options (with & without)
•  1536 x 106 calculations
Calculation sequence
Viruses in raw sewage
Treatment efficiency
UV disinfection, if present
Plume dispersion model
(including inactivation)
Bioaccumulation Virus concentration
Duration of swim
at beach
Virus concentrations
in shellfish
Water ingestion/
inhalation rate
Meal size
Number of viruses
ingested
Dose-response
relationships
Number of viruses
ingested or inhaled
Proportion of
population infected
Dose-response
relationships
Dose-response models
• Constant susceptibility—simple exponential
(d = average dose, Prinf = infection prob)
Prinf ( d ) = 1 - e
- rd
• Variable susceptibility—“beta-Poisson”

d
Prinf ( d )  1 - 1  
 
-
• Calculations performed using “@RISK” (an Excel plug-in)
Occasion 1, Individual
1
Volume
ingested
Dose
Ingestion rate
Probability
of infection
Microorg. concn
Frequency
Frequency
Prob(inf)
Frequency
Duration
Binomial
distribution
Infected?
Occasion 1, Individual
2
Volume
ingested
Dose
Ingestion rate
Probability
of infection
Microorg. concn
Frequency
Frequency
Prob(inf)
Frequency
Duration
Binomial
distribution
Infected?
Occasion 1, Individual
3
Volume
ingested
Dose
Ingestion rate
Probability
of infection
Microorg. concn
Frequency
Frequency
Prob(inf)
Frequency
Duration
Binomial
distribution
Infected?
Occasion 1, Individual 1000
Volume
ingested
Dose
Ingestion rate
Probability
of infection
Microorg. concn
Frequency
Frequency
Prob(inf)
Frequency
Duration
Binomial
distribution
Infected?
Sum the cases
Occasion 2, Individual
1
Volume
ingested
Dose
Ingestion rate
Probability
of infection
Microorg. concn
Frequency
Frequency
Prob(inf)
Frequency
Duration
Binomial
distribution
Infected?
Occasion 2, Individual
2
Volume
ingested
Dose
Ingestion rate
Probability
of infection
Microorg. concn
Frequency
Frequency
Prob(inf)
Frequency
Duration
Binomial
distribution
Infected?
Occasion 2, Individual
3
Volume
ingested
Dose
Ingestion rate
Probability
of infection
Microorg. concn
Frequency
Frequency
Prob(inf)
Frequency
Duration
Binomial
distribution
Infected?
Characterising the results
• Risk percentiles—percent of time the risk is below
a stated value
• IIR—Individual Infection Risk (total number of
calculated infections divided by total number of
exposures)
Results
South New Brighton
RAW SHELLFISH CONSUMPTION: NORMAL NONCONSERVATIVE ROTAVIRUS
Summer
2 km
no UV
Winter
3 km
UV
no UV
2 km
UV
no UV
3 km
UV
no UV
UV
Min
0
0
0
0
0
0
0
0
50%ile
0
0
0
0
0
0
0
0
90%ile
0
0
0
0
0
0
0
0
95%ile
0
0
0
0
1
0
0
0
98%ile
0
0
0
0
4
1
2
1
99%ile
0
0
0
0
6
2
3
1
99.9%ile
1
1
0
0
15
6
7
3
Max
2
1
0
0
16
7
8
4
IIR(%)
0.0005
0.0002
0.0000
0.0000
0.0052
0.0089
0.0032
Integers are cases per 1000 exposures
0.0244
IIR: Normal influent, South Brighton
adenovirus, swim
2 km, no UV
2 km, UV
3 km, no UV
3 km, UV
Summer
0.0001
0.0000
0.0000
0.0000
0.0034
0.0002
0.0016
0.0005
Winter
Numbers are percentages. MfE/MoH (2003) guidelines: <0.3% = “Very good”.
IIR: Normal influent, South Brighton
rotavirus, shellfish
2 km, no UV
2 km, UV
3 km, no UV
3 km, UV
Summer
0.0005
0.0002
0.0000
0.0000
0.0244
0.0052
0.0089
0.0032
Winter
Numbers are percentages.
IIR: Outbreak influent, South Brighton
adenovirus, swim
2 km, no UV
2 km, UV
3 km, no UV
3 km, UV
Summer
0.0568
0.0179
0.0009
0.0003
2.1135
0.5552
1.0959
0.3016
Winter
Numbers are percentages. MfE/MoH (2003) guidelines: 1.9 - 3.9% = “Fair” - “Poor”.
IIR: Outbreak influent, South Brighton
rotavirus, shellfish
2 km, no UV
2 km, UV
3 km, no UV
3 km, UV
Summer
0.3882
0.1034
0.0033
0.0005
4.9911
2.1668
2.3916
1.1779
Winter
Numbers are percentages.
IIR: Outbreak influent, South Brighton
hepatitis A, shellfish
2 km, no UV
2 km, UV
3 km, no UV
3 km, UV
Summer
0.0343
0.0107
0.0000
0.0001
0.9441
0.2477
0.4633
0.1733
Winter
Numbers are percentages.
