biost 536 lecture 14 1 lecture 14 – stratified models outline description of stratified models ...

BIOST 536 Lecture 14 1

Lecture 14 – Stratified Models Outline

Description of stratified models Exact and approximate conditional likelihood Comparison to standard logistic regression models Comparison to Mantel-Haenszel

Stratification may be done after the fact

Many cases and controls in the same stratum (no longer 1 case matched to m controls)

Underlying model still :

logit (P log (P P X X Xji ji ji ji1 ji2 jik) / ( )) ... 1 1 2 j k

for the j th stratum, i th person in that stratum

Stratified conditional model : “Remove” 1 2, ,..., J from the model

Can be compared to the standard logistic regression model to see if the coefficients are similar

Don’t have to think very hard about how to model the confounder, but do need to be conscious of the

number of strata (every stratum has to have at least one case or the entire stratum is lost)


Leisure world data Suppose they stratified on 5-year age groups post hoc, rather

than forming matched sets

Multiple cases and controls in each stratum Need a minimum of one case and one control per stratum Cases are compared to only controls within their stratum and

the results accumulated over all strata

stratum outcome Age age group gall estrogen 1 1 74 70-74 0 1 1 0 75 70-74 1 0 1 0 72 70-74 0 0 1 1 74 70-74 1 0 1 0 75 70-74 0 1 2 1 67 65-69 0 1 2 0 67 65-69 1 1 2 1 68 65-69 0 0 2 0 67 65-69 1 1 2 1 65 65-69 0 0 2 1 67 65-69 1 1 2 0 68 65-69 0 1


Underlying likelihoodStrata will vary in size and will have multiple cases and controls,

e.g. rj cases and mj controls in Stratum j

Suppose that nj = rj + mj total number in stratum

Exact conditional likelihood contribution from each stratum is a tad ugly :

e

e

X

All r cases

X

r peopleAll ways to chooser people from n people

j

j

j j

The denominator has many terms ( nj choose rj )

e.g. Suppose a set has 3 cases (observations 1, 2, 3) and 17 controls i=4, …,20

XXX

XXX

mkj

321

e

e

with X i = covariate vector for person i

With the summation over the 20 C 3 = 1,140 ways to choose j, k, m

Stata uses this exact likelihood - can be a little slow


LikelihoodBreslow-Peto approximation :

e

e

X

All r cases

X

i=1 to n

rj

i

j

j

Numerator the same, but denominator has been simplified

In our example with three cases, 17 controls

1 2 3

1 2 20

X X X

3X X X...

e

e e e

Numerator the same, but denominator has been simplified

If there are few cases in each stratum, the approximation works well

In general, but not always, exact approximation

Some packages use the approximation as the default (SAS, Egret),

but can do the exact method


Example Leisure WorldStratify by age into 5–year age groups . summ age Variable | Obs Mean Std. Dev. Min Max -------------+-------------------------------------------------------- age | 315 70.76508 6.251271 57 83 . gen agegrp=floor((age-55)/5)+1 . tabulate agegrp case | case agegrp | 0 1 | Total -----------+----------------------+---------- 1 | 4 1 | 5 2 | 43 12 | 55 3 | 60 15 | 75 4 | 77 21 | 98 5 | 37 6 | 43 6 | 31 8 | 39 -----------+----------------------+---------- Total | 252 63 | 315

Conditional logistic regression on six age groups . clogit case estrogen gall , group(agegrp) note: multiple positive outcomes within groups encountered. Conditional (fixed-effects) logistic regression Number of obs = 315 LR chi2(2) = 45.30 Prob > chi2 = 0.0000 Log likelihood = -123.28176 Pseudo R2 = 0.1552 ------------------------------------------------------------------------------ case | Coef. Std. Err. z P>|z| [95% Conf. Interval] -------------+---------------------------------------------------------------- estrogen | 2.073984 .4309608 4.81 0.000 1.229316 2.918652 gall | 1.233267 .38524 3.20 0.001 .4782104 1.988323 ------------------------------------------------------------------------------


Example Leisure World. clogit, or Conditional (fixed-effects) logistic regression Number of obs = 315 LR chi2(2) = 45.30 Prob > chi2 = 0.0000 Log likelihood = -123.28176 Pseudo R2 = 0.1552 ------------------------------------------------------------------------------ case | Odds Ratio Std. Err. z P>|z| [95% Conf. Interval] -------------+---------------------------------------------------------------- estrogen | 7.956459 3.428922 4.81 0.000 3.418892 18.51631 gall | 3.432425 1.322307 3.20 0.001 1.613185 7.303279 ------------------------------------------------------------------------------

