lecture 9 - Amazon Web Serviceslecturecontent.s3.amazonaws.com/pdf/15167.pdf · Lecture 9 . 2 Lecture Set Overview ! Testing for effect modification and estimating different outcome

Effect Modification and Non-Linear Associations: Regression Based Approaches

Lecture 9

2

Lecture Set Overview

n  Testing for effect modification and estimating different outcome/ predictor associations for different levels of a potential effect modifier via the use of interaction terms in regression

n  Conceptualizing non-linearity as a type of effect-modification and showing another way to model it in a regression context (without categorizing the continuous predictor)

Section A

Regression With Interaction Terms

4

Learning Objectives

n  Describe the “interaction term” approach to estimating separate outcome/predictor associations for different levels of an effect modifier, and for testing for effect modification

5

Assessing EM By Presenting Stratified Results

n  Example 1: Suicide Outcomes and Sexual Identity1

1 Pinney T, Millman S. Asian/Pacific Islander Adolescent Sexual Orientation and Suicide Risk in Guam. American Journal of Public Health (2004) . Vol 4, No 7. pps 1204-1206.

6


n  Example 1: Suicide Outcomes and Sexual Identity

7


n  Example 1: Suicide Outcomes and Sexual Identity

Table: Relationship Between Attempted Suicide and Sexual Orientation,

Presented Separately by Sex.

Odds Ratio of Suicide Attempt1

Unadjusted Adjusted2

Males 5.01 (2.13, 11.78) 5.06 (1.65, 15.55)

Females 2.65 (1.17, 6.00) 2.17 (0.84, 5.60)

1 Compares the odds of having had at least one suicide attempt for

youth how identify as homosexual compared to youth who do not identify as homosexual

2 Adjusted for ethnicity, relationship abuse, alcohol abuse and markers of depression


n  Example 22: Coffee Drinking and Mortality (multivariate: adjusted for body-mass index; race or ethnic group; level of education; alcohol consumption; the number of cigarettes smoked per day, use or nonuse of pipes or cigars, and time of smoking cessation (<1 year, 1 to <5 years, 5 to <10 years, or ≥10 years before baseline); health status; diabetes (yes vs. no); marital status; physical activity; total energy intake; consumption of fruits, vegetables, red meat, white meat, and saturated fat; use or nonuse of vitamin supplements; and use or nonuse of postmenopausal hormone therapy)

2 Freedman N, et al. Association of Coffee Drinking with Total and Cause-Specific Mortality. New England Journal of Medicine (2012). 366 (20) 1891-1904.

8


n  Example 22: Coffee Drinking and Mortality

9

Another Approach

n  Sometimes, however, the researcher may want to estimate separate associations for one predictor only, and pool the estimates for the other predictors across all groups. For example:

-  Estimate the sex-specific associations between wages and years of education, after adjusting for other factors (across both males and females)

-  Estimate age specific associations between mortality and race in dialysis patients, after adjusting for other factors (using data from all age groups combined)

n  This can be done by including an “interaction term” in a multiple regression model

10

11

Assessing EM With an Interaction Term

n  Example 1: Hourly Wages and Years of Education

Table 1: Unadjusted and Adjusted Linear Regression Slopes:Years of Education(outcome is Hourly Wages, n= 534)Index Slope (95% CI) Estimate is Adjusted for: A 0.75 (0.59. 0.91) (unadjusted)B 0.75 (0.59. 0.91) SexC 0.76 (0.60. 0.92) Sex, Union MembershipD 0.47 (0.29, 0.65) Sex, Union Membership, Job TypeE 0.49 (0.31, 0.67) Sex, Union Membership, Job Type, Sector, Marital Status

n  Results from Model C, plotted separately for males and females

12


05

1015

Hou

rly W

ages

(US

$)

0 5 10 15 20Years of Education

Males Females

Adjusted for sex, and union membership

Plotted Separately By SexRegression Estimates of Hourly Wages for Years of Education

n  The slope of each sex-specific regression line is the same: this is the slope of years of education from the MLR with years of education and sex (0.76), and other adjustment variables

n  The difference between the estimated wages for males and females is the same at each value of years of education: this is the difference in hourly wages between males and females adjusting for years of education (and other adjustment variables): in this example, this difference is $1.89

n  Similar graphics could be shown for the other models

13


n  Once sex has been adjusted for, the wages/years of education relationship is the same in each level of sex

n  Once years of education has been adjusted for, the wages/sex relationship is same at each value of years of education

n  Suppose, however, we are interested in investigating whether the relationship between wages and years of education is modified by sex (after adjustment for union membership)

14


n  We could stratify the sample by sex, and estimate two different regressions of wages on years of education after adjusting for union membership

n  Results:

Slope of Years of Education (95% CI) Males 0.71 (0.50, 0.91) Females 0.84 (0.62, 1.06)

15


16


n  Approach 2: add an interaction term between years of education and sex to the model that includes the other adjustment variables

n  Here’s how it works: -  Add an interaction term to the model which already include

years of education ( x1 ) and sex ( x2 ) as predictors, as well as the other predictors

n  This interaction term, x3 , can be created by taking the product of x1 and x2 ; x3 =x1*x2

n  New model with interaction term

Where x1= years of education, x2 =sex (1 for females), and x3= interaction term (x1*x2)

n  What is value of x3 for:

-  Males? -  Females?