Statistical modelling can reveal
important information gaps
•
•
•
•
•
•
•
Bioaccumulation factors for NZ shellfish
Dose-response for norovirus (new study published)
Detailed exposure data (ingestion rates etc.)
Constancy of virulence?
Campylobacter in shellfish?
Better methods for uncertainty analysis
Better models for illness, cf. infection
Conclusions
• Longer outfall no UV still has higher risk than shorter
outfall with UV
• But risks low
• What if UV doesn’t work 24/7 (technology
breakdown, power outage,…)
• Decision: longer outfall, no UV
Semi-Quantitative approach
Use when hazards and exposures are less well-defined
and more widespread
Paradigm is:
Risk score = Likelihood x Consequences
Use scores as a relative measure of risk.
Use panel of “experts”; may solicit list of hazards from
affected community
Hazards
•
•
•
•
Pathogens (from humans and animals)
Chemicals
Algal toxins
Physical objects
“End-points” (exposures)
•
•
•
•
Recreational contact
Drinking water consumption
Consumptions of aquatic organisms
Food? (more difficult)
The delivery chain
• Can be called “hazardous event”
• How does the hazard get from its origin to the point
of exposure?
Likelihood
Probability of an exposure event (for at least one
person) in a year (cf. any year) to a sufficient degree to
cause harm. Scores:
0
1
2
4
6
8
10
Impossible
Extremely unlikely
Very unlikely
Unlikely
Even
Likely
Very likely
0
1
1 – 5%
6 – 40%
41 – 60%
61 – 95%
>95%
Consequences
Scale#
Severity*
Duration*
1: <1%
2: 1–5%
3: 5–10%
4: 10–20%
5: >20%
1: Asymptomatic
2: Discomfort
3: Visit doctor
4: Hospitalisation
5: Death
1: Day
2: Week
3: Month
4: Year
5: Permanent
#
Percent of total community
* Refers to health effect
Typical results
Exposure
Recreational
Water
Recreational
Water
Recreational
Water
Recreational
Water
Recreational
Water
Recreational
Water
Recreational
Water
Recreational
Water
Recreational
Water
Recreational
Water
Recreational
Water
Recreational
Water
Recreational
Water
Recreational
Water
Recreational
Water
Drinking-water
Population
Score
Hazardous event
Normal
250
Toxic algal bloom (marine) – inhalation
Normal
250
Strong rips and current in bathing areas
Normal
240
Urban stormwater discharge in streams and beaches
Normal
240
Bird defecation into freshwater margins
Normal
200
Toxic algal blooms (f/w) – inhalation
Normal
200
Algae from overflow of oxidation ponds – inhalation
Normal
200
Algae released from farm dams etc. – inhalation
Susceptible
200
Bather shedding of infectious organisms
Susceptible
200
Urban stormwater discharge in streams and beaches
Susceptible
200
Bird defecation into coastal waters
Normal
180
Dry weather sewage overflows in streams and beaches
Normal
180
Cuts from naturally-occurring objects (oyster shells etc.)
Normal
160
Bather shedding of infectious organisms
Normal
160
Slipping on slimy surfaces
Susceptible
150
Dry weather sewage overflows in streams and beaches
Mainland
125
Toxic algal blooms (f/w) – ingestion
Conclusions
• Use QRA for well-defined “local” problems
• Use semi-quantitative methods for broader-scale
problems
• Risk assessment identifies many knowledge gaps,
some need urgent attention
• Most difficult gap often the “delivery chain”
• Can update assessments with new data
• Especially useful in ranking risks
EXAMPLE
Compliance with Drinking Water Standards
How to assess compliance with microbial limits?
• Can’t sample everything
• Need high assurance that supply isn’t contaminated
in some assessment period; can’t be fully assured
• MoH then said: “We want to be 95% confident that
the water is uncontaminated for 95% of the time.
What should the compliance rule be?”
What kind of a question is this?
• Bayesian
– It asks about the probability of an hypothesis, given
data that we will collect
– Frequentist (“classical” methods) ask about the
probability of data assuming an hypothesis to be true
• Precautionary (not “permissive”)
– Benefit of doubt goes to the consumer, not to the
supplier
• One-sided
– Hypothesis to be tested is breach, not compliance
Results
Results
Policy Implications
• Results in Table 8.2 now incorporated into
2005 Drinking-water Standards for New
Zealand
– http://www.moh.govt.nz/moh.nsf/0/12F2D7FFADC900A4CC256
FAF0007E8A0/$File/drinkingwaterstandardsnz-2005.pdf
EXAMPLE
Effect of microbial contamination on
swimmers’ health
Epidemiological study at 7 NZ beaches
Main Findings
• Using generalized regression models
– Evidence of respiratory illness effects related to
microbial contamination
– Human- and animal-waste impacted beaches
not separable in terms of health effects
– Both were separable from “control” beaches
Policy implications
• Human and animal wastes no longer
distinguished in terms of health risks
• Result incorporated into new guidelines
– http://www.mfe.govt.nz/publications/water/microbiological-quality-jun03/