Advantages and inconveniences of the Cox model compared with the logistic model: application to a study of risk factors of nursing cow infertility

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Advantages and inconveniences of the Cox model

compared with the logistic model: application to a study

of risk factors of nursing cow infertility

F Bugnard, C Ducrot, D Calavas

To cite this version:

Advantages

and

inconveniences

of the

Cox

model

compared

with

the

_logistic

model:

_application

to

a

_study

of

risk factors of

_nursing

cow

_infertility

F

_Bugnard

C

Ducrot D Calavas

Centre

_{d’Écopathologie}

Animale,

26, rue de la Baisse, 69100 Villeurbanne, France

Summary ―The

survival Cox model and the

logistic

model were

_compared

on a data set obtained from an

ecopathological survey

relative to the risk factors of

_nursing

cow

infertility.

The risk factors

resulting

from the 2 models were the same. The Cox model has the

_advantage

of

_preserving

the variable in its

original quantitative

form, and of

_using

a maximum of information. However, very restrictive condi-tions of

_application

of this model make its use rather limited.

ecopathology

I survival

_{model 1 logistic}

model

Résumé -

_Avantages

et inconvénients du modèle de Cox par rapport au modèle

_{logistique :}

application

à l’étude des facteurs de

_risque

de l’infécondité des vaches allaitantes. Le modèle de survie de Cox et le modèle

_logistique

ont été

_comparés

sur

_{un jeu}

de données issu d’une

enquête

d’écopathologie

relative aux facteurs de

_risque

de l’infécondité des vaches allaitantes. Les facteurs de

risque

issus des 2 modèles étaient les mêmes. Le modèle de Cox

présente l’avantage

de conserver à la variable étudiée sa forme

_{quantitative d’origine}

et d’utiliser le maximum d’information.

Cepen-dant,

les conditions

d’application

très restrictives de ce modèle sont une limite à son utilisation.

écopathologielmodèle

de survielmodèle

_logistique

INTRODUCTION

Logistic

regression

is

_widely

used

for

investi-gation

of risk factors in

_epidemiology

in

gen-eral and in the

Centre

_{d’Ecopathologie}

Ani-male in

_particular.

There are 2 main reasons

for this.

_Firstly,

the

_pathologies

studied are

often characterized as absent or

_present

and we wish to

_explain

the risk of an occur-rence

of

the disease. In this sense, the

logis-*

Correspondence

and

reprints

tic model is

_appropriate

as it allows the

pre-diction of the

_probability

of an event. On the other

hand,

an

_advantage

of the

_logistic

model is that the

_parameters

can be

inter-preted

as

_being

the

_logarithms

of the odds ratios of

_explanatory

variables.

However,

when the

_pathology

studied is character-ized

_by

a time interval

_{(eg, calving}

interva-land

_{breeding-to-conception period),}

logis-tic

_regression

seems less suitable.

Indeed,

of

time into discrete

classes,

which leads to

an

_important

_{loss of information and poses} a

_problem

of which

_group

end

_points

to

choose. The survival models

and,

in

partic-ular the Cox

model,

appear,

therefore,

to

be better

_adapted.

These

models,

initially

created for cancer research in order to

_study

the survival

time of

_patients

after

_therapy,

allow the

_study

of

time intervals without divi-sion into

classes.

In

_agreement

with

the

logistic

model,

the

_parameters

of the Cox model can be

_{easily interpreted}

since

_they

are

the

_logarithms

of

the relative risks of

explanatory

variables. To

compare

their

results,

both the

_logistic

and the Cox model

were

_applied

to the same data set collected

in a

_survey

carried out

_by

the

Centre

d’Eco-pathologie

Animale in order to

_highlight

the risk

factors

of

_nursing

cow

_infertility.

MATERIALS AND METHODS

The data were collected

_during

a _{survey carried}

out from 1987 to 1989

_by

the Centre

d’fco-pathologie

Animale in 116 farms in the

Rh6nes-Alpes

and Centre

_regions

and in the district of Yonne

(Ducrot, 1993).

In the course of this

period,

3 583

_mating

cows were

_individually

monitored after their first

_{calving during}

the survey. The

infertility

of cows was determined

_by

the

_calving

interval. The covariates introduced in the

logis-tic and the Cox model were selected from the

hypotheses

of the risk factors with the

_help

of univariate

_analyses

_(x

₂

_,

_Logrank

and Wilcoxon

test)

and

_only

those with a 20%

significant

link with

infertility

were

_kept.

The 20% level was cho-sen in order not to eliminate covariates which had little effect on _{the disorder when studied}

sep-arately,

but, if associated,

might

have a

_strong

influence upon the disorder

_(Hosmer

and

Lemeshow,

1989).

