Biostat 513

Discussion Section Week 4

4/21/10 & 4/22/10

Agenda

• HW 2 comments (briefly)

• Review

• Plots: log-odds and probability estimates

• Exercises:

– WCGS data

– Prostate cancer data

Homework 2 Comments

• “Estimate” vs. “test”:– Estimate (e.g. OR, RR, risk …)

• Include point estimate, include 95% CI (if possible)

– Test (e.g. M-H test, test of homogeneity, Wald test,

likelihood ratio test…)

• Include test statistic and p value

Homework 2 Comments

• For matched/paired data we estimate a

conditional OR– Conditional on the matching variables

– Interpretation: OR within a matched pair

• A Mantel-Haenszel OR estimate adjusts

for one or more potential confounders– Be sure to state what you’re adjusting for.

Review

ReviewIn logistic regression, we are modeling the log

odds of disease as a linear combination of the

predictors (X1, X2, ...).

When using the regression model to estimate odds

or risks (probabilities), remember that:

exp(log odds) = odds

odds = probability / (1- probability)

probability = odds / (1 + odds)

Plots

PlotsStata code:* Fit model

logit lbw age

* Generate estimated probabilities from model

predict prob

* Generate estimated log odds from model

predict logodds, xb

Plots: estimated log odds

twoway (lowess logodds age) (scatter logodds age, msymbol(s) col(black)) , ///

scheme(s1mono) xtitle(Age (years)) ytitle(Estimated Logit(P(LBW))) ///

legend(off)

Plots: estimated probabilities

twoway (lowess prob age) (lfit prob age) ///

(scatter prob age, msymbol(s) col(black)) , ///

scheme(s1mono) xtitle(Age (years)) ytitle(Estimated P(LBW)) ///

legend(off)

Exercises• Note connections between estimates obtained

from logistic regression estimates and from other

analyses (2 x 2 table, Mantel-Haenszel

methods)

• Estimate ORs from logistic regression output

– Note about Stata “logit” and “logistic” commands

• Assess effect modification using logistic

regression output

ExercisesWestern Collaborative Group Study (WCGS)

This prospective cohort study recruited healthy men aged 39-59

years during 1960-61. Subjects were followed over time for up to

9 years for incident coronary heart disease (CHD). The exposure

of interest here is behavior pattern (type A vs. type B).

Variables of interest:

case: 0 = no CHD; 1 = CHD

typeA: 0 = type B; 1 = type A

smoke:0/1/2/3 = nonsmoker/ 1-20 cigs / 21-30 cigs / > 30 cigs

ExercisesWestern Collaborative Group Study (WCGS)

1. What is the estimated crude (unadjusted) odds ratio

(OR) of CHD comparing type A to type B men using:

a) an appropriate Stata epitab command :

2.373

b) logistic regression analysis:

exp(0.864) = 2.373

ExercisesWestern Collaborative Group Study (WCGS)

2. What is the estimated OR of CHD comparing type A to

type B men, adjusted for smoking status, using:

a) Mantel-Haenszel analysis :

2.249

b) logistic regression analysis:

exp(0.815) = 2.259

ExercisesWestern Collaborative Group Study (WCGS)

3. Do we have any evidence that smoking status is an

effect modifier? Support your answer with estimates

and/or p values if appropriate

The estimated OR of CHD comparing Type A to Type B

individuals ranges from 1.71 in heavy smokers to 2.94 in

non-smokers. However, using the Breslow-Day test of

homogeneity, we fail to reject the null, with a p value of

0.500. Thus, we do not have evidence that the OR of CHD

depends on smoking status (i.e. do not have evidence that

smoking modifies the association between behavior type and

CHD)

ExercisesWestern Collaborative Group Study (WCGS)

4. What is the estimated OR of CHD comparing a type A

man who smokes > 30 cigs/day to a type A man who is

a non-smoker?

exp(0.775) = 2.17

ExercisesWestern Collaborative Group Study (WCGS)

5. What is the estimated OR of CHD comparing a type A

man who smokes > 30 cigs/day to a type B man who is

a non-smoker?