Compare to original matched sets analysis . clogit case estrogen gall , group(set) Conditional (fixed-effects) logistic regression Number of obs = 315 LR chi2(2) = 45.05 Prob > chi2 = 0.0000 Log likelihood = -78.871308 Pseudo R2 = 0.2221 ------------------------------------------------------------------------------ case | Coef. Std. Err. z P>|z| [95% Conf. Interval] -------------+---------------------------------------------------------------- estrogen | 2.114785 .4397942 4.81 0.000 1.252804 2.976766 gall | 1.274654 .4108678 3.10 0.002 .4693683 2.079941 ------------------------------------------------------------------------------ . clogit, or Conditional (fixed-effects) logistic regression Number of obs = 315 LR chi2(2) = 45.05 Prob > chi2 = 0.0000 Log likelihood = -78.871308 Pseudo R2 = 0.2221 ------------------------------------------------------------------------------ case | Odds Ratio Std. Err. z P>|z| [95% Conf. Interval] -------------+---------------------------------------------------------------- estrogen | 8.287802 3.644927 4.81 0.000 3.500144 19.62424 gall | 3.577465 1.469865 3.10 0.002 1.598984 8.003994 ------------------------------------------------------------------------------


Alcohol/tobacco exampleCompare the unconditional model to the exact and approximate conditional models . list in 1/10 +----------------------------+ | age alc tob cc | |----------------------------| 1. | 35-44 40-79 10-19 0 | 2. | 35-44 0-39 0-9 0 | 3. | 35-44 120+ 0-9 1 | 4. | 25-34 40-79 0-9 0 | 5. | 35-44 0-39 0-9 0 | 6. | 35-44 40-79 30+ 0 | 7. | 35-44 0-39 20-29 0 | 8. | 35-44 0-39 0-9 0 | 9. | 35-44 0-39 10-19 0 | 10. | 25-34 0-39 10-19 0 | +----------------------------+

First fit an unconditional analysis . xi: logistic cc i.age alc tob i.age _Iage_1-6 (naturally coded; _Iage_1 omitted) Logistic regression Number of obs = 975 LR chi2(7) = 276.83 Prob > chi2 = 0.0000 Log likelihood = -356.32774 Pseudo R2 = 0.2798 ------------------------------------------------------------------------------ cc | Odds Ratio Std. Err. z P>|z| [95% Conf. Interval] -------------+---------------------------------------------------------------- _Iage_2 | 5.909744 6.426261 1.63 0.102 .7014092 49.79273 _Iage_3 | 33.46817 34.99045 3.36 0.001 4.312349 259.7467 _Iage_4 | 58.15342 60.6019 3.90 0.000 7.542841 448.3484 _Iage_5 | 98.41494 103.5183 4.36 0.000 12.52333 773.3965 _Iage_6 | 98.71308 108.572 4.18 0.000 11.433 852.2935 alc | 2.908565 .3051819 10.18 0.000 2.367915 3.572657 tob | 1.552015 .1493572 4.57 0.000 1.285231 1.874178 ------------------------------------------------------------------------------


Alcohol/tobacco exampleNow change to a conditional logistic regression with six age strata (exact likelihood) . clogit cc alc tob , group(age) or note: multiple positive outcomes within groups encountered. Conditional (fixed-effects) logistic regression Number of obs = 975 LR chi2(2) = 154.69 Prob > chi2 = 0.0000 Log likelihood = -343.51645 Pseudo R2 = 0.1838 ------------------------------------------------------------------------------ cc | Odds Ratio Std. Err. z P>|z| [95% Conf. Interval] -------------+---------------------------------------------------------------- alc | 2.883635 .3009606 10.15 0.000 2.350183 3.538172 tob | 1.546565 .1481706 4.55 0.000 1.281793 1.86603 ------------------------------------------------------------------------------

No age OR's are given

Compare the OR's from the unconditional and conditional methods

Analysis for the approximate conditional analysis not shown

Alcohol OR Alcohol CI Tobacco OR Tobacco CI

Unconditional 2.91 2.37 - 3.57 1.55 1.29 - 1.87

Cond - exact 2.88 2.35 - 3.54 1.55 1.28 - 1.87

Cond - approx 1.83 1.60 - 2.10 1.26 1.11 - 1.44

Little difference between the unconditional and exact conditional analyses - few age strata

Approximate values are not close to exact because there are many cases in each of the age strata

- exact method is more appropriate


Alcohol/tobacco example

Effect modification

Consider effect modification of the tobacco effect by age after fitting group linear terms

for tobacco and alcohol

1 6 1 2logit (p) ( 1) ... ( 6)age age tob alc with 1 6, ..., conditioned out

versus

1 6 1 2

3 4 5 6 7

logit (p) ( 1) ... ( 6)

*( 2) *( 3) *( 4) *( 5) *( 6)

age age tob alc

tob age tob age tob age tob age tob age

with 1 6, ..., conditioned out

Perform a likelihood ratio test to determine if the association of tobacco use and disease is constant

