lecture 4lecturecontent.s3.amazonaws.com/pdf/15024.pdfLecture 4 2 Learning Objectives ! In this set of lectures we will: - Formally define confounding and give explicit examples of

Confounding

Lecture 4

2

Learning Objectives

n  In this set of lectures we will: -  Formally define confounding and give explicit examples of it’s

impact -  Define adjustment and adjusted estimates conceptually -  Begin a discussion of the analytics of adjustment

Section A

Confounding: A Formal Definition, and Some Examples

4

Learning Objectives

n  Formally define confounding

n  Establish conditions which can results in the confounding of an outcome/exposure relationship

n  Demonstrate the potential effects of confounding via examples

5

Confounding (Lurking Variable)

n  Consider results from the following (fictitious) study: -  This study was done to investigate the association between

smoking and a certain disease in male and female adults -  210 smokers and 240 non-smokers were recruited for the study

Results for All Subjects

Smokers Non Smokers TOTALS

Disease 52 64 116

No Disease 158 176 334

TOTALS 210 240 450

0.9364/24021052

pp

RRsmokersnon

smokers ≈==−ˆˆˆ

0.91)p(1p

)p-(1pRO

smokersnonsmokersnon

smokerssmokers ≈×

×=

−=

−− 6415817652

ˆˆˆˆ

n  Smoking is protective against disease?

n  Most of the smokers are male and non-smokers are female

6

What’s Going On?

All Subjects


Male 160 40 200

Female 50 200 250

TOTALS 210 240 450

n  Smoking is protective against disease?

n  Further, most of the persons with disease are female

7

What’s Going On?

All Subjects

Disease No Disease TOTALS

Male 33 167 200

Female 83 167 250

TOTALS 116 324 450

n  A picture?

8

What’s Going On?

Disease

Smoking Sex

n  The comparison of disease risk between smokers and non-smokers is potentially distorted or negated by the disproportionate percentage of males among the smokers

9

What’s Going On?

n  The original outcome of interest is DISEASE

n  The original exposure of interest is SMOKING

n  In this sample, SEX is related to both the outcome and exposure -  This relationship is possible impacting overall relationship

between DISEASE and SMOKING

n  How can we look at relationship between DISEASE and SMOKING removing any possible “interference” from SEX? -  On approach – look at DISEASE and SMOKING relationship

separately for males and females 10

What’s Going On?

n  Is smoking related to disease in males?

11

Example

Results for MALES


Disease 29 4 33


TOTALS 160 40 200

1.84/40

16029p

pRR

smokersnon male

smokersmalemales ≈==

−ˆˆˆ

213143629

)p(1p)p-(1p

ROsmokersnon malesmokersnon male

smokersmale smokersmalemales ≈

×

×=

−=

−− ˆˆˆˆ

n  Is smoking related to disease in females?

12

Example

Results for FEMALES


Disease 23 60 83


TOTALS 50 200 250

1.560/200p

pRR

smokersnon female

smokersfemalefemales ≈==

−

5023ˆˆˆ

26027

14023)p(1p

)p-(1pRO

smokersnon femalesmokersnon female

smokersfemale smokersfemalefemales ≈

×

×=

−=

−− ˆˆˆˆ

13

Smoking, Disease and Sex

n  A recap -  The overall (sometimes called crude, unadjusted) relationship

(RR) between smoking and disease was nearly 1 (risk difference nearly 0)

-  The sex specific results showed similar positive associations

between smoking and disease

MALES: FEMALES:

(note, for the moment we are not considering statistical significance, just using estimates to illustrate point)

02.0ˆˆˆ −== smokers-nonsmokers p-p0.93;RR

08.0ˆˆ;8.1ˆ ≈= smokers-non male smokersmale p-pRR

16.0ˆˆ;5.1ˆ ≈= smokers-non female smokersfemale p-pRR

14

Simpson’s Paradox

n  The nature of an association can change (and even reverse direction) or disappear when data from several groups are combined to form a single group

n  An association between an exposure X and an outcome Y can be confounded by another lurking (hidden) variable Z (or variables Z1, Z2, …)

15

Confounding (Lurking Variable)

n  A confounder Z (or set of confounders Z1…Zp) distorts the true relation between X and Y

n  This can happen if Z is related both to X and to Y

X Y

Z

n  A picture

16

Y

X Z

What’s Going On?

