1. Önder Ergönül, MD, MPH Koç University, School of Medicine
Correlation
Önder Ergönül, MD, MPH
Koç University, School of Medicine
Summer Course on Research Methodology and Ethics in Medical Sciences
June 10-21, 2017, Istanbul

14. Example

20. Multivariate Analysis
Önder Ergönül, MD, MPH
Koç University, School of Medicine
Summer Course on Research Methodology and Ethics in Medical Sciences
June 17-28, 2019, Istanbul

21. Background

22. %
1989
Descriptive only 27 12 13
Statistics tables (contingency tables) 36 53
Epidemiologic measures (relatif risk, odds) 10 22 35
Survival analysis 11 32 61
Multivariate analysis (regression) 5 14 51
Power analysis 3 39
Horton NJ, Switzer SS. NEJM 2005; 353.

23. Multivariate analyses
Regression
Dependent variable (outcome)
Linear
Continuous
Logistic
Dichotomous
Cox
Poisson

24. Confounder
Variable A
Outcome
Variable B

25. Confounder fatality Severity index APACHE score
Acinetobacter infection
fatality
Severity index
APACHE score

26. Control of the confounders
Randomization
Stratification
Adjustment by multivariate analysis

27. Odds Ratio p = probability (or proportion)
The lower bound is 0, and the upper bound is 1.
Probability of success: Pr(y = 1) = p
Probability of failure: Pr(y = 0) = 1 – p

28. What is the p of success or failure?
Total
1 - p p
(1 - p) + p = 1
.25 = 1 - p
.75 = p
1 = (1 - p) + p
Odds = p/(1-p) = .75/ ( ) = .75/.25 = 3
Odds Ratio= pA/(1-pA)
pB/(1-pB)

29. Relative Risk = DVT Heparin 8 92 Plasebo 18 82 Riskheparin= 8/100 =
0.08
Riskplasebo=
18/100 =
0.18
Risk plasebo
0.18 =
Relative risk =
= 2.25
Risk heparin
0.08

30. Odds Ratio = DVT Heparin 8 92 Plasebo 18 82 Oddsheparin= 8/92 = 0.087
Oddsplasebo=
18/82 =
0.22
Odds plasebo
0.22 =
Odds ratio =
= 2.53
Odds heparin
0.087

31. Comparing risks and odds
0.05 or 5%
0.053
0.1 or 10%
0.11
0.2 or 20%
0.25
0.3 or 30%
0.43
0.4 or 40%
0.67
0.5 or 50% 1
0.6 or 60%
1.5
0.7 or 70%
2.3
0.8 or 80% 4
0.9 or 90% 9
0.95 or 95% 19

32. Confounder OC MI
Smoking

33. Oral contraceptives (OC) and myocardial infarction (MI)
Case-control study, unstratified data
OC MI Controls OR
Yes
No Ref.
Total

34. Oral contraceptives (OC) and myocardial infarction (MI)
Case-control study, unstratified data
Smoking MI Controls OR
Yes
No Ref.
Total

35. Odds ratio for OC adjusted for smoking = 4 .5

36. Regression

37. From Correlation to Regression
Lineer Reg
Logistic Reg
TheSimplest case:
Y depends linearly on X
E(Y) = ß0+ ß1x
ß0, ß1: PARAMETERS

38. Regression
X: Predictor or independent variable Y: outcome or dependent variable Y
Y= a+bx
All 3 of them possible best summarze closest to the observed data how to defıne = square vertical distances between the points and the line then select a line that minimizes the sum of squared distances least squares estımates X

39. Scatterplot and Linear Regression

40. Lineer Regression E(Y) = ß0 + ß1 x Weight = -44,16 + 0,55 * height
Parameters:
ß0: INTERCEPT
ß1: SLOPE, regression coefficient

41. SBP (mm Hg)
Age (years)

42. Simple linear regression
Relation between 2 continuous variables (SBP and age)
Regression coefficient b1
Measures association between y and x
Amount by which y changes on average when x changes by one unit
Least squares method y
Slope x

43. Age and signs of coronary heart disease (CD)
Logistic regression
Age and signs of coronary heart disease (CD)

44. Logistic function
Probability of disease x

45. Why is it called “Logistic” regression?
It uses the logit transformation.
The logistics transformation can be interpreted as the logarithm of the odds of success vs. failure.