over age group . est store A


Alcohol/tobacco example. xi: clogit cc alc tob i.age*tob, group(age) i.age _Iage_1-6 (naturally coded; _Iage_1 omitted) i.age*tob _IageXtob_# (coded as above) note: tob dropped because of collinearity note: multiple positive outcomes within groups encountered. note: _Iage_2 omitted because of no within-group variance. note: _Iage_3 omitted because of no within-group variance. Etc Conditional (fixed-effects) logistic regression Number of obs = 975 LR chi2(7) = 157.59 Prob > chi2 = 0.0000 Log likelihood = -342.0694 Pseudo R2 = 0.1872 ------------------------------------------------------------------------------ cc | Coef. Std. Err. z P>|z| [95% Conf. Interval] -------------+---------------------------------------------------------------- alc | 1.068209 .1053501 10.14 0.000 .8617269 1.274692 tob | -.1220555 .8373177 -0.15 0.884 -1.763168 1.519057 _IageXtob_2 | .4033857 .9070478 0.44 0.657 -1.374395 2.181167 _IageXtob_3 | .6426227 .856762 0.75 0.453 -1.0366 2.321845 _IageXtob_4 | .6927818 .8530814 0.81 0.417 -.979227 2.364791 _IageXtob_5 | .3139141 .8673739 0.36 0.717 -1.386108 2.013936 _IageXtob_6 | .4352042 .9089803 0.48 0.632 -1.346365 2.216773 ------------------------------------------------------------------------------ . lrtest . A Likelihood-ratio test LR chi2(5) = 2.89 (Assumption: A nested in .) Prob > chi2 = 0.7163

No evidence of effect modification of age on the tobacco effect

Now check for effect modification by age on the alcohol effect


. xi: clogit cc alc tob i.age*alc, group(age) i.age _Iage_1-6 (naturally coded; _Iage_1 omitted) i.age*alc _IageXalc_# (coded as above) Conditional (fixed-effects) logistic regression Number of obs = 975 LR chi2(7) = 161.79 Prob > chi2 = 0.0000 Log likelihood = -339.96555 Pseudo R2 = 0.1922 ------------------------------------------------------------------------------ cc | Coef. Std. Err. z P>|z| [95% Conf. Interval] -------------+---------------------------------------------------------------- alc | 13.05114 724.2372 0.02 0.986 -1406.428 1432.53 tob | .4321314 .0965203 4.48 0.000 .242955 .6213078 _IageXalc_2 | -11.88859 724.2373 -0.02 0.987 -1431.368 1407.59 _IageXalc_3 | -11.6854 724.2372 -0.02 0.987 -1431.164 1407.793 _IageXalc_4 | -12.11288 724.2372 -0.02 0.987 -1431.592 1407.366 _IageXalc_5 | -12.26953 724.2372 -0.02 0.986 -1431.748 1407.209 _IageXalc_6 | -11.42697 724.2374 -0.02 0.987 -1430.906 1408.052 ------------------------------------------------------------------------------

Estimates are nonsensical – do not have enough data in the separate age categories to do this

Try a different way of assessing effect modification . gen alcage=alc*age . xi: clogit cc alc tob alcage , group(age) note: multiple positive outcomes within groups encountered. Conditional (fixed-effects) logistic regression Number of obs = 975 LR chi2(3) = 156.21 Prob > chi2 = 0.0000 Log likelihood = -342.75918 Pseudo R2 = 0.1856 ------------------------------------------------------------------------------ cc | Coef. Std. Err. z P>|z| [95% Conf. Interval] -------------+---------------------------------------------------------------- alc | 1.523761 .3955047 3.85 0.000 .7485858 2.298936 tob | .4328816 .0960843 4.51 0.000 .2445598 .6212033 alcage | -.1186174 .0963783 -1.23 0.218 -.3075154 .0702805 ------------------------------------------------------------------------------ . lrtest . A Likelihood-ratio test LR chi2(1) = 1.51 (Assumption: A nested in .) Prob > chi2 = 0.2184


Alcohol/tobacco example

No evidence of effect modification of age on the tobacco effect using the model

1 6 1 2 3logit (p) ( 1) ... ( 6) *age age tob alc tob age

This is a simpler form of effect modification that assumes the log OR for tobacco

changes linearly with age group

Interaction terms can be modeled in a simpler way than main effects, but not vice-versa

Test whether alcohol and tobacco interact . gen alctob=alc*tob . xi: clogit cc alc tob alctob , group(age) or Conditional (fixed-effects) logistic regression Number of obs = 975 LR chi2(3) = 155.78 Prob > chi2 = 0.0000 Log likelihood = -342.97434 Pseudo R2 = 0.1851 ------------------------------------------------------------------------------ cc | Odds Ratio Std. Err. z P>|z| [95% Conf. Interval] -------------+---------------------------------------------------------------- alc | 3.521209 .7743686 5.72 0.000 2.288229 5.418565 tob | 1.929266 .4469089 2.84 0.005 1.225219 3.037879 alctob | .9031615 .0875702 -1.05 0.293 .7468497 1.092189 ------------------------------------------------------------------------------ . lrtest . A Likelihood-ratio test LR chi2(1) = 1.08 (Assumption: A nested in .) Prob > chi2 = 0.2978

No evidence of an interaction between the two grouped linear variables


Alcohol/tobacco exampleReturn to two variable model Conditional (fixed-effects) logistic regression Number of obs = 975 LR chi2(2) = 154.69 Prob > chi2 = 0.0000 Log likelihood = -343.51645 Pseudo R2 = 0.1838 ------------------------------------------------------------------------------ cc | Coef. Std. Err. z P>|z| [95% Conf. Interval] -------------+---------------------------------------------------------------- alc | 1.059052 .1043685 10.15 0.000 .8544933 1.26361 tob | .4360365 .0958063 4.55 0.000 .2482596 .6238133