17

What is the Solution for Confounding?

n  If you DON’T KNOW what the potential confounders are, there’s not much you can do after the study is over -  Randomization is the best protection -  Randomization eliminates the potential links between the

exposure of interest and potential confounders Z1, Z2,..Z3

n  If you can’t randomize but KNOW what the potential confounders are there are statistical methods to help control (adjust for confounders) -  Potential confounders must be measured as part of study

18

Randomization Minimizes Threat of Confounding

n  How/Why does randomization minimize the threat of confounding?

19

Example 2: Arm Circumference and Height

n  An observational study to estimate association between arm circumference and height in Nepali children -  150 randomly selected subjects, ages [0, 12) months, had arm

circumference, weight and height measured -  This study is observational—it is not possible to randomize

subjects to height groups!

20


n  The data -  Arm circumference range: 7.3–15.6 cm -  Height range: 40.9–73.3 cm -  Weight range: 1.6 – 9.9 kg

n  Scatterplot: arm circumference by height

21


810

1214

16

Arm

Circ

umfe

renc

e (c

m)

40 50 60 70 80Height (cm)

With Regression Line

Nepalese Children < 12 Months (n= 150)Arm Circumference versus Height

45.016.07.2ˆ

21

=

+=

Rxy

n  Notice, perhaps not surprisingly:

22


4050

6070

80

Hei

ght

(cm

)2 4 6 8 10

Weight (kg)

Nepalese Children < 12 Months (n= 150)Height versus Weight

810

1214

16

Arm

Circ

umfe

renc

e (c

m)

2 4 6 8 10Weight (kg)

Nepalese Children < 12 Months (n= 150)Arm Circumference versus Weight

70.02 =R 86.02 =R

n  Scatterplot: arm circumference by height, after adjusting for weight

23


Arm

Circ

umfe

renc

e (c

m)

Height (cm)


16.0)(1̂ −=heightβ

24

Example 3: South African Study

n  A longitudinal study from South Africa: birth cohort, followed up five years after birth1 : Participation by medical aid status at birth, all baseline participants

1 Morell C. Simpson's Paradox: An Example From a Longitudinal Study in South Africa. Journal of Statistical Education (1999)

95% CI: 0.53 to 0.92

All Subjects

Medical Aid No Medical Aid TOTAL

Follow-Up Participation

46 370 416

No Follow-Up Participation

195 979 1,164

TOTAL 241 1,349 1,590

0.70.270.19

370/1,34924146

pp

RRaid medical no

aid medicalup-follow ≈===

ˆˆˆ

25


n  A longitudinal study from South Africa: birth cohort, followed up five years after birth : Participation by medical aid status at birth, Black participants

95% CI: 0.76 to 1.36

Black Subjects



36 368 404


91 957 1,048

TOTAL 127 1,325 1,452

1.0.280.28

368/1,32512736

p̂p̂RR̂

Blackaid medical no

Blackaid medical Blackup-follow ≈===

26


n  A longitudinal study from South Africa: birth cohort, followed up five years after birth : Participation by medical aid status at birth, White participants

95% CI: 0.25 to 4.5

White



10 2 12


104 22 126

TOTAL 114 24 138

1.05.083

0.0882/24

11410pp

RRWhite aid medical no

White aid medicalWhite up-follow ≈===

ˆˆˆ

27


n  Whats going on?