The selected factors were: the characteristics of the cow and the calf

_(parity,

breed of the cow, number, sex,

weight,

and pre-sentation of

_calves);

the

_calving

conditions

(calv-ing difficulty,

characterized

by

the

type

of

inter-vention,

retained

_placenta,

characterized

by

the fact that the

_placenta

is not eliminated

sponta-neously

within 24 h of

calving,

and acute metritis with

_general

_symptoms

_appearing

within 2 d of

calving);

and the

_mating

conditions

_(calving

to

turning-out-to-graze period,

whether or not the cows turned out to graze before

_mating,

cleanli-ness score of the cows at the time of

_turning

them out to graze measured on 4 anatomic zones of the hind

_quarters

(Faye

and Barnouin,

1985),

type

of basal ration after

_{calving, quantity}

of

concen-trates distributed after

_{calving, fattening}

score at the time of

_turning

out to _{graze determined}

_by

manual

handling (Agabriel

etal,

1986), fattening

variation

_{during winter).}

Two additional data ele-ments were taken into consideration as

adjust-ing

factors: the

_{geographic region}

and the

type

of winter

_housing.

In the

following,

the vector

_zi=

_{= (Zj}

_1,

_Z

_j

_2,

_...,

_z

_i

_P

₎

refers to the p covariates measured for the

indi-vidual j U

1, ...,

n).

The

_logistic

model

The

_calving

interval was divided into 2 discrete classes; the cows were considered infertile if the

calving

interval was more than 368

d,

or if

_they

were not fertilized, or if

_they

were fertilized

be-latedly,

ie if the zootechnical

objective

of one calf per year was not reached. Thus, we tried to

explain

an event Y coded as

_present

or absent

(infertile/fertile).

).

If

_P(Z)

is the

_probability

of a cow

_being

infertile

knowing

Z=

_{= (}

_Z1’

z2, ...,

zp),

the

logistic

model is

defined as follows:

where b

_{(bo, b}

₁

_{, ...,}

_bp)

is the vector of the model

parameters.

The

_logistic

model was

_adjusted

to the data

using

the maximum likelihood estimation method. The

resulting

model was

_{analyzed by}

_comparing

globally

all estimated parameters

b

i

to 0,

using

the likelihood ratio test, and also

_{by comparing}

each

_parameter

to 0,

using

the Wald test

(Hosmer

and

Lemeshow,

1989).

The

parameters

not

sig-nificantly

different from 0 at the 5% level were eliminated.

Finally,

the

goodness

of fit of the final model was determined

_by

the

_X

₂

test of Hosmer r

and Lesmeshow

₍₁₉₈₉₎

which compares the num-ber of the

_predicted

_infertility

cases to the number of the observed cases. This

analysis

was carried out with the SAS software, LOGISTIC

procedure

The Cox model

The

_calving

interval was studied without

being

divided into classes. The studied interval was from

_calving

to the end of the _survey

_period

for the cows that did not have a second

calving

before the end of the survey

(censored cows).

Let the hazard function

_h(t,Z)

be the

proba-bility

of

calving

at the time tfor a cow

_knowing

Z

=

(

z

i,

_z2,...,

_zp).

The Cox model is:

where b’

= (b’

1’

b’

2

,

...,

b’p)

is the vector of the

model

parameters.

These

parameters

are esti-mated

_by

the maximum likelihood method

_using

the Cox

_partial

likelihood;

l1o(t)

remains an unspec-ified function which does not need to be esti-mated

(Cox

and

Oakes,

1984).

The

analysis

was

performed

in 2

_stages.

The utilization of the Cox model is bound to the veri-fication of the

_proportional

hazards

assumption

(which

means, in other

words,

that the effect of the factors must be

independent

of

_time),

and so the first

_stage

consisted in

verifying

this

_{hypothesis by}

representing

the

_logarithm

of the

_negative

loga-rithm of the survival function of each class as a function of time for each covariate

(fig 1

Accord-ing

to the

proportional

hazards

_assumption

the

curves must not cross each other

(Lee

et

al,

1989).

Covariates

_presenting

_{many classes did} not

_verify

this

_hypothesis,

and so we

_grouped

classes in order to obtain variables

_satisfying

this

application

condition. For the cleanliness score, the curves of the 2 classes did cross each other

(fig 2).

Since

_grouping

was

_impossible,

this vari-able was

_integrated

in the model but the esti-mated

parameters

could not be

_interpreted.

In the course of a second

_stage,

the Cox model was

applied

to the whole data set. As for the

logistic

model, the estimated

_parameters

were

_analyzed

by

the likelihood ratio test and the Wald test

(Cox

and

Oakes,

1984).