Compute the estimated linear predictor

1 1 2 2 kˆ ˆ ˆX X ... Xk

which in this case is

0.436 tobacco 1.059 alcohol . predict xb, xb . sum xb Variable | Obs Mean Std. Dev. Min Max ---------+----------------------------------------------------- xb | 975 2.732436 1.110127 1.495088 5.980353

If 1 1 2 2 kˆ ˆ ˆ1 X X ... X 0k for all observations then we can use a special test

for deciding if the covariates act multiplicatively on the OR (Barlow, 1985)

Compute 1 1 2 2 k 1 1 2 2 kˆ ˆ ˆ ˆ ˆ ˆ1 X X ... X log 1 X X ... Xk kz and fit z

as another covariate ; if z is not statistically significant then the multiplicative model is OK


Alcohol/tobacco example. gen z=(1+xb)*log(1+xb) . clogit cc alc tob z, group(age) Conditional (fixed-effects) logistic regression Number of obs = 975 LR chi2(3) = 155.47 Prob > chi2 = 0.0000 Log likelihood = -343.12662 Pseudo R2 = 0.1847 ------------------------------------------------------------------------------ cc | Coef. Std. Err. z P>|z| [95% Conf. Interval] -------------+---------------------------------------------------------------- alc | 2.368205 1.48855 1.59 0.112 -.5492987 5.285709 tob | .9711536 .6152065 1.58 0.114 -.234629 2.176936 z | -.5048729 .5714515 -0.88 0.377 -1.624897 .6151514 ------------------------------------------------------------------------------ . lrtest . A Likelihood-ratio test LR chi2(1) = 0.78 (Assumption: A nested in .) Prob > chi2 = 0.3772

No evidence against the multiplicative model


Binary exposure variableTesting a binary exposure only with stratification

Suppose we are interested only in the relationship of low alcohol (0-79) to high alcohol (80+ g) and its

relationship to esophageal cancer

How do we control for age group ?

- Many choices for computing the OR adjusting for age (ordinary logistic with age as an explicit covariate;

conditional logistic with age as an implicit covariate; Mantel-Haenszel)

Unconditional logistic regression - Fit age group and add alcohol(binary) to the model and test with a LR test . gen alcbin=(alc>2) . xi: logistic cc i.age i.age _Iage_1-6 (naturally coded; _Iage_1 omitted) Logistic regression Number of obs = 975 LR chi2(5) = 121.04 Prob > chi2 = 0.0000 Log likelihood = -434.22195 Pseudo R2 = 0.1223 ------------------------------------------------------------------------------ cc | Odds Ratio Std. Err. z P>|z| [95% Conf. Interval] -------------+---------------------------------------------------------------- _Iage_2 | 5.447368 5.777946 1.60 0.110 .6812858 43.55562 _Iage_3 | 31.67665 32.24812 3.39 0.001 4.307063 232.9685 _Iage_4 | 52.6506 53.37904 3.91 0.000 7.218137 384.0445 _Iage_5 | 59.66981 60.74305 4.02 0.000 8.114154 438.7995 _Iage_6 | 48.22581 50.98864 3.67 0.000 6.071737 383.0417 ------------------------------------------------------------------------------ . est store A


Binary exposure variable. xi: logistic cc i.age alcbin i.age _Iage_1-6 (naturally coded; _Iage_1 omitted) Logistic regression Number of obs = 975 LR chi2(6) = 200.57 Prob > chi2 = 0.0000 Log likelihood = -394.46094 Pseudo R2 = 0.2027 ------------------------------------------------------------------------------ cc | Odds Ratio Std. Err. z P>|z| [95% Conf. Interval] -------------+---------------------------------------------------------------- _Iage_2 | 4.675303 4.983382 1.45 0.148 .5787862 37.76602 _Iage_3 | 24.50217 25.06914 3.13 0.002 3.298423 182.0131 _Iage_4 | 40.99664 41.75634 3.65 0.000 5.56895 301.8028 _Iage_5 | 52.81958 54.03823 3.88 0.000 7.111389 392.3155 _Iage_6 | 52.57232 55.99081 3.72 0.000 6.519386 423.9432 alcbin | 5.311584 1.007086 8.81 0.000 3.662981 7.702174 ------------------------------------------------------------------------------ . lrtest . A Likelihood-ratio test LR chi2(1) = 79.52 (Assumption: A nested in .) Prob > chi2 = 0.0000

Highly significant result - this is the maximum likelihood estimate

( See page 144, Breslow and Day)

However, note there is some evidence for a poor model fit . lfit Logistic model for cc, goodness-of-fit test number of observations = 975 number of covariate patterns = 12 Pearson chi2(5) = 9.32 Prob > chi2 = 0.0970


Now try conditional logistic regression . clogit cc alcbin, group(age) or note: multiple positive outcomes within groups encountered. Conditional (fixed-effects) logistic regression Number of obs = 975 LR chi2(1) = 79.01 Prob > chi2 = 0.0000 Log likelihood = -381.35643 Pseudo R2 = 0.0939 ------------------------------------------------------------------------------ cc | Odds Ratio Std. Err. z P>|z| [95% Conf. Interval] -------------+---------------------------------------------------------------- alcbin | 5.250918 .9913914 8.78 0.000 3.626815 7.6023 ------------------------------------------------------------------------------