n  Race -  Majority of sample Black subjects (91%)

n  Race and follow-up participation -  26% of Black subjects completed follow-up as compared to 9% of

White subjects

n  Race and medical aid -  9% of Black subjects had medical aid compared to 83% of White

subjects

28


n  Recap

29

Example 4: “Batch Effects” In Lab Based Analyses

n  Lab based results can be influenced by the technician, the laboratory used, the time of day, the temperature in the lab etc..

n  If the goal of a study is to ascertain differences in lab measures between groups (for example diseased and non-diseased), and the group is associated with at least some of the above characteristics, then there can be confounding

30

Summary

n  In non-randomized studies, outcome/exposures relationships of interest may be confounded by other variables

n  In order to confound an outcome/exposure relationship, a variable must be related to both the outcome and exposure

Section B

Adjusted Estimates: Presentation, Interpretation and Utility for Assessing Confounding

32

Learning Objectives

n  Understand how to interpret estimates of association that have been adjusted to control for a confounder

n  Compare/contrast the comparisons being made by unadjusted and adjusted association estimates

33

Adjustment

n  Adjustment is a method for making comparable comparisons between groups in the presence of a confounder/confounding variables

n  We will discuss the basics of the mechanics behind adjustment in the next lecture section

34

Example 1: Fictitious Study





Disease 52 64 116


TOTALS 210 240 450

0.9364/24021052

pp

RRsmokersnon


35


n  This relative risk is being influenced by the difference sex distributions among smokers and non-smokers

n  This relative risk compares all smokers to all non-smokers in the sample without taking any other factors into account: this is called the unadjusted or crude estimated association between disease and smoking

36


n  Adjustment provides a mechanism for estimating an outcome/exposure relationship after removing the potential distortion or negation that comes from a confounder or multiple confounders

n  In the fictional example, for example, the relationship between disease and smoking can be adjusted for sex

n  Frequently, the presentation of results from non-randomized studies will include a table of unadjusted and adjusted measures of association

n  Example: table of relative risks

37


Table 2: Unadjusted and Adjusted Relative Risks of Disease

Unadjusted Adjusted1

Non-‐Smoker ref refSmoker 0.93 (0.68, 1.27) 1.57 (1.12, 2.20)

1 adjusted for sex

n  Unadjusted estimated relative risk, 0.93

n  Adjusted estimated relative risk, 1.57

38


n  Comparing unadjusted and adjusted associations to assess confounding

39


Table 2: Unadjusted and Adjusted Relative Risks of Disease

Unadjusted Adjusted1

Non-‐Smoker ref refSmoker 0.93 (0.68, 1.27) 1.57 (1.12, 2.20)

1 adjusted for sex

40


n  An observational study to estimate association between arm circumference and height in Nepali children -  150 randomly selected subjects, ages [0, 12) months, had arm

circumference, weight and height measured -  This study is observational—it is not possible to randomize

subjects to height groups!

41


n  The data -  Arm circumference range: 7.3–15.6 cm -  Height range: 40.9–73.3 cm -  Weight range: 1.6 – 9.9 kg

n  Frequently, the presentation of results from non-randomized studies will include a table of unadjusted and adjusted measures of association

n  Example: table of linear regression slopes

42


Table 2: Regression Slopes for Arm CircumferenceUnadjusted Adjusted

Height (cm) 0.16 (0.13, 0.19) -‐0.16 (-‐0.21, -‐0.11)Weight (kg) 0.80 (0.72, 0.89) 1.40 (1.21, 1.60)

n  Unadjusted linear regression slope estimate for height,

n  Adjusted linear regression slope estimated for height,

43


16.0ˆ =heightβ

16.0ˆ −=heightβ

n  Comparing unadjusted and adjusted associations to assess confounding

44


Table 2: Regression Slopes for Arm CircumferenceUnadjusted Adjusted

Height (cm) 0.16 (0.13, 0.19) -‐0.16 (-‐0.21, -‐0.11)Weight (kg) 0.80 (0.72, 0.89) 1.40 (1.21, 1.60)

Example 3: Academic Physician Salaries1

n  From abstract

1 Jagsi R, et al. Gender Differences in the Salaries of Physician Researchers. Journal of the American Medical Association (2012); 307(22); 2410-2417.

