Choosing Regression Models

Regression models 1
Choosing regression models
An elementary introduction
Stephen Senn

Explanation
• I am not presenting these things because I
think you don’t know them
• I am presenting them because the people
you work with don’t know them
• And you need to explain these things to
them
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Outline
• Basic considerations in modelling
• Choosing predictors
• Transformation of the predictor(s)
• Transformation of the outcome
• Advice
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Basic considerations
Thinking before you model
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Some Modelling Tasks
• Choose a generally suitable probability model
• Choose a set of suitable predictors
• Consider whether these need to be transformed
• Consider whether the outcome needs to be transformed
• Choose a technique for fitting the model
• Fit the model
• Assess goodness of fit of model
• Make causal inferences
• Issue predictions

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Factors Affecting Choice of
Model
• Purpose of model
– Causal, predictive, classification
• Design of study
– Designed experiment, observational study, survey,
• Temporal sequence
• Prior knowledge
• Type of data
– Continuous measurements, binary, ordinal, counts, censored life-
times
• Case ascertainment
• Results of model fitting

Preliminaries
• Choosing good regression models is not a question
of throwing some data at a stepwise selection
algorithm
• Two things are important
– Being clear about the purpose
– Insight (which in turn is based on)
• Experience
• Understanding
• Logic
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Two Extremes
Causal analysis
• The putative causal factor(s) must
be in the model
• Other factors are in the model
because they help us understand
the causal factor(s)
• They are of no interest in
themselves
• We pay particular attention to the
significance of the putative causal
factor(s)
Predictive modelling
• We are trying to find predictors of
some outcome
• It is their joint value as predictors
that is important
• We simply want the most
predictive model
• We compare entire models to
judge which is best
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Example
• Modelling the effect of treatment in a clinical trial
• Treatment must be in any model whether or not it
is significant
• Other factors will be in the model to help me
improve my estimate of the effect of treatment
– They are of little interest in themselves
– They are nearly always predetermined
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Does Smoking Cause Lung Cancer?
A Tale of Two Statisticians
Works in public health
• I wish to establish whether it
is causal
• If so I can warn smokers to
quit and this will benefit
their health
• It is important for me to rule
out possible confounding
factors
Works in life insurance
• I don’t care if it is causal or
not
• The data show that smokers
are much more likely to get
lung cancer
• That’s enough for me to
take account of it in setting
the premiums
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Warning
• Regression models are there to help you use
your insight, experience and prior
knowledge to understand your datasets
• They are not a substitute for scientific
understanding

Choosing predictors
It’s not just a matter of significance

An Example
• Multicentre trial of asthma comparing formoterol, salbutamol and
placebo for their effects on forced expiratory volume in one second
(FEV1).
• Randomisation stratified by steroid use (yes/no) and centre
• Sex, age, height of patient and baseline FEV1 also measured
• Definitely in the model
– Blocking factors: centre & steroid use
– Treatment factor (3 levels: formoterol, salbutamol, placebo)
• Possibly in the model
– Covariates: sex, age, height of patient and baseline FEV1
– NB sex, age, height are very predictive of baseline FEV1 also therefore if
you put them all in the model none may be significant
– This does not matter

Temporal Sequence I
• If we are interested in causal inferences it is
usually inappropriate to include variables
that were measured later in a model than
putative causal variables that were
measured earlier.
• The later variables cannot have caused the
earlier variables and so should not be
included.

Example
• It is desired to study whether the type of school attended
(private or state school) affects students’ chances of
success in final degree examinations at university
• Data are obtained for a large group of students
• In addition to information on degree results and type of
school attended, information is obtained on
– sex of student,
– high school results
– parents’ income
• Which of these factors is it inappropriate to include in the
model and why?

Temporal Sequence II
• The same does not apply if the purpose of
the model is simply classification
• It may then be helpful to have factors in the
model even if they are measured after the
“outcome variable of interest”
• Indeed they can be included even if they
have been “caused” by the variable of
interest

Example
• We wish to develop a model for classifying
patients who present with abdominal pain as
either suffering from appendicitis or non-
specific abdominal pain
• We use location of pain, degree of pain,
absence/presence of nausea, body
temperature as “predictor” variable
– Even though these are consequences of rather
than causes of appendicitis

Prior Knowledge
• Frequently when fitting models we already have strong opinions about
the effect of some factors even if we are ignorant about others.
– For example we may be examining the effect of a previously
unstudied environmental exposure on health
– we know, however, that age is an important determinant of health
• We will tend to put factors we believe are important in the model
irrespective of their significance according to the current data set.
• Similarly, implicitly, there will always be a host of factors we believe
are irrelevant.
• We will not put these in the model on prior grounds

19
Type of Data
and Choice of Basic Model
Type of Data
• Continuous
measurement
• Count data
• Binary data
• Ordered categorical
• Censored lifetimes
• Multinomial
Possible Basic Model
• General linear model
(Normal outcomes)
• Poisson regression
• Logistic regression
• Proportional odds
• Proportional hazards
• Log-linear
Regression models

Case Ascertainment
• The way in which data are obtained (ascertained) can
affect the way that we build a model
• For example in a case-control study we sample by outcome
(cases and controls) and then measure how these two differ
by exposure
– Example
• Case: lung cancer, Control: other cancer
• Exposure: smoker versus non-smoker
• We cannot model relative risk using such data
• We can only model (log) odds ratios
• For a cohort study where we sample by exposure we could
model either