Pretty close to the unconditional estimate - the conditional estimate will also

be very close to the Mantel-Haenszel estimate . mhodds cc alcbin age Mantel-Haenszel estimate of the odds ratio Comparing alcbin==1 vs. alcbin==0, controlling for age ---------------------------------------------------------------- Odds Ratio chi2(1) P>chi2 [95% Conf. Interval] ---------------------------------------------------------------- 5.157623 85.01 0.0000 3.494918 7.611359 ---------------------------------------------------------------- Consider another binary exposure variable (0-9 cigarettes versus more) . gen tobbin=(tob>1) if tob~=. . tabulate tobbin cc | cc tobbin | 0 1 | Total -----------+----------------------+---------- 0 | 447 78 | 525 1 | 328 122 | 450 -----------+----------------------+---------- Total | 775 200 | 975


Binary exposure variableFirst fit a conditional logistic regression model . clogit cc tobbin, group(age) or Conditional (fixed-effects) logistic regression Number of obs = 975 LR chi2(1) = 22.65 Prob > chi2 = 0.0000 Log likelihood = -409.53599 Pseudo R2 = 0.0269 ------------------------------------------------------------------------------ cc | Odds Ratio Std. Err. z P>|z| [95% Conf. Interval] -------------+---------------------------------------------------------------- tobbin | 2.24614 .3874419 4.69 0.000 1.60181 3.149652 ------------------------------------------------------------------------------

Now consider testing for effect modification by age - similar to testing

for heterogeneity with the Mantel-Haenszel estimate . xi: clogit cc tobbin i.age*tobbin, group(age) or Conditional (fixed-effects) logistic regression Number of obs = 975 LR chi2(5) = 26.96 Prob > chi2 = 0.0001 Log likelihood = -407.38213 Pseudo R2 = 0.0320 ------------------------------------------------------------------------------ cc | Odds Ratio Std. Err. z P>|z| [95% Conf. Interval] -------------+---------------------------------------------------------------- tobbin | 5.18e+07 3.02e+11 0.00 0.998 0 . _IageXtobb~2 | 8.65e-08 .0005046 -0.00 0.998 0 . _IageXtobb~3 | 5.13e-08 .0002995 -0.00 0.998 0 . _IageXtobb~4 | 4.87e-08 .0002845 -0.00 0.998 0 . _IageXtobb~5 | 2.67e-08 .0001557 -0.00 0.998 0 . _IageXtobb~6 | 4.02e-08 .0002346 -0.00 0.998 0 .

Model gives nonsensical estimates – must be an age stratum with too sparse data


Mantel-Haenszel estimator . cc cc tobbin, by(age) age | OR [95% Conf. Interval] M-H Weight -----------------+------------------------------------------------- 25-34 | . 0 . 0 (exact) 35-44 | 4.512048 .8251473 45.2992 .8341709 (exact) 45-54 | 2.671614 1.270052 5.808779 5.061033 (exact) 55-64 | 2.536216 1.387893 4.681213 7.644628 (exact) 65-74 | 1.385399 .6734822 2.833404 7.31677 (exact) 75+ | 2.121212 .4669621 9.705495 1.5 (exact) -----------------+------------------------------------------------- Crude | 2.131567 1.533217 2.970895 (exact) M-H combined | 2.263103 1.609809 3.181516 ------------------------------------------------------------------- Test of homogeneity (Tarone) chi2(5) = 3.92 Pr>chi2 = 0.5615 Test that combined OR = 1: Mantel-Haenszel chi2(1) = 22.53 Pr>chi2 = 0.0000

Consider eliminating the first age group – changes the overall crude OR

and the Mantel-Haenszel OR . cc cc tobbin if age>1, by(age) age | OR [95% Conf. Interval] M-H Weight -----------------+------------------------------------------------- 35-44 | 4.512048 .8251473 45.2992 .8341709 (exact) 45-54 | 2.671614 1.270052 5.808779 5.061033 (exact) 55-64 | 2.536216 1.387893 4.681213 7.644628 (exact) 65-74 | 1.385399 .6734822 2.833404 7.31677 (exact) 75+ | 2.121212 .4669621 9.705495 1.5 (exact) -----------------+------------------------------------------------- Crude | 2.066549 1.47703 2.897909 (exact) M-H combined | 2.236111 1.589326 3.146107 ------------------------------------------------------------------- Test of homogeneity (M-H) chi2(4) = 3.18 Pr>chi2 = 0.5281 Test that combined OR = 1: Mantel-Haenszel chi2(1) = 21.72 Pr>chi2 = 0.0000


