Upload
others
View
11
Download
0
Embed Size (px)
Citation preview
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