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## Adjusting for extraneous factors

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Adjusting for extraneous factors

Topics for today

- More on logistic regression analysis for binary data and how it relates to the Wolf and Mantel-Haenszel estimates of a common odds ratios
- Interpreting logistic regression analysis
- Estimating a common risk ratio in the presence of a stratification factor. Connection to Poisson regression.

Readings

- Jewell Chapter 9

Regression-based stratified analysis for Berkeley data

data berkeley;

input stratum male a b ;

cards;

1 1 512 313

1 0 89 19

2 1 353 207

2 0 17 8

3 1 120 205

3 0 202 391

4 1 138 279

4 0 131 244

5 1 53 138

5 0 94 299

6 1 22 351

6 0 24 317

run;

data berkeley; set berkeley;

n=a+b;

procgenmod;

class stratum;

model a/n=male stratum/dist=binomial;

run;

Stratified analysis

Standard 95% Conf Chi-

Parameter DF Estimate Error Limits Square Pr > ChiSq

Intercept 1 -2.6246 0.1577 -2.9337 -2.3154 276.88 <.0001

male 1 -0.0999 0.0808 -0.2583 0.0586 1.53 0.2167

stratum 1 1 3.3065 0.1700 2.9733 3.6396 378.38 <.0001

stratum 2 1 3.2631 0.1788 2.9127 3.6135 333.12 <.0001

stratum 3 1 2.0439 0.1679 1.7149 2.3729 148.24 <.0001

stratum 4 1 2.0119 0.1699 1.6788 2.3449 140.18 <.0001

stratum 5 1 1.5672 0.1804 1.2135 1.9208 75.44 <.0001

stratum 6 0 0.0000 0.0000 0.0000 0.0000 . .

Scale 0 1.0000 0.0000 1.0000 1.0000

Wolf estimate: -0.0746 SE= 0.0822 CI: (-0.2357, 0.0866)

Mantel-Haenszel (have to take logs of this estimate): -0.1002 CI: (-0.2931, 0.0615)

Lets talk about the rest of these regression results … what do they mean?

Interpreting the logistic regression model for the Berkeley data

We are fitting the following model:

Based on the fitted model, we can predict the admission probabilities in each cell

More on the Berkeley logistic regression analysis

In addition to providing an estimate of the overall gender effect, the logistic regression analysis allows us to compare admission rates between the departments.

Based on the observed data, what is the log odds ratio for admission to department 5 versus department 6 for females?

What about for males?

What about department 1 versus department 6 for males? Females?

Another example

We can add additional factors into the logistic regression model so as to obtain an estimate of the log-odds ratio, adjusting for these additional factors.

Example, smoking in the Epilepsy study. Lets look in SAS:

procfreq ;

table one3*cig2 /chisq;

run;

Standard Wald 95% Confidence Chi-

Parameter DF Estimate Error Limits Square Pr > ChiSq

Intercept 1 -3.1396 0.2229 -3.5765 -2.7028 198.41 <.0001

DRUG 1 1 1.0384 0.2876 0.4748 1.6020 13.04 0.0003

DRUG 2 1 -0.2944 0.6275 -1.5243 0.9355 0.22 0.6390

DRUG 3 0 0.0000 0.0000 0.0000 0.0000 . .

Scale 0 1.0000 0.0000 1.0000 1.0000

Standard Wald 95% Confidence Chi-

Parameter DF Estimate Error Limits Square Pr > ChiSq

Intercept 1 -3.3872 0.2435 -3.8644 -2.9100 193.55 <.0001

DRUG 1 1 1.0712 0.2939 0.4952 1.6472 13.29 0.0003

DRUG 2 1 -0.3596 0.6337 -1.6016 0.8824 0.32 0.5704

DRUG 3 0 0.0000 0.0000 0.0000 0.0000 . .

CIG2 1 1.0721 0.3131 0.4585 1.6857 11.73 0.0006

Scale 0 1.0000 0.0000 1.0000 1.0000

Why don’t drug estimates change much??

Hint – look at association between drug and smoking

procfreq ;

table one3*cig2 /chisq;

run;

Relative Risks

We’ve talked about estimating odds ratios while adjusting for another factor. Several approaches:

- Cochran-Mantel-Haenszel test
- Wolf estimate of the adjusted logodds ratio
- Mantel-Haenszel estimate of adjusted odds ratio
- Logistic regression

Lets turn now to analogous consideration for risk ratios or relative risks

Example from Jewell Table 9.2

Relationship between behavior (type A vs type B personality) and coronary heart disease events (see p82 in Jewell for description). Unadjusted RR was 2.2 with 95% CI of (1.72, 2.87).

Since weight is important, we need to adjust for it too

CHD example

Jewell table 9.7 provides the weights for the Wolf and Mantel-Haenszel methods

Lets look at using Poisson regression to do the adjustment.

Fitting the CHD model in SAS

data chd;

input weight behavior a b;

cards;

1 1 22 253

1 0 10 305

2 1 21 235

2 0 10 270

3 1 29 297

3 0 21 297

4 1 47 248

4 0 19 253

5 1 59 378

5 0 19 361

Run;

data cdh; set chd;

lnn=log(a+b);

procgenmod;

model a=behavior/dist=poisson offset=lnn;

procgenmod;

class weight;

model a=behavior weight/dist=poisson offset=lnn;

run;

Notice the inclusion of the offset term corresponding to log of the sample size in each cell

Lets analyze the arsenic data in SAS

data mlung; set mlung;

latrisk=log(atrisk);

procgenmod;

class conc;

model events= conc /dist=poisson offset=latrisk;

where conc=0 | conc>900;

run;

procgenmod;

class age conc;

model events= conc age /dist=poisson offset=latrisk;

where conc=0 | conc>900;

run;

data mlung;

input village conc age atrisk events;

cards;

1 0 22.5 2956638 14

1 0 27.5 2175046 26

………………………

35 544 57.5 208 0

35 544 62.5 179 1

35 544 67.5 152 0

35 544 72.5 139 0

35 544 77.5 61 0

35 544 82.5 59 0

…………………..

43 934 67.5 154 2

43 934 72.5 87 0

43 934 77.5 30 0

43 934 82.5 41 0

run;

Results

Unadjusted analysis

Adjusted analysis

Why does the concentration effect change? How do we interpret the age effects? How to add in all the concentrations?

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