Eliminate the first age group in the conditional logistic regression and

check for interaction . clogit cc tobbin if age>1, group(age) or Conditional (fixed-effects) logistic regression Number of obs = 859 LR chi2(1) = 21.84 Prob > chi2 = 0.0000 Log likelihood = -405.1876 Pseudo R2 = 0.0262 ------------------------------------------------------------------------------ cc | Odds Ratio Std. Err. z P>|z| [95% Conf. Interval] -------------+---------------------------------------------------------------- tobbin | 2.219142 .3838213 4.61 0.000 1.581109 3.114643 ------------------------------------------------------------------------------ . est store A . xi: clogit cc tobbin i.age*tobbin if age>1, group(age) or Conditional (fixed-effects) logistic regression Number of obs = 859 LR chi2(5) = 25.11 Prob > chi2 = 0.0001 Log likelihood = -403.55349 Pseudo R2 = 0.0302 ------------------------------------------------------------------------------ cc | Odds Ratio Std. Err. z P>|z| [95% Conf. Interval] -------------+---------------------------------------------------------------- tobbin | 1.382556 .4672329 0.96 0.338 .712886 2.681298 _IageXtobb~2 | 3.24064 2.854645 1.33 0.182 .576523 18.21566 _IageXtobb~3 | 1.923551 .9431359 1.33 0.182 .7357855 5.028705 _IageXtobb~4 | 1.827305 .8127669 1.36 0.175 .7641966 4.369351 _IageXtobb~6 | 1.507227 1.12128 0.55 0.581 .3507032 6.477652 ------------------------------------------------------------------------------ . lrtest . A Likelihood-ratio test LR chi2(4) = 3.27 (Assumption: A nested in .) Prob > chi2 = 0.5140

Test of interaction is testing homogeneity of the OR across age strata -

similar to M-H result above


Note that Stata has made the 5th age group the referent

Compute the OR's for each age cell . lincom tobbin+ _IageXtobbi_2 , or ( 1) [cc]tobbin + [cc]_IageXtobbi_2 = 0 ------------------------------------------------------------------------------ cc | Odds Ratio Std. Err. z P>|z| [95% Conf. Interval] -------------+---------------------------------------------------------------- (1) | 4.480365 3.644705 1.84 0.065 .9096505 22.06746 ------------------------------------------------------------------------------ . lincom tobbin+ _IageXtobbi_3 , or ( 1) [cc]tobbin + [cc]_IageXtobbi_3 = 0 ------------------------------------------------------------------------------ cc | Odds Ratio Std. Err. z P>|z| [95% Conf. Interval] -------------+---------------------------------------------------------------- (1) | 2.659416 .9447269 2.75 0.006 1.325582 5.335386 ------------------------------------------------------------------------------ . lincom tobbin+ _IageXtobbi_4 , or ( 1) [cc]tobbin + [cc]_IageXtobbi_4 = 0 ------------------------------------------------------------------------------ cc | Odds Ratio Std. Err. z P>|z| [95% Conf. Interval] -------------+---------------------------------------------------------------- (1) | 2.526351 .7305863 3.20 0.001 1.433302 4.452968 ------------------------------------------------------------------------------ . lincom tobbin+ _IageXtobbi_6 , or ( 1) [cc]tobbin + [cc]_IageXtobbi_6 = 0 ------------------------------------------------------------------------------ cc | Odds Ratio Std. Err. z P>|z| [95% Conf. Interval] -------------+---------------------------------------------------------------- (1) | 2.083825 1.381044 1.11 0.268 .5685019 7.638192


We are not adjusting for alcohol use – could consider explicitly including it in the model

or stratifying by Age and Alcohol level

However, the latter includes all 2-way combinations of Age and alcohol (potentially 24 strata)

Check the number of cases and the denominators if we did this – would lose too many strata

due to insufficient number of cases . tabulate age alc if cc==1 | alc age | 0-39 40-79 80-119 120+ | Total -----------+--------------------------------------------+---------- 25-34 | 0 0 0 1 | 1 35-44 | 1 4 0 4 | 9 45-54 | 1 20 12 13 | 46 55-64 | 12 22 24 18 | 76 65-74 | 11 25 13 6 | 55 75+ | 4 4 2 3 | 13 -----------+--------------------------------------------+---------- Total | 29 75 51 45 | 200 . tabulate age alc | alc age | 0-39 40-79 80-119 120+ | Total -----------+--------------------------------------------+---------- 25-34 | 61 45 5 5 | 116 35-44 | 89 80 20 10 | 199 45-54 | 78 81 39 15 | 213 55-64 | 89 84 43 26 | 242 65-74 | 71 53 29 8 | 161 75+ | 27 12 2 3 | 44 -----------+--------------------------------------------+---------- Total | 415 355 138 67 | 975

Probably would have to model alcohol use rather than use it as another stratifying variable


Example

Data on condom use collected as part of a nationwide survey of males ages 19-39

Sample was stratified such that low income groups and minorities were oversampled

Sampling fractions not included here, but can be accomodated by Stata using pweights

(precision can be enhanced by appropriate weighting)

Below we use data on men with multiple "current" partners, but only use the first partner

here so that each male appears only once

Outcome: condom used for the prevention of disease

Covariates:

Have information on male's age, race, and education and same information on the first female partner

Male age both continuous and grouped (19-24, 25-29, 30-34, 35-39)

Female age both continuous and grouped (<24, 25-29, 30-34, 35+)

Education by number of years completed and categorized into three tertiles (low, medium, high)


Also have information on attitudes and beliefs of the male about condom use

including barriers and benefits

Believed personal susceptibilty to Aids (continuous, high = more likely )

Belief in severity of Aids (continuous, high=more severe )

Barriers (high = greater barrier) for (1) AIDS suspicion, (2) hassle of carrying condoms,