45

n  Unadjusted linear regression slope estimate for sex (1=M, 0 = F)

n  Adjusted linear regression slope estimated for sex (1=M, 0 = F)

( after adjustment for specialty, academic rank, leadership positions, publications, and research time)

46

Example 3: Academic Physician Salaries

764,32$ˆ =sexβ

399,13$ˆ =sexβ

n  Unadjusted linear regression slope estimate for sex (1=M, 0 = F)

n  Adjusted linear regression slope estimated for sex (1=M, 0 = F)

( after adjustment for specialty, academic rank, leadership positions, publications, and research time)

47

Example 3: Academic Physician Salaries

764,32$ˆ =sexβ

399,13$ˆ =sexβ

n  Adjustment is a method for making comparable comparisons between groups in the presence of a confounder/confounding variables

n  The group comparisons made by adjusted associations are more specific than those made by unadjusted (crude) associations

n  Contrasting crude and adjusted association estimates is useful for identifying confounding

48

Summary

Section C

Adjusted Estimates: The General Idea Behind the Computations

50

Learning Objectives

n  Gain some insight conceptually as to how adjusted estimates are computed

51






Disease 52 64 116


TOTALS 210 240 450

0.9364/24021052

pp

RRsmokersnon


52

Example 1 :Smoking, Disease and Sex

n  A recap -  The overall (sometimes called crude, unadjusted) relationship

(RR) between smoking and disease was nearly 1 (risk difference nearly 0)

-  The sex specific results showed similar positive associations

between smoking and disease

MALES: FEMALES:

(note, for the moment we are not considering statistical significance, just using estimates to illustrate point)

0.93;RR̂ =

;8.1RR̂ =;5.1RR̂ =

53

Example 1: How to Adjust for Confounding?

n  Stratify when Z is categorical -  Look at tables separately -  For our example, separate tables for males and females -  Take weighted average of stratum specific estimates Ex: To get a sex adjusted relative risk for the smoking disease

relationship we could weight the sex-specific relative risks by numbers of males and females

femalesmales

femalesfemalesmalesmalesadjusted sex nn

RRnRRnRR

+

×+×=

ˆˆˆ

1.6250200

1.52501.8200RR adjusted sex ≈+

×+×=ˆ

54

Example 1: How to Adjust for Confounding?

n  There are better ways than this to take such a weighted average (weighting by standard error, for example), but this just illustrates the concept

n  Confidence intervals can be computed for these adjusted measures of association

n  Multiple regression (in this case, logistic) will be a very useful tool for performing adjustment

n  Scatterplot: arm circumference by height

55


810

1214

16

Arm

Circ

umfe

renc

e (c

m)

40 50 60 70 80Height (cm)

With Regression Line


45.016.07.2ˆ

21

=

+=

Rxy

n  IDEA Scatterplots: arm circumference by height, stratified by weight values

56


n  The adjusted association between Y and X, adjusted for a single potential confounder Z can be estimated by: -  Stratifying on Z (hard to operationalize is Z is continous) -  Estimate the Y/X relationship for each strata of Z -  Take a weighted estimate of all Z strata specific Y/X

associations

n  Idea can be generalized to estimating the adjusted association between Y and X, adjusted for a multiple potential confounders Z1, Z2, ….Zc

57

Summary

n  Multiple regression methods will make the adjustment process easy and straightforward

58

Summary

Documents

lecture 4lecturecontent.s3.amazonaws.com/pdf/15024.pdfLecture 4 2 Learning Objectives ! In this set of lectures we will: - Formally define confounding and give explicit examples of