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Lecture 15: Logistic Regression: Inference and link functions

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### Lecture 15:Logistic Regression: Inference and link functions

BMTRY 701Biostatistical Methods II

More on our example

> pros5.reg <- glm(cap.inv ~ log(psa) + gleason, family=binomial)

> summary(pros5.reg)

Call:

glm(formula = cap.inv ~ log(psa) + gleason, family = binomial)

Coefficients:

Estimate Std. Error z value Pr(>|z|)

(Intercept) -8.1061 0.9916 -8.174 2.97e-16 ***

log(psa) 0.4812 0.1448 3.323 0.000892 ***

gleason 1.0229 0.1595 6.412 1.43e-10 ***

---

Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

(Dispersion parameter for binomial family taken to be 1)

Null deviance: 512.29 on 379 degrees of freedom

Residual deviance: 403.90 on 377 degrees of freedom

AIC: 409.9

What is a good multiple regression model?

- Principles of model building are analogous to linear regression
- We use the same approach
- Look for significant covariates in simple models
- consider multicollinearity
- look for confounding (i.e. change in betas when a covariate is removed)

Multiple regression model proposal

- Gleason, logPSA, Volume, Digital Exam result, detection in RE
- But, what about collinearity? 5 choose 2 pairs.

gleason log.psa. vol

gleason 1.00 0.46 -0.06

log.psa. 0.46 1.00 0.05

vol -0.06 0.05 1.00

Categorical pairs

> dpros.dcaps <- epitab(dpros, dcaps)

> dpros.dcaps$tab

Outcome

Predictor 1 p0 2 p1 oddsratio lower upper

1 95 0.2802360 4 0.09756098 1.000000 NA NA

2 123 0.3628319 9 0.21951220 1.737805 0.5193327 5.815089

3 84 0.2477876 12 0.29268293 3.392857 1.0540422 10.921270

4 37 0.1091445 16 0.39024390 10.270270 3.2208157 32.748987

Outcome

Predictor p.value

1 NA

2 4.050642e-01

3 3.777900e-02

4 1.271225e-05

> fisher.test(table(dpros, dcaps))

Fisher\'s Exact Test for Count Data

data: table(dpros, dcaps)

p-value = 2.520e-05

alternative hypothesis: two.sided

Categorical vs. continuous

- t-tests and anova: means by category

> summary(lm(log(psa)~dcaps))

Coefficients:

Estimate Std. Error t value Pr(>|t|)

(Intercept) 1.2506 0.1877 6.662 9.55e-11 ***

dcaps 0.8647 0.1632 5.300 1.97e-07 ***

---

Residual standard error: 0.9868 on 378 degrees of freedom

Multiple R-squared: 0.06917, Adjusted R-squared: 0.06671

F-statistic: 28.09 on 1 and 378 DF, p-value: 1.974e-07

> summary(lm(log(psa)~factor(dpros)))

Coefficients:

Estimate Std. Error t value Pr(>|t|)

(Intercept) 2.1418087 0.0992064 21.589 < 2e-16 ***

factor(dpros)2 -0.1060634 0.1312377 -0.808 0.419

factor(dpros)3 0.0001465 0.1413909 0.001 0.999

factor(dpros)4 0.7431101 0.1680055 4.423 1.28e-05 ***

---

Residual standard error: 0.9871 on 376 degrees of freedom

Multiple R-squared: 0.07348, Adjusted R-squared: 0.06609

F-statistic: 9.94 on 3 and 376 DF, p-value: 2.547e-06

Categorical vs. continuous

> summary(lm(vol~dcaps))

Coefficients:

Estimate Std. Error t value Pr(>|t|)

(Intercept) 22.905 3.477 6.587 1.51e-10 ***

dcaps -6.362 3.022 -2.106 0.0359 *

---

Residual standard error: 18.27 on 377 degrees of freedom

(1 observation deleted due to missingness)

Multiple R-squared: 0.01162, Adjusted R-squared: 0.009003

F-statistic: 4.434 on 1 and 377 DF, p-value: 0.03589

> summary(lm(vol~factor(dpros)))

Coefficients:

Estimate Std. Error t value Pr(>|t|)

(Intercept) 17.417 1.858 9.374 <2e-16 ***

factor(dpros)2 -1.638 2.453 -0.668 0.505

factor(dpros)3 -1.976 2.641 -0.748 0.455

factor(dpros)4 -3.513 3.136 -1.120 0.263

---

Residual standard error: 18.39 on 375 degrees of freedom

(1 observation deleted due to missingness)

Multiple R-squared: 0.003598, Adjusted R-squared: -0.004373

F-statistic: 0.4514 on 3 and 375 DF, p-value: 0.7164

Categorical vs. continuous

> summary(lm(gleason~dcaps))

Coefficients:

Estimate Std. Error t value Pr(>|t|)

(Intercept) 5.2560 0.1991 26.401 < 2e-16 ***

dcaps 1.0183 0.1730 5.885 8.78e-09 ***

---

Residual standard error: 1.047 on 378 degrees of freedom

Multiple R-squared: 0.08394, Adjusted R-squared: 0.08151

F-statistic: 34.63 on 1 and 378 DF, p-value: 8.776e-09

> summary(lm(gleason~factor(dpros)))