(3) worry about condom failure

Benefits (high = greater benefit) for (1) STD/AIDS prevention, (2) improving sex,

(3) improving relationship

Variable | Obs Mean Std. Dev. Min Max ---------+----------------------------------------------------- condom | 137 .459854 .5002146 0 1 educ | 137 13.21168 2.726317 6 24 age | 137 28.05109 5.997942 19 39 suscept | 137 .5258199 1.315653 -.7697575 5.175257 severity | 137 .3047982 .963961 -3.35639 1.255085 barrier1 | 137 .2069278 1.161492 -1.355876 2.746918 barrier2 | 137 -.301471 .9886608 -1.877701 2.031423 barrier3 | 137 .3502592 1.093366 -2.494576 2.403311 benefit1 | 137 .1175456 .9908533 -4.091574 1.137231 benefit2 | 137 -.3094053 1.056781 -1.949218 2.353029 benefit3 | 137 .1619384 1.034456 -2.989232 2.331082 ptage | 137 26.46715 5.7061 17 45 pteduc | 137 13.32117 2.452293 8 24 typepar | 137 2.065693 .5174084 1 3 edugrp | 137 1.642336 .7147318 1 3

pt prefix is for the (first) partner


Check for an age effect on condom use . logistic condom age Logistic regression Number of obs = 137 LR chi2(1) = 8.11 Prob > chi2 = 0.0044 Log likelihood = -90.463046 Pseudo R2 = 0.0429 ------------------------------------------------------------------------------ condom | Odds Ratio Std. Err. z P>|z| [95% Conf. Interval] -------------+---------------------------------------------------------------- age | .9191588 .0280178 -2.77 0.006 .8658531 .9757463 ------------------------------------------------------------------------------

Significant age effect (linear) with use decreasing with age - check to see if the linear age model is sufficient

. fracpoly logistic condom age, compare -> gen double Iage__1 = X^-2-.1270867769 if e(sample) -> gen double Iage__2 = X^-2*ln(X)-.131082712 if e(sample) (where: X = age/10) Logistic regression Number of obs = 137 LR chi2(2) = 8.75 Prob > chi2 = 0.0126 Log likelihood = -90.144149 Pseudo R2 = 0.0463 ------------------------------------------------------------------------------ condom | Odds Ratio Std. Err. z P>|z| [95% Conf. Interval] -------------+---------------------------------------------------------------- Iage__1 | 4.78e-07 7.34e-06 -0.95 0.343 4.23e-20 5416774 Iage__2 | 4.53e+21 1.56e+23 1.45 0.147 2.42e-08 8.46e+50 ------------------------------------------------------------------------------ Deviance: 180.29. Best powers of age among 44 models fit: -2 -2. --------------------------------------------------------------- age df Deviance Dev. dif. P [*] Powers --------------------------------------------------------------- Not in model 0 189.038 8.750 0.068 Linear 1 180.926 0.638 0.888 1 m = 1 2 180.665 0.377 0.828 3 m = 2 4 180.288 -- -- -2 -2 --------------------------------------------------------------- [*] P-value from deviance difference comparing reported model with m = 2 model


Linear age term seems sufficient; check survey variables . logistic condom age suscept Logistic regression Number of obs = 137 LR chi2(2) = 9.46 Prob > chi2 = 0.0088 Log likelihood = -89.786959 Pseudo R2 = 0.0501 ------------------------------------------------------------------------------ condom | Odds Ratio Std. Err. z P>|z| [95% Conf. Interval] -------------+---------------------------------------------------------------- age | .9207313 .0281974 -2.70 0.007 .8670913 .9776895 suscept | .8528191 .117933 -1.15 0.250 .6503507 1.11832

The more susceptible to AIDS - the less likely to use condoms (!!) . logistic condom age severity Logistic regression Number of obs = 137 LR chi2(2) = 10.52 Prob > chi2 = 0.0052 Log likelihood = -89.259015 Pseudo R2 = 0.0557 ------------------------------------------------------------------------------ condom | Odds Ratio Std. Err. z P>|z| [95% Conf. Interval] -------------+---------------------------------------------------------------- age | .924917 .0284517 -2.54 0.011 .8708004 .9823968 severity | .7475505 .1430667 -1.52 0.128 .5137325 1.087787 ------------------------------------------------------------------------------

The greater the belief that AIDS is severe - the less likely to use condoms (!!) . logistic condom age barrier1 barrier2 barrier3 Logistic regression Number of obs = 137 LR chi2(4) = 10.74 Prob > chi2 = 0.0296 Log likelihood = -89.148431 Pseudo R2 = 0.0568 ------------------------------------------------------------------------------ condom | Odds Ratio Std. Err. z P>|z| [95% Conf. Interval] -------------+---------------------------------------------------------------- age | .9188923 .0283186 -2.74 0.006 .865032 .9761062 barrier1 | .7901184 .1295718 -1.44 0.151 .5729324 1.089635 barrier2 | 1.149433 .2147997 0.75 0.456 .7969219 1.657874 barrier3 | .9401959 .1583129 -0.37 0.714 .6759127 1.307814 ------------------------------------------------------------------------------

