1

Introduction to Modeling

Beyond the Basics (Chapter 7)

2

Content

• Simple and multiple linear regression• Simple logistic regression

– The logistic function– Estimation of parameters– Interpretation of coefficients

• Multiple logistic regression– Interpretation of coefficients– Coding of variables

3

How can we analyse these data?

Age SBP Age SBP Age SBP 22 131 41 139 52 128 23 128 41 171 54 105 24 116 46 137 56 145 27 106 47 111 57 141 28 114 48 115 58 153 29 123 49 133 59 157 30 117 49 128 63 155 32 122 50 183 67 176 33 99 51 130 71 172 35 121 51 133 77 178 40 147 51 144 81 217

Table 1 Age and systolic blood pressure (SBP) among 33 adult women

4

80

100

120

140

160

180

200

220

20 30 40 50 60 70 80 90

SBP (mm Hg)

Age (years)

adapted from Colton T. Statistics in Medicine. Boston: Little Brown, 1974

5

Simple linear regression• Relation between 2 continuous variables (SBP and age)

• Regression coefficient 1– Measures association between y and x– Amount by which y changes on average when x changes by one unit– Least squares method

y

x

11xβα(y) Slope

6

Multiple linear regression

• Relation between a continuous variable and a set ofi continuous or categorical variables

• Partial regression coefficients i

– Amount by which y changes on average when xi changes by one unit and all the other xis remain constant

– Measures association between xi and y adjusted for all other xi

• Example– SBP versus age, weight, height, etc

xβ ... xβ xβαy ii2211

7

Multiple linear regression

Dependent Independent variablesPredicted Predictor variablesResponse variable Explanatory variablesOutcome variable Covariables

xβ ... xβ xβα y ii2211

8

Multivariate analysis

Model Outcome

Linear regression continousPoisson regression countsCox model survivalLogistic regression binomial......

• Choice of the tool according to study, objectives, and the variables

– Control of confounding– Model building, prediction

9

Logistic regression

• Models the relationship between a set of variables xi– dichotomous (eat : yes/no)– categorical (social class, ... )– continuous (age, ...)

and

– dichotomous variable Y

• Dichotomous (binary) outcome most common situation in biology and epidemiology

10

How can we analyse these data?

Table 2 Age and signes of Coronary Heart Disease (CHD) , 33 women

CHD CHD CHD

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How can we analyse these data?

• Comparison of the mean age of diseased and non-diseased women

– Non-diseased: 38.6 years– Diseased: 58.7 years (p<0.0001)

• Linear regression?

12

Dot-plot: Data from Table 2

13

NO

YES

Y = -0.527 + 0.20 x AGE

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Table 3 - Prevalence (%) of signs of CHD according to age group

15

0

20

40

60

80

100

0 1 2 3 4 5 6 7

Age group

CHD

(%)

20-29 30-39 40-49 50-59 60-69 70-79 80-89

Dot-plot: Data from Table 3

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Dot-plot: Data from Table 3

0

20

40

60

80

100

0 2 4 6 8

Diseased %

Age (years)

P1-P

17

Dot-plot: Data from Table 3

-100-80-60-40-20

020406080

100

0 2 4 6 8

Diseased %

Age (years)

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The logistic function (2)

logit of P(y|x)

{

19

The logistic function (1)

0.0

0.2

0.4

0.6

0.8

1.0Probability of disease

x

20

The logistic function (2)

logit of P(y|x)

{

21

The logistic function (3)

• Advantages of the logit– Simple transformation of P(y|x)– Linear relationship with x

– Can be continuous (Logit between - to + )– Known binomial distribution (P between 0 and 1)– Directly related to the notion of odds of disease

βxαP-1

P ln

eP-1

P βxα

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Interpretation of (1)

eP-1

P βxα

23

Practice 1. MI and Hyperhomocysteinemia?

Hyper Homocysteinemia

no yes Total

control 62 21 83

case 42 41 83

Total 104 62 166

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Practice 1

Normal Homocysteine

HighHomocysteine

MI (%) 40.38 66.13

Odds 0.68 1.95

Ln(Odds) -0.39 0.67

βxαP-1

P ln

25

βxαP-1

P ln

• Normal HC X = 0 ln(Odds)= + x 0 = ln(Odds) ……. = -0.39

• High HC X=1 ln(Odds)= + x 1 = ln(Odds)- ……. = 0.67 - (-0.39) = 1.06

• OR ? = e = 2.88• SE = 0.33 How can you interpret /OR?

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Interpretation of (2)

• = increase in log-odds for a one unit increase in x• Test of the hypothesis that =0 (Wald test)

• Interval testing

df) (1 Variance(

2β)

β2

OR

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If you run Linear Regression …

Y = .04 + 0.257 x High HC

% MI in High HC = 66.13% MI in Normal HC = 40.38

Diff = 25.7 %1

What is your interpretation about 1 ?

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Example

• Age (<55 and 55+ years) and risk of developing coronary heart disease (CD)

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• Results of fitting Logistic Regression Model

Age 2.094 0.841- Age βαP-1

P ln 1

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Interpretation of (1)

eP-1

P βxα

31

Multiple logistic regression

• More than one independent variable– Dichotomous, ordinal, nominal, continuous …

• Interpretation of i – Increase in log-odds for a one unit increase in xi with all the

other xis constant– Measures association between xi and log-odds adjusted for

all other xi

ii2211 xβ ... xβ xβαP-1

P ln

32

Multiple logistic regression

• Effect modification– Can be modelled by including interaction terms

1132211 xxβ xβ xβαP-1

P ln

33

Reference

• Hosmer DW, Lemeshow S. Applied logistic regression.Wiley & Sons, New York, 1989