Introduction to Multivariate analysis

Multivariate Analysis
Prof. Dr. Jamalludin Ab Rahman MD MPH
Department of Community Medicine
Kulliyyah of Medicine

Smoking & lung cancer
Good case-control study associating lung cancer to smoking (Wynder EL, Graham E.
Tobacco smoking as a possible etiologic factor in bronchiogenic carcinoma: a study of 684
proven cases. JAMA 1950;143:329-36.)
Tobacco dust (not smoke) might be causing the elevated incidence of lung tumours among German tobacco
workers. (Hermann Rottmann in Würzburg 1898)
Difference Causal
Smoking might be related to lung cancer, but lung cancer is still rare (Adler I. Primary Malignant
Growths of the Lungs and Bronchi. London: Longmans, 1912:22)
86 lung cancers patients were likely smoked (Müller FH. Tabakmissbrauch und Lungencarcinom.
Zeitschrift für Krebsforschung 1939;49:57–85.)
Smoking 35 sticks per day increase risk to 40 times (Doll R,
Hill AB. The mortality of doctors in relation to their smoking
habits. BMJ 1954;1:1451–5.)
Animal study associating cigarette smoke tar with cancer (Wynder E,
Graham EA, Croninger AB. Experimental production of carcinoma with
cigarette tar. Cancer Res 1953;13:855–66)
8November2016(C)JamalludinAbRahman2015
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It is about relationship
Analysis of the
relationships
between two or
more variables.
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Multivariate?
 Multivariate - general
term – multiple IV
 May involved multiple DV
DVIV IV
DV DV
IV
IV
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Outcome
Exposure
Exposure Exposure
Exposure
Exposure
Effect modifier or Moderator
Confounder
Mediator
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6

7

Exercise & fitness
Low Moderate High
Is there any difference % between
Low & Moderate intensity?
How big is the difference %
between Low & Moderate?
Is there any pattern now?
What is your conclusion?
Fitness level
Exercise intensity
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Physical activity & blood pressure
Time SBP
30 140
40 145
85 130
90 143
100 130
110 120
110 110
120 120
130 110
130 109
140 98
150 100
140 110
135 120
160 100
160 96
170 100
200 89
200 100
240 80
y = -0.3287x + 155.89
R² = 0.8508
0
20
40
60
80
100
120
140
160
0 50 100 150 200 250 300
SBP(mmHg)
Time (minutes/week)
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The 3rd variable
Outcome
Exposure
Exposure
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The 3rd factors can be a...
1. Confounder
2. Mediator or intervening factor
3. Moderator or effect modifier (interaction)
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The 3rd variable
Outcome
Exposure
Exposure
Confounder
Confounder
influence a
relationship
(between two
variables) but it is
not a part of the
pathway
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The 3rd variable
Outcome
Exposure
Exposure
Moderator
When an
exposure has
different effects
on disease at
different values
of a variable
(interaction)
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Stress vs. MS vs. Coping mechanism
Multiple
sclerosis
new
lesions
Coping
Mech.
Stress
Mohr, D. C., Goodkin, D. E., Nelson, S., Cox, D., & Weiner, M. (2002).
Moderating Effects of Coping on the Relationship Between Stress and the
Development of New Brain Lesions in Multiple Sclerosis. Psychosom Med,
64(5), 803-809.
OR = 1.62, p = 0.009
Distraction (OR=0.69, p=0.009),
instrumental (OR=0.77, p=0.081),
emotional preoccupation (OR=1.46, p=0.088)
& palliative (NS)
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The 3rd variable
Outcome
Exposure
Exposure
Mediator
Mediator
influence a
relationship
(between two
variables) and it
is also a part of
the pathway
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Why multivariate?
 Multi-factorial – which are the significant factors?
 Multiple outcomes
 Multiple unit of measurements
 Exploration of associations
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Regression
 Most common multivariate
technique
 Best line to fit the data - OLS
 Many types:
 Linear regression
 Logistic regression
 & many others!!
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x
y
𝑦 = 𝛽0 + 𝛽1 𝑥
𝛽0

Regression equation
 e.g. Linear regression
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Dependent Var
Intercept
Coefficient for Var x1
Explanatory Var x1
Error/Residual
𝑌 = 𝛽𝑜 + 𝛽1 𝑥1 + 𝛽2 𝑥2 + ⋯ 𝛽 𝑛 𝑥 𝑛 + 𝜀

Example #1
Arterial BP = Constant + Age + Body weight + Pulse rate +
Stress + Residual
𝑌 = 𝛽𝑜 + 𝛽1 𝑥1 + 𝛽2 𝑥2 + ⋯ 𝛽 𝑛 𝑥 𝑛 + 𝜀
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Logistic regression
 𝐿𝑛
𝑃
1−𝑃
= 𝛽𝑜 + 𝛽1 𝑥1 + ⋯ + 𝛽 𝑛 𝑥 𝑛
 P = Probability that Y=1 i.e. Event occurs
 1-P = Event not occur
 𝐿𝑛
𝑃
1−𝑃
= Ln of OR = Logit
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R2 = 0.986, meaning 99% of variation in ABP is
explained by Age, Body Weight, Pulse Rate &
Stress (F(4,15)=291.948), P<0.001)
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Main Result
Arterial BP = 17.3 + 0.6(Age) + 0.9(Body weight) +
0.09(Pulse rate) + 0.01(Stress)
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Example #2
 Snoring and risk of cardiovascular disease in women . Hu 2000.
From The Nurses’ Health Study. Cohort. Baseline, N=71,779
women 40 to 65 years old and without diagnosed CVD or cancer
in 1986. Till 31st May 1994.
 CVD = Snoring + Age +Smoking + BMI + Alcohol + Physical
Activity + Menopausal status + Family history of MI + DM +
Cholesterol + Hours sleeping + Sleeping position
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28

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Testing assumptions for linear
regression
1. Linearity between DV & IVs – Scatter plot residuals vs. predicted
2. Normality – Histogram of residuals
3. No outliers – Casewise diagnostics (within +/- 2SD), Cook’s D (for
influential points, <1), leverage point (< 4/n)
4. Homogeneity – Scatter plot
5. Independence (no autocorrelation) – Durbin Watson 1.5-2.5 (or
some says 0-4)
6. No multicollinearity – Tolerance > 0.1, VIF <10
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Residuals
x
y
Residual = Observed - Predicted
𝑦 = 𝛽0 + 𝛽1 𝑥
𝛽0
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Residual statistics - Linearity
Predicted
Residuals Residuals
Linear Non linear
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Residual statistics - Normality
Predicted
Residuals Residuals
Normal distribution Not Normal
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Residual statistics - Heterogeneity
Predicted
Residuals Residuals
Homogenous Heterogenous
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Influential data
x
y A B
C
A = Outlier, still within the
range of x, large residual value
B & C = Leverage points
B = Good leverage, it won’t
impact the regression line
C = Bad leverage. It will
change the regression line
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Type of multivariate tests
Dependent
Variables
Independent
Variables
Test
1 – Cont ≥ 2 – All Cont Linear Regression
1 – Cont ≥ 2 – All Cat ANOVA
1 – Cont ≥ 2 – Cont + Cat ANCOVA
> 1 – Cont All Cat MANOVA
> 1 – Cont Cat + Cont MANCOVA
1 – Dichotomous ≥ 2 – Cont + Cat Binary Logistic Regression
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Introduction to Multivariate analysis

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