17

xs) and slopesother (ˆˆˆˆˆ 3322110 ++++= xxxy ββββ


n  Results

n  Slope of years of education Males (x2=0, x3=0) Females (x2=1, x3=x1*1)

18

xs) and slopesother (14.069.370.040.0ˆ 321 ++−++= xxxy


xs) and slopesother (70.040.0ˆ 1 ++= xy

xs) and slopesother ()1*(14.069.370.040.0ˆ 11 ++−++= xxyxs) and slopesother (14.070.069.340.0ˆ 11 +++−+= xxy

xs) and slopesother ()14.070.0(69.340.0ˆ 1 +++−+= xy

n  Results from model with interaction term plotted separately for males and females

19


05

1015

Hou

rly W

ages

(US

$)

0 5 10 15 20Years of Education

Males Females

Adjusted for sex and union membership, with an interaction term for sex and years of education

Plotted Separately By SexRegression Estimates of Hourly Wages for Years of Education

n  Results

n  Testing formally if slope of x3 is statistically significant is called “a test of interaction”

Ho: β3=0 HA:β3≠0

-  In this example, the p-value is 0.38: there is not a statistically significant interaction between years of education and sex after adjusting for union membership status

20

xs) and slopesother (14.069.370.040.0ˆ 321 ++−++= xxxy


n  Example: Mortality in patients with primary biliary cirrhosis (Mayo clinic data)

n  Randomized trial: patients randomized to receive DPCA or placebo

n  Results: (unadjusted) HR (DPCA to placebo) 1.06 (0.75, 1.50)

n  Question: Is effect of drug modified by age of patient?

21


n  Age of patient: categorized into quartiles

Quartile 1: < 42 years Quartile 2: [42,50) Quartile 3: [50, 57) Quartile 4: >= 57 years

n  To investigate whether age modifies the effect of the drug, we will need to fit a Cox model that includes drug, the age quartile indicators, and interaction terms between drug and each of the age quartile indicators

22


n  Model Results

n  For each of the age quartiles:

-  Age quartile 1: -  Age quartile 2: -  Age quartile 3: -  Age quartile 4:

23


)x*-0.25(x)x*0.10(x)x*0.28(x 1.22x0.58x0.02x

0.07x- [t]ˆ ) hazardln(

413121

432

1o

++

+++

++= λ

1o 0.07x- [t]ˆ ) hazardln( += λ

)1*(28.002.00.07x- [t]ˆ ) hazardln( 11o x+++= λ)1*(10.058.00.07x- [t]ˆ ) hazardln( 11o x+++= λ

)1*(25.022.10.07x- [t]ˆ ) hazardln( 11o x−+++= λ

n  For each of the age quartiles:

-  Age quartile 1:




24


1o 0.07x- [t]ˆ ) hazardln( += λ

1o 0.28)x(-0.07 02.0[t]ˆ ) hazardln( +++= λ

1o .10)x0.07- (58.0[t]ˆ ) hazardln( +++= λ

1o )25.0(-0.07 22.1[t]ˆ ) hazardln( x−+++= λ

n  HRs comparing drug to placebo for each of the age quartiles:

-  Age quartile 1: 0.94 (0.38, 2.33) -  Age quartile 2: 1.24 (0.59, 2.69) -  Age quartile 3: 1.03 (0.54, 1.97) -  Age quartile 4: 0.73 (0.40, 1.34)

25


n  Testing for an interaction between age and treatment

-  Resulting p-value: 0.74

26


)x*-0.25(x)x*0.10(x)x*0.28(x 1.22x0.58x0.02x

0.07x- [t]ˆ ) hazardln(

413121

432

1o

++

+++

++= λ

n  Conclusion: the effect of the drug is not modified by age (although the results looked promising for the oldest age quartile, this was not significant after accounting for sampling variability)

27


n  The inclusion of interaction terms in a multiple regression model (linear, logistic, or Cox) allows for the assessment of effect modification with (or without) adjustment for potentially multiple other predictors

28


Section B

Examples of Interaction Terms in Published Research

30

Learning Objectives

n  Exposure to several examples of the use of interaction terms in published analyses

31

Linear Regression With Interaction Term

n  Example 1: Depression and PTSD1

1 Chemtob C, et al. Maternal Posttraumatic Stress Disorder and Depression in Pediatric Primary Care Association With Child Maltreatment and Frequency of Child Exposure to Traumatic Events. JAMA Pediatrics (2013) .