Coefficients:

Estimate Std. Error t value Pr(>|t|)

(Intercept) 5.9798 0.1060 56.402 < 2e-16 ***

factor(dpros)2 0.4217 0.1403 3.007 0.00282 **

factor(dpros)3 0.4890 0.1511 3.236 0.00132 **

factor(dpros)4 0.9636 0.1795 5.367 1.40e-07 ***

---

Residual standard error: 1.055 on 376 degrees of freedom

Multiple R-squared: 0.07411, Adjusted R-squared: 0.06672

F-statistic: 10.03 on 3 and 376 DF, p-value: 2.251e-06

Lots of “correlation” between covariates

- We should expect that there will be insignificance and confounding.
- Still, try the ‘full model’ and see what happens

Full model results

> mreg <- glm(cap.inv ~ gleason + log(psa) + vol + dcaps +

factor(dpros), family=binomial)

>

> summary(mreg)

Coefficients:

Estimate Std. Error z value Pr(>|z|)

(Intercept) -8.617036 1.102909 -7.813 5.58e-15 ***

gleason 0.908424 0.166317 5.462 4.71e-08 ***

log(psa) 0.514200 0.156739 3.281 0.00104 **

vol -0.014171 0.007712 -1.838 0.06612 .

dcaps 0.464952 0.456868 1.018 0.30882

factor(dpros)2 0.753759 0.355762 2.119 0.03411 *

factor(dpros)3 1.517838 0.372366 4.076 4.58e-05 ***

factor(dpros)4 1.384887 0.453127 3.056 0.00224 **

---

Null deviance: 511.26 on 378 degrees of freedom

Residual deviance: 376.00 on 371 degrees of freedom

(1 observation deleted due to missingness)

AIC: 392

What next?

- Drop or retain?
- How to interpret?

Likelihood Ratio Test

- Recall testing multiple coefficients in linear regression
- Approach: ANOVA
- We don’t have ANOVA for logistic
- More general approach: Likelihood Ratio Test
- Based on the likelihood (or log-likelihood) for “competing” nested models

Likelihood Ratio Test

- Ho: small model
- Ha: large model
- Example:

Estimating the log-likelihood

- Recall that we use the log-likelihood because it is simpler (back to linear regression)
- MLEs:
- Betas are selected to maximize the likelihood
- Betas also maximize the log-likelihood
- If we plus the estimated betas, we get our ‘maximized’ log-likelihood for that model
- We compare the log-likelihoods from competing (nested) models

Likelihood Ratio Test

- LR statistic = G2 = -2*(LogL(H0)-LogL(H1))
- Under the null: G2 ~ χ2(p-q)
- If G2 < χ2(p-q),1-α, conclude H0
- If G2 > χ2(p-q),1-αconclude H1

LRT in R

- -2LogL = Residual Deviance
- So, G2 = Dev(0) - Dev(1)
- Fit two models:

> mreg1 <- glm(cap.inv ~ gleason + log(psa) + vol + factor(dpros),

+ family=binomial)

> mreg0 <- glm(cap.inv ~ gleason + log(psa) + vol, family=binomial)

> mreg1

Coefficients:

(Intercept) gleason log(psa) vol

-8.31383 0.93147 0.53422 -0.01507

factor(dpros)2 factor(dpros)3 factor(dpros)4

0.76840 1.55109 1.44743

Degrees of Freedom: 378 Total (i.e. Null); 372 Residual

(1 observation deleted due to missingness)

Null Deviance: 511.3

Residual Deviance: 377.1 AIC: 391.1

> mreg0

Coefficients:

(Intercept) gleason log(psa) vol

-7.76759 0.99931 0.50406 -0.01583

Degrees of Freedom: 378 Total (i.e. Null); 375 Residual

(1 observation deleted due to missingness)

Null Deviance: 511.3

Residual Deviance: 399 AIC: 407

Testing DPROS

- Dev(0) – Dev(1) =
- p – q =
- χ2(p-q),1-α, =
- Conclusion?
- p-value?

More in R

qchisq(0.95,3)

-2*(logLik(mreg0) - logLik(mreg1))

1-pchisq(21.96, 3)

> anova(mreg0, mreg1)

Analysis of Deviance Table

Model 1: cap.inv ~ gleason + log(psa) + vol

Model 2: cap.inv ~ gleason + log(psa) + vol + factor(dpros)

Resid. Df Resid. Dev Df Deviance

1 375 399.02

2 372 377.06 3 21.96

>

Notes on LRT

- Again, models have to be NESTED
- For comparing models that are not nested, you need to use other approaches
- Examples:
- AIC
- BIC
- DIC
- Next time….

For next time, read the following article

Low Diagnostic Yield of Elective Coronary Angiography

Patel, Peterson, Dai et al.

NEJM, 362(10). pp. 2886-95

March 11, 2010

http://content.nejm.org/cgi/content/short/362/10/886?ssource=mfv

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