46. { Transformation α = log odds of disease in unexposed
β = log odds ratio associated with being exposed
eβ = odds ratio
logit of P(y|x) {

47. Fitting equation to the data
Linear regression: Least squares
Logistic regression: Maximum likelihood
Likelihood function
Estimates parameters a and b
Practically easier to work with log-likelihood

48. Maximum likelihood Iterative computing Results
Choice of an arbitrary value for the coefficients (usually 0)
Computing of log-likelihood
Variation of coefficients’ values
Reiteration until maximisation (plateau)
Results
Maximum Likelihood Estimates (MLE) for  and 
Estimates of P(y) for a given value of x

49. Why Do We do Logistic Regression so much?
Predict the likelihood of discrete outcomes
Group membership
Binary outcome (disease/no disease)
Quite Flexible Statistical Assumptions
No need for assumptions about the distributions of the predictor variables.
Predictors do not have to be normally distributed
Does not have to be linearly related.
Does not have to have equal variance within each group.
Very good in giving Odds Ratio

50. Construction of Model

51. Construction of Model
Perform univariate statistics (outliers, distribution, gaps)
Perform univariate analysis
Transform nominal independent variables to dichotomous (dummied) variable

52. Occupation
Nominal
Dummy
Farmer
Nurse 1
Housewife 2
Physician 3 4
Policeman 5

53. Construction of Model Multicollinearity
4. Run a correlation matrix. If any pair of independent variables are correlated at > 0.90 (multicollinearity), decide which one to keep and which one to exclude.
More practical way is to consider “biologic relation”
(smoking, carrying matches, and cancer)

54. Diagnosing confounder
Outcome
Variable A
Variable B
Confounder
Fatality
Acinetobacter
infection
APACHE
score

55. How to choose variables?
Kitchen sink
Inclusion of significant variables
Forward selection
Backward selection

56. An example
Outcome : Deep Vein Thrombosis Independent variables : Heparin, gender, Coronary Heart Disease, aspirin use
Y= a+b1x1+ b2x2 + b3x3 + b4x4
DVT=a+b1(heparin)+ b2(female) + b3(CAD)+ b4(aspirin)
0.5
1.5 3
0.6

57. OR p-değeri 95% CI Heparin use 0.5 0.003 0.15-0.72 Female 1.48 0.504
Logistic regression Number of obs = 10
LR chi2(4) = 0.97
Prob > chi2 =
Log likelihood = Pseudo R =
DVT | Odds Ratio Std. Err z P>|z| [95% Conf Interval]
Heparin |
Kadin |
KAH |
aspirin | OR
p-değeri
95% CI
Heparin use
0.5
0.003
Female
1.48
0.504
CHD
3.06
0.009
Aspirin use
0.58
0.622
Bır sonraki slyta baska regresyonlarda OR ın degısecegını koy

58. Assessment of the Model

59. Methods for measuring how well a model accounts for the outcome
Multiple lineer regression
Multiple logistic regression
Proportional hazard analysis
Model accounts for outcome better than chance
F test
Likelihood ratio test
(LR)
Quantitative/qualitative assessment of how well model accounts for outcome R2
R2 (rarely used)
Comparison of estimated to observed value
Hosmer-Lemeshov
Comparsion of estimated to observed value
Prediction of outcome NA
Sensitivity, Specificity, Accuracy,
c index

60. LR and p value of the test
For both logistic regression and proportional hazard analysis:
If chi square of the LR is large,
the p value will be small, and the null hypothesis can be rejected.

61. R2
R2 is a quantitative measure of how well the independent variables account for the outcome When R2 is multiplied by 100, it can be thought of as the percentage of the variance in the dependent variable explained by the independent variable

62. Goodness of Fit r² is between 0 and 1 r² = 1 perfect fit
(x does not add any information to y)
r² > 0.8 pretty good fit
r² ≈ 0.3 what we see usually !
r² ≈ 0 poor fit

64. Interactions How can interactions be taken into consideration?
Let X and Y be two explanatory variables, then
include X·Y in the model.
Is X a categorical variable with dummy variables
(X1,...Xm-1), then X·Y = (X1 ·Y,…,Xm-1 ·Y).
By interactions the multiplicative character of the OR
is changed.

65. TIME-TO EVENT TIME-TO DEATH
SURVIVAL ANALYSIS
TIME-TO EVENT
TIME-TO DEATH

66. Exposure and outcome
exposure NO
exposure

67. Person-Time Jan Feb March April May June Total Time at risk A 3 months
Total person time
3+6+2=11

68. The Reason for Survival Analysis
Censored cases
During the follow up:
expected outcome did not happen for the case
lossed from follow up or dropped
All the patients do not need to start together.

69. Survival Analysis
life table
Kaplan-Meier curve
Cox regresyon

70. Kaplan-Meier Curve
Duration = Time to event (time until an event occurs)
“Event” = Fatality, survived, relaps...
months
Number of the patients 50 49 44 42 40 39 2 4 6 8
Log-rank test compares 2 curves statistically

71. Kaplan-Meier Curve The number of patients treatment placebo ay 50 4944 42 40 39
treatment
placebo 2 4 6 8

72. Cox Regression Hazard ratio Logistic regression Number of obs = 10
LR chi2(4) = 0.97
Prob > chi2 =
Log likelihood = Pseudo R =
DVT | Odds Ratio Std. Err z P>|z| [95% Conf Interval]
Heparin |
Kadin |
KAH |
aspirin |
Cox regression
Hazard ratio