32


n  Example 1: Depression and PTSD

33



34



35


n  Example 1: Depression and PTSD: Interaction model for outcome of psychological abuse score for child

where

321 02.026.015.0ˆˆ xxxy o −+++= β

313

21

* and scale, PTSD ,scale depression

score, abuse calpsychologi mean ˆ

xxxxx

y

=

==

=

36


n  Let’s look at several examples of the relationship between psychological abuse score and depression for specific values of PTSD PTSD= 3:

PTSD = 4:

PTSD=10:

321 02.026.015.0ˆˆ xxxy o −+++= β

)*3(02.0)3(26.015.0ˆˆ 11 xxy o −+++= β

)*4(02.0)3(26.015.0ˆˆ 11 xxy o −+++= β

)*10(02.0)3(26.015.0ˆˆ 11 xxy o −+++= β

37


n  Overall Interpretation The association between psychological abuse scores and depression decreases with increasing PTSD scores

38



39



40

Cox Regression With Interaction Term

n  Example 2: Second Smoke and Fiber Consumption2

2 Clark M, et al. Dietary fiber intake modifies the association between secondhand smoke exposure and coronary heart disease mortality among Chinese non-smokers in Singapore. Nutrition (2013) Vol 29: 1304–1309

41


n  Example 2: Second Smoke and Fiber Consumption2

2 Clark M, et al. Dietary fiber intake modifies the association between secondhand smoke exposure and coronary heart disease mortality among Chinese non-smokers in Singapore. Nutrition (2013) Vol 29: 1304–1309

42


n  Example 3: Association of Race and Age With Survival Among Patients Undergoing Dialysis3

3 Kucircka K, et al. Association of Race and Age With Survival Among Patients Undergoing Dialysis. Journal of the American Medical Association. (2011) 306;6.

43


n  Example 3: Association of Race and Age With Survival Among Patients Undergoing Dialysis3

3 Kucircka K, et al. Association of Race and Age With Survival Among Patients Undergoing Dialysis. Journal of the American Medical Association. (2011) 306;6.

44


n  Methods Section Excerpts

45


n  Methods Section Excerpts

Seven Age Groups 18-30 years 31-40 years 41-50 years 51-60 years 61-70 years 71-80 years > 80 years

46


n  Model used by authors

betas) and sx'other ( )*(ˆ)*(ˆ)*(ˆ)*(ˆ)*(ˆ)*(ˆ

ˆˆˆˆˆˆ

ˆ[t]ˆ death) of hazardln(

7113611251114110319218

776655443322

11o

++++++

++++++

++=

xxxxxxxxxxxx

xxxxxx

x

ββββββ

ββββββ

βλ

47


n  Close-up of Table 2

48

Mentioning Investigation of Interaction

n  Many articles will mention that the researchers investigated interaction, even if no interactions were found or reported

Mentioning Investigation of Effect Modification

n  From abstract4

4Jagsi R, et al. Gender Differences in the Salaries of Physician Researchers. Journal of the American Medical Association (2012); 307(22); 2410-2417.

49

Mentioning Investigation of Effect Modification

n  “We explored pairwise interactions between gender and the other characteristics”

50

Section C

Non-linear Relationships with Continuous Predictors in Regression: the Spline Approach

52

Learning Objectives

n  Get a brief overview of another method for handling non-linearity in a regression setting, that allows for a piecewise approaching to estimating the relationship between an outcome and a continuous predictor

810

1214

1618

Arm

Circ

umfe

renc

e (c

m)

0 5 10 15 20Weight (kg)

1,000 Children, 0-60 Months OldArm Circumference and Weight

Example 1

n  Scatterplot: Arm Circumference and Weight, 1,000 Nepalese Children 0-60 months

53

Example 1

n  Scatterplot : Linear or not?