The covariates were elimi-nated from the model in the same _way as in the

logistic

model

(parameters

not

_{significantly}

dif-ferent from 0 at the 5%

_level).

This

_analysis

was carried out with the SAS

software,

PHREG pro-cedure

(SAS

Institute

Inc,

1991

).

The methods that allow an estimation of the

goodness

of fit of the Cox model have not

_yet

been

_{completely developed. They}

are based

mainly

on

_graphical

methods and were not

imple-mented in this

study.

RESULTS

The 2 final models

_gave

the same risk

sig-nificant for

_parity,

acute

metritis,

and clean-liness score. For retained

_placenta,

the 2

models

_{gave almost}

the same P values

(logistic

model: P =

0.04;

Cox model: P =

0.0521

).

However,

there are 2 differences. In the

logistic

model,

for the variable

_’calving

diffi-culty’,

the

_parameters

of

the

classes

’as-sistance with

calf

_{puller (1 person)’}

and ’forcible extraction’ were not

_{significantly}

different from

0,

but

_they

were in the Cox model. On the

_contrary,

for

the variable

’calv-ing

to

_{turning-out-to-graze period’ the}

param-eters of the classes ’< 1

month’,

’1-2

months’ and ’2-3 months’ were not

signifi-cantly

different

from 0 in the

Cox

model,

but

they

were in the

_logistic

model.

Finally,

for the

2

_adjusting

variables,

it

was observed that:

_(i)

for the

_type

of winter

housing,

the same results were obtained in the 2

models;

and

_(ii)

for the

_geographic

region,

the classes ’Loire’ and ’Centre/ Yonne’ were

_positioned

in the same _way

relative

_{to ’Rh6ne-Alpes’ (Loire being}

more

favorable,

’Centre/Yonne’ less

_favorable).

However,

’Loire’ was significant only for the

logistic

model,

while ’Centre/Yonne’ was

significant

only

for the Cox model.

DISCUSSION

Considering

risk

factors,

the results of both

models

were

_equivalent.

In our

_example,

the loss of information due to the division into classes of the

_calving

interval did not

seem to influence the results.

The

Cox

model has 3 main

_advantages.

First,

it allows us to take into account all the

cows even if

_they

were not observed for the

same

_period

of time and even if the

stud-ied event

_(next

_calving)

did not take

_place

before

the end _{of the survey. This}

advan-tage

was

_arbitrarily

blurred to a certain

extent in our

_example

insofar as the cows

that did not have a second

_calving

could be

integrated

in the

_logistic

model

_{(as they}

were

declared

_infertile).

Second,

it does not

_require

the division of the studied

variable

into discrete classes. This is an

_advantage

as it allows us to

_keep

eliminates the

_always

difficult

_problem,

of which

_group

end

_points

to choose.

Finally,

it allows us to

formulate

the

results

_according

to

the increase

in the stud-ied time interval in d when the factor is

pre-sent

_(Lee

et al,

1989).

For

_example,

acute

metritis increases the

_calving

interval

_by

23 d. For

_{popularization,}

this

_way

of

pre-senting

the results is more

_{comprehensible}

than the

_presentation

in terms of the odds ratio or even

of

relative risk.

Despite

all these

_advantages

the use of

the

Cox

model is

very

limited

_by

its

very

restrictive

_application

conditions

_(the

haz-ards should be

_{proportional).}

_Indeed,

we

noticed

in our

_example

that if the studied variable consisted of a

_large

number of

classes,

the hazards were

_rarely

propor-tional all

_along

the curve. This leads to _group

classes. In extreme cases, this condition

can

_prevent

us from

_integrating

certain vari-ables into the model.

However,

problems

only

arise if the devi-ation from the

_proportional

hazards

assump-tion is

_large

and the

_frequency

of the studied

event is

_{high (the}

second

_calving,

in our

case).

In this case estimated

_parameters

are

_greatly

influenced

_(Ingram

and

Klein-man,

1989)

and therefore cannot be

inter-preted.

As for the power of the 2

models,

the Cox model and the

_logistic

model seem to

be

_{statistically equivalent}

for

_identifying

the risk

factors

when the

_percentage

of the stud-ied events is

low,

whereas the Cox model is

more

_powerful

when the

_percentage

is

_high

(Annesi

et al,

1989).

The conclusion is that when the

_follow-up

time is

_sufficiently

short

or the survival rate is

_high,

the choice of model must

_depend

on convenience

_(for

the

_{user), availability}

of

_computer

software,

and

_expertise.

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J, Giraud JM, Petit M

₍₁₉₈₆₎

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T,

Lellouch J

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logistic regression

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proportional

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_longitudinal

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Cox

_DR,

Oakes D

(1984) Analysis

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Chapman

and Hall, London, 201 p Ducrot C

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