Social Status: Longer life expectancy for Oscar winners
A study of actors and actresses found that Oscar winners lived, on average,
almost four years longer than nominees who went home empty-handed, reports
the March issue of the Harvard Health Letter. Actors aren’t the only people who
reap benefits. Dr. Donald Redelmeier of Toronto’s Sunnybrook and Women’s
College Health Sciences Centre found that Oscar-winning directors live longer
than non-winners, and male directors live 4.5 years longer on average than
actors. These findings add to a large body of evidence delineating connections
between social status and health and longevity, reports the Harvard Health
Letter. Redelmeier theorizes that an Oscar on the mantel moves the winner up
the Hollywood pecking order. Winners find it easier to get work, and when they
do, they’re better appreciated and better paid.

Not Harvard Health Publications
A study has shown that getting a telegram from The Queen can add 20 years to
your life
An extensive study of individuals who have received telegrams from The
Queen has shown that an astonishing proportion of them have lived to be 100.
Age at death of a control group of non-recipients was typically 20 years less.
Researchers have postulated that esteem is an important determinant of health
Joked lead researcher, Prof Morton Gullible, ‘our advice to her Majesty is send
yourself a telegram, Ma'am’

Results of Model Fitting
• Statisticians have developed a number of
techniques for assessing the adequacy of various
models using the data in hand
– Standard errors, significance tests on coefficients
– Analysis of variance/ deviance on factors
– Goodness of fit generally
– Residual plots
– AIC, BIC
• These are important tools but are by no means the
only tools for assessing the adequacy of a model

Transforming predictors
The X Files

Luxembourg Temperature Example
Data on temperatures in Luxembourg
Month Normal temperatures deg C
January 0.6
February 1.4
March 4.7
April 7.7
May 12.4
June 15.1
July 17.5
August 17.3
September 13.5
October 8.9
November 4.0
December 1.8

Modelling the temperature
Note that in the yearly rhythm, January follows December even
though January is point 1 and December point 12.
The data are periodic and we need a model that reflects this.
The simplest periodic pattern is a sine wave.
𝑌 = 𝛼 + βsin 𝑋 + 𝜃
 = level (the average temperature)
 = amplitude (the difference max to average)
 = phase (governs point at which maximum is reached)

Fitting a sine wave
A sine wave model can be fitted by using the fact that
sin 𝑋 + 𝜃 = cos 𝜃 sin 𝑋 + sin 𝜃 cos 𝑋
This is linear in sin 𝑋, cos 𝑋. Hence by regressing Y on two
variables sin 𝑋, cos 𝑋 we can obtain a periodic fit.
Note that X must be transformed from linear to angular
measure. So we can write
𝑋 = 360 × 𝑚𝑜𝑛𝑡ℎ 12
if we measure in degrees or
𝑋 = 2Π × 𝑚𝑜𝑛𝑡ℎ 12
radians

3 parameters
fit 12 points
rather well

Transforming the outcome
Being wise about Ys

An Example of a One-way Layout
• Four experimental p38 kinase inhibitors
• Vehicle and marketed product as controls
• Thrombaxane B2 (TXB2) is used as a
marker of COX-1 activity
• Six rats per group were treated for a total of
36 rats
• At the end of the study rats are sacrificed
and TXB2 is measured.

GenStat® ANOVA
(Original data)
Analysis of variance
Variate: TXB2 𝜎2
2
Source of variation d.f. s.s. m.s. v.r. F pr.
Treatment 5 184596. 36919. 6.31 <.001
Residual 30 175439. 5848.
Total 35 360035.
𝜎1
2
𝜎2
2
𝜎1
2
A2WAY [TREATMENTS=Treatment] TXB2

GenStat plot of
residuals

GenStat ANOVA
(log transformed)
A2WAY [TREATMENTS=Treatment] logTXB2
Analysis of variance
Variate: logTXB2
Source of variation d.f. s.s. m.s. v.r.
Treatment 5 62.6760 12.5352 40.09
Residual 30 9.3800 0.3127
Total 35 72.0559
Signal to noise ratio is
now much higher

GenStat plot
of residuals

Homogeneity of Variances
(Bartlett’ Test: GenStat)
Untransformed
*** Bartlett's Test for homogeneity of variances ***
Chi-square 50.87 on 5 degrees of freedom: probability <
0.001
Log-transformed
*** Bartlett's Test for homogeneity of variances ***
Chi-square 8.95 on 5 degrees of freedom: probability 0.111

Data-filtering examples
or find the flaw
• A 20 year follow-up study of women in an English village
found higher survival amongst smokers than non-smokers
• Transplant receivers on highest doses of cyclosporine had
higher probability of graft rejection than on lower doses
• Left-handers observed to die younger on average than
right-handers
• Obese infarct survivors have better prognosis than non-
obese

Advice
Statistics is a way of improving your thinking, not a substitute for it

Advice
• Think before you model
• Purpose is key
– Causal
– Predictive
– Classification
• Think about time
• Think about case ascertainment
• Testing is a small part of discerning
• Don’t use stepwise regression as a substitute for
understanding

Choosing Regression Models

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