exp(0.815) *exp(0.775) = 2.259 * 2.170 = 4.902

To make the calculation explicit, our model is:

log odds = β0 + β1*type + β2*smoke1 + β3*smoke2 + β4* smoke3

Estimated odds for person 1= exp(β0+ β1 + β4)

Estimated odds for person 2 = exp(β0)

OR = = exp(β1 ) *exp( β4) = 2.259 * 2.1700 1 4

0

ˆ ˆ ˆexp( )

ˆexp( )

^^

^^

^

^

ExercisesWestern Collaborative Group Study (WCGS)

6. [EXTRA] What is the estimated OR of CHD comparing a

type B man who smokes > 30 cigs/day to a type A man

who smokes 1-20 cigs/day?

Our model is:

log odds = β0 + β1*type + β2*smoke1 + β3*smoke2 + β4* smoke3

Estimated odds for person 1= exp(β0+ β4)

Estimated odds for person 2 = exp(β0 + β1 + β2)

OR = = exp( β4 – β1 – β2) = 0.643

^

^^

^ ^

0 4

0 1 2

ˆ ˆexp( )

ˆ ˆ ˆexp( )

^ ^^

ExercisesProstate Cancer Study

This cross-sectional study included 53 patients presenting with

prostatic cancer who had undergone laporotomy to ascertain the

extent to which cancer had spread to the lymph nodes. The data

set includes several variables that are indicative of nodal

involvement. The predictor of interest here is tumor size.

Variables of interest:

nodal: 0 = nodal involvement absent;

1 = nodal involvement present

tsize: 0 = small; 1 = large

tgrade: 0 = less serious; 1 = critical

ExercisesProstate Cancer Study

1. What is the estimated crude OR of nodal involvement

comparing pts. with large tumors to pts. with small

tumors? Use:

a) An appropriate estimate from the table of Mantel-

Haenszel results:

5.25

b) An appropriate logistic regression analysis:

exp(1.658) = 5.25

ExercisesProstate Cancer Study

2. What is the estimated OR of nodal involvement

comparing patients with large tumors to patients with

small tumors amongst those with less serious tumor

grade? Use:

a. The table of Mantel-Haenszel results : 13.3

b. an appropriate logistic regression analysis:

The model is:

log odds = β0 + β1*tsize + β2*tgrade + β3*tsize*tgrade

Here, we have tgrade = 0, so the model becomes:

log odds = β0 + β1*tsize

So the estimated OR is exp(β1) = exp(2.588) = 13.3^

ExercisesProstate Cancer Study

3. What is the estimated OR of nodal involvement comparing

patients with large tumors to patients with small tumors

amongst those with critical tumor grade? Use:

a. the table of Mantel-Haenszel results : 0.762

b. an appropriate logistic regression analysis:

The model is:

log odds = β0 + β1*tsize + β2*tgrade + β3*tsize*tgrade

Here, we have tgrade = 1, so the model becomes:

log odds = β0 + β1*tsize + β2+ β3*tsize

(β0 + β2) + (β1 + β3) *tsize

So the estimated OR is exp(β1 + β3) = exp(2.588 + (-2.860))

= 0.762

^ ^

ExercisesProstate Cancer Study

4. Do we have any evidence for tumor grade as an effect

modifier (in other words, does the association between

tumor size and nodal involvement depend on tumor

grade)? Support your answer with a relevant test(s) and

p value(s).

Estimated ORs are 13.3 and 0.76 in those with less serious

and critical tumor grade, respectively.

• M-H analysis: Using the B-D test of homogeneity, we reject the null

(p = 0.036) and conclude that these ORs are different.

• Logistic regression: Using the Wald test for the interaction term, we

reject the null (p = 0.043) and conclude that these ORs are

different.

Thus, we do have evidence for effect modification by tumor grade.

Exercises: summaryConnections between Mantel-Haenszel (M-H) and

logistic regression estimates:

M-H Logistic regression

Combined OR ≈ Adjusted OR from main

effects model

Test of combined OR=1 ≈ Wald test of βexposure = 0 from

adjusted (main effects) model

Stratum-specific ORs ≈ Stratum-specific ORs,

from model with interaction

Test of homogeneity ≈ Wald test of βinteraction = 0