54

810

1214

1618

Arm

Circ

umfe

renc

e (c

m)

0 5 10 15 20Weight (kg)


Example 1: Investigating Nonlinearity

n  Option 1: categorize weight into 4 groups

55

810

1214

1618

Arm

Circ

umfe

renc

e (c

m)

0 5 10 15 20Weight (kg)



n  Option 2: fit a curve ( )

56

810

1214

1618

Arm

Circ

umfe

renc

e (c

m)

0 5 10 15 20Weight (kg)


21211

ˆˆˆˆ xxy o βββ ++=


n  Option 3: More than 1 Line (spline)

57

810

1214

1618

Arm

Circ

umfe

renc

e (c

m)

0 5 10 15 20Weight (kg)



n  Linear spline approach -  Allows for nonlinearity to be investigated via fitting lines with

differing slopes across the predictor range -  Researcher can pick points where line slope can change -  Slope changes can be estimated at multiple points

n  Effect modification analogy -  Non-linearity occurs when an outcome/predictor relationship is

modified by the predictor -  Ex: relationship between AC and weight depends on weight

58



59

810

1214

1618

Arm

Circ

umfe

renc

e (c

m)

0 5 10 15 20Weight (kg)



n  Estimated association with a spline at 5 kg

where and

60

+−−++= )5(86.017.125.6ˆ 11 xxy weight1 =x

⎩⎨⎧

≥−

<=− +

5x if )5(5x if 0

)5(11

11 xx


n  Why does this work? -  When x1 <5

-  When x1≥5

61

117.125.6ˆ xy +=

1

11

11

11

-0.86)x(1.17 10.75 -0.86x1.17x)(-0.86)(-56.25

)(-0.86)(-5-0.86x1.17x 6.25 )5(86.017.125.6ˆ

++=

+++=

+++=

−−++= xxy



62

810

1214

1618

Arm

Circ

umfe

renc

e (c

m)

0 5 10 15 20Weight (kg)



n  Testing for change in slope at 5 kg

63

+−−++= )5(86.017.125.6ˆ 11 xxy


n  NHANEs Data: Obesity and age: lowess plot of ln(odds) of obesity versus age

64

-3-2

-10

1

ln(O

dds

of B

eing

Obe

se)

20 40 60 80Age (years)

bandwidth = .8

Logit transformed smoothLowess smoother


n  NHANEs Data: Obesity and age: lowess plot of ln(odds) of obesity versus age Fit a model that allows for changes in slope at 40 and 60, plus adjustment for HDL levels and sex where

65

5544

131211

ˆ ˆ

)60(ˆ)40(ˆˆˆobesity) of oddsln(

xx

xxxo

ββ

ββββ

+

+−+−++= ++

F)(1sex x (mg/dL), HDL,(years) age 541 === xx


n  Adjusted ln(odds ratios) for age age ln(OR) for one year difference in age < 40 40-60 >= 60

66

5544

131211

ˆ ˆ

)60(ˆ)40(ˆˆˆobesity) of oddsln(

xx

xxxo

ββ

ββββ

+

+−+−++= ++

321ˆˆˆ βββ ++

21ˆˆ ββ +

1̂β


n  Results

67

Table 1: Logistic Regression Results for Predictors of ObesityOdds Ratio (95% CI)

Predictor Unadjusted Adjusted1

Age (years)< 40 years 1.04 (1.03, 1.05) 1.04 (1.02, 1.05)40-‐60 years 0.97 (0.95, 0.99) 1.00 (0.99, 1.01)>= 60 years 0.96 (0.94, 0.98) 0.99 (0.95, 1.01)

1 adjusted for HDL levels and sex


n  Soda consumption and physical education classes1

1 Chen H, and Wang Y. Influence of School Beverage Environment on the Association of Beverage Consumption With Physical Education Participation Among US Adolescents. American Journal of Public Health (2013). 103 (11)

68


n  Soda consumption and physical education classes

69



70



71



n  Model 4: Where:

x1 = number days of moderate to vigorous activity x2 = number of days participating in physical education class

72

xs and slopesother )3(61.018.026.0ˆˆ 221 +−+−+−+= +xxxy oβ

⎩⎨⎧

≥−

<=− +

3x if )3(3x if 0

)3(21

22 xx


n  Model 4: When x2<3:

73


xs and slopesother 18.026.0ˆˆ 21 +−+−+= xxy oβ


n  Model 4: When x2≥3:

74


xs and slopesother )3(61.018.026.0ˆˆ 221 +−+−+−+= xxxy oβ

xs and slopesother )61.0(361.018.026.0ˆˆ 221 +−+−+−+= xxxy oβ

xs and slopesother )61.018.0(26.0)61.0(3ˆˆ 21 ++−+−+−= xxy oβ

75

Summary

n  Linear splines offer an alternative to categorizing a continuous predictor when investigating and/or handling potential non-linearity in an outcome/exposures association estimated with regression (simple or multiple)

n  This approach is useful when the “per unit” change in a measure of association (mean difference, odds ratio, hazard ratio) is of scientific interest, but the association is not necessarily linear on the regression scale

Documents

lecture 9 - Amazon Web Serviceslecturecontent.s3.amazonaws.com/pdf/15167.pdf · Lecture 9 . 2 Lecture Set Overview ! Testing for effect modification and estimating different outcome