None of the barriers appears to affect condom use


. logistic condom age benefit1 benefit2 benefit3 Logistic regression Number of obs = 137 LR chi2(4) = 13.00 Prob > chi2 = 0.0113 Log likelihood = -88.017402 Pseudo R2 = 0.0688 ------------------------------------------------------------------------------ condom | Odds Ratio Std. Err. z P>|z| [95% Conf. Interval] -------------+---------------------------------------------------------------- age | .917169 .0288622 -2.75 0.006 .8623094 .9755188 benefit1 | .9553348 .1838037 -0.24 0.812 .6552195 1.392914 benefit2 | 1.137751 .1982079 0.74 0.459 .8086474 1.600793 benefit3 | 1.493691 .2881765 2.08 0.038 1.023385 2.18013 ------------------------------------------------------------------------------ . logistic condom age benefit3 Logistic regression Number of obs = 137 LR chi2(2) = 12.43 Prob > chi2 = 0.0020 Log likelihood = -88.305042 Pseudo R2 = 0.0657 ------------------------------------------------------------------------------ condom | Odds Ratio Std. Err. z P>|z| [95% Conf. Interval] -------------+---------------------------------------------------------------- age | .917197 .0285421 -2.78 0.005 .8629273 .9748797 benefit3 | 1.453072 .2693064 2.02 0.044 1.010483 2.089514

Only one of the benefits appears to affect condom use

"Improving the relationship" increases condom use

Consider partner's age as an additional predictor to male’s age


. logistic condom age ptage Logistic regression Number of obs = 137 LR chi2(2) = 8.33 Prob > chi2 = 0.0155 Log likelihood = -90.355054 Pseudo R2 = 0.0441 ------------------------------------------------------------------------------ condom | Odds Ratio Std. Err. z P>|z| [95% Conf. Interval] -------------+---------------------------------------------------------------- age | .9305389 .0374007 -1.79 0.073 .8600478 1.006808 ptage | .9804494 .0416898 -0.46 0.642 .9020512 1.065661 ------------------------------------------------------------------------------

Neither age effect is significant in a Wald test - possible collinearity? . corr age ptage (obs=137) | age ptage -------------+------------------ age | 1.0000 ptage | 0.6715 1.0000

Reparametrize using age difference . gen agediff=age-ptage . logistic condom age agediff Logistic regression Number of obs = 137 LR chi2(2) = 8.33 Prob > chi2 = 0.0155 Log likelihood = -90.355054 Pseudo R2 = 0.0441 ------------------------------------------------------------------------------ condom | Odds Ratio Std. Err. z P>|z| [95% Conf. Interval] -------------+---------------------------------------------------------------- age | .9123463 .0316291 -2.65 0.008 .8524135 .9764929 agediff | 1.01994 .043369 0.46 0.642 .9383845 1.108585 ------------------------------------------------------------------------------

Model has the same predictive effect as previous model

– age difference in ages does not predict


Now consider strafying by age group

Use conditional logistic regression . clogit condom benefit3 , group(agegrp) Conditional (fixed-effects) logistic regression Number of obs = 137 LR chi2(1) = 4.20 Prob > chi2 = 0.0404 Log likelihood = -80.818459 Pseudo R2 = 0.0253 ------------------------------------------------------------------------------ condom | Coef. Std. Err. z P>|z| [95% Conf. Interval] -------------+---------------------------------------------------------------- benefit3 | .3644537 .182804 1.99 0.046 .0061645 .722743 ------------------------------------------------------------------------------

Almost the same as directly fitting age

Now stratify by age and education simultaneously

(Note: Sometimes stratify on variables for which you want to disguise the effect

for reasons of sensitivity or confidentiality) . table agegrp edugrp ---------------------------- | edugrp agegrp | 1 2 3 ----------+----------------- 1 | 21 24 4 2 | 19 10 3 3 | 16 6 9 4 | 12 10 3 ----------------------------


. gen ageeduc=3*(agegrp-1)+edugrp . table ageeduc condom ---------------------- | condom ageeduc | 0 1 ----------+----------- 1 | 11 10 2 | 10 14 3 | 1 3 4 | 11 8 5 | 2 8 6 | 2 1 7 | 11 5 8 | 3 3 9 | 4 5 10 | 9 3 11 | 7 3 12 | 3 ----------------------

Ageeduc = 12 will drop out since no members use condoms . clogit condom benefit3 , group(ageeduc) note: 1 group (3 obs) dropped because of all positive or all negative outcomes. Conditional (fixed-effects) logistic regression Number of obs = 134 LR chi2(1) = 3.06 Prob > chi2 = 0.0803 Log likelihood = -69.653107 Pseudo R2 = 0.0215 ------------------------------------------------------------------------------ condom | Coef. Std. Err. z P>|z| [95% Conf. Interval] -------------+---------------------------------------------------------------- benefit3 | .3190356 .1864274 1.71 0.087 -.0463554 .6844266 ------------------------------------------------------------------------------

Slightly reduced OR for benefit3 - now technically ns; Strata are too small

In this case the unconditional analysis might be preferred since the age effect is of scientific interest

biost 536 lecture 14 1 lecture 14 – stratified models outline description of stratified models ...

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