Claire Ferguson1, Adrian Bowman1, Marian Scott1 and Laurence Carvalho2

1University of Glasgow2Centre for Ecology & Hydrology

A Case Study of Loch Leven

http://www.tulbol.demon.co.uk/art/banatvewe.jpg

Additive and Varying-Coefficient Models

Dataset:

1968 - 2002

150 variables measured,

covering…

• Physics

• Lake chemistry

• Lake biology

• Weather

Loch Leven

Log SRP

Years

Lo

g(S

RP

, m

ug

/l)

1970 1980 1990 2000

-10

12

34

Log TP

Years

Lo

g(T

P,

mu

g/l)

1970 1980 1990 20003

.54

.04

.55

.05

.5

Log Chlorophyll

Years

Lo

g(C

hlo

rop

hyl

l, m

ug

/l)

1970 1980 1990 2000

12

34

5

Water Temperature

Years

Wa

ter

Tem

pe

ratu

re,

oC

1970 1980 1990 2000

05

10

15

20

Log Daphnia

Years

Lo

g(D

ap

hn

ia,

ind

/l)

1970 1980 1990 2000

-4-2

02

4

Log NO3N

Years

Lo

g(N

O3

N,

mg

/l)

1970 1980 1990 2000

-4-2

0

Loch Leven – the Data

The EC Water Framework Directive (WFD) 2000 states that there should be ‘good status’ in all shallow waters by 2016.

Research:

Investigating statistical techniques to explore the combined impacts of climate and nutrients on water quality.

Motivation

tt3t

ttt

ta

)N)-(NO)temp(water

)(SRP)()(month)(year

lchlorophyl

(log))(log(

log

)log(

654

321

mmDaphniam

mmm

t

chlorophylla

Response

year, month, SRP, Daphnia, water temp, NO3-N

Covariates

Additive Model 1988 - 2002

a) estimate of m1(year)

year

m1

(ye

ar)

1990 1994 1998 2002

-10

12

b) estimate of m2(month)

month

m2

(mo

nth

)

2 4 6 8 10 12-1

01

2

c) estimate of m3(log SRP)

log(SRP,mug/l)

m3

(lo

g(S

RP

))

1 2 3 4

-10

12

d) estimate of m4(log Daphnia)

log(Daphnia, individuals/l)

m4

(lo

g(D

ap

hn

ia))

-4 -2 0 2 4

-10

12

e) estimate of m5(water temp)

water temp,oC

m5

(wa

ter

tem

p)

5 10 15 20

-10

12

f) estimate of m6(log NO3-N)

log(NO3-N,mg/l)

m6

(lo

g(N

O3

-N))

-4 -3 -2 -1 0 1

-10

12

Chlorophyll

Should a covariate be linear as opposed to nonparametric?

Is a nonparametric covariate significant?

ttt

ttt

xmxmH

xxmH

)()(:

)(:

22111

2110

ttt

tt

xmxmH

xmH

)()(:

)(:

22111

110

Additive Models - Inference

Variable F-Test ‘linear effect’

p-value

F-Test ‘no effect’

p-value

year 0.632 0.634

month - 0.212

log(SRP) 0.068 0.016

log(Daphnia)

1 0.951

water temp 0.059 0.090

log(NO3-N) 0.651 0.153

Inference

Mainly linear relationships between variables.

Little evidence of trend and seasonality.

Important ecological relationships identified.

The nature of trends and relationships can be explored simultaneously.

For chlorophylla (in this time period)-

Summary of Additive Models

Linear Regression Model

k

jijiji XY

1

Varying-coefficient Model (VCM)

k

jijiji XmmY

1)()(

VCM – Case Study

tt mm )SRPlog()()()chlalog(

